Zero-point energy - Knowledge

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FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work. Such a theory has met with resistance: Macdonald (1962) and Harris (1971) claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982) and Kleen (1986), argued that the Johnson noise of a resistor connected to an antenna must satisfy Planck's thermal radiation formula, thus the noise must be zero at zero temperature and FDT must be invalid. Kiss (1988) pointed out that the existence of the zero-point term may indicate that there is a renormalization problem—i.e., a mathematical artifact—producing an unphysical term that is not actually present in measurements (in analogy with renormalization problems of ground states in quantum electrodynamics). Later, Abbott et al. (1996) arrived at a different but unclear conclusion that "zero-point energy is infinite thus it should be renormalized but not the 'zero-point fluctuations'". Despite such criticism, FDT has been shown to be true experimentally under certain quantum, non-classical conditions. Zero-point fluctuations can, and do, contribute towards systems which dissipate energy. A paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000 showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the

16373:. In fact, it has been argued that these early vacuum fluctuations led to the expansion of the universe and in turn have guaranteed the non-equilibrium conditions necessary to drive order from chaos, as without such expansion the universe would have reached thermal equilibrium and no complexity could have existed. With the continued accelerated expansion of the universe, the cosmos generates an energy gradient that increases the "free energy" (i.e. the available, usable or potential energy for useful work) which the universe is able to use to create ever more complex forms of order. The only reason Earth's environment does not decay into an equilibrium state is that it receives a daily dose of sunshine and that, in turn, is due to the sun "polluting" interstellar space with entropy. The sun's fusion power is only possible due to the gravitational disequilibrium of matter that arose from cosmic expansion. In this essence, the vacuum energy can be viewed as the key cause of the structure throughout the universe. That humanity might alter the morphology of the vacuum energy to create an energy gradient for useful work is the subject of much controversy. 16728:, but with nuclear mass density, dineutrons are nearly ideal sources of gravitational waves at X-ray and gamma-ray frequencies. The dynamical interplay, mediated by nuclear forces, between the electrically neutral dineutrons and the electrically charged core nucleus is the fundamental mechanism by which nuclear vibrations can be converted to a rotational state of dineutrons with emission of gravitational waves. Gravity and gravitational waves are well described by General Relativity, that is not a quantum theory, this implies that there is no Zero-point energy for gravity in this theory, therefore dineutrons will emit gravitational waves like any other known source of gravitational waves. In Fontana and Binder paper, nuclear species with dynamical instabilites, related to the Zero-point energy of the electromagnetic field and nuclear forces, and possessing dineutrons, will emit gravitational waves. In experimental physics this approach is still unexplored. 8071:

arbitrary data point, but rather changes in data points, so adding or subtracting a constant (even if infinite) should be allowed. However this is not the whole story, in reality energy is not so arbitrarily defined: in general relativity the seat of the curvature of spacetime is the energy content and there the absolute amount of energy has real physical meaning. There is no such thing as an arbitrary additive constant with density of field energy. Energy density curves space, and an increase in energy density produces an increase of curvature. Furthermore, the zero-point energy density has other physical consequences e.g. the Casimir effect, contribution to the Lamb shift, or anomalous magnetic moment of the electron, it is clear it is not just a mathematical constant or artifact that can be cancelled out.

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no effect on Heisenberg equations of motion without latter consequence. Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory. For instance, in general relativity the zero of energy (i.e. the energy density of the vacuum) contributes to a cosmological constant of the type introduced by Einstein in order to obtain static solutions to his field equations. The zero-point energy density of the vacuum, due to all quantum fields, is extremely large, even when we cut off the largest allowable frequencies based on plausible physical arguments. It implies a cosmological constant larger than the limits imposed by observation by about 120 orders of magnitude. This "cosmological constant problem" remains one of the greatest unsolved mysteries of physics.

16220:, which fully subsumes the standard Maxwell vectorial equations largely used today. In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered". It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in 9029: 16773: 2038:

nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space ... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space

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from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons" these will leak out of the system. So again a cut-off is permissible, almost necessary. The question here becomes one of consistency since the very high energy quanta will act as a mass source and start curving the geometry.

12568: 9643:{\displaystyle {\begin{aligned}\mathbf {\dot {x}} &=(i\hbar )^{-1}={\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\\\mathbf {\dot {p}} &=(i\hbar )^{-1}{\begin{aligned}&={\tfrac {1}{2}}\nabla \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}-m\omega _{0}^{2}\mathbf {\dot {x}} \\&=-{\frac {1}{m}}\left\left-{\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\times \nabla \times \left-m\omega _{0}^{2}\mathbf {\dot {x}} \\&={\frac {e}{c}}(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}\end{aligned}}} 1864: 12167: 1859:

is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.

2199:, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well. 1229: 16787: 15273:

of the most important consequences is that, even in the vacuum, the Maxwell equations have to be exchanged by more complicated formulas. In general, it will be not possible to separate processes in the vacuum from the processes involving matter since electromagnetic fields can create matter if the field fluctuations are strong enough. This leads to highly complex nonlinear interaction – gravity will have an effect on the light at the same time the light has an effect on gravity. These effects were first predicted by Werner Heisenberg and

2173:). In the 1970s experiments were being performed to test aspects of quantum optics and showed that the rate of spontaneous emission of an atom could be controlled using reflecting surfaces. These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to 3128: 12563:{\displaystyle {\begin{aligned}{\frac {1}{4\pi }}\left\langle \mathbf {E} _{0}^{2}(t)\right\rangle &={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\sum _{\mathbf {k'} \lambda '}{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}{\sqrt {\frac {2\pi \hbar \omega _{k'}}{V}}}\times \left\langle a_{\mathbf {k} \lambda }(0)a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right\rangle \\&={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\left({\frac {2\pi \hbar \omega _{k}}{V}}\right)\\&=\int _{0}^{\infty }dw\,\rho _{0}(\omega )\end{aligned}}} 19178: 1178: 30784: 30177: 2100: 8581: 15579:, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory". From such an approach it is possible to derive the Casimir Effect without reference to a fluctuating field. Such a derivation was first given by Schwinger (1975) for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978). in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". 14726: 9949: 30904: 16260:, which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 16354: 1533: 13986: 14516: 31332: 29916: 5098: 14211: 1814: 1524:, which ruled out the idea of a stationary aether altogether. To scientists of the period, it seemed that a true vacuum in space might be created by cooling and thus eliminating all radiation or energy. From this idea evolved the second concept of achieving a real vacuum: cool a region of space down to absolute zero temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved. 31356: 5250:

operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.

1992: 8287: 14672: 15614: 13624: 12609:. We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-point field re-emerges as the homogeneous solution for the field equation. A charged particle in the vacuum will therefore always see a zero-point field of infinite density. This is the origin of one of the infinities of quantum electrodynamics, and it cannot be eliminated by the trivial expedient dropping of the term 9653: 1988:

Einstein had obtained in 1909. However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning". Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928, performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".

31368: 1423: 31344: 13654: 4840: 14689:, who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move. 13991: 16749:

errors in the methodology and that they found the results reasonable. In the paper, the authors identify and discuss nine potential sources of experimental errors, including rogue air currents, leaky electromagnetic radiation, and magnetic interactions. Not all of them could be completely ruled out, and further peer-reviewed experimentation is needed in order to rule these potential errors out.

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evolutionary biology to even consciousness, technology and civilizations are themselves examples of thermodynamic dissipative systems; nature having naturally selected these structures to accelerate entropy flow within the universe to an ever-increasing degree. For example, it has been estimated that human body is 10,000 times more effective at dissipating energy per unit of mass than the sun.

10896: 4832: 16286:. However, it has been questioned in a series of journal articles whether the quantum mechanically canonised London equations can be given a purely classical derivation. Bostick, for instance, has claimed to show that the London equations do indeed have a classical origin that applies to superconductors and to some collisionless plasmas as well. In particular it has been asserted that the 2088:(1947). As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field. But in light of renormalisation being able to remove some zero-point infinities from calculations, not all physicists were comfortable attributing zero-point energy any physical meaning, viewing it instead as a mathematical artifact that might one day be eliminated. In 10275: 16701:

perturbations and the aerodynamic resistance of viscous drag exerted on the interior of the vehicle is compared to the Lorentz force exerted by the zero-point field (a Casimir-like force is exerted on the exterior by unbalanced zero-point radiation pressures). They find that the optimized negative energy required for an Alcubierre drive is where it is a saucer-shaped vehicle with

14547:), which has the unusual property of a non-zero amplitude in its ground state (zero-point) energy after renormalization; i.e., a non-zero vacuum expectation value. It can have this effect because of its unusual "Mexican hat" shaped potential whose lowest "point" is not at its "centre". Below a certain extremely high energy level the existence of this non-zero vacuum expectation 16330:. This occurs because the "organised" dissipative energy of the vortex configuration comprising the ions and electrons far exceeds the "disorganised" dissipative random thermal energy. The transition from disorganised fluctuations to organised helical structures is a phase transition involving a change in the condensate's energy (i.e. the ground state or zero-point energy) but 7402: 13234: 16452:(Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved." 43: 5236:, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with 1967:. This equation explained the new, non-classical fact that an electron confined to be close to a nucleus would necessarily have a large kinetic energy so that the minimum total energy (kinetic plus potential) actually occurs at some positive separation rather than at zero separation; in other words, zero-point energy is essential for atomic stability. 8576:{\displaystyle {\begin{aligned}{\frac {1}{V}}\sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \omega _{k}&={\frac {2}{8\pi ^{3}}}\int d^{3}k{\tfrac {1}{2}}\hbar \omega _{k}\\&={\frac {4\pi }{4\pi ^{3}}}\int dk\,k^{2}\left({\tfrac {1}{2}}\hbar \omega _{k}\right)\\&={\frac {\hbar }{2\pi ^{2}c^{3}}}\int d\omega \,\omega ^{3}\end{aligned}}} 2668: 16576:

field – change in the motion of a body on a pivot, detection of a transverse voltage in a semiconductor crystal, and a change in the specific heat of a crystal material having spin-aligned nuclei. There are no publicly available independent tests verifying Wallace's devices. Such an effect if any would be small. Referring to Wallace's patents, a

1503:, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were transmitted in empty space was considered evidence that their associated aethers were part of the fabric of space itself. However Maxwell noted that for the most part these aethers were 15439:, rather than the cosmological constant. No evidence of quintessence is yet available, but it has not been ruled out either. It generally predicts a slightly slower acceleration of the expansion of the universe than the cosmological constant. Some scientists think that the best evidence for quintessence would come from violations of Einstein's 14401:

cannot have one without the other. In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point (vacuum) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails.

2857: 9944:{\displaystyle {\begin{aligned}m\mathbf {\ddot {x}} &=\mathbf {\dot {p}} -{\frac {e}{c}}\mathbf {\dot {A}} \\&=-{\frac {e}{c}}\left+{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \\&=e\mathbf {E} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \end{aligned}}} 6448: 1854:. With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated with it had returned, with Einstein stating in 1920: 4609: 10062: 2025:

is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state ...

10651: 10434: 1337:. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy and make it completely vanish. If the vacuum has no intrinsic, absolute value of energy it will not gravitate. It was believed that as the universe expands from the aftermath of the 16350:) and where the overall system structure is independent of a reductionist or deterministic view, that "classical" macroscopic order can also causally affect quantum phenomena. Furthermore, the pair production of Beltrami vortices has been compared to the morphology of pair production of virtual particles in the vacuum. 13981:{\displaystyle {\begin{aligned}\left&\approx {\frac {2i\hbar e^{2}}{3\pi mc^{3}}}\omega _{0}^{3}\int _{-\infty }^{\infty }{\frac {dx}{x^{2}+\tau ^{2}\omega _{0}^{6}}}\\&=\left({\frac {2i\hbar e^{2}\omega _{0}^{3}}{3\pi mc^{3}}}\right)\left({\frac {\pi }{\tau \omega _{0}^{3}}}\right)\\&=i\hbar \end{aligned}}} 5427: 5093:{\displaystyle {\begin{aligned}H_{F}-\left\langle 0|H_{F}|0\right\rangle &={\tfrac {1}{2}}\hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right)-{\tfrac {1}{2}}\hbar \omega \\&=\hbar \omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)-{\tfrac {1}{2}}\hbar \omega \\&=\hbar \omega a^{\dagger }a\end{aligned}}} 15252: 8282: 14206:{\displaystyle {\begin{aligned}&\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)\\&\mathbf {\ddot {x}} \approx -\omega _{0}^{2}\mathbf {x} (t)&&\mathbf {\overset {...}{x}} \approx -\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}} 8106:. The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electric and magnetic fields do not have definite values. We can imagine them to be fluctuating about their mean value of zero. 15431:, launched on 1 July 2023, will map galaxies up to 10 billion light years away. By seeing how dark energy influences their arrangement and shape, the mission will allow scientists to see if the strength of dark energy has changed. If dark energy is found to vary throughout time it would indicate it is due to 2490: 16748:

and that the thrust may be due to particles pushing off the quantum vacuum. While peer review doesn't guarantee that a finding or observation is valid, it does indicate that independent scientists looked over the experimental setup, results, and interpretation and that they could not find any obvious

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article in 1980 stated "Although the Wallace patents were initially ignored as cranky, observers believe that his invention is now under serious but secret investigation by the military authorities in the USA. The military may now regret that the patents have already been granted and so are available

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predict that Lorentz invariance is not an exact symmetry of nature. It is predicted the speed at which light travels through the vacuum depends on its direction, polarization and the local strength of the magnetic field. There have been a number of inconclusive results which claim to show evidence of

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Early experimental tests from the 1950s onwards gave positive results showing the force was real, but other external factors could not be ruled out as the primary cause, with the range of experimental error sometimes being nearly 100%. That changed in 1997 with Lamoreaux conclusively showing that the

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is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So

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From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest state is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum

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of the atom, but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem

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There is a weighty argument to be adduced in favour of the aether hypothesis. To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view ... according to the general theory of relativity space

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and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a

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In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter. The fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in quantum electrodynamics. One

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which in turn gives rise to the Higgs mechanism and triggers the acquisition of mass by those particles interacting with the field. The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. This effect occurs because scalar field components of the Higgs field are "absorbed"

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The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved. it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the

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In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. An atom, for instance,

4563:. The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" 2156:

in electric circuits. The fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication

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The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one

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of the lattice ions. Quoting from their third paper: "It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a time-dependent applied magnetic vector potential field, a detectable gravitoelectric field." The claimed size

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which states: "The patents are written in a very believable style which include part numbers, sources for some components, and diagrams of data. Attempts were made to contact Wallace using patent addresses and other sources but he was not located nor is there a trace of what became of his work. The

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in electric circuits. Fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of

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In 2009, Munday et al. published experimental proof that (as predicted in 1961) the Casimir force could also be repulsive as well as being attractive. Repulsive Casimir forces could allow quantum levitation of objects in a fluid and lead to a new class of switchable nanoscale devices with ultra-low

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Zero-point energy has many observed physical consequences. It is important to note that zero-point energy is not merely an artifact of mathematical formalism that can, for instance, be dropped from a Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has

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Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or

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about zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his prediction of an attractive

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The idea that "empty" space can have an intrinsic energy associated with it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. It is often argued that the entire universe is completely bathed in the zero-point radiation, and as such it can add only some constant amount to

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were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down. This revived discussion that Einstein's cosmological constant, long disregarded by physicists as being equal to zero, was in fact some small positive value. This would indicate empty space

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has commented that "When Einstein came up with the theory of general relativity 100 years ago, he had no idea that it would be used for navigational systems. The consequences of this discovery probably will also have to be realised on a longer timescale." The team found that visible light from the

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of the difference between the energies of the 1s and 2s levels, and amounts to 1,058 MHz in frequency units. A small part of this shift (27 MHz ≈ 3%) arises not from fluctuations of the electromagnetic field, but from fluctuations of the electron–positron field. The creation of (virtual)

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We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a

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provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of BO and BO: the isotopic difference in the transition frequencies between the ground vibrational states of two different electronic levels would vanish if there were no zero-point energy, in

16524:(that would deal with the role of quantum phenomena like zero-point energy) does not yet exist. Speculative papers explaining a relationship between zero-point energy and gravitational shielding effects have been proposed, but the interaction (if any) is not yet fully understood. According to the 16463:, he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)) to say that the Casimir effect is a conservative force and therefore even though such an engine can exploit the Casimir force for useful work it cannot produce more output energy than has been input into the system. 14643:

which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged and thus the field has a nonzero vacuum expectation value. Interaction with the vacuum energy filling the space prevents certain forces from propagating over long distances (as it does in a

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In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy. In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the

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theory to develop an experimental apparatus for generating and detecting a secondary gravitational field, which he named the kinemassic field (now better known as the gravitomagnetic field). In his three patents, Wallace describes three different methods used for detection of the gravitomagnetic

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What we have here is an example of a "fluctuation-dissipation elation". Generally speaking if a system is coupled to a bath that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand we

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Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field

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However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field

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Froning and Roach (2002) put forward a paper that builds on the work of Puthoff, Haisch and Alcubierre. They used fluid dynamic simulations to model the interaction of a vehicle (like that proposed by Alcubierre) with the zero-point field. Vacuum field perturbations are simulated by fluid field

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directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boeing to work with Podkletnov had been blocked by Russian government, lso commenting that "The

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dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by:

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he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist an infinite zero-point energy term. He was able to obtain the same fluctuation formula that

1959:. For the forces between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would tend to shrink into nothing or to diminish indefinitely in size." The resolution to this puzzle came in 1926 when 1511:

To those who maintained the existence of a plenum as a philosophical principle, nature's abhorrence of a vacuum was a sufficient reason for imagining an all-surrounding aether ... Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to

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A 2008 patent by Haisch and Moddel details a device that is able to extract power from zero-point fluctuations using a gas that circulates through a Casimir cavity. A published test of this concept by Moddel was performed in 2012 and seemed to give excess energy that could not be attributed to

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electron-positron field. The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it.

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It is often argued that the entire universe is completely bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be

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section. This is a consequence of viewing electromagnetism as a U(1) gauge theory, which topologically does not allow the complex interaction of a field with and on itself. In higher symmetry groups and in reality, the vacuum is not a calm, randomly fluctuating, largely immaterial and passive

15277:

in 1936 and independently the same year by Victor Weisskopf who stated: "The physical properties of the vacuum originate in the "zero-point energy" of matter, which also depends on absent particles through the external field strengths and therefore contributes an additional term to the purely

1329:

calculations. Physical measurements will therefore reveal only deviations from this value. For many practical calculations zero-point energy is dismissed by fiat in the mathematical model as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of

8070:

This leads to the second question. Divergent or not, finite or infinite, is the zero-point energy of any physical significance? The ignoring of the whole zero-point energy is often encouraged for all practical calculations. The reason for this is that energies are not typically defined by an

2019:

of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started. In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state.

16528:, rotating matter can generate a new force of nature, known as the gravitomagnetic interaction, whose intensity is proportional to the rate of spin. In certain conditions the gravitomagnetic field can be repulsive. In neutron stars for example, it can produce a gravitational analogue of the 16142:

as a linear system with no overall observable consequence. For example, in the case of the Casimir effect, Lamb shift, and so on these phenomena can be explained by alternative mechanisms other than action of the vacuum by arbitrary changes to the normal ordering of field operators. See the

11454: 7234: 4264:

In QFT the zero-point energy of the vacuum state is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum

2503: 16268:

for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations" or "order out of chaos". It has been argued by some that all emergent order in the universe from galaxies, solar systems, planets, weather, complex chemistry,

16669:

of the generated force has been disputed by some but defended by others. In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all. Li is reported to have left the University of Alabama in 1999 to found the company

15150:

It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine-structure constant is an increasing function of energy. It has been shown that at energies of the order of the

15049:

The fine-structure constant is the coupling constant of quantum electrodynamics (QED) determining the strength of the interaction between electrons and photons. It turns out that the fine-structure constant is not really a constant at all owing to the zero-point energy fluctuations of the

14330: 6674: 12973: 8979: 25729: 13619:{\displaystyle {\begin{aligned}&=\left+\left\\&=\left\\&=\left({\frac {i\hbar e^{2}}{2\pi ^{2}mc^{3}}}\right)\left({\frac {8\pi }{3}}\right)\int _{0}^{\infty }{\frac {d\omega \,\omega ^{4}}{\left(\omega ^{2}-\omega _{0}^{2}\right)^{2}+\tau ^{2}\omega ^{6}}}\end{aligned}}} 1217:, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes; it is a consequence of the 4335:. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a 6319: 2778: 11252: 2157:

of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work. FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.

16244:, which elaborates Maxwell's equations is violated by these force free vortices. These apparent violations are due to the fact that the traditional conservation laws in classical and quantum electrodynamics (QED) only display linear U(1) symmetry (in particular, by the extended 16555:

that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic permeability of iron, making it an effective core (i.e. the doughnut of iron) in a transformer, whose properties are dependent on magnetic permeability. In 1966

11033: 10046: 1345:

and all matter in the universe should begin to decelerate. This possibility was ruled out in 1998 by the discovery that the expansion of the universe is not slowing down but is in fact accelerating, meaning empty space does indeed have some intrinsic energy. The discovery of

13085: 11888: 6108: 16185:

SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states, symmetry breaking, chaos and

2003:'s theory of emission and absorption (1927) was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created: 16277:

which occur at absolute zero temperature. These are quantum phase transitions which are driven by EM field fluctuations as a consequence of zero-point energy. A good example of a spontaneous phase transition that are attributed to zero-point fluctuations can be found in

1042:, and physics currently lacks a full theoretical model for understanding zero-point energy in this context; in particular, the discrepancy between theorized and observed vacuum energy in the universe is a source of major contention. Yet according to Einstein's theory of 15534:

The mechanism for inflation is unclear, it is similar in effect to dark energy but is a far more energetic and short lived process. As with dark energy the best explanation is some form of vacuum energy arising from quantum fluctuations. It may be that inflation caused

16425:

is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature. In 1984

10296: 15045: 16408:

literature causing ridicule around the subject. Despite rejection by the scientific community, harnessing zero-point energy remains an interest of research, particularly in the US where it has attracted the attention of major aerospace/defence contractors and the

15589:

has shown the necessity of the vacuum field for the formal consistency of QED. Modern physics does not know any better way to construct gauge-invariant, renormalizable theories than with zero-point energy and they would seem to be a necessity for any attempt at a

6929: 5261: 1901:

led them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934), that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924

16282:. Superconductivity is one of the best known empirically quantified macroscopic electromagnetic phenomena whose basis is recognised to be quantum mechanical in origin. The behaviour of the electric and magnetic fields under superconductivity is governed by the 5212: 17452:

Nernst, Walther (1916). "Über einen Versuch, von quantentheoretischen Betrachtungen zur Annahme stetiger Energieänderungen zurückzukehren" [On an attempt to return from quantum-theoretical considerations to the assumption of constant energy changes].

8145: 2359: 16601:

solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of vacuum space..." In the same year

4394:

have been superseded by quantum electrodynamics (QED). By considering the zero-point energy that arises from QED it is possible to gain a characteristic understanding of zero-point energy that arises not just through electromagnetic interactions but in all

1829:

of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to

2137:. The generic name for both van der Waals and Casimir forces is dispersion forces, because both of them are caused by dispersions of the operator of the dipole moment. The role of relativistic forces becomes dominant at orders of a hundred nanometers. 16224:

oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries. It has often been argued that quaternions are not compatible with special relativity, but multiple papers have shown ways of incorporating relativity.

4827:{\displaystyle {\begin{aligned}H_{cl}&={\frac {p^{2}}{2m}}+{\tfrac {1}{2}}m\omega ^{2}{q}^{2}\\&={\tfrac {1}{2}}\hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right)\\&=\hbar \omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)\end{aligned}}} 2133:

force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956) in the case of plane parallel

8983:

This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate

2376: 10270:{\displaystyle {\begin{aligned}\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} &\approx {\frac {e}{m}}\mathbf {E} \\&\approx \sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}\lefte_{\mathbf {k} \lambda }\end{aligned}}} 1480:

invented the first vacuum pump and the first testable scientific ideas began to emerge. It was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum.

12147: 1296:

that contains energy can be thought of as having mass to create particles. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces; it treats every single point of space as a

15255: 1685:. This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one 25760: 16470:

solicited research proposals in the area of Casimir Effect Enhancement (CEE). The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir force.

7407: 5735: 15254: 14800:, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy; this effect is called the Lamb shift. The shift of about 15259: 15258: 15253: 10891:{\displaystyle a_{\mathbf {k} \lambda }(t)=a_{\mathbf {k} \lambda }(0)e^{-i\omega _{k}t}+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\int _{0}^{t}dt'\,e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} (t')e^{i\omega _{k}\left(t'-t\right)}} 16697:

physical principles – and Podkletnov's device is not the only one – appear to be valid... There is basic science there. They're not breaking the laws of physics. The issue is whether the science can be engineered into something workable"

8689: 6012: 2120:– short-range, attractive forces that exist between neutral atoms and molecules. One of Casimir's colleagues, Theo Overbeek, realized that the theory that was used at the time to explain Van der Waals forces, which had been developed by 7013: 6196: 15260: 16511:

Despite efforts to reconcile quantum mechanics and thermodynamics over the years, their compatibility is still an open fundamental problem. The full extent that quantum properties can alter classical thermodynamic bounds is unknown

14216: 13229: 10643: 7527: 6453: 1713:

is the Planck constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy." In a series of papers from 1911 to 1913, Planck found the average energy of an oscillator to be:

12644:

The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory:

15583:(2005) has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED." 12648: 8848: 1809: 16507:

There have been a growing number of papers showing that in some instances the classical laws of thermodynamics, such as limits on the Carnot efficiency, can be violated by exploiting negative entropy of quantum fluctuations.

11257: 2677: 11038: 15574:

There has been a long debate over the question of whether zero-point fluctuations of quantized vacuum fields are "real" i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory?

8664: 5526: 2128:, Casimir discovered that the interaction between two neutral molecules could be correctly described only if the fact that light travels at a finite speed was taken into account. Soon afterwards after a conversation with 11547: 2168:

interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave nonintuitive predictions such as that an atom's spontaneous emission could be driven by field of effectively constant frequency

152: 11918:. in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole. 10915: 9954: 7918: 5631: 12980: 11783: 2111:

showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called Casimir effect. At the time, Casimir was studying the properties of

11595:

is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation

6021: 13239: 12036: 7397:{\displaystyle \int _{V}d^{3}r\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )\cdot \mathbf {A} _{\mathbf {k} '\lambda '}^{\ast }(\mathbf {r} )=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}} 18634:

Eisenschitz, R. & London, F. (1930). "Über das Verhältnis der Van der Waalsschen Kräfte zu den homöopolaren Bindungskräften" [On the relationship of van der Waals forces to homeopolar binding forces].

17690:

Heisenberg, W. (1925). "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" [On quantum-theoretical reinterpretation of kinematic and mechanical relationships]. In Blum, Walter;

20566:

Wesson, Paul S. "Cosmological constraints on the zero-point electromagnetic field." Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 378, Sept. 10, 1991, p. 466-470. Research supported by NSERC. 378 (1991):

16272:

One may query what this has to do with zero-point energy. Given the complex and adaptive behaviour that arises from nonlinear systems considerable attention in recent years has gone into studying a new class of

1248:

The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its

14496:

but lacks an electric charge. Gluons therefore participate in the strong interaction in addition to mediating it, making QCD significantly harder to analyze than QED (quantum electrodynamics) as it deals with

4245:, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose 14946: 5885: 1634: 6679: 2177:(CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited") or amplified. Amplification was first predicted by Purcell in 1946 (the 16430:

published work showing how a "vacuum-fluctuation battery" could be constructed; the battery can be recharged by making the electrical forces slightly stronger than the Casimir force to reexpand the plates.

15308:. Yet, in theory, large nonlinear self-interaction of light due to quantum fluctuations should lead to this principle being measurably violated if the interactions are strong enough. Nearly all theories of 15257: 15146: 14440:

in order to maintain the canonical commutation relation. This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.

2663:{\displaystyle \left\langle {\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}\right\rangle \left\langle {\frac {1}{2m}}{\hat {p}}^{2}\right\rangle \geq \left({\frac {\hbar }{4}}\right)^{2}{\frac {k}{m}}\,.} 2888:

The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the

16743:

which makes use of the Casimir effect for propulsion. In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time. The paper suggests that the zero-point field acts as

15416:

There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum, but this guess is known to be off by 120

5117: 7532: 2221: 1916: 24881: 4550:

implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for

2195:

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle. Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and

27478:; Filippenko; Challis; Clocchiatti; Diercks; Garnavich; Gilliland; Hogan; Jha; et al. (1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". 15511:

caused by zero-point energy arising in the microscopic inflationary period, later became magnified to a cosmic size, becoming the gravitational seeds for galaxies and structure in the Universe (see

8043: 23258: 6443:{\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} \lambda }e^{i\mathbf {k} \cdot \mathbf {r} }\,,\quad \lambda ={\begin{cases}1\\2\end{cases}}} 28219:

Wilson, C. M.; Johansson, G.; Pourkabirian, A.; Simoen, M.; Johansson, J. R.; Duty, T.; Nori, F.; Delsing, P. (2011). "Observation of the Dynamical Casimir Effect in a Superconducting Circuit".

13996: 13659: 12653: 12172: 10067: 9658: 9204: 9034: 8292: 5266: 4845: 4614: 1468:

the concept of emptiness had absolute character: it was the distinction between existence and nonexistence. Debate about the characteristics of the vacuum were largely confined to the realm of

16298:. Others have also pointed out this connection, Fröhlich has shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter ( 1086:, only true at very high energies, and no one has been able to show a theory where zero-point cancellations occur in the low-energy universe we observe today. This discrepancy is known as the 15278:

Maxwellian field energy". Thus strong magnetic fields vary the energy contained in the vacuum. The scale above which the electromagnetic field is expected to become nonlinear is known as the

2029:

Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by

15197: 16629:

if it travels between two Casimir plates, an example of negative energy. In the most general sense, the exotic matter needed to create wormholes would share the repulsive properties of the

7978: 3079: 3001:, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit 25073: 12056: 24636: 5662: 14404:

The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of

8837:

This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220

16637:

proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand (see

10429:{\displaystyle {\dot {a}}_{\mathbf {k} \lambda }=i\omega _{k}a_{\mathbf {k} \lambda }+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\mathbf {\dot {x}} \cdot e_{\mathbf {k} \lambda }} 5890: 21874:"On the Controversy over Whether Classical Systems Like Plasmas Can Behave Like Superconductors (Which Have Heretofore Been Supposed to Be Strictly Quantum Mechanically Dominated)" 1377:. These effects are usually called "radiative corrections". In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such as 24703: 11691: 5422:{\displaystyle {\begin{aligned}H_{F}&={\frac {1}{8\pi }}\int d^{3}r\left(\mathbf {E} ^{2}+\mathbf {B} ^{2}\right)\\&={\frac {k^{2}}{2\pi }}|\alpha (t)|^{2}\end{aligned}}} 25124:

Abbott, D.; Davis, B. R.; Phillips, N. J.; Eshraghian, K. (1996). "Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums".

15317:

by finding a rotation of the polarization plane of light coming from distant galaxies. The first concrete evidence for vacuum birefringence was published in 2017 when a team of

6113: 16560:

was first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence of

2053:(1948), who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field". This new theory, which Dirac coined 26859: 14933: 24788: 11778: 10528: 1350:

is best explained by zero-point energy, though it still remains a mystery as to why the value appears to be so small compared to the huge value obtained through theory – the

28158:

White, H.; March, P.; Lawrence, J.; Vera, J.; Sylvester, A.; Brady, D.; Bailey, P. (2016). "Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum".

16708:

In 2009, Giorgio Fontana and Bernd Binder presented a new method to potentially extract the Zero-point energy of the electromagnetic field and nuclear forces in the form of

15256: 14739:

The quantum fluctuations of the electromagnetic field have important physical consequences. In addition to the Casimir effect, they also lead to a splitting between the two

24830: 8277:{\displaystyle \sum _{\mathbf {k} \lambda }\longrightarrow \sum _{\lambda }\left({\frac {1}{2\pi }}\right)^{3}\int d^{3}k={\frac {V}{8\pi ^{3}}}\sum _{\lambda }\int d^{3}k} 16652:

published a heavily debated journal article claiming a specific type of rotating superconductor could shield gravitational force. Independently of this, from 1991 to 1993

7524:

This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations:

28415: 22081:

Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010). "Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order".

10465:

we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say,

4548: 4273:

interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying

2020:

Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac:

24847: 14397:

at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.

1717: 24746: 2116:. These are viscous materials, such as paint and mayonnaise, that contain micron-sized particles in a liquid matrix. The properties of such solutions are determined by 26504: 16705:

electromagnetic fields. The EM fields distort the vacuum field perturbations surrounding the craft sufficiently to affect the permeability and permittivity of space.

15361:

currently exists that can make such measurements, there are several proposed X-ray telescopes that may soon be able to verify the result conclusively such as China's

14438: 11707:

is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at

4475: 30818: 15353:

or plasma, the effect should have been no more than 1%. Definitive proof would require repeating the observation at other wavelengths and on other neutron stars. At

24813: 14813:

electron–positron pairs has the effect of screening the Coulomb field and acts as a vacuum dielectric constant. This effect is much more important in muonic atoms.

13092: 8586: 5635:

We wish to "quantize" the electromagnetic energy of free space for a multimode field. The field intensity of free space should be independent of position such that

5452: 2485:{\displaystyle {\sqrt {\left\langle \left({\hat {x}}-x_{0}\right)^{2}\right\rangle }}{\sqrt {\left\langle {\hat {p}}^{2}\right\rangle }}\geq {\frac {\hbar }{2}}\,,} 16316: 11469: 1265:-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical 24984: 24763: 22878:

Boukobza, E.; Ritsch, H. (2013). "Breaking the Carnot limit without violating the second law: A thermodynamic analysis of off-resonant quantum light generation".

7866: 5557: 26023: 24873: 14558:

by the massive bosons as degrees of freedom, and couple to the fermions via Yukawa coupling, thereby producing the expected mass terms. The expectation value of

25090: 23780:

Contract F04611-88-C-0014, Astronautics Laboratory (AFSC), Air Force Space Technology Center, Space Systems Division, Air Force Systems Command, Edwards AFB, CA

13628:

For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e.,

8062:

There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume

1321:). All these fields have zero-point energy. Recent experiments support the idea that particles themselves can be thought of as excited states of the underlying 11975: 1520:

in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led to

24893: 14492:

of quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates the

7922:

This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor

4261:(ZPF), which is the lowest energy state of a particular field. The vacuum can be viewed not as empty space, but as the combination of all zero-point fields. 28389: 28372: 24389:

Li, N.; Noever, D.; Robertson, T.; Koczor, R.; Brantley, W. (August 1997). "Static Test for a Gravitational Force Coupled to Type II YBCO Superconductors".

7515:{\displaystyle H_{F}=\sum _{\mathbf {k} \lambda }\hbar \omega _{k}\left(a_{\mathbf {k} \lambda }^{\dagger }a_{\mathbf {k} \lambda }+{\tfrac {1}{2}}\right)} 28456: 16590:

concept can be somewhat justified on general relativistic grounds since rotating frames of time varying fields are expected to emit gravitational waves."

15304:

Both Einstein's theory of special and general relativity state that light should pass freely through a vacuum without being altered, a principle known as

16103: 16040: 24963: 8828:{\displaystyle \int _{\omega _{1}}^{\omega _{2}}d\omega \rho _{0}(\omega )={\frac {\hbar }{8\pi ^{2}c^{3}}}\left(\omega _{2}^{4}-\omega _{1}^{4}\right)} 29160: 18722: 5788: 2124:

in 1930, did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with

1562: 27051:; Aldering; Goldhaber; Knop; Nugent; Castro; Deustua; Fabbro; Goobar; et al. (1999). "Measurements of Ω and Λ from 42 High-Redshift Supernovae". 24938: 17189: 15229:

depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of

15221:. The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining 29952: 27416: 15098: 7229:{\displaystyle \mathbf {A} (\mathbf {r} t)=\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\lefte_{\mathbf {k} \lambda }} 2884:

causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed.

1023:

in physics since some systems can detect the existence of this energy. However, this aether cannot be thought of as a physical medium if it is to be

22246:

Forays into "free energy" inventions and perpetual-motion machines using ZPE are considered by the broader scientific community to be pseudoscience.

19686:

Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. (1956). "Direct measurement of molecular attraction between solids separated by a narrow gap".

18808:

Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. (1956). "Direct measurement of molecular attraction between solids separated by a narrow gap".

30214: 1871: 1153:

interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In

82: 25001: 23363: 16677:

grant for $ 448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were made public.

29713: 22461: 14325:{\displaystyle \mathbf {\ddot {x}} +\tau \omega _{0}^{2}\mathbf {\dot {x}} +\omega _{0}^{2}\mathbf {x} \approx {\frac {e}{m}}\mathbf {E} _{0}(t)} 7845:{\displaystyle {\begin{aligned}\left&=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}\\\left&=\left=0\end{aligned}}} 6669:{\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} ,t)={\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\lefte_{\mathbf {k} \lambda }} 28406: 16543:, published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved. Since all atoms have 16322:

or the Planck constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology of

14679:

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by

29038: 24856: 18082:

Jordan, P.; Pauli, W. (1928). "Zur Quantenelektrodynamik ladungsfreier Felder" [On the quantum electrodynamics of charge-free fields].

15594:. Nevertheless, as pointed out by Jaffe, "no known phenomenon, including the Casimir effect, demonstrates that zero point energies are “real”" 12968:{\displaystyle {\begin{aligned}\left&=\left\\&=U^{\dagger }(t)\leftU(t)\\&=i\hbar U^{\dagger }(t)U(t)\\&=i\hbar \end{aligned}}} 8974:{\displaystyle H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}+{\tfrac {1}{2}}m\omega _{0}^{2}\mathbf {x} ^{2}+H_{F}} 4214: 30811: 30637: 30447: 29361: 28808: 15907: 418: 18263:

Weisskopf, Viktor (1935). "Probleme der neueren Quantentheorie des Elektrons" [Problems of the new quantum theory of the electron].

10279:

Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for

7983: 27415:

Riek, C.; Seletskiy, D. V.; Moskalenko, A. S.; Schmidt, J. F.; Krauspe, P.; Eckart, S.; Eggert, S.; Burkard, G.; Leitenstorfer, A. (2015).

23710:

Ritter, R. C.; Winkler, L. I.; Gillies, G. T. (1993). "Search for anomalous spin-dependent forces with a polarized-mass torsion pendulum".

11449:{\displaystyle \mathbf {E} _{RR}(t)=-{\frac {4\pi e}{V}}\sum _{\mathbf {k} \lambda }\int _{0}^{t}dt'\left\cos \omega _{k}\left(t'-t\right)} 2181:) and has been experimentally verified. This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom. 24772: 24476:

Froning, H.; Roach, R. (2002). "Preliminary Simulations of Vehicle Interactions with the Quantum Vacuum by Fluid Dynamic Approximations".

19799:

Hunklinger, S.; Geisselmann, H.; Arnold, W. (1972). "A Dynamic Method for Measuring the Van der Waals Forces between Macroscopic Bodies".

16656:

and Douglas Torr published a number of articles about gravitational effects in superconductors. One finding they derived is the source of

8110:

can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by

2773:{\displaystyle \left\langle {\hat {H}}\right\rangle \geq V_{0}+{\frac {\hbar }{2}}{\sqrt {\frac {k}{m}}}=V_{0}+{\frac {\hbar \omega }{2}}} 30529: 30488: 29127: 25107: 15807: 15444: 11247:{\displaystyle \mathbf {E} _{0}(t)=i\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}\lefte_{\mathbf {k} \lambda }} 3297: 2096:

he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality".

919: 24711: 19146:

Jaynes, E. T.; Cummings, F. W. (1963). "Comparison of quantum and semiclassical radiation theories with application to the beam maser".

25065: 24822: 23211: 22371:"Extraction of Zero-Point Energy from the Vacuum: Assessment of Stochastic Electrodynamics-Based Approach as Compared to Other Methods" 16520:

The use of zero-point energy for space travel is speculative and does not form part of the mainstream scientific consensus. A complete

16475:

another source. However it has not been conclusively shown to be from zero-point energy and the theory requires further investigation.

11965:

as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at

4315:

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as

29884: 6018:

can assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function:

31348: 25031: 19834:

Van Blokland, Peter H. G. M.; Overbeek, J. Theodoor G. (1978). "Van der Waals forces between objects covered with a chromium layer".

15169: 11028:{\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} ={\frac {e}{m}}\mathbf {E} _{0}(t)+{\frac {e}{m}}\mathbf {E} _{RR}(t)} 10041:{\displaystyle \mathbf {\dot {A}} ={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} ^{3}\,.} 3337: 26721:"Evidence for vacuum birefringence from the first optical-polarimetry measurement of the isolated neutron star RX J1856.5−3754" 25377: 24839: 20941: 13080:{\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)} 11883:{\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)} 7925: 1341:, the energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe; 30804: 30267: 29210: 24738: 24091:

Tajmar, M.; Plesescu, F.; Marhold, K. & de Matos, C. J. (2006). "Experimental Detection of the Gravitomagnetic London Moment".

21970: 21929: 21888: 6103:{\displaystyle \mathbf {A} _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }} 3023: 1889:'s laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of 1512:

convey sensations from one part of our bodies to another, and so on, till a space had been filled three or four times with aethers.

170: 24282:

Harris, Edward G. (1999). "Comments on "Gravitoelectric-Electric Coupling via Superconductivity" by Douglas G. Torr and Ning Li".

22272: 21302:

Solitons and Chaos in Periodic Nonlinear Optical Media and Lasers: Advanced Electromagnetism: Foundations, Theory and Applications

31434: 29896: 23772: 18678:

London, F. (1930). "Zur Theorie und Systematik der Molekularkräfte" [On the theory and systematics of molecular forces].

16096: 4489:

appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator

626: 28423: 24797: 18240: 15293: 12053:

is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero:

30292: 29580: 29153: 29006: 25321:

Bressi, G.; Carugno, G.; Onofrio, R.; Ruoso, G. (2002). "Measurement of the Casimir Force between Parallel Metallic Surfaces".

16736: 16645:

and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive.

15827: 2967:

associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the

399: 53:

and does not freeze regardless of temperature at standard atmospheric pressure due to zero-point energy. When cooled below its

23155: 18306:

Welton, Theodore Allen (1948). "Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field".

1444:, "the empty"; i.e., space independent of body. He believed this concept violated basic physical principles and asserted that 30627: 30396: 29945: 29514: 28511: 28111: 27875: 27848: 27765: 27727: 27689: 27389: 27365: 27111: 27030: 26549: 26514: 26474: 26398: 25996: 25930: 25901: 25825: 25643: 25548: 25266: 24755: 24493: 23076: 22151: 21841: 21833:

Finite Size Scaling for Criticality of the Schrödinger Equation: Solving the Schrödinger Equation: Has Everything Been Tried?

21762: 21408: 21310: 20594: 19431: 18844: 18444: 18223: 18142: 18058: 17948: 17716: 15902: 15314: 2493: 582: 28138: 16381:

Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (

4265:

electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with

30519: 30207: 29437: 28311:

Woods, R. C.; Cooke, S. G.; Helme, J.; Caldwell, C. H. (2001). "Gravity modification by high-temperature superconductors".

25948:

Goy, P.; Raimond, J. M.; Gross, M.; Haroche, S. (1983). "Observation of Cavity-Enhanced Single-Atom Spontaneous Emission".

25670: 25082: 24438: 15757: 15490: 15398: 1675:

in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.

1091: 505: 19003: 16606:

launched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them".

15040:{\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}={\frac {q_{e}^{2}}{4\pi \varepsilon _{0}\hbar c}}\approx {\frac {1}{137}}} 4072: 30257: 27981:

Urban, M.; Couchot, F.; Sarazin, X.; Djannati-Atai, A. (2013). "The Quantum Vacuum as the Origin of the Speed of Light".

27525:

Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. (2014). "Nanoscale Heat Engine Beyond the Carnot Limit".

22241: 20840: 19748:

Tabor, D.; Winterton, R. H. S. (1968). "Surface Forces: Direct Measurement of Normal and Retarded Van der Waals Forces".

16674: 16045: 15762: 15337: 11599: 9951:

since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative

6924:{\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} ,t)={\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\left} 2016: 1082:

has so far found no evidence to support it. Moreover, it is known that if supersymmetry is valid at all, it is at most a

18565:

Casimir, Hendrik Brugt Gerhard; Polder, Dirk (1948). "The Influence of Retardation on the London–Van der Waals Forces".

17512:

Bennewitz, Kurt; Simon, Franz (1923). "Zur Frage der Nullpunktsenergie" [On the question of zero-point energy].

4277:, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field. 1668:. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900. 30375: 28733: 28690: 28650: 28381: 28364: 27811: 27651: 26869: 26636: 26208: 24976: 23912: 23501:

Barker, B. M.; O'Connell, R. F. (1979). "The gravitational interaction: Spin, rotation, and quantum effects-a review".

22178: 21814: 21737: 21623: 21268:

De Haas, W. J.; Van Alphen, P. M. (1930). "The dependance of the susceptibility of diamagnetic metals upon the field".

16400:

proposed that useful energy could be obtained from free space, or what was assumed at that time to be an all-pervasive

16089: 10472:) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator 4207: 1019:). All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an 26062: 22301:

Forward, Robert L. (1985). "Extracting electrical energy from the vacuum by cohesion of charged foliated conductors".

16192:

What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by

8845:

linear dipole oscillator in the vacuum. The Hamiltonian for the oscillator plus the field with which it interacts is:

1325:, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field. 31419: 31336: 30787: 30180: 29183: 29146: 28801: 28448: 26929: 24901: 22666: 21787: 21685: 21391: 21337: 20810: 20234: 20209: 16581:

for anyone to read." A further reference to Wallace's patents occur in an electric propulsion study prepared for the

15605: 15242: 14466: 5207:{\displaystyle :H_{F}:\equiv \hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right):\equiv \hbar \omega a^{\dagger }a} 165: 15349:

star had undergone linear polarisation of around 16%. If the birefringence had been caused by light passing through

14477:

in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of

8079:

The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which

2808:

A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly

1058:

shows any such energy to be exceptionally weak. One proposal that attempts to address this issue is to say that the

31404: 30717: 30524: 30467: 30287: 30052: 29938: 29293: 18536: 16050: 15792: 15519:). Many physicists also believe that inflation explains why the Universe appears to be the same in all directions ( 4253:

and quarks), force fields whose quanta are bosons (i.e. photons and gluons) and a Higgs field whose quantum is the

3842: 2354:{\displaystyle {\hat {H}}=V_{0}+{\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}+{\frac {1}{2m}}{\hat {p}}^{2}\,,} 956: 254: 28282:

Woods, R. C. (2005). "Manipulation of gravitational waves for communications applications using superconductors".

26720: 26234:

Kostelecký, V. Alan; Mewes, M. (2009). "Electrodynamics with Lorentz-violating operators of arbitrary dimension".

20202:

Gauge Theories in Particle Physics: A Practical Introduction: Volume 1: From Relativistic Quantum Mechanics to QED

16772: 5258:

From Maxwell's equations, the electromagnetic energy of a "free" field i.e. one with no sources, is described by:

30677: 30580: 30560: 30320: 30200: 29975: 29851: 29560: 29555: 29278: 23579:

O'Connell, R. F.; Rasband, S. N. (1971). "Lense-Thirring Type Gravitational Forces Between Disks and Cylinders".

17896:

Schrödinger, Erwin (1926). "Quantisierung als Eigenwertproblem" [Quantization as an eigenvalue problem].

17193: 16488: 15362: 15246: 14636:. It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. 4501: 3140: 2207: 2145: 1834:

of the same year Einstein declared zero-point energy "dead as a doornail". Zero-point energy was also invoked by

1517: 1250: 1066:

has positive zero-point energy and thus these energies somehow cancel out each other. This idea would be true if

912: 361: 341: 209: 20: 16369:

The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause of

16240:

by creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particular

15562:

examined the cosmological implications of assuming that zero-point energy is real. Among numerous difficulties,

30483: 30242: 30237: 29863: 29535: 28954: 28040: 19364: 16740: 16732: 16456: 15997: 15512: 15504: 2174: 1825:. In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the 601: 331: 21229:"New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance" 20052:

Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, Lev P. (1961). "General Theory of Van der Waals' Forces".

19198: 31414: 31399: 31372: 31317: 30928: 29841: 29618: 29540: 29323: 29193: 24993: 24955: 24439:"Annual Report on Cooperative Agreements and Other Transactions Entered into During FY2001 Under 10 USC 2371" 23930: 22263: 21651: 19862: 19645: 16582: 16177: 16160:

and a rich nonlinear structure. There are many observed nonlinear physical electromagnetic phenomena such as

15684: 14548: 4431: 4312:, the vacuum energy is one possible explanation for the cosmological constant and the source of dark energy. 4200: 3405: 3254: 3214: 2072:

of the electron. Discrepancies between these experiments and Dirac's theory led to the idea of incorporating

1382: 1351: 1298: 1087: 641: 379: 279: 28398: 25486: 22420: 22058: 19903:

Mohideen, Umar; Roy, Anushree (1998). "Precision Measurement of the Casimir Force from 0.1 to 0.9 μm".

31409: 30863: 30756: 30632: 30282: 29575: 29509: 29504: 29475: 29188: 28794: 24946: 22610:

Grau, G.; Kleen, W. (1982). "Comments on zero-point energy, quantum noise and spontaneous-emission noise".

16540: 16525: 16233: 16060: 15717: 3942: 3872: 3244: 1876: 577: 572: 543: 394: 175: 20838:

Ehrenberg, W; Siday, RE (1949). "The Refractive Index in Electron Optics and the Principles of Dynamics".

17972:

Born, M.; Heisenberg, W.; Jordan, P. (1926). "Zur Quantenmechanik. II" [On quantum mechanics II].

15458:, but an analogous problem to the cosmological constant problem (or the problem of constructing models of 7010:

shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write:

1261:). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of their 31444: 30751: 30545: 30102: 29985: 29643: 29550: 29091: 25980:

Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics

21956:"Recent Experimental Results of The Plasma-Focus Group at Darmstadt, West Germany: A Review and Critique" 15524: 15508: 3400: 3342: 2990: 2215: 2012: 611: 356: 346: 34: 28198: 27582:

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

24662: 24113: 16392:

Nevertheless, the allure of free energy has motivated such research, usually falling in the category of

31424: 31108: 31017: 30325: 29919: 29681: 29489: 29460: 28922: 28895: 27799: 27639: 20461: 16745: 16401: 16131: 14645: 13988:

the necessity of the vacuum field can also be appreciated by making the small damping approximation in

12595:

since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with

5246:

and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.

3249: 1492:

as a substitute for a true void was the most prevalent theory of the time. According to the successful

905: 557: 528: 24184:

Li, Ning; Torr, D. G. (1992). "Gravitational effects on the magnetic attenuation of superconductors".

21730:

Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations

21110:

Chiao, Raymond Y.; Wu, Yong-Shi (1986). "Manifestations of Berry's Topological Phase for the Photon".

20442: 14911: 4052: 1209:, for example, arises from the intensity of random particle motion caused by kinetic energy (known as 1157:

this combination of fields is called the vacuum state, its associated zero-point energy is called the

31439: 31394: 31057: 31032: 30493: 30123: 30062: 29990: 29703: 29570: 29494: 29455: 29412: 29386: 29343: 29236: 26719:

Mignani, R. P.; Testa, V.; González Caniulef, D.; Taverna, R.; Turolla, R.; Zane, S.; Wu, K. (2017).

26289:

Kostelecký, V. Alan; Mewes, M. (2013). "Constraints on Relativity Violations from Gamma-Ray Bursts".

25009:

Powell, C. S. (1994). "Unbearable Lightness: A New Theory May Explain Why Objects Tend to Stay Put".

22492: 16551:

gravitational field that acts against the force of gravity could be generated by materials that have

16439: 16358: 16197: 16012: 15837: 15822: 15802: 12142:{\displaystyle \langle \mathbf {E} _{0}(t)\rangle =\langle \Psi |\mathbf {E} _{0}(t)|\Psi \rangle =0} 11752: 11459: 5104:(or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted 3390: 2161: 2057:(QED), predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources. 1851: 1461: 1457: 1047: 562: 523: 476: 451: 374: 234: 25808:

Drexhage, K. H. (1974). "IV Interaction of Light with Monomolecular Dye Layers". In Wolf, E. (ed.).

25292:

Bostick, W. H.; Prior, W.; Grunberger, L.; Emmert, G. (1966). "Pair Production of Plasma Vortices".

25146: 25099: 24864: 22937:

Abah, Obinna; Lutz, Eric (2014). "Efficiency of heat engines coupled to nonequilibrium reservoirs".

21702: 20862: 6420: 4304:, resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason 1863: 31227: 30981: 30971: 30827: 30729: 29770: 29750: 29740: 29730: 29686: 29261: 29028: 28720: 16861: 16386: 16335: 16261: 16161: 16022: 15872: 15580: 4228: 3877: 3507: 3352: 3317: 1449: 1370: 1273:

does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy.

1162: 646: 20682:

Schwinger, Julian; DeRaad, Lester L.; Milton, Kimball A. (1978). "Casimir effect in dielectrics".

16459:

has highlighted that he believes such devices hinge on the assumption that the Casimir force is a

5730:{\displaystyle \mathbf {A} _{0}(\mathbf {r} )=e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }} 3867: 31449: 31429: 31073: 30601: 30457: 30426: 30401: 30340: 30016: 29961: 29465: 29391: 28343: 28141:[On the Elecrodynamics of the Vacuum on the Basis of the Quantum Theory of the Electron] 27580:

Rugh, S. E.; Zinkernagel, H. (2002). "The Quantum Vacuum and the Cosmological Constant Problem".

23852: 22231: 21566: 17498: 17366:[Some arguments for the assumption of a molecular agitation at the absolute zero point]. 15752: 14940: 14822: 4152: 3622: 3430: 3425: 3287: 2975: 2054: 1821:

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant

1453: 1357:

Many physical effects attributed to zero-point energy have been experimentally verified, such as

1138: 815: 533: 441: 27: 29879: 29427: 23983:

Podkletnov, E.; Nieminen, R. (1992). "A possibility of gravitational force shielding by bulk YBa

23904: 23810:

Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering

18134: 6209:

is chosen to be a unit vector which specifies the polarization of the field mode. The condition

1964: 1484:

Late in the 19th century, however, it became apparent that the evacuated region still contained

491: 389: 155: 30768: 30692: 30687: 30622: 30550: 30452: 30355: 30345: 30330: 30247: 29790: 29565: 29545: 29470: 29338: 29112: 28974: 25141: 20857: 16661: 16598: 16586: 16501: 16343: 16339: 16327: 16295: 16253: 16209: 16002: 15992: 15937: 15927: 15712: 15664: 15566:

requires that such energy not gravitate, so it cannot be similar to electromagnetic radiation.

15466:

theory predicts that scalar fields should acquire large masses again due to zero-point energy.

15459: 15432: 6007:{\displaystyle \left(k_{x},k_{y},k_{z}\right)={\frac {2\pi }{L}}\left(n_{x},n_{y},n_{z}\right)} 4396: 4391: 4266: 4022: 4002: 3502: 3435: 3347: 3312: 1537: 1146: 820: 538: 366: 336: 299: 30138: 29381: 28435: 27351: 26899: 24227:

Torr, Douglas G.; Li, Ning (1993). "Gravitoelectric-electric coupling via superconductivity".

23177:

Podkletnov, Evgeny; Modanese, Giovanni (2001). "Impulse Gravity Generator Based on Charged YBa

22068: 21661: 21189: 19576: 18183:

Weinberg, Steven (1977). "The Search for Unity: Notes for a History of Quantum Field Theory".

17940: 17796:"On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus" 2064:

technology made it possible to take more precise measurements of the shift of the levels of a

1374: 446: 31124: 31103: 31037: 30741: 30736: 30575: 30570: 30503: 30406: 30370: 30360: 30315: 29815: 29328: 29308: 29117: 29086: 28949: 28870: 28399:"Research in a Vacuum: DARPA Tries to Tap Elusive Casimir Effect for Breakthrough Technology" 27229:(1912b). "La loi du rayonnement noir et l'hypothèse des quantités élémentaires d'action". In 25163:

Barton, G.; Scharnhorst, K. (1993). "QED Between Parallel Mirrors: Light Signals Faster Than

24331:"Gravitomagnetic Fields in Rotating Superconductors to Solve Tate's Cooper Pair Mass Anomaly" 22421:"Research in a Vacuum: DARPA Tries to Tap Elusive Casimir Effect for Breakthrough Technology" 20885:

Aharonov, Y; Bohm, D (1959). "Significance of electromagnetic potentials in quantum theory".

20878: 18250: 17604:

Dugdale, J. S.; Simon, F. E. (1953). "Thermodynamic Properties and Melting of Solid Helium".

16657: 16633:, dark energy or zero-point radiation of the vacuum. Building on the work of Thorne, in 1994 16630: 16617:

can exist in spacetime only if they are threaded by quantum fields generated by some form of

16561: 16323: 16287: 16055: 16017: 15832: 15812: 15747: 15622: 15619: 15475: 15440: 15425: 14892: 14493: 14423: 14410:, the spectral energy density of the vacuum field must be proportional to the third power of 4387: 3912: 3772: 3468: 3332: 3194: 2893:

of the system. The professional physics literature tends to measure frequency, as denoted by

2190: 2008: 1406: 1334: 1258: 1218: 1094:. Many physicists believe that "the vacuum holds the key to a full understanding of nature". 1004: 289: 274: 23896: 23060:

Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems

22646: 20227:

Advanced School on Electroweak Theory: Hadron Colliders, the Top Quark, and the Higgs Sector

18215: 17932: 15357:

wavelengths the polarization from the quantum fluctuations should be near 100%. Although no

10050:

For nonrelativistic motion we may neglect the magnetic force and replace the expression for

5100:

without affecting any physical predictions of the theory. The new Hamiltonian is said to be

1679:

boundaries of finite cells in phase space, where their energies became integer multiples of

1445: 31093: 30868: 30682: 30416: 30252: 30223: 29846: 29775: 29720: 29450: 29273: 29231: 29076: 28900: 28860: 28592: 28526: 28482: 28291: 28238: 28055: 28000: 27943: 27902: 27893:

Scully, M. O. (2001). "Extracting Work from a Single Thermal Bath via Quantum Negentropy".

27599: 27544: 27497: 27431: 27248: 27206: 27138: 27070: 26981: 26827: 26795: 26745: 26675: 26586: 26361: 26308: 26253: 26159: 26100: 26038: 25957: 25890:

Einstein, A. (1995). Klein, Martin J.; Kox, A. J.; Renn, Jürgen; Schulmann, Robert (eds.).

25854: 25788: 25744: 25692: 25589: 25540:

Thinking About Nothing : Otto Von Guericke and The Magdeburg Experiments on the Vacuum

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Heisenberg, W.; Euler, H. (1936). "Folgerungen aus der Diracschen Theorie des Positrons".

25042: 24141:

Li, N.; Torr, D. G. (1991). "Effects of a gravitomagnetic field on pure superconductors".

23928:

Alcubierre, Miguel (1994). "The warp drive: hyper-fast travel within general relativity".

22741:

Galve, Fernando; Lutz, Eric (2009). "Nonequilibrium thermodynamic analysis of squeezing".

19993: 16404:. Others have since claimed to exploit zero-point or vacuum energy with a large amount of 16301: 14459:

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a

4241:

In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of

3239: 1960: 1907:

contrast to the observed spectra. Then just a year later in 1925, with the development of

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The Collected Papers of Albert Einstein Vol. 5 The Swiss Years: Correspondence, 1902–1914

23844: 23424:

Ross, D. K. (1983). "The London equations for superconductors in a gravitational field".

22134:

Chaisson, Eric J. (2005). "2 Non-equilibrium Thermodynamics in an Energy-Rich Universe".

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which goes to zero at the two edges of the well. The energy of the particle is given by:

2084:(1936), and first successfully applied to calculate a finite value for the Lamb shift by 2076:

into QED to deal with zero-point infinities. Renormalization was originally developed by

1979:

published the first attempt to quantize the electromagnetic field. In a joint paper with

1932: 1497: 1489: 1237: 1186: 1020: 952: 948: 705: 513: 431: 259: 239: 191: 28596: 28530: 28486: 28295: 28242: 28059: 28004: 27947: 27906: 27603: 27548: 27501: 27435: 27210: 27142: 27074: 26985: 26831: 26799: 26749: 26679: 26590: 26416:

Kragh, H. (2012). "Preludes to Dark Energy: Zero-Point Energy and Vacuum Speculations".

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has been observed for several years now in precision experiment in high-energy physics.

10638:{\displaystyle i\hbar {\dot {U}}=HU\,,\quad U^{\dagger }(t)=U^{-1}(t)\,,\quad U(0)=1\,.} 6985:. This gives the vector potential for a plane wave mode of the field. The condition for 3882: 3612: 2978:. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a 31194: 31184: 31022: 30933: 30380: 30158: 30133: 30057: 29891: 29760: 29658: 29366: 29313: 29205: 29107: 28984: 28979: 28932: 28618: 28582: 28270: 28228: 28185: 28087: 28024: 27990: 27969: 27623: 27589: 27568: 27534: 27513: 27487: 27463: 27238: 27100: 27086: 27060: 27019: 27005: 26971: 26907: 26843: 26771: 26735: 26699: 26610: 26576: 26567:

Leuchs, G.; Sánchez-Soto, L. L. (2013). "A Sum Rule For Charged Elementary Particles".

26451: 26425: 26340: 26298: 26277: 26243: 26185: 26149: 26124: 26090: 25920: 25878: 25844: 25800: 25716: 25682: 25610: 25579: 25567: 25525: 25473: 25439: 25425:"Is There Evidence for Cosmic Anisotropy in the Polarization of Distant Radio Sources?" 25408: 25364: 25330: 25243: 24371: 24345: 24299: 24252: 24092: 24051: 23965: 23939: 23897: 23825: 23783: 23692: 23658: 23526: 23343: 23324: 23237: 23190: 23159: 23133: 23124: 23034: 23003: 22991: 22972: 22946: 22811: 22785: 22658: 22402: 22382: 22349: 22116: 22090: 21528: 21299:

Feng, J. H.; Kneubühl, F. K. (1995). Barrett, Terence William; Grimes, Dale M. (eds.).

21092: 21014: 20922: 20896: 20757: 20731: 20664: 20600: 20536: 20502: 20029: 19938: 19912: 19781: 19672: 19392: 19113: 19079: 19042: 18703: 18660: 18505: 18288: 18192: 18127: 18107: 17997: 17795: 17672: 17629: 17586: 17537: 17364:"Einige Argumente für die Annahme einer molekularen Agitation beim absoluten Nullpunkt" 16792: 16725: 16709: 16626: 16547:, gravitational permeability may be able to differ from material to material. A strong 16460: 16449: 16362: 15912: 15842: 15782: 15563: 15418: 15305: 14701: 14498: 14489: 5446: 4328: 4324: 4309: 4177: 4102: 4032: 3917: 3902: 3832: 3797: 3787: 3747: 3662: 3552: 3512: 3302: 3264: 3219: 3207: 3184: 3179: 2211: 2048: 1903: 1657: 1521: 1330: 1071: 1043: 1032: 944: 426: 351: 284: 196: 27611: 27129:

Pinto, F. (1999). "Engine cycle of an optically controlled vacuum energy transducer".

26352:

Kowitt, Mark (1994). "Gravitomagnetism and magnetic permeability in superconductors".

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Callen, Herbert; Welton, Theodore A. (1951). "Irreversibility and Generalized Noise".

17753: 16757:

The concept of Zero-point energy used as an energy source has been an element used in

5775:

To achieve the desired normalization we pretend space is divided into cubes of volume

1333:

the absolute energy value of space is not an arbitrary constant and gives rise to the

1240:

an average energy equal to the frequency of oscillation multiplied by one-half of the

1141:(e.g., electromagnetic interactions between photons, electrons and the vacuum) or the 979:, the universe can be thought of not as isolated particles but continuous fluctuating 31355: 31302: 31267: 31174: 31098: 30431: 30118: 30087: 29901: 29810: 29780: 29708: 29671: 29666: 29648: 29613: 29603: 29318: 29283: 29169: 29122: 29071: 29016: 28996: 28882: 28766: 28747: 28739: 28729: 28704: 28696: 28686: 28664: 28656: 28646: 28610: 28542: 28498: 28494: 28262: 28254: 28189: 28125: 28117: 28107: 28091: 28079: 28016: 27973: 27961: 27918: 27881: 27871: 27854: 27844: 27825: 27817: 27807: 27779: 27771: 27761: 27741: 27733: 27723: 27703: 27695: 27685: 27665: 27657: 27647: 27615: 27560: 27455: 27447: 27403: 27395: 27385: 27361: 27338: 27330: 27310: 27302: 27282: 27260: 27252: 27117: 27107: 27090: 27036: 27026: 27009: 26997: 26943: 26935: 26925: 26883: 26875: 26865: 26847: 26807: 26763: 26707: 26691: 26650: 26642: 26632: 26614: 26602: 26555: 26545: 26528: 26520: 26510: 26488: 26480: 26470: 26455: 26443: 26404: 26394: 26377: 26332: 26324: 26281: 26269: 26222: 26214: 26204: 26177: 26172: 26137: 26128: 26116: 26054: 26010: 26002: 25992: 25936: 25926: 25907: 25897: 25882: 25870: 25821: 25708: 25657: 25649: 25639: 25632: 25615: 25554: 25544: 25517: 25487:"Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force" 25465: 25356: 25280: 25272: 25262: 25235: 25192: 25188: 24605: 24587: 24558:"The Close-Packed-Spheron Model of Atomic Nuclei and ITS Relation to the Shell Model" 24538: 24489: 24303: 24256: 24209: 24166: 24016: 23961: 23908: 23877: 23829: 23735: 23684: 23565: 23530: 23445: 23072: 23039: 22976: 22860: 22815: 22703: 22662: 22647: 22631: 22561: 22258: 22174: 22147: 22051: 21837: 21810: 21783: 21758: 21733: 21681: 21619: 21597: 21532: 21423: 21387: 21333: 21306: 21174: 21135: 21096: 21075:

Pancharatnam, S. (1956). "Generalized theory of interference, and its applications".

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Senitzky, I. R. (1960). "Dissipation in Quantum Mechanics. The Harmonic Oscillator".

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which satisfies the Helmholtz equation, transversality, and the "box normalization":

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is determined by the degeneracy of the ground state. Many systems, such as a perfect

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Aharonov, Y.; Anandan, J. (1987). "Phase change during a cyclic quantum evolution".

21018: 20761: 20540: 20073: 19942: 19117: 18789:

Lifshitz, E. M. (1956). "The theory of molecular Attractive Forces between Solids".

18770:

Lifshitz, E. M. (1954). "The Theory of Molecular Attractive Forces between Solids".

18741: 17703:] (in German). Berlin, Heidelberg: Springer (published 1985). pp. 382–396. 17633: 16396:. As long ago as 1889 (before quantum theory or discovery of the zero point energy) 16228:

A good example of nonlinear electromagnetics is in high energy dense plasmas, where

11780:

that is formally the same as the classical equation for a linear dipole oscillator:

3852: 31272: 31164: 31144: 31139: 31134: 31129: 30986: 30966: 30923: 30888: 30858: 30702: 30153: 30036: 29628: 29623: 29480: 29376: 29061: 29001: 28927: 28600: 28534: 28490: 28351: 28316: 28299: 28246: 28175: 28167: 28071: 28063: 28008: 27951: 27910: 27791: 27753: 27715: 27677: 27627: 27607: 27572: 27556: 27552: 27505: 27439: 27242: 27214: 27146: 27078: 26989: 26917: 26895: 26855: 26835: 26815: 26803: 26783: 26753: 26683: 26594: 26435: 26369: 26344: 26320: 26316: 26261: 26167: 26108: 26046: 25984: 25965: 25893:

The Collected Papers of Albert Einstein Vol. 4 The Swiss Years: Writings, 1912–1914

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in 1915: "There would be a very real difficulty in supposing that the (force) law

31297: 31252: 30918: 30835: 30724: 30462: 30411: 30097: 30082: 29980: 29858: 29785: 29765: 29735: 29698: 29693: 29598: 29422: 29081: 28991: 28937: 28837: 28776: 28716: 28680: 28640: 28099: 28036: 28012: 27838: 27379: 27355: 27048: 26687: 26626: 26598: 26464: 26198: 25978: 25538: 25420: 25256: 24968: 23753: 23116: 22168: 21831: 21804: 21675: 21546:

Kauffmann, T.; Sun, Wen IyJ (1993). "Quaternion mechanics and electromagnetism".

21381: 21327: 21300: 20887: 20800: 19713:

Sparnaay, M. J. (1958). "Measurements of attractive forces between flat plates".

18949:

Johnson, John Bertrand (1928). "Thermal Agitation of Electricity in Conductors".

18400: 18354: 17708: 16758: 16622: 16532:, but the force produced in such an example is theorized to be exceedingly weak. 16529: 16444: 16245: 16165: 16157: 15950: 15867: 15772: 15463: 15410: 15322: 15309: 15298: 15279: 14905: 14832: 14686: 14510: 11730:

is the source field, the field generated by the dipole and acting on the dipole.

4576:

The zero-point energy of the field arises formally from the non-commutativity of

4354: 4305: 4242: 4117: 4112: 4082: 4077: 4047: 4007: 3977: 3957: 3862: 3757: 3752: 3717: 3702: 3677: 3657: 3637: 3567: 3557: 3478: 3440: 3357: 3322: 3154: 3132: 3127: 3086: 2998: 2108: 2073: 2069: 2043: 1839: 1641: 1500: 1277: 1241: 1210: 1028: 980: 890: 760: 740: 486: 326: 28303: 28139:"Über die Elektrodynamik des Vakuums auf Grund des Quantentheorie des Elektrons" 26663: 25969: 25461: 25378:"Casimir Forces and Quantum Electrodynamical Torques: Physics and Nanomechanics" 23873: 23410: 22488: 22462:"Test of zero-point energy emission from gases flowing through Casimir cavities" 22195: 22016: 21053: 20524: 20131:

Hawton, M. (1993). "Self-consistent frequencies of the electron–photon system".

19934: 19281: 19256: 19225: 19101: 19030: 16712:. In the spheron model of the nucleus, proposed by the two times Nobel laureate 16641:). The ship would then ride this wave inside a region of flat space, known as a 1177: 31360: 31307: 31189: 31078: 30667: 30642: 30421: 30128: 29836: 29805: 29795: 29417: 29407: 29241: 29011: 28966: 28944: 28832: 28676: 27795: 27794:(1991). "The Physical Significance of the Vacuum State of a Quantum Field". In 27635: 26818:(1983). "Does the Electromagnetic Mass of an Electron Depend on Where It Is?". 26265: 25485:

Chan, H. B.; Aksyuk, V. A.; Kleiman, R. N.; Bishop, D. J.; Capasso, F. (2001).

24442: 23731: 23680: 23320: 22899: 22855: 22830: 22762: 22112: 21253: 21228: 21153:

Josephson, B. D. (1962). "Possible new effects in superconductive tunnelling".

21131: 20753: 19365:"Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" 18395: 18391: 17876: 16801: 16778: 16665: 16594: 16544: 16484: 16422: 16393: 16382: 16279: 16265: 16236:

by increasing the energy gradient within the electromagnetic field and violate

16205: 16135: 15674: 15669: 15635: 15559: 15449: 15287: 14844: 14797: 14729: 14666: 14554: 14359:

would be exponentially dampened, and commutators like would approach zero for

4336: 4274: 4142: 4092: 4012: 3997: 3982: 3962: 3932: 3927: 3922: 3907: 3892: 3857: 3847: 3777: 3712: 3682: 3617: 3582: 3572: 3547: 3418: 3385: 3259: 2203: 2178: 2170: 2141: 2089: 1976: 1886: 1847: 1831: 1402: 1322: 1266: 1202: 1055: 825: 785: 765: 735: 715: 665: 631: 481: 471: 264: 50: 28780: 28770: 27264: 27150: 26993: 26921: 26439: 25988: 25866: 25404: 24295: 23614:

Peres, Asher (1978). "Test of equivalence principle for particles with spin".

23487: 23473: 23459: 23092:

Noever, David; Bremner, Christopher (1999). "Large-scale Sakharov condition".

23068: 22526: 22446: 21213: 20586: 18454: 17861: 17822: 17780: 17726: 13224:{\displaystyle \left=\delta _{\mathbf {kk'} }^{3},\delta _{\lambda \lambda '}} 12977:

We can calculate from the formal solution of the operator equation of motion

8659:{\displaystyle \rho _{0}(\omega )={\frac {\hbar \omega ^{3}}{2\pi ^{2}c^{3}}}} 5659:

for each mode of the field. The mode function satisfying these conditions is:

5521:{\displaystyle \left(\nabla ^{2}+k^{2}\right)\mathbf {A} _{0}(\mathbf {r} )=0} 5226:. Since zero-point energy is intimately connected to the non-commutativity of 4407:

In the quantum theory of the electromagnetic field, classical wave amplitudes

3897: 2993:, a system at absolute zero temperature exists in its ground state; thus, its 1301:. According to QFT the universe is made up of matter fields, whose quanta are 31388: 31232: 31214: 31199: 31179: 31083: 31052: 30883: 30763: 30697: 30672: 30272: 29755: 29608: 29499: 29333: 29303: 29256: 29021: 28751: 28708: 28614: 28546: 28502: 28431: 28258: 28129: 28083: 28020: 27885: 27858: 27829: 27669: 27619: 27451: 27407: 27342: 27314: 27286: 27234: 27230: 27218: 27121: 27040: 27001: 26947: 26887: 26786:(1981). "Radiation reaction and the nonrelativistic theory of the electron". 26767: 26695: 26622: 26606: 26559: 26532: 26492: 26447: 26408: 26328: 26273: 26181: 26120: 26014: 25940: 25919:

Einstein, A. (1993). Klein, Martin J.; Kox, A. J.; Schulmann, Robert (eds.).

25911: 25874: 25558: 25469: 25284: 25196: 24591: 24542: 24205: 23808:

Allen, J. E. (2005). "Aeronautics-1903; aerospace-2003; ? ? 2103".

23635: 22682:

Kiss, L. B. (1988). "To the problem of zero-point energy and thermal noise".

22575:

Harris, I. A. (1971). "Zero-point fluctuations and thermal-noise standards".

22322: 21427: 20989:

Berry, M. V. (1984). "Quantal Phase Factors Accompanying Adiabatic Changes".

20532: 20488:"Quintessence and the Rest of the World: Suppressing Long-Range Interactions" 20154: 19889: 19441: 18911:

Nyquist, Harry (1928). "Thermal Agitation of Electric Charge in Conductors".

18854: 18749: 18233: 18152: 18068: 18042: 17958: 17917: 17438: 17387: 16856: 16713: 16681: 16618: 16577: 16496: 16492: 16405: 16334:. This is an example of zero-point energy having multiple stable states (see 16241: 16181: 15962: 15895: 15797: 15777: 15455: 15283: 14793: 11542:{\displaystyle \mathbf {E} _{RR}(t)={\frac {2e}{3c^{3}}}\mathbf {\ddot {x}} } 6977:

are photon annihilation and creation operators for the mode with wave vector

4232: 4147: 4137: 4127: 4122: 4042: 4027: 3827: 3822: 3652: 3647: 3607: 3577: 3542: 3532: 3463: 3458: 3009: 2153: 2149: 2093: 2065: 1882: 1826: 1362: 1270: 1254: 1214: 1158: 1067: 1059: 960: 885: 880: 810: 780: 750: 621: 567: 294: 269: 147:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 58: 46: 28786: 28668: 28605: 28570: 28180: 28067: 27956: 27931: 27783: 27745: 27707: 27443: 27161:(1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum". 26904:

Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy

26758: 26654: 26373: 26226: 26050: 25661: 25513: 24809: 24784: 24162: 24090: 23821: 22776:

Dillenschneider, R.; Lutz, E. (2009). "Energetics of quantum correlations".

21995:

Edwards, W. Farrell (1981). "Classical Derivation of the London Equations".

20918: 20579:

Physical Reality and Mathematical Description Is the Zero-Point Energy Real?

19562: 18327: 14693:

Casimir force was real. Results have been repeatedly replicated since then.

7913:{\displaystyle \sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \omega _{k}} 5626:{\displaystyle \int d^{3}r\left|\mathbf {A} _{0}(\mathbf {r} )\right|^{2}=1} 1817:

Einstein's official 1921 portrait after receiving the Nobel Prize in Physics

1236:, so that it never comes to a complete stop. Zero-point radiation gives the 31169: 30991: 30893: 30796: 30707: 30617: 30335: 30277: 30262: 29638: 29251: 29246: 29046: 28266: 27965: 27922: 27564: 27459: 26959: 26711: 26381: 26336: 25835:

Dunne, G. V. (2012). "The Heisenberg-Euler Effective Action: 75 years on".

25756: 25712: 25619: 25521: 25360: 25255:

Bordag, M; Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (2009).

25239: 25061: 24609: 24213: 24170: 23881: 23739: 23688: 23600: 23043: 22864: 21366: 21139: 21061: 21010: 20038: 19500: 19325: 19290: 19241: 19159: 19109: 18932: 18586: 18501: 18480:

Bethe, Hans Albrecht (1947). "The Electromagnetic Shift of Energy Levels".

18421: 18376: 18349: 18015:

Einstein, Albert (1909). "Zum gegenwärtigen Stand des Strahlungsproblems".

17625: 16866: 16851: 16841: 16836: 16831: 16806: 16572: 16557: 16552: 16397: 16319: 16221: 16201: 16117: 15972: 15957: 15922: 15882: 15689: 15536: 15436: 15326: 15265: 14740: 14482: 14478: 5101: 4270: 4236: 4182: 4107: 3992: 3967: 3952: 3887: 3767: 3762: 3642: 3632: 3627: 3592: 3587: 3562: 3517: 3395: 3362: 3189: 3169: 2890: 2851: 2121: 2077: 1917:

Quantum theoretical re-interpretation of kinematic and mechanical relations

1386: 1150: 972: 875: 870: 805: 790: 755: 249: 54: 29930: 28355: 26058: 24680: 24637:"Propulsion on an Interstellar Scale – the Quantum Vacuum Plasma Thruster" 24616: 24582: 23342:

Swain, John (2010). "Gravitatomagnetic Analogs of Electric Transformers".

22596: 20162: 19777: 18970: 18894: 16353: 14725: 14414:

in order for to hold. In the case of a dissipative force proportional to

12164:. however, the energy density associated with the free field is infinite: 1813: 1133:) can be used when referring to a specific vacuum field, for instance the 1046:, any such energy would gravitate, and the experimental evidence from the 31289: 31007: 30976: 30956: 30903: 30365: 30192: 30077: 30067: 30031: 30006: 29676: 28442:. Vol. 77, no. 1088. pp. 298–300 – via Google Books. 27932:"Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence" 27492: 27375: 27322: 27065: 26976: 26955: 26500: 26154: 25687: 25627: 25444: 25335: 24912:(19 November 2011). "Out of the Ether: The Changing Face of the Vacuum". 24909: 24889: 24699: 22397: 22370: 20507: 19847: 19699: 19084: 18821: 18341: 16739:

Laboratories announced that they had successfully validated the use of a

16603: 16007: 15852: 15847: 15722: 15679: 15659: 15654: 15626: 15379: 15345: 15318: 14789: 14680: 14551: 14485: 12031:{\displaystyle |\Psi \rangle =|{\text{vac}}\rangle |\psi _{D}\rangle \,,} 4254: 4162: 4097: 4057: 4017: 3972: 3837: 3742: 3737: 3722: 3707: 3692: 2125: 2099: 1835: 1665: 1473: 1398: 1378: 1347: 1206: 1063: 1051: 840: 795: 730: 685: 28473:

Boyer, T. H. (1970). "Quantum Zero-Point Energy and Long-Range Forces".

28320: 28250: 26095: 24485: 24097: 24056: 23195: 23101: 22540:

MacDonald, D. K. C. (1962). "On Brownian Movement and irreversibility".

22348:

Scandurra, M. (2001). "Thermodynamic properties of the quantum vacuum".

21468:

Rocher, E. Y. (1972). "Noumenon: Elementary entity of a new mechanics".

21353:

Josephs, H.J. (1959). "The Heaviside papers found at Paignton in 1957".

20020: 19917: 18196: 14543:

The Standard Model hypothesises a field called the Higgs field (symbol:

14515: 4257:. The matter and force fields have zero-point energy. A related term is 1842:

would cause a reduction in the intensity of the diffracted radiation in

1559:, e.g., a vibrating atomic unit, as a function of absolute temperature: 1532: 31204: 31042: 30878: 30143: 30072: 29066: 29056: 28147:

Kongelige Danske Videnskabernes Selskab, Mathematisk-fysiske Meddelelse

27594: 27475: 27294: 27272: 27226: 27190: 27174: 27158: 26839: 26703: 26112: 24511:"Electromagnetic to Gravitational wave Conversion via Nuclear Holonomy" 24248: 23522: 23242: 22831:"Effects of reservoir squeezing on quantum systems and work extraction" 22354: 22143: 21524: 21088: 20736: 20660: 19388: 19304:

Purcell, E. M. (1946). "Proceedings of the American Physical Society".

19257:"Inhibited Spontaneous Emission in Solid-State Physics and Electronics" 18699: 18656: 18284: 18103: 17993: 17533: 16821: 16816: 16811: 16717: 16610: 16213: 16139: 15817: 14733: 14720: 14454: 4381: 4361: 4316: 3817: 3732: 3597: 3537: 3229: 3164: 2983: 2134: 2129: 2085: 2000: 1927: 1923: 1822: 1672: 1548: 1469: 1366: 1142: 1134: 830: 800: 720: 695: 690: 675: 28075: 25601: 25313: 25155: 24756:"Scientists Catch "Virtual Particles" Hopping In and Out of Existence" 24534: 24367: 24350: 23944: 23663: 23364:"Physicist Predicts Gravitational Analogue Of Electrical Transformers" 23285: 23138: 23025: 22030:

Fröhlich, H. (1966). "Macroscopic wave functions in superconductors".

21489: 20462:"The Dark Universe Is Waiting. What Will the Euclid Telescope Reveal?" 19820: 19769: 19233: 19038: 17192:(Press release). Niels Bohr Institute. 19 January 2007. Archived from 16168:, Aharonov–Anandan, Pancharatnam and Chiao–Wu phase rotation effects, 15527:

radiation is distributed evenly, why the Universe is flat, and why no

1422: 31262: 31088: 30873: 30498: 29530: 29226: 28917: 28569:

Lahteenmaki, P.; Paraoanu, G. S.; Hassel, J.; Hakonen, P. J. (2013).

28538: 28171: 25230: 25205: 24391: 24114:"On the Mechanism for a Gravity Effect using Type II Superconductors" 23299:

Forward, R. L. (1961). "General Relativity for the Experimentalist".

23156:

10.1002/1521-3889(200105)10:5<393::AID-ANDP393>3.0.CO;2-Z

23058: 22238: 21503:

Imaeda, K. (1976). "A new formulation of classical electrodynamics".

20279: 20267: 19065:"Extraction of Work from a Single Thermal Bath in the Quantum Regime" 17668: 17581: 17556: 16291: 16186: 16121: 15967: 15862: 15629:

produced by colliding protons decaying into hadron jets and electrons

15520: 15405: 15358: 14700:

An interesting hypothetical side effect of the Casimir effect is the

14683: 13231:

and that equal-time particle and field operators commute, we obtain:

4339: 4332: 4320: 3017: 2061: 1649: 1435: 1393:. Active areas of research include the effects of virtual particles, 1390: 1293: 1039: 321: 29138: 28206:. JANNAF Joint Propulsion Meeting; 5–9 December 2011; Huntsville, AL 27194: 24773:"Physicists Observe Weird Quantum Fluctuations Of Empty Space—Maybe" 23544:

O'Connell, R. F. (1970). "The gravitational field of the electron".

21641:

Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse

20819: 20780: 17837: 17414: 17363: 16385:) or uncompensated momentum; such efforts are seen as tantamount to 16318:= electric charge density / mass density), without involving either 11693:

satisfied by the field in the (source free) vacuum. For this reason

30596: 30026: 30011: 28197:

White, H.; March, P.; Williams, N.; O'Neill, W. (5 December 2011).

27930:

Scully, M. O.; Zubairy, M. S.; Agarwal, G. S.; Walther, H. (2003).

27509: 27256: 27082: 26740: 26466:

The Weight of the Vacuum : A Scientific History of Dark Energy

23389:

DeWitt, Bryce S. (1966). "Superconductors and Gravitational Drag".

23008: 20901: 18129:

QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga

17647:

Mulliken, Robert S. (1924). "The band spectrum of boron monoxide".

17061: 16721: 16070: 15917: 15857: 15787: 15727: 15548: 15540: 15479: 15383: 15268:(left) becomes linearly polarised as it travels through the vacuum. 14704:, a hypothetical phenomenon in which light signals travel slightly 8583:

or in other words the spectral energy density of the vacuum field:

7236:

for the total vector potential in free space. Using the fact that:

4350: 4250: 3782: 2196: 1991: 1980: 1890: 1465: 1342: 1338: 1302: 1282: 1233: 1190: 992: 968: 700: 28660: 28587: 28233: 28121: 27995: 27539: 27399: 26939: 26912: 26646: 26581: 26484: 26430: 26391:

Quantum Generations: A History of Physics in the Twentieth Century

26303: 26248: 26218: 25849: 25584: 25276: 23842: 23348: 22951: 22790: 22387: 22095: 19863:"Demonstration of the Casimir Force in the 0.6 to 6 μm Range" 19448: 15613: 15487:

Why does the observable universe have more matter than antimatter?

15095:

are the standard deviations of position and momentum states that:

14564:

in the ground state (the vacuum expectation value or VEV) is then

4586:. This is true for any harmonic oscillator: the zero-point energy 31312: 31279: 31257: 31237: 18609:

Proceedings of the Royal Netherlands Academy of Arts and Sciences

16689: 16153: 15152: 14671: 5880:{\displaystyle \mathbf {A} (x+L,y+L,z+L,t)=\mathbf {A} (x,y,z,t)} 5216:

In other words, within the normal ordering symbol we can commute

4246: 2994: 2113: 1629:{\displaystyle \varepsilon ={\frac {h\nu }{e^{h\nu /(kT)}-1}}\,,} 988: 963:, atoms and molecules retain some vibrational motion. Apart from 28743: 28700: 28568: 27821: 27775: 27760:. Reading, Massachusetts: Advanced Book Program, Perseus Books. 27737: 27722:. Reading, Massachusetts: Advanced Book Program, Perseus Books. 27699: 27684:. Reading, Massachusetts: Advanced Book Program, Perseus Books. 27661: 27334: 27306: 26879: 26524: 26006: 25730:"The Quantum Theory of the Emission and Absorption of Radiation" 25653: 24704:"NASA Team Claims 'Impossible' Space Engine Works—Get the Facts" 20051: 12572:

The important point of this is that the zero-point field energy

31247: 31242: 30848: 30021: 27357:

Advanced Electromagnetism: Foundations, Theory and Applications

24939:"Weighty Implications: NASA Funds Controversial Gravity Shield" 21226: 19992:

Munday, J. N.; Capasso, Federico; Parsegian, V. Adrian (2009).

17881:(3rd ed.). Cambridge: Cambridge University Press. p. 17034: 17032: 16702: 16685: 16548: 16229: 15544: 15392: 15141:{\displaystyle \Delta _{x}\Delta _{p}\geq {\frac {1}{2}}\hbar } 14732:

of energy levels in hydrogen – relativistic corrections to the

2943:. In these terms, an example of zero-point energy is the above 2672:

The expectation value of the energy must therefore be at least

1431: 1314: 1306: 1232:

Zero-point radiation continually imparts random impulses on an

1194: 1106: 1012: 996: 984: 940: 26861:

The Quantum Vacuum: An Introduction to Quantum Electrodynamics

25669:

Decca, R. S.; López, D.; Fischbach, E.; Krause, D. E. (2003).

25376:

Capasso, F.; Munday, J. N.; Iannuzzi, D.; Chan, H. B. (2007).

23903:(2nd ed.). New York: Cambridge University Press. p. 22990:

Gardas, Bartłomiej; Deffner, Sebastian; Saxena, Avadh (2016).

20720:

Jaffe, R. L. (2005). "Casimir effect and the quantum vacuum".

20422: 19639: 19637: 19610: 18997: 18995: 17169: 17167: 17165: 15286:, thus in principle a rotation of the polarization frame (the 31222: 30843: 28200:

Eagleworks Laboratories: Advanced Propulsion Physics Research

28196: 28157: 24874:"Quantum Particles Seen Distorting Light From a Neutron Star" 24823:"Something from Nothing? A Vacuum Can Yield Flashes of Light" 24798:"Astronomers Spot Signs of Weird Quantum Distortion in Space" 24686: 24622: 24030:

Rounds, Frederic N. (1998). "Anomalous Weight Behavior in YBa

23760:. Vol. 85, no. 1194. 14 February 1980. p. 485. 20639:

Schwinger, Julian (1975). "Casimir effect in source theory".

20581:. Dordrecht: D. Reidel Publishing Company. pp. 124–132. 19197:

Hulet, Randall G.; Hilfer, Eric S.; Kleppner, Daniel (1985).

17491:

Einstein, Albert (1922). Jeffery, G. B.; Perrett, W. (eds.).

16564:. In 1983, Dewitt's work was substantially expanded by Ross. 16467: 16148:

substance, but at times can be viewed as a turbulent virtual

15395:

not cause a large cosmological constant? What cancels it out?

15365:(HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE). 15354: 15341: 15330: 14420:, by contrast, the fluctuation force must be proportional to 14391:. With the vacuum field included, however, the commutator is 1894: 1439: 1318: 1310: 1269:. This results in motion even at absolute zero. For example, 1253:

which also describes the system as a harmonic oscillator, or

1016: 1008: 1000: 25671:"Measurement of the Casimir Force between Dissimilar Metals" 25566:

Correa, L. A.; Palao, J. P.; Alonso, D.; Adesso, G. (2014).

22906: 22521:(Thesis for: MS). Advisors: Fernandes, Luis & Amaro, F. 21616:

Energy potential: Toward a new electro-magnetic field theory

20243: 20112: 19129: 19127: 17494:

Sidelights on Relativity: Ether and the Theory of Relativity

17184: 17182: 17029: 15368: 2915:. This leads to a convention of writing the Planck constant 1919:" the zero-point energy was derived from quantum mechanics. 28685:(1954 reprint 3rd ed.). New York: Dover Publications. 25634:

Superforce: The Search for a Grand Unified Theory of Nature

21576: 21409:"Tesla's Nonlinear Oscillator-Shuttle-Circuit (OSC) Theory" 21280: 20080: 19798: 19685: 19634: 19002:

Koch, Roger H.; Van Harlingen, D. J.; Clarke, John (1981).

18992: 18807: 18602:"On the attraction between two perfectly conducting plates" 18350:"Fine Structure of the Hydrogen Atom by a Microwave Method" 18212:

Spontaneous emission and laser oscillation in microcavities

17394: 17162: 16720:

are among the components of this structure. Similarly to a

16515: 16491:(FDT) which was originally formulated in classical form by 16435: 16075: 6436: 4369: 2148:(FDT) which was originally formulated in classical form by 1262: 1079: 964: 42: 27195:"Über die Begründung das Gesetzes des schwarzen Strahlung" 26506:

Black-Body Theory and the Quantum Discontinuity, 1894-1912

25779:

Drexhage, K. H. (1970). "Monomolecular Layers and Light".

22918: 22710: 19961: 17239: 17215: 17073: 17002: 14519:

The potential for the Higgs field, plotted as function of

11749:

we obtain an equation for the Heisenberg-picture operator

8668:

The zero-point energy density in the frequency range from

2974:

If more than one ground state exists, they are said to be

1846:

even as the temperature approached absolute zero. In 1916

1430:

Zero-point energy evolved from historical ideas about the

30853: 27352:"Foundational Electrodynamics and Beltrami Vector Fields" 26902:. In Greenberger, D.; Hentschel, K.; Weinert, F. (eds.). 24072: 23845:"Wormholes, Time Machines, and the Weak Energy Condition" 22645:

Kleen, W. (1985). "Thermal noise and zero-point-energy".

22519:

Study of atomic energy shifts induced by Casimir cavities

22268: 22136:

Non-equilibrium Thermodynamics in an Energy-Rich Universe

20410: 20169: 19973: 19598: 19124: 19062: 17179: 17051: 17049: 17047: 16965: 16963: 16410: 8838: 7980:

shows. The summation becomes approximately the integral:

5785:

and impose on the field the periodic boundary condition:

5772:

satisfied for the Coulomb gauge in which we are working.

1075: 26203:(2005 ed.). Mineola, New York: Dover Publications. 20304: 20181: 19949: 19460: 18982: 18980: 16567:

From 1971 to 1974 Henry William Wallace, a scientist at

15435:, where observed acceleration is due to the energy of a 14629:. The measured value of this parameter is approximately 1038:

The notion of a zero-point energy is also important for

28106:(2nd ed.). Cambridge: Cambridge University Press. 27417:"Direct Sampling of Electric-Field Vacuum Fluctuations" 27354:. In Barrett, Terence William; Grimes, Dale M. (eds.). 27179:

Verhandlungen der Deutschen Physikalischen Gesellschaft

27163:

Verhandlungen der Deutschen Physikalischen Gesellschaft

24388: 23649:

Obukhov, Yuri N. (2001). "Spin, gravity, and inertia".

22232:"Zero-Point Energy: Can We Get Something From Nothing?" 21449: 21386:. Cornell: Cornell University Press. pp. 165–166. 20940:

Altshuler, B. L.; Aronov, A. G.; Spivak, B. Z. (1981).

20255: 19994:"Measured long-range repulsive Casimir–Lifshitz forces" 19622: 19001: 17019: 17017: 16980: 16978: 16913: 16911: 16909: 16896: 16894: 16892: 16504:, by exploiting certain quantum mechanical properties. 15539:, the hypothetical physical processes that produced an 15503:

just after the Big Bang. It explains the origin of the

15282:. At this point the vacuum has all the properties of a 8074: 4386:

The oldest and best known quantized force field is the

2842:, requires solving for the ground state of the system. 1472:, it was not until much later on with the beginning of 1117:. Sometimes used interchangeably with it are the terms 28571:"Dynamical Casimir Effect in a Josephson Metamaterial" 25385:

IEEE Journal of Selected Topics in Quantum Electronics

24739:"It's Confirmed: Matter is Merely Vacuum Fluctuations" 24419: 20768: 20577:

Enz, Charles P. (1974). Enz, C. P.; Mehra, J. (eds.).

20398: 19991: 19524: 19522: 19507: 19332: 18461: 17044: 16960: 15404:

In the late 1990s it was discovered that very distant

9214: 8917: 8478: 8391: 8321: 8038:{\displaystyle {\frac {4\pi hV}{c^{3}}}\int v^{3}\,dv} 7886: 7496: 5253: 5040: 5020: 4965: 4905: 4804: 4710: 4663: 4308:

is needed to make sense of quantum field theories. In

2971:

of the Hamiltonian of the system in the ground state.

2513: 2254: 2068:, now known as the Lamb shift, and measurement of the 1276:

Given the equivalence of mass and energy expressed by

26962:(2003). "The Cosmological Constant and Dark Energy". 25976: 23236:

Lano, R. P. (1996). "Gravitational Meissner Effect".

22284: 22282: 21836:. Singapore: Imperial College Press. pp. 91–92. 21639:

Noether, E. (1918). "Invariante Variationsprobleme".

20939: 20837: 20825: 20786: 20681: 20285: 20273: 19412:

Q is for Quantum: An Encyclopedia of Particle Physics

18977: 17417:[Interference of X-rays and thermal motion]. 17302: 17280: 17278: 17227: 17114: 17112: 17067: 16304: 15447:

in space or time. Scalar fields are predicted by the

15264:

Light coming from the surface of a strongly magnetic

15172: 15101: 14949: 14914: 14426: 14219: 13994: 13657: 13237: 13095: 12983: 12651: 12170: 12059: 11978: 11786: 11755: 11602: 11472: 11260: 11041: 10918: 10654: 10531: 10299: 10065: 9957: 9656: 9032: 8851: 8692: 8589: 8290: 8148: 7986: 7928: 7869: 7530: 7410: 7242: 7016: 6682: 6456: 6322: 6116: 6024: 5893: 5791: 5665: 5560: 5455: 5264: 5120: 4843: 4612: 4504: 4434: 3026: 2680: 2506: 2379: 2224: 1850:

proposed that empty space was filled with zero-point

1838:, who noted that zero-point energy of the atoms of a 1720: 1565: 1257:, that fluctuates between various energy states (see 85: 25100:"Does Dark Energy Spring From the 'Quantum Vacuum?'" 23843:

Thorne, Kip; Michael Morris; Ulvi Yurtsever (1988).

23709: 23578: 21700: 20362: 20097: 20095: 19833: 19166: 19058: 19056: 18166: 18164: 18162: 17971: 17878:

The mathematical theory of electricity and magnetism

17754:"On the Constitution of Atoms and Molecules, Part I" 17733: 17014: 16990: 16975: 16906: 16889: 16768: 16361:. On the left, the dramatic expansion occurs in the 11551:

The total field acting on the dipole has two parts,

11466:

is regarded as the "observed" mass then we can take

1464:

were not made of atoms, but were continuous. To the

24857:"Anti-Gravity Propulsion Comes "Out of the Closet"" 24478:

AIAA 38th Joint Propulsion Conference & Exhibit

24263: 23500: 23057:Gemmer, Jochen; Michel, M.; Mahler, Günter (2009). 22989: 22775: 22722: 20555:

Origins: Fourteen Billion Years of Cosmic Evolution

19519: 19344: 19196: 19063:Allahverdyan, A. E.; Nieuwenhuizen, Th. M. (2000). 18516: 18398:(1948). "On the Intrinsic Moment of the Electron". 17415:"Interferenz von Röntgenstrahlen und Wärmebewegung" 17343: 17290: 17263: 17205: 17203: 14939:)) we can form a dimensionless quantity called the 1705:, as an additional term dependent on the frequency 16:

Lowest possible energy of a quantum system or field

27099: 27018: 26906:. Berlin, Heidelberg: Springer. pp. 864–866. 26664:"Inertia: Does Empty Space Put Up the Resistance?" 26566: 26138:"Polarization evolution in strong magnetic fields" 26021: 25631: 24840:"Science: Can Photons Travel 'Faster Than Light'?" 24310: 23982: 23189:Superconductor with Composite Crystal Structure". 23176: 23056: 22329: 22279: 22170:Cosmic Evolution: The Rise of Complexity in Nature 21806:Cosmic Evolution: The Rise of Complexity in Nature 21596: 20942:"The Aaronov-Bohm effect in disordered conductors" 20884: 20553:Tyson, Neil deGrasse and Donald Goldsmith (2004), 20374: 19199:"Inhibited Spontaneous Emission by a Rydberg Atom" 18126: 17939:. Cambridge: Cambridge University Press. pp. 17557:"Behaviour of Condensed Helium near Absolute Zero" 17275: 17251: 17152: 17119: 17109: 16310: 15191: 15140: 15039: 14927: 14432: 14324: 14205: 13980: 13618: 13223: 13079: 12967: 12562: 12141: 12030: 11882: 11772: 11685: 11541: 11448: 11246: 11027: 10890: 10637: 10428: 10269: 10040: 9943: 9642: 8973: 8827: 8658: 8575: 8276: 8037: 7972: 7912: 7844: 7514: 7396: 7228: 6923: 6668: 6442: 6190: 6102: 6006: 5879: 5729: 5625: 5520: 5421: 5206: 5092: 4826: 4542: 4469: 3073: 2772: 2662: 2484: 2353: 1970: 1803: 1671:The concept of zero-point energy was developed by 1628: 146: 28509: 28313:AIAA 37th Joint Propulsion Conference and Exhibit 27360:. Singapore: World Scientific. pp. 217–249. 27047: 26728:Monthly Notices of the Royal Astronomical Society 26631:(3rd ed.). Oxford: Oxford University Press. 26142:Monthly Notices of the Royal Astronomical Society 25162: 24509:Fontana, Giorgio; Binder, Bernd (16 March 2009). 24457: 24044:Proc. NASA Breakthrough Propulsion Phys. Workshop 23212:"Antigravity machine weighed down by controversy" 23114: 23094:AIAA 35th Joint Propulsion Conference and Exhibit 22459: 22368: 22138:. Understanding Complex Systems. pp. 21–31. 21752: 21332:. Cornell: Cornell University Press. p. 17. 21267: 21220: 20428: 20102: 20092: 19053: 18633: 18159: 16923: 16688:'s advanced research and development facility in 16413:as well as in China, Germany, Russia and Brazil. 15597: 11958:the free field is the only field in existence at 955:in their lowest energy state as described by the 31386: 24962:. Vol. 145, no. 1966. pp. 30–33. 24916:. Vol. 212, no. 2839. pp. 50–52. 21727: 20386: 18772:Journal of Experimental Theoretical Physics USSR 17200: 17097: 16416: 16164:(AB) and Altshuler–Aronov–Spivak (AAS) effects, 13638:. Then the integrand above is sharply peaked at 6450:in terms of which the vector potential becomes: 6225:means that there are two independent choices of 4402: 4375: 4349:, it is sometimes said that the contribution of 2370:is the minimum of the classical potential well. 1931:of classical mechanics was nicely summarized by 1309:and quarks), and force fields, whose quanta are 28575:Proceedings of the National Academy of Sciences 27579: 27524: 27299:Physikalische Abhandlungen und Vorträge. Vol. 2 27278:Vorlesungen über die Theorie der Wärmestrahlung 27097: 26288: 26233: 26080: 24562:Proceedings of the National Academy of Sciences 22912: 21677:Patterns and Interfaces in Dissipative Dynamics 21594: 21187: 21031: 20331: 20327: 20249: 19616: 19454: 19356: 18437:H. A. Kramers: Between Tradition and Revolution 18209: 18041: 17173: 17085: 16752: 15192:{\displaystyle \alpha \approx {\frac {1}{129}}} 14465:vacuum state, characterized by a non-vanishing 14448: 10293:is found similarly from the Hamiltonian to be: 2982:which acts non-trivially on a ground state and 1999:Building on the work of Heisenberg and others, 28715: 28413: 26718: 26022:Haisch, B.; Rueda, A.; Puthoff, H. E. (1994). 25977:Greiner, W.; Müller, B.; Rafelski, J. (2012). 25375: 25291: 25169:Journal of Physics A: Mathematical and General 24328: 22877: 22267:. Chedd-Angier Production Company. 1997–1998. 21582: 20344: 20086: 19747: 19145: 18340: 17511: 16613:et al. published work showing how traversable 16597:'s then Rocket Propulsion Laboratory (RPL) at 14711:between two closely spaced conducting plates. 7973:{\displaystyle {\frac {8\pi v^{2}dv}{c^{3}}}V} 5753:in order to have the transversality condition 30812: 30208: 29946: 29154: 28816: 28802: 28414:Matthews, R.; Sample, I. (1 September 1996). 28218: 27929: 27865: 27634: 26544:(2012 ed.). Cambridge University Press. 26462: 26196: 26024:"Inertia as a Zero-Point-Field Lorentz Force" 25565: 25320: 25254: 25123: 24974: 24508: 23543: 23117:"Inertial mass and the quantum vacuum fields" 23091: 22924: 22716: 21077:Proceedings of the Indian Academy of Sciences 20715: 20713: 20118: 19967: 19569: 19133: 18944: 18942: 18906: 18904: 18868: 18866: 18864: 18834: 18736:(9). Institute of Physics Publishing: 29–32. 18564: 18246: 17930: 17361: 17245: 17221: 17156: 17079: 17038: 16625:. In 1993 Scharnhorst and Barton showed that 16097: 4208: 3104:corresponds to the ground-state energy), and 3074:{\displaystyle {\frac {h^{2}n^{2}}{8mL^{2}}}} 913: 30826: 28416:"'Anti-Gravity' Device Gives Science a Lift" 28310: 27980: 27870:. Cambridge UK: Cambridge University Press. 27474: 26954: 26820:International Journal of Theoretical Physics 25668: 25419: 24977:"Scientists Examine Nothing, Find Something" 24698: 24475: 24078: 22649:Noise in Physical Systems and 1/F Noise 1985 22259:"Zero-point energy, on season 8 , episode 2" 21753:Prigogine, Ilya; Stengers, Isabelle (1984). 21545: 21305:. Singapore: World Scientific. p. 438. 21298: 21227:K. v. Klitzing; G. Dorda; M. Pepper (1980). 21074: 20479: 20416: 20323: 20225:Quigg, C (1998). Espriu, D; Pich, A (eds.). 20199: 19979: 19254: 18872: 18051:The Historical Development of Quantum Theory 17935:The Stability of Matter in Quantum Mechanics 17603: 17148: 16946: 16571:was issued with three patents. Wallace used 16290:in the plasma focus display the same paired 15391:Why does the large zero-point energy of the 14651: 12581:does not affect the Heisenberg equation for 12130: 12090: 12084: 12060: 12021: 12003: 11987: 4280:Each point in space makes a contribution of 2214:giving its energy) can be approximated as a 2210:of a general system (the quantum-mechanical 1111: 1097: 1062:has a negative zero-point energy, while the 141: 115: 29960: 27414: 27025:(8th ed.). New York: Alfred A. Knopf. 25568:"Quantum-enhanced absorption refrigerators" 25484: 22829:Huang, X. L.; Wang, Tao; Yi, X. X. (2012). 22828: 21567:"QUATERNIONS AND THREE TEMPORAL DIMENSIONS" 21355:Proceedings of the IEE - Part C: Monographs 21190:"The discovery of tunnelling supercurrents" 20802:Topological Foundations of Electromagnetism 20449:. Vol. 232, no. 3097. p. 35. 19955: 19902: 18537:"Exclusion principle and quantum mechanics" 18081: 17895: 15481: 15385: 4368:, or the zero-point energy to the particle 1161:and the average energy value is called the 30902: 30819: 30805: 30222: 30215: 30201: 29953: 29939: 29161: 29147: 28809: 28795: 28771:the issue of vacuum energy and dark energy 27758:Particles, Sources, and Fields: Volume III 27752: 27714: 27676: 26135: 25947: 24991: 23927: 23115:Haisch, B.; Rueda, A.; Dobyns, Y. (2001). 22460:Dmitriyevaa, Olga; Moddel, Garret (2012). 22369:Moddel, Garret; Dmitriyevaa, Olga (2009). 21830:Kais, Sabre (2011). Popelier, Paul (ed.). 21599:An Introduction to Magneto-Fluid Mechanics 20805:. Singapore: World Scientific. p. 2. 20710: 20626: 20622: 20618: 20310: 19646:"The Mystery of the Cosmological Constant" 19534: 19410:Gribbin, J. R. (1998). Gribbin, M. (ed.). 19362: 19338: 18939: 18901: 18861: 18723:"The Casimir effect: a force from nothing" 18390: 17689: 17455:Verhandlungen der Deutschen Physikalischen 16332:without any associated rise in temperature 16104: 16090: 15612: 14816: 10525:is the time evolution operator satisfying 8142:is infinite. We can make the replacement: 4327:require the energy to be small enough for 4215: 4201: 3126: 2007:. Dirac described the quantization of the 920: 906: 28722:Structured Vacuum: Thinking About Nothing 28604: 28586: 28379: 28232: 28179: 28136: 27994: 27955: 27720:Particles, Sources, and Fields: Volume II 27593: 27538: 27491: 27177:(1911). "Eine neue Strahlungshypothese". 27064: 26975: 26911: 26757: 26739: 26580: 26539: 26429: 26302: 26247: 26171: 26153: 26094: 25925:. Princeton: Princeton University Press. 25896:. Princeton: Princeton University Press. 25848: 25837:International Journal of Modern Physics A 25686: 25609: 25583: 25443: 25334: 25229: 25203: 25145: 24936: 24736: 24663:"Nasa validates 'impossible' space drive" 24599: 24581: 24425: 24349: 24329:Tajmar, Martin; de Matos, Clovis (2006). 24111: 24096: 24055: 23943: 23773:"Electric Propulsion Study: Final Report" 23662: 23347: 23241: 23194: 23137: 23033: 23007: 22950: 22854: 22789: 22740: 22609: 22539: 22516: 22396: 22386: 22353: 22347: 22173:. Harvard University Press. p. 216. 22094: 22080: 21809:. Harvard University Press. p. 139. 21252: 21152: 20900: 20861: 20735: 20638: 20506: 20404: 20261: 20187: 20175: 20028: 19916: 19860: 19479:Wheeler, John Archibald (1955). "Geons". 19280: 19083: 18720: 18467: 18375: 18262: 17580: 17023: 16376: 16144: 15369:Speculated involvement in other phenomena 15321:looked at the light coming from the star 13528: 12536: 12024: 10799: 10631: 10608: 10559: 10034: 8558: 8461: 8028: 6404: 2656: 2478: 2347: 2164:was developed describing the system of a 1622: 1536:Planck in 1918, the year he received the 1405:, a reason for the observed value of the 1027:such that there is no contradiction with 28344:"Moving Mirrors Make Light From Nothing" 28341: 28098: 28035: 27682:Particles, Sources, and Fields: Volume I 27098:Peskin, M. E.; Schroeder, D. V. (1995). 26661: 25918: 25889: 25807: 25778: 24953: 24729: 24660: 23209: 22936: 22225: 22223: 22221: 22193: 22166: 22133: 22029: 21802: 21548:Annales de la Fondation Louis de Broglie 21416:Annales de la Fondation Louis de Broglie 20557:, W. W. Norton & Company, pp. 84–85. 20204:(4th ed.). CRC Press. p. 343. 19712: 19540: 19184: 19172: 18788: 18769: 18203: 18182: 18124: 18014: 17646: 17490: 17471: 17400: 17127: 17123: 17008: 16969: 16942: 16539:, a physicist and aerospace engineer at 16516:Space travel and gravitational shielding 16352: 16134:, quantum electrodynamics (QED) and the 15292: 15250: 14724: 14670: 14514: 2855: 2373:The uncertainty principle tells us that 2184: 2098: 2015:with the introduction of the concept of 1990: 1862: 1812: 1531: 1527: 1421: 1417: 1227: 1176: 975:also has these properties. According to 41: 28675: 28510:Bulsara, A. R.; Gammaitoni, L. (1996). 27244:La Théorie du Rayonnement et les Quanta 27225: 27189: 27102:An Introduction To Quantum Field Theory 27016: 26894: 26854: 26814: 26782: 25083:"Vacuum Energy: Something For Nothing?" 24753: 24702:; Greshko, Michael (21 November 2016). 24555: 23894: 23770: 23648: 23298: 23256: 22300: 21994: 21953: 21912: 21871: 21782:(1998 ed.). Vintage. p. 308. 21638: 21455: 21406: 21352: 21286: 21109: 20798: 20774: 20485: 20356: 20326:and Kostelecký and Mewes ( 20301:for a historical review of the subject. 19628: 19604: 19528: 19478: 19466: 19409: 19350: 19303: 18986: 18948: 18910: 18600:Casimir, Hendrik Brugt Gerhard (1948). 18599: 18434: 18133:. Princeton University Press. pp. 17497:. New York: Methuen & Co. pp. 17329: 17325: 17308: 17296: 17055: 16917: 16627:the speed of a photon will be increased 15569: 15236: 5554:and assume it is normalized such that: 4606:appears when we write the Hamiltonian: 4573:associated with these classical modes. 3111: 2144:and Theodore Welton proved the quantum 31387: 30293:Two-dimensional conformal field theory 29007:Atomic, molecular, and optical physics 28638: 28430: 28396: 28362: 27892: 27866:Scully, M. O.; Zubairy, M. S. (1997). 27790: 27293: 27271: 27173: 27157: 26621: 26463:Kragh, H. S.; Overduin, J. M. (2014). 26351: 25626: 25536: 25060: 25008: 24908: 24888: 24281: 24269: 24226: 24183: 24140: 24029: 23388: 23210:Matthews, Robert (21 September 1996). 22992:"Non-hermitian quantum thermodynamics" 22728: 22574: 21963:International Journal of Fusion Energy 21922:International Journal of Fusion Energy 21881:International Journal of Fusion Energy 21857:"Classical Physics Makes a Comeback". 21777: 21673: 21502: 21467: 20459: 20130: 19643: 18677: 18305: 17451: 17362:Einstein, Albert; Stern, Otto (1913). 17337: 17333: 17321: 17269: 17257: 17233: 17135: 17131: 16996: 16984: 16954: 16929: 16900: 16495:(1928) as an explanation for observed 16434:In 1999, Pinto, a former scientist at 15828:de Sitter invariant special relativity 15808:N = 4 supersymmetric Yang–Mills theory 15445:variation of the fundamental constants 14891:i.e. a measure of the strength of the 11939:According to our earlier equation for 10436:in the electric dipole approximation. 2152:(1928) as an explanation for observed 2042:This view was also later supported by 30800: 30196: 29934: 29168: 29142: 28790: 28645:(6th ed.). Boston: McGraw-Hill. 28472: 28446: 28281: 27836: 27321: 27128: 26509:. New York: Oxford University Press. 26418:Archive for History of Exact Sciences 26415: 26388: 25834: 25727: 25543:. San Francisco: Saint Austin Press. 25080: 25029: 25004:from the original on 7 February 2017. 24871: 24837: 24634: 24316: 23807: 23613: 23341: 22644: 22335: 22288: 22218: 22196:"Peeking inside an electron's screen" 21657: 21613: 20988: 20826:Greiner, Müller & Rafelski (2012) 20787:Greiner, Müller & Rafelski (2012) 20719: 20440: 20380: 20352: 20298: 20286:Greiner, Müller & Rafelski (2012) 20274:Greiner, Müller & Rafelski (2012) 20224: 20200:Aitchison, Ian; Hey, Anthony (2012). 20106: 19513: 18534: 18522: 18479: 18384: 18170: 18053:. Vol. 6. Springer. p. 57. 17874: 17739: 17554: 17412: 17284: 17068:Greiner, Müller & Rafelski (2012) 16950: 16152:that can have complex vortices (i.e. 14644:superconducting medium; e.g., in the 12042:is the vacuum state of the field and 6316:. Thus we define the mode functions: 2060:Throughout the 1940s improvements in 1709:, which was greater than zero (where 31343: 27374: 27349: 27327:Introductory Quantum Electrodynamics 27247:. Paris: Gauthier-Villars. pp. 26499: 25206:"Dark energy: Staring into darkness" 25097: 25039:Aviation Week & Space Technology 24854: 24820: 24687:White, March, Lawrence et al. (2016) 24623:White, March, Williams et al. (2011) 24463: 23754:"Antigravity Not So Crazy After All" 23423: 23235: 22681: 22275:from the original on 1 January 2006. 22251: 22064: 21829: 21379: 21325: 21181: 20368: 18334: 17835: 17793: 17751: 17480:] (in German). Berlin: Springer. 17349: 17209: 17103: 16478: 16440:Jet Propulsion Laboratory at Caltech 15758:6D (2,0) superconformal field theory 15297:Wide field view of the neutron star 8075:Necessity of the vacuum field in QED 2986:with the Hamiltonian of the system. 1926:had proposed what is now called the 31367: 28465: 28041:"The Cosmological Constant Problem" 27806:. Oxford: Oxford University Press. 27646:. Oxford: Oxford University Press. 27301:. Braunschweig: Vieweg & Sohn. 26197:Itzykson, C.; Zuber, J.-B. (1980). 26136:Heyl, J. S.; Shaviv, N. J. (2000). 25261:. Oxford: Oxford University Press. 25110:from the original on 29 March 2014. 24987:from the original on 31 March 2013. 24966:from the original on 13 April 2016. 24937:Lucentini, J. (28 September 2000). 24795: 24770: 22242:National Ground Intelligence Center 22032:Proceedings of the Physical Society 21728:Nicolis, G.; Prigogine, I. (1977). 21709:. Royal Swedish Academy of Sciences 21703:"The Nobel Prize in Chemistry 1977" 21595:Ferraro, V .; Plumpton, C. (1961). 20841:Proceedings of the Physical Society 20576: 20443:"The Universe is Flat as a Pancake" 20392: 20348: 19688:Quarterly Reviews, Chemical Society 19586:. Particle Data Group. 12 July 2012 19577:"Higgs bosons: theory and searches" 18835:Mahanty, J.; Ninham, B. W. (1976). 18810:Quarterly Reviews, Chemical Society 18546:. Royal Swedish Academy of Sciences 17931:Lieb, E. H.; Seiringer, R. (2009). 17091: 15763:Noncommutative quantum field theory 15625:particle detector data depicting a 15505:large-scale structure of the cosmos 15491:(more unsolved problems in physics) 15469: 15399:(more unsolved problems in physics) 15338:National Institute for Astrophysics 14501:to characterize such interactions. 11686:{\displaystyle \left\mathbf {E} =0} 5254:Electromagnetic field in free space 4323:. Other scientists specializing in 2017:creation and annihilation operators 13: 28409:from the original on 2 March 2015. 28380:Cartlidge, E. (17 November 2011). 28329: 25812:. Vol. 12. pp. 163–232. 25801:10.1038/scientificamerican0370-108 24737:Battersby, S. (20 November 2008). 23503:General Relativity and Gravitation 22659:10.1016/B978-0-444-86992-0.50072-2 21603:. Oxford: Oxford University Press. 21564: 19673:10.1038/scientificamerican0588-106 18210:Yokoyama, H.; Ujihara, K. (1995). 17697:Wissenschaftliche Originalarbeiten 17120:Haisch, Rueda & Puthoff (1994) 16232:occur which seemingly violate the 15290:) can be observed in empty space. 15113: 15103: 13787: 13782: 13514: 12525: 12127: 12093: 11984: 11651: 11641: 11609: 10016: 9986: 9976: 9783: 9555: 9450: 9365: 9225: 8284:the zero-point energy density is: 7404:we find the field Hamiltonian is: 5462: 3014:particle in a one-dimensional well 1397:, the difference (if any) between 452:Sum-over-histories (path integral) 138: 112: 68:Part of a series of articles about 14: 31461: 30788:Template:Quantum mechanics topics 30181:Template:Quantum mechanics topics 28781:the cosmological constant problem 28760: 28459:from the original on 27 May 2017. 28392:from the original on 30 May 2017. 28382:"How to Turn Darkness into Light" 28375:from the original on 30 May 2017. 28365:"Light Pulled Out of Empty Space" 28334: 27840:Encyclopedia of Nonlinear Science 26714:– via Calphysics Institute. 25093:from the original on 27 May 2017. 25076:from the original on 27 May 2017. 25066:"What Is The Physics Of Nothing?" 25023:10.1038/scientificamerican0594-27 24954:Matthews, R. (25 February 1995). 24884:from the original on 29 May 2017. 24850:from the original on 30 May 2017. 24833:from the original on 30 May 2017. 24816:from the original on 29 May 2017. 24791:from the original on 27 May 2017. 24766:from the original on 29 May 2017. 24749:from the original on 27 May 2017. 24441:. DOD. p. 66. Archived from 22229: 21976:from the original on 3 April 2016 21935:from the original on 3 April 2016 21894:from the original on 3 April 2016 20229:. World Scientific. p. 143. 18214:. Boca Raton: CRC Press. p. 15243:Lorentz-violating electrodynamics 15135: 15015: 14675:Casimir forces on parallel plates 14660: 14639:The Higgs mechanism is a type of 14481:. In technical terms, gluons are 13971: 13867: 13722: 13433: 12958: 12911: 12482: 12328: 12298: 11094: 10535: 10162: 9170: 9062: 8489: 8402: 8332: 7897: 7854:Clearly the least eigenvalue for 7439: 7064: 6730: 6504: 5431:We introduce the "mode function" 5185: 5137: 5067: 5051: 4992: 4976: 4916: 4776: 4721: 4342:of positive and negative energy. 2845: 1193:can be thought of as having some 89: 31366: 31354: 31342: 31331: 31330: 30783: 30782: 30176: 30175: 30053:Anomalous magnetic dipole moment 29915: 29914: 26173:10.1046/j.1365-8711.2000.03076.x 25638:. New York: Simon and Schuster. 25041:. pp. 50–53. Archived from 24821:Choi, C. Q. (12 February 2013). 24754:Bennett, J. (30 November 2016). 24692: 24661:Hambling, David (31 July 2014). 24654: 24635:Maxey, Kyle (11 December 2012). 24628: 24549: 24502: 24469: 24431: 24382: 24322: 24275: 24220: 24177: 24134: 24105: 24084: 24023: 23976: 23921: 23888: 23836: 23801: 23764: 23746: 23703: 23642: 23607: 23572: 23537: 23494: 23480: 23466: 23452: 23417: 23382: 23356: 23335: 23292: 23250: 23229: 23203: 23170: 23108: 23085: 23050: 22983: 22930: 22871: 22822: 22769: 22734: 22675: 22638: 22603: 22568: 22533: 22510: 22453: 22439: 22413: 22362: 22341: 22294: 22187: 22160: 22127: 22074: 22023: 21988: 21947: 21915:"The Morphology of the Electron" 21906: 21865: 21850: 21823: 21796: 21771: 21746: 21721: 21694: 21667: 21632: 21607: 21588: 21558: 21539: 21496: 21461: 21400: 21373: 21346: 21319: 21292: 21261: 21146: 21103: 21068: 21025: 20982: 20933: 20831: 20792: 20675: 20632: 20611: 20570: 20560: 20547: 20453: 20434: 20337: 20316: 20291: 20218: 20193: 20124: 20045: 19985: 19896: 19854: 19827: 19792: 19741: 19706: 19679: 17606:Proceedings of the Royal Society 17153:Leuchs & Sánchez-Soto (2013) 17122:for proposal and Matthews ( 16785: 16771: 16421:A common assumption is that the 16264:was awarded to thermodynamicist 16130:The mathematical models used in 15793:Mathematical universe hypothesis 15551:, but this is far from certain. 15547:and antibaryons produced in the 15053:The Heisenberg inequality where 14928:{\displaystyle \varepsilon _{0}} 14303: 14284: 14259: 14256: 14226: 14223: 14193: 14190: 14160: 14157: 14154: 14150: 14132: 14104: 14101: 14074: 14053: 14050: 14047: 14043: 14031: 14006: 14003: 13185: 13181: 13135: 13107: 13058: 13037: 13034: 13031: 13027: 13015: 12990: 12987: 12460: 12392: 12367: 12270: 12254: 12197: 12103: 12065: 11861: 11840: 11837: 11834: 11830: 11818: 11793: 11790: 11757: 11673: 11533: 11530: 11475: 11379: 11376: 11361: 11315: 11263: 11235: 11178: 11125: 11075: 11044: 11003: 10969: 10950: 10925: 10922: 10824: 10821: 10806: 10688: 10661: 10439:In deriving these equations for 10417: 10402: 10399: 10346: 10315: 10254: 10220: 10193: 10143: 10123: 10101: 10076: 10073: 10024: 10007: 10004: 9980: 9964: 9961: 9933: 9907: 9897: 9894: 9874: 9856: 9830: 9820: 9817: 9792: 9774: 9771: 9754: 9751: 9714: 9711: 9689: 9686: 9670: 9667: 9626: 9623: 9595: 9585: 9582: 9562: 9546: 9543: 9511: 9508: 9475: 9438: 9420: 9392: 9353: 9335: 9295: 9292: 9253: 9235: 9190: 9151: 9148: 9132: 9114: 9082: 9043: 9040: 8948: 8898: 8880: 8311: 8155: 7876: 7795: 7762: 7712: 7684: 7633: 7624: 7574: 7546: 7483: 7463: 7430: 7358: 7349: 7333: 7306: 7299: 7287: 7274: 7268: 7217: 7200: 7192: 7157: 7142: 7134: 7107: 7045: 7026: 7018: 6907: 6899: 6848: 6830: 6822: 6773: 6704: 6691: 6685: 6657: 6640: 6632: 6597: 6582: 6574: 6547: 6478: 6465: 6459: 6398: 6390: 6372: 6344: 6331: 6325: 6164: 6154: 6148: 6094: 6086: 6071: 6043: 6033: 6027: 5846: 5793: 5721: 5713: 5698: 5682: 5668: 5599: 5585: 5505: 5491: 5339: 5324: 2805:at which the system oscillates. 1955:held down to the zero values of 1213:). As temperature is reduced to 957:Heisenberg uncertainty principle 29128:Timeline of physics discoveries 28682:The Quantum Theory of Radiation 28363:Brooks, M. (16 November 2011). 28160:Journal of Propulsion and Power 27983:The European Physical Journal D 26569:The European Physical Journal D 25167:, or Amplified by the Vacuum". 25116: 25081:Shiga, D. (28 September 2005). 24992:Pilkington, M. (17 July 2003). 24975:O'Carroll, E. (25 March 2013). 24556:Pauling, Linus (October 1965). 24042:Compounds at Low Temperature". 20641:Letters in Mathematical Physics 20460:Miller, Katrina (1 July 2023). 20334:) for an overview of this area. 20103:Barton & Scharnhorst (1993) 20074:10.1070/PU1961v004n02ABEH003330 19836:J. Chem. Soc., Faraday Trans. 1 19472: 19403: 19297: 19248: 19190: 19139: 18828: 18801: 18782: 18763: 18714: 18671: 18627: 18593: 18558: 18528: 18473: 18428: 18299: 18256: 18176: 18118: 18075: 18035: 18008: 17965: 17924: 17889: 17868: 17829: 17787: 17745: 17683: 17640: 17597: 17548: 17505: 17484: 17465: 17445: 17406: 17355: 17314: 17141: 16489:fluctuation-dissipation theorem 15363:Hard X-ray Modulation Telescope 14796:which was not predicted by the 11773:{\displaystyle \mathbf {x} (t)} 10898:and therefore the equation for 10612: 10563: 6408: 2146:fluctuation-dissipation theorem 1971:Quantum field theory and beyond 1409:and the nature of dark energy. 1399:inertial and gravitational mass 1371:magnetic moment of the electron 21:Quantum vacuum (disambiguation) 31435:Non-equilibrium thermodynamics 29864:Relativistic quantum mechanics 28397:Marcus, A. (12 October 2009). 27557:10.1103/PhysRevLett.112.030602 26393:. Princeton University Press. 26321:10.1103/PhysRevLett.110.201601 25258:Advances in the Casimir Effect 25126:IEEE Transactions on Education 24872:Crane, L. (30 November 2016). 24284:Foundations of Physics Letters 24229:Foundations of Physics Letters 21270:Proc. Netherlands R. Acad. Sci 17474:Äther und Relativitäts-Theorie 16935: 16741:Quantum Vacuum Plasma Thruster 16487:and Welton proved the quantum 16208:. The original equations used 15998:Causal dynamical triangulation 15598:Chaotic and emergent phenomena 15513:galaxy formation and evolution 15373: 14504: 14348:in this equation the operator 14319: 14313: 14142: 14136: 14090: 14084: 13698: 13692: 13676: 13670: 13405: 13399: 13378: 13372: 13346: 13340: 13314: 13308: 13289: 13283: 13262: 13256: 13164: 13158: 13122: 13116: 13074: 13068: 12942: 12936: 12930: 12924: 12895: 12889: 12878: 12872: 12856: 12850: 12839: 12833: 12805: 12799: 12793: 12787: 12774: 12768: 12752: 12746: 12740: 12734: 12728: 12722: 12692: 12686: 12670: 12664: 12553: 12547: 12421: 12415: 12382: 12376: 12218: 12212: 12123: 12119: 12113: 12097: 12081: 12075: 12007: 11994: 11980: 11877: 11871: 11767: 11761: 11494: 11488: 11282: 11276: 11198: 11192: 11140: 11134: 11060: 11054: 11022: 11016: 10985: 10979: 10841: 10830: 10703: 10697: 10676: 10670: 10622: 10616: 10605: 10599: 10580: 10574: 10240: 10234: 10208: 10202: 10019: 9998: 9558: 9537: 9200: 9186: 9174: 9164: 9092: 9078: 9066: 9056: 8744: 8738: 8606: 8600: 8164: 8049:. It diverges proportional to 7824: 7818: 7782: 7776: 7736: 7730: 7699: 7693: 7603: 7597: 7561: 7555: 7337: 7329: 7291: 7283: 7177: 7171: 7122: 7116: 7033: 7022: 6911: 6879: 6868: 6862: 6834: 6802: 6788: 6782: 6714: 6700: 6617: 6611: 6562: 6556: 6488: 6474: 6348: 6340: 6168: 6160: 6047: 6039: 5874: 5850: 5839: 5797: 5686: 5678: 5603: 5595: 5509: 5501: 5405: 5400: 5394: 5387: 4885: 4870: 2691: 2601: 2539: 2447: 2398: 2335: 2280: 2231: 2175:cavity quantum electrodynamics 1781: 1772: 1608: 1599: 1137:which specifically deals with 1090:and it is one of the greatest 602:Relativistic quantum mechanics 134: 127: 108: 1: 30929:Spontaneous symmetry breaking 29842:Quantum statistical mechanics 29619:Quantum differential calculus 29541:Delayed-choice quantum eraser 29324:Symmetry in quantum mechanics 28104:Lectures on Quantum Mechanics 27915:10.1103/PhysRevLett.87.220601 27612:10.1016/S1355-2198(02)00033-3 25818:10.1016/S0079-6638(08)70266-X 25705:10.1103/PhysRevLett.91.050402 25353:10.1103/PhysRevLett.88.041804 24930:10.1016/S0262-4079(11)62858-3 24413:10.1016/S0921-4534(97)01462-7 24121:NASA Technical Reports Server 23931:Classical and Quantum Gravity 22264:Scientific American Frontiers 21701:The Nobel Foundation (1977). 20250:Heisenberg & Euler (1936) 19735:10.1016/S0031-8914(58)80090-7 19617:Peskin & Schroeder (1995) 19455:Peskin & Schroeder (1995) 17875:Jeans, James Hopwood (1915). 17174:Rugh & Zinkernagel (2002) 16878: 16522:quantum theory of gravitation 16417:Casimir batteries and engines 15685:Cosmological constant problem 14904:is the absolute value of the 14714: 11733:Using the above equation for 4403:Redefining the zero of energy 4376:Quantum electrodynamic vacuum 3255:Spontaneous symmetry breaking 3215:Symmetry in quantum mechanics 3093:is the mass of the particle, 1516:Moreever, the results of the 1352:cosmological constant problem 1181:Kinetic energy vs temperature 1092:unsolved mysteries in physics 1088:cosmological constant problem 951:, quantum systems constantly 642:Quantum statistical mechanics 28495:10.1016/0003-4916(70)90027-8 28447:Yirka, B. (2 October 2015). 28342:Brumfiel, G. (3 June 2011). 28284:Physica C: Superconductivity 26808:10.1016/0375-9601(81)90191-2 26688:10.1126/science.263.5147.612 24796:Cho, A. (30 November 2016). 24017:10.1016/0921-4534(92)90055-H 23997:Physica C: Superconductivity 23566:10.1016/0375-9601(70)90022-8 22704:10.1016/0038-1098(88)91020-4 22632:10.1016/0038-1101(82)90204-0 22562:10.1016/0031-8914(62)90019-8 21618:. New York: Campaigner Pub. 21175:10.1016/0031-9163(62)91369-0 20799:Barrett, Terence W. (2008). 20704:10.1016/0003-4916(78)90172-0 18125:Schweber, Silvan S. (1994). 17709:10.1007/978-3-642-61659-4_26 17478:Aether and relativity theory 16753:Zero-point energy in fiction 16541:Hughes Research Laboratories 16526:general theory of relativity 16234:second law of thermodynamics 15718:Cosmic censorship hypothesis 15554: 14449:Quantum chromodynamic vacuum 11458:It can be shown that in the 2919:with a bar through its top ( 57:, it exhibits properties of 7: 30752:Quantum information science 29976:Euler–Heisenberg Lagrangian 29644:Quantum stochastic calculus 29634:Quantum measurement problem 29556:Mach–Zehnder interferometer 29092:Quantum information science 28304:10.1016/j.physc.2005.10.003 26628:The Quantum Theory of Light 25970:10.1103/PhysRevLett.50.1903 25462:10.1103/PhysRevLett.79.2394 25204:Battersby, Stephen (2016). 25030:Scott, W. B. (March 2004). 24112:Robertson, Glen A. 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AC Gravity was awarded a 16357:Artist's conception of the 15525:cosmic microwave background 15509:quantum vacuum fluctuations 15482:Unsolved problem in physics 15411:negative pressure or energy 15386:Unsolved problem in physics 15247:Euler–Heisenberg Lagrangian 15199:rather than the low-energy 14494:electromagnetic interaction 8990:and the canonical momentum 4543:{\displaystyle \left\neq 1} 2991:third law of thermodynamics 2216:quantum harmonic oscillator 1551:derived the average energy 1518:Michelson–Morley experiment 1299:quantum harmonic oscillator 1172: 947:system may have. Unlike in 612:Quantum information science 35:Zero point (disambiguation) 10: 31466: 31109:Spin gapless semiconductor 31018:Nearly free electron model 28923:Classical electromagnetism 28642:Concepts of Modern Physics 28436:"The Ether Transmogrified" 28013:10.1140/epjd/e2013-30578-7 26864:. Boston: Academic Press. 26599:10.1140/epjd/e2013-30577-8 26266:10.1103/PhysRevD.80.015020 25728:Dirac, Paul A. M. (1927). 25189:10.1088/0305-4470/26/8/024 25098:Wall, M. (27 March 2014). 24838:Chown, M. (7 April 1990). 24771:Cho, A. (1 October 2015). 24515:AIP Conference Proceedings 23962:10.1088/0264-9381/11/5/001 23732:10.1103/PhysRevLett.70.701 23681:10.1103/PhysRevLett.86.192 23446:10.1088/0305-4470/16/6/026 23321:10.1109/JRPROC.1961.287932 22900:10.1103/PhysRevA.87.063845 22856:10.1103/PhysRevE.86.051105 22808:10.1209/0295-5075/88/50003 22763:10.1103/PhysRevA.79.055804 22684:Solid State Communications 22517:Henriques, Carlos (2014). 22425:www.scientificamerican.com 22167:Chaisson, Eric J. (2002). 22113:10.1103/PhysRevB.82.155138 22052:10.1088/0370-1328/87/1/137 21861:. London. 14 January 1982. 21803:Chaisson, Eric J. (2002). 21254:10.1103/PhysRevLett.45.494 21132:10.1103/PhysRevLett.57.933 20872:10.1088/0370-1301/62/1/303 20754:10.1103/PhysRevD.72.021301 20324:Carroll & Field (1997) 19255:Yablonovitch, Eli (1987). 18721:Lambrecht, Astrid (2002). 17701:Original Scientific Papers 16947:Peebles & Ratra (2003) 16675:U.S. Department of Defense 16442:in Pasadena, published in 16411:U.S. Department of Defense 16132:classical electromagnetism 16115: 15473: 15377: 15240: 14820: 14718: 14664: 14508: 14452: 14445:driving zero-point field. 12641:in the field Hamiltonian. 4418:are replaced by operators 4379: 4226: 3250:Explicit symmetry breaking 3108:is the width of the well. 2849: 2188: 1440: 1412: 32: 25: 19:For related articles, see 18: 31326: 31288: 31213: 31157: 31117: 31066: 31058:Density functional theory 31033:electronic band structure 31000: 30949: 30942: 30911: 30900: 30834: 30777: 30660: 30610: 30589: 30538: 30512: 30476: 30440: 30389: 30308: 30301: 30230: 30167: 30111: 30045: 29999: 29991:Path integral formulation 29968: 29910: 29872: 29824: 29704:Quantum complexity theory 29682:Quantum cellular automata 29657: 29589: 29523: 29436: 29400: 29387:Path integral formulation 29354: 29219: 29176: 29100: 29037: 28965: 28881: 28853: 28825: 28048:Reviews of Modern Physics 27151:10.1103/PhysRevB.60.14740 26994:10.1103/RevModPhys.75.559 26964:Reviews of Modern Physics 26922:10.1007/978-3-540-70626-7 26440:10.1007/s00407-011-0092-3 25989:10.1007/978-3-642-82272-8 25867:10.1142/S0217751X12600044 25405:10.1109/JSTQE.2007.893082 24981:Christian Science Monitor 24855:Cook, N. (29 July 2002). 23895:Wheeler, J Craig (2007). 23069:10.1007/978-3-540-70510-9 22527:10.13140/RG.2.1.4297.1608 21214:10.1103/RevModPhys.46.251 20949:Pisma Zh. Eksp. Teor. Fiz 20587:10.1007/978-94-010-2274-3 20486:Carroll, Sean M. (1998). 19861:Lamoreaux, S. K. (1997). 18742:10.1088/2058-7058/15/9/29 18017:Physikalische Zeitschrift 17862:10.1080/14786441308635031 17823:10.1080/14786441308634993 17781:10.1080/14786441308634955 17472:Einstein, Albert (1920). 16726:suitable rotational state 16387:perpetual motion machines 16359:expansion of the universe 16256:need not always apply to 16178:De Haas–Van Alphen effect 16013:Canonical quantum gravity 15838:Black hole thermodynamics 15823:Doubly special relativity 15606:Beyond the Standard Model 15325:, the closest discovered 14652:Experimental observations 11972:is therefore of the form 5653:should be independent of 3406:Bargmann–Wigner equations 3012:of the ground state of a 2923:) to denote the quantity 1852:electromagnetic radiation 1496:aether theory based upon 1098:Etymology and terminology 1048:expansion of the universe 939:) is the lowest possible 31420:Condensed matter physics 31228:Bogoliubov quasiparticle 30972:Quantum spin Hall effect 30864:Bose–Einstein condensate 30828:Condensed matter physics 30159:Photon-photon scattering 29771:Quantum machine learning 29751:Quantum key distribution 29741:Quantum image processing 29731:Quantum error correction 29581:Wheeler's delayed choice 29029:Condensed matter physics 28631: 28388:. Institute of Physics. 27804:The Philosophy of Vacuum 27644:The Philosophy of Vacuum 27281:. Leipzig: J. A. Barth. 27219:10.1002/andp.19123420403 24969:Via Calphysics Institute 24863:. London. Archived from 24206:10.1103/PhysRevB.46.5489 23636:10.1103/PhysRevD.18.2739 22323:10.1103/PhysRevB.30.1700 21188:B. D. Josephson (1974). 20429:Perlmutter et al. 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B. (1997). 25323:Physical Review Letters 24956:"Nothing Like a Vacuum" 24810:10.1126/science.aal0437 24785:10.1126/science.aad4655 24296:10.1023/A:1021621425670 24163:10.1103/PhysRevD.43.457 23853:Physical Review Letters 23822:10.1243/095441005X30252 23771:Cravens, D. L. (1990). 23712:Physical Review Letters 23651:Physical Review Letters 23391:Physical Review Letters 23257:Forward, R. L. (1963). 22612:Solid-State Electronics 21997:Physical Review Letters 21680:. Springer. p. 3. 21407:Barrett, T. W. (1991). 21380:Hunt, Bruce J. (2005). 21326:Hunt, Bruce J. (2005). 21233:Physical Review Letters 21112:Physical Review Letters 21034:Physical Review Letters 20919:10.1103/PhysRev.115.485 20495:Physical Review Letters 19905:Physical Review Letters 19870:Physical Review Letters 19563:10.1103/PhysRev.119.670 19261:Physical Review Letters 19206:Physical Review Letters 19148:Proceedings of the IEEE 19072:Physical Review Letters 19011:Physical Review Letters 18328:10.1103/PhysRev.74.1157 16583:Astronautics Laboratory 16328:Type II superconductors 16296:Type II superconductors 15426:European Space Agency's 15336:Roberto Mignani at the 14941:fine-structure constant 14823:Fine-structure constant 14817:Fine-structure constant 14433:{\displaystyle \omega } 14334:Without the free field 9026:of the oscillator are: 4479:The classical quantity 4470:{\displaystyle \left=1} 4331:to agree with observed 3431:Electroweak interaction 3426:Quantum electrodynamics 3401:Wheeler–DeWitt equation 3288:Background field method 2055:quantum electrodynamics 1488:. The existence of the 1139:quantum electrodynamics 1104:zero-point energy (ZPE) 357:Leggett–Garg inequality 28:Zero Point (photometry) 30769:Quantum thermodynamics 30693:On shell and off shell 30688:Loop quantum cosmology 30530:N = 4 super Yang–Mills 30489:N = 1 super Yang–Mills 30356:Scalar electrodynamics 30346:Quantum chromodynamics 30248:Conformal field theory 30224:Quantum field theories 29791:Quantum neural network 29113:Nobel Prize in Physics 28975:Relativistic mechanics 28137:Weisskopf, V. (1936). 26540:Le Bellac, M. (2006). 26469:. New York: Springer. 26083:Zeitschrift für Physik 25757:10.1098/rspa.1927.0039 25537:Conlon, T. E. (2011). 23601:10.1038/physci232193a0 23301:Proceedings of the IRE 22913:Roßnagel et al. (2014) 21778:Gleick, James (1987). 21674:Pismen, L. M. 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New York: Springer. 18422:10.1103/PhysRev.73.412 18377:10.1103/PhysRev.72.241 18084:Zeitschrift für Physik 17974:Zeitschrift für Physik 17842:Philosophical Magazine 17803:Philosophical Magazine 17761:Philosophical Magazine 17626:10.1098/rspa.1953.0105 17514:Zeitschrift für Physik 16662:type II superconductor 16599:Edwards Air Force Base 16587:Edwards Air Force Base 16502:laws of thermodynamics 16457:University of Colorado 16377:Purported applications 16366: 16344:Topological degeneracy 16340:Quantum critical point 16312: 16254:laws of thermodynamics 16156:vis-à-vis particles), 16140:electromagnetic vacuum 16003:Causal fermion systems 15993:Loop quantum cosmology 15938:Large extra dimensions 15928:Supersymmetry breaking 15833:Causal fermion systems 15460:cosmological inflation 15301: 15269: 15193: 15142: 15041: 14929: 14736: 14676: 14646:Ginzburg–Landau theory 14540: 14434: 14326: 14207: 13982: 13620: 13225: 13081: 12969: 12564: 12143: 12032: 11884: 11774: 11687: 11543: 11450: 11248: 11029: 10892: 10639: 10430: 10271: 10042: 9945: 9644: 8975: 8829: 8660: 8577: 8278: 8039: 7974: 7914: 7846: 7516: 7398: 7230: 6925: 6670: 6444: 6192: 6104: 6008: 5881: 5731: 5627: 5522: 5423: 5208: 5094: 4828: 4544: 4471: 4397:quantum field theories 4319:claimed it is, like a 4267:quantum chromodynamics 3436:Quantum chromodynamics 3313:Effective field theory 3075: 2885: 2860:The zero-point energy 2774: 2664: 2486: 2355: 2104: 2103:Hendrik Casimir (1958) 2040: 2027: 1996: 1885:(1923), who worked at 1868: 1861: 1818: 1805: 1630: 1544: 1538:Nobel Prize in Physics 1514: 1427: 1379:multiple stable states 1245: 1182: 1165:(VEV) also called its 1147:quantum chromodynamics 1112: 148: 62: 31104:Topological insulator 31038:Anderson localization 30742:Quantum hydrodynamics 30737:Quantum hadrodynamics 30361:Scalar chromodynamics 30124:Breit–Wheeler process 30063:Klein–Nishina formula 29816:Quantum teleportation 29344:Wave–particle duality 29118:Philosophy of physics 28719:; Muller, B. (1985). 28356:10.1038/news.2011.346 27837:Scott, Alwyn (2006). 27053:Astrophysical Journal 26759:10.1093/mnras/stw2798 26662:Matthews, R. (1994). 25064:(22 September 2016). 24892:(22 September 1994). 24861:Jane's Defence Weekly 24730:Articles in the press 24583:10.1073/pnas.54.4.989 23488:U.S. patent 3,823,570 23474:U.S. patent 3,626,605 23460:U.S. patent 3,626,606 23368:MIT Technology Review 22447:U.S. patent 7,379,286 22194:Peterson, I. (1997). 21614:White, Carol (1977). 21583:Bostick et al. (1966) 20345:Mignani et al. (2017) 20087:Capasso et al. (2007) 18971:10.1103/PhysRev.32.97 18895:10.1103/PhysRev.83.34 17413:Debye, Peter (1913). 16461:nonconservative force 16356: 16320:quantum phase factors 16313: 16018:Semiclassical gravity 15813:Twistor string theory 15803:Randall–Sundrum model 15620:Large Hadron Collider 15476:Inflation (cosmology) 15441:equivalence principle 15409:exerted some form of 15296: 15263: 15194: 15143: 15042: 14930: 14893:electromagnetic force 14887:(the electromagnetic 14728: 14674: 14518: 14435: 14327: 14208: 13983: 13621: 13226: 13082: 12970: 12565: 12144: 12033: 11885: 11775: 11688: 11544: 11451: 11249: 11030: 10893: 10640: 10431: 10272: 10043: 9946: 9645: 8976: 8830: 8661: 8578: 8279: 8040: 7975: 7915: 7847: 7517: 7399: 7231: 6926: 6671: 6445: 6193: 6105: 6009: 5882: 5732: 5628: 5523: 5424: 5209: 5095: 4829: 4545: 4472: 4388:electromagnetic field 3391:Klein–Gordon equation 3333:LSZ reduction formula 3097:is the energy state ( 3076: 2859: 2775: 2665: 2487: 2356: 2202:Near the bottom of a 2191:Uncertainty principle 2185:Uncertainty principle 2162:Jaynes–Cummings model 2102: 2035: 2022: 2009:electromagnetic field 1994: 1866: 1856: 1816: 1806: 1631: 1535: 1528:Second quantum theory 1509: 1425: 1418:Early aether theories 1407:cosmological constant 1335:cosmological constant 1259:wave-particle duality 1231: 1219:uncertainty principle 1180: 1076:Large Hadron Collider 971:, the empty space of 959:. Therefore, even at 342:Elitzur–Vaidman 332:Davisson–Germer 149: 45: 31415:Mathematical physics 31400:Quantum field theory 30982:Aharonov–Bohm effect 30869:Fermionic condensate 30713:Quantum fluctuations 30683:Loop quantum gravity 30253:Lattice field theory 29847:Quantum field theory 29776:Quantum metamaterial 29721:Quantum cryptography 29451:Consistent histories 29077:Mathematical physics 28728:. H. Deutsch: Thun. 28512:"Tuning in to Noise" 28420:The Sunday Telegraph 27480:Astronomical Journal 27384:. New York: DK Pub. 27329:. London: Longmans. 26200:Quantum Field Theory 23426:Journal of Physics A 22925:Correa et al. (2014) 22717:Abbott et al. (1996) 22653:. pp. 331–332. 22398:10.3390/atoms7020051 21954:Bostick, W. (1985). 21913:Bostick, W. (1985). 21872:Bostick, W. (1985). 21437:on 13 September 2016 20617:See Schwinger ( 19968:Bressi et al. (2002) 19848:10.1039/F19787402637 19700:10.1039/QR9561000295 19134:Scully et al. (2003) 18822:10.1039/qr9561000295 18435:Dresden, M. (1987). 17836:Bohr, Niels (1913). 17794:Bohr, Niels (1913). 17752:Bohr, Niels (1913). 17080:Bordag et al. (2009) 16658:gravitomagnetic flux 16562:fluxoid quantization 16553:nonlinear properties 16311:{\displaystyle \mu } 16302: 16216:notation, a kind of 16145:alternative theories 15983:Loop quantum gravity 15878:Gauge theory gravity 15743:Grand Unified Theory 15738:Unified field theory 15733:Theory of everything 15695:Neutrino oscillation 15570:Alternative theories 15543:(imbalance) between 15531:have been observed. 15344:who led the team of 15237:Vacuum birefringence 15170: 15099: 14947: 14912: 14549:spontaneously breaks 14424: 14217: 13992: 13655: 13235: 13093: 13089:Using the fact that 12981: 12649: 12168: 12057: 11976: 11784: 11753: 11600: 11470: 11258: 11039: 10916: 10652: 10529: 10297: 10063: 9955: 9654: 9030: 8849: 8690: 8587: 8288: 8146: 7984: 7926: 7867: 7528: 7408: 7240: 7014: 6680: 6454: 6320: 6114: 6022: 5891: 5789: 5663: 5558: 5453: 5262: 5118: 4841: 4610: 4502: 4432: 3474:Theory of everything 3328:Lattice field theory 3298:Correlation function 3120:Quantum field theory 3112:Quantum field theory 3024: 3003:negative temperature 2678: 2504: 2500:terms above satisfy 2377: 2222: 2118:Van der Waals forces 2013:harmonic oscillators 2005:spontaneous emission 1965:Schrödinger equation 1718: 1563: 1395:quantum entanglement 1359:spontaneous emission 1155:quantum field theory 1119:zero-point radiation 977:quantum field theory 607:Quantum field theory 519:Consistent histories 156:Schrödinger equation 83: 33:For other uses, see 31410:Concepts in physics 31373:Physics WikiProject 31048:tight binding model 31028:Fermi liquid theory 31013:Free electron model 30962:Quantum Hall effect 30943:Electrons in solids 30747:Quantum information 30351:Quartic interaction 30139:Delbrück scattering 30093:Vacuum polarization 30017:Faddeev–Popov ghost 29832:Quantum fluctuation 29801:Quantum programming 29761:Quantum logic gates 29746:Quantum information 29726:Quantum electronics 29201:Classical mechanics 29052:Atmospheric physics 28891:Classical mechanics 28819:branches of physics 28639:Beiser, A. (2003). 28597:2013PNAS..110.4234L 28531:1996PhT....49c..39B 28487:1970AnPhy..56..474B 28434:(2 February 1978). 28403:Scientific American 28321:10.2514/6.2001-3363 28296:2005PhyC..433..101W 28251:10.1038/nature10561 28243:2011Natur.479..376W 28060:1989RvMP...61....1W 28005:2013EPJD...67...58U 27948:2003Sci...299..862S 27907:2001PhRvL..87v0601S 27604:2002SHPMP..33..663R 27549:2014PhRvL.112c0602R 27502:1998AJ....116.1009R 27436:2015Sci...350..420R 27211:1912AnP...342..642P 27143:1999PhRvB..6014740P 27075:1999ApJ...517..565P 27021:The Road to Reality 26986:2003RvMP...75..559P 26900:"Zero-Point Energy" 26832:1983IJTP...22..323M 26800:1981PhLA...82..225M 26750:2017MNRAS.465..492M 26680:1994Sci...263..612M 26591:2013EPJD...67...57L 26366:1994PhRvB..49..704K 26313:2013PhRvL.110t1601K 26258:2009PhRvD..80a5020K 26164:2000MNRAS.311..555H 26105:1936ZPhy...98..714H 26043:1994PhRvA..49..678H 25962:1983PhRvL..50.1903G 25859:2012IJMPA..2760004D 25793:1970SciAm.222c.108D 25781:Scientific American 25749:1927RSPSA.114..243D 25697:2003PhRvL..91e0402D 25594:2014NatSR...4E3949C 25506:2001Sci...291.1941C 25500:(5510): 1941–1944. 25454:1997PhRvL..79.2394C 25397:2007IJSTQ..13..400C 25345:2002PhRvL..88d1804B 25306:1966PhFl....9.2078B 25222:2016Natur.537S.201B 25181:1993JPhA...26.2037B 25138:1996ITEdu..39....1A 25048:on 26 February 2017 25011:Scientific American 24994:"Zero Point Energy" 24949:on 9 November 2000. 24922:2011NewSc.212Q..50D 24904:on 17 January 1999. 24827:Scientific American 24714:on 22 November 2016 24708:National Geographic 24574:1965PNAS...54..989P 24527:2009AIPC.1103..524F 24486:10.2514/6.2002-3925 24405:1997PhyC..281..260L 24360:2006AIPC..813.1415T 24241:1993FoPhL...6..371T 24198:1992PhRvB..46.5489L 24155:1991PhRvD..43..457L 24079:Woods et al. (2001) 24066:1997physics...5043R 24050:: physics/9705043. 24009:1992PhyC..203..441P 23954:1994CQGra..11L..73A 23899:Cosmic Catastrophes 23866:1988PhRvL..61.1446M 23724:1993PhRvL..70..701R 23673:2001PhRvL..86..192O 23628:1978PhRvD..18.2739P 23593:1971NPhS..232..193O 23558:1970PhLA...32..402O 23515:1979GReGr..11..149B 23438:1983JPhA...16.1331R 23403:1966PhRvL..16.1092D 23313:1961PIRE...49..892F 23278:1963AmJPh..31..166F 23148:2001AnP...513..393H 23102:10.2514/6.1999-2146 23018:2016NatSR...623408G 22961:2014EL....10620001A 22892:2013PhRvA..87f3845B 22847:2012PhRvE..86e1105H 22800:2009EL.....8850003D 22755:2009PhRvA..79e5804G 22696:1988SSCom..67..749K 22624:1982SSEle..25..749G 22597:10.1049/el:19710095 22589:1971ElL.....7..148H 22554:1962Phy....28..409M 22481:2012PhPro..38....8D 22315:1984PhRvB..30.1700F 22105:2010PhRvB..82o5138C 22044:1966PPS....87..330F 22009:1981PhRvL..47.1863E 21732:. Wiley-Blackwell. 21517:1976NCimB..32..138I 21482:1972JMP....13.1919R 21289:, pp. 453–454. 21245:1980PhRvL..45..494K 21206:1974RvMP...46..251J 21167:1962PhL.....1..251J 21124:1986PhRvL..57..933C 21046:1987PhRvL..58.1593A 21003:1984RSPSA.392...45B 20961:1981JETPL..33...94A 20911:1959PhRv..115..485A 20854:1949PPSB...62....8E 20746:2005PhRvD..72b1301J 20696:1978AnPhy.115....1S 20653:1975LMaPh...1...43S 20517:1998PhRvL..81.3067C 20417:Riess et al. (1998) 20347:for experiment and 20147:1993PhRvA..48.1824H 20066:1961SvPhU...4..153D 20021:10.1038/nature07610 20013:2009Natur.457..170M 19980:Decca et al. (2003) 19927:1998PhRvL..81.4549M 19882:1997PhRvL..78....5L 19813:1972RScI...43..584H 19762:1968Natur.219.1120T 19756:(5159): 1120–1121. 19727:1958Phy....24..751S 19665:1988SciAm.258e.106A 19653:Scientific American 19555:1960PhRv..119..670S 19493:1955PhRv...97..511W 19457:, pp. 786–791. 19424:1999qqep.book.....G 19381:1927ZPhy...43..172H 19318:1946PhRv...69Q.674. 19273:1987PhRvL..58.2059Y 19218:1985PhRvL..55.2137H 19094:2000PhRvL..85.1799A 19023:1981PhRvL..47.1216K 18963:1928PhRv...32...97J 18925:1928PhRv...32..110N 18887:1951PhRv...83...34C 18692:1930ZPhy...63..245L 18649:1930ZPhy...60..491E 18579:1948PhRv...73..360C 18494:1947PhRv...72..339B 18414:1948PhRv...73..412F 18368:1947PhRv...72..241L 18320:1948PhRv...74.1157W 18277:1935NW.....23..631W 18265:Naturwissenschaften 18096:1928ZPhy...47..151J 18029:1909PhyZ...10..185E 17986:1926ZPhy...35..557B 17910:1926AnP...385..437S 17854:1913PMag...26..857B 17815:1913PMag...26..476B 17773:1913PMag...26....1B 17661:1924Natur.114..349M 17618:1953RSPSA.218..291D 17573:1934Natur.133Q.529S 17526:1923ZPhy...16..183B 17431:1913AnP...348...49D 17403:, pp. 563–565. 17380:1913AnP...345..551E 17149:Urban et al. (2013) 17011:, pp. 270–285. 16827:Quantum fluctuation 16761:and related media. 16710:gravitational waves 16664:material is due to 16631:inflationary energy 16258:dissipative systems 16212:'s more expressive 16174:Quantum Hall effect 15753:Kaluza–Klein theory 15549:very early universe 15517:structure formation 15452:of particle physics 15419:orders of magnitude 15284:birefringent medium 15275:Hans Heinrich Euler 15166:90 GeV, that: 14997: 14937:vacuum permittivity 14499:nonlinear equations 14490:strong interactions 14282: 14252: 14186: 14130: 14029: 13950: 13894: 13840: 13791: 13773: 13574: 13518: 13199: 13157: 13013: 12529: 12414: 12211: 11816: 11462:field, if the mass 11338: 11191: 10948: 10787: 10483:evolves in time as 10233: 10099: 9931: 9854: 9619: 9504: 9288: 8945: 8819: 8801: 8721: 8045:for high values of 7817: 7775: 7647: 7596: 7476: 7372: 7328: 7170: 6861: 6610: 5445:that satisfies the 4392:Maxwell's equations 4366:vacuum fluctuations 4359:elementary particle 4347:perturbation theory 4249:are fermions (e.g. 3453:Incomplete theories 2496:of the kinetic and 2114:colloidal solutions 2033:who in 1935 wrote: 1933:James Hopwood Jeans 1426:James Clerk Maxwell 1401:, variation in the 1375:Delbrück scattering 1187:classical mechanics 1123:ground state energy 1007:, whose quanta are 949:classical mechanics 395:Stern–Gerlach 192:Classical mechanics 31445:Physical paradoxes 30934:Critical phenomena 30633:Nambu–Jona-Lasinio 30561:Higher dimensional 30468:Wess–Zumino–Witten 30258:Noncommutative QFT 30134:Compton scattering 29885:in popular culture 29667:Quantum algorithms 29515:Von Neumann–Wigner 29495:Objective collapse 29206:Old quantum theory 29108:History of physics 27199:Annalen der Physik 27106:. Addison-Wesley. 26840:10.1007/BF02082897 26389:Kragh, H. (2002). 26113:10.1007/BF01343663 26089:(11–12): 714–732. 25810:Progress in Optics 25766:on 25 October 2016 25572:Scientific Reports 24249:10.1007/BF00665654 23756:. Patents Review. 23523:10.1007/BF00756587 23125:Annalen der Physik 22996:Scientific Reports 22144:10.1007/11672906_2 21755:Order out of Chaos 21525:10.1007/BF02726749 21505:Il Nuovo Cimento B 21089:10.1007/BF03046050 20970:on 4 November 2016 20661:10.1007/BF00405585 20466:The New York Times 19956:Chan et al. (2001) 19389:10.1007/BF01397280 18839:. Academic Press. 18700:10.1007/BF01421741 18657:10.1007/BF01341258 18346:Retherford, Robert 18285:10.1007/BF01492012 18104:10.1007/BF02055793 17994:10.1007/BF01379806 17898:Annalen der Physik 17693:Rechenberg, Helmut 17555:Simon, F. (1934). 17534:10.1007/BF01327389 17419:Annalen der Physik 17368:Annalen der Physik 16793:mathematics portal 16450:thought experiment 16367: 16363:inflationary epoch 16308: 16230:vortical phenomena 15913:Superstring theory 15843:Unparticle physics 15783:Superstring theory 15713:Brans–Dicke theory 15564:general relativity 15529:magnetic monopoles 15306:Lorentz invariance 15302: 15270: 15189: 15138: 15037: 14983: 14925: 14737: 14702:Scharnhorst effect 14677: 14541: 14430: 14322: 14268: 14238: 14203: 14201: 14172: 14116: 14015: 13978: 13976: 13936: 13880: 13826: 13774: 13759: 13616: 13614: 13560: 13504: 13221: 13175: 13128: 13077: 12999: 12965: 12963: 12560: 12558: 12515: 12468: 12385: 12287: 12262: 12195: 12139: 12028: 11880: 11802: 11770: 11683: 11539: 11460:radiation reaction 11446: 11324: 11323: 11244: 11172: 11083: 11025: 10934: 10888: 10773: 10635: 10426: 10267: 10265: 10214: 10151: 10085: 10038: 9941: 9939: 9917: 9840: 9640: 9638: 9634: 9605: 9490: 9274: 9223: 8971: 8931: 8926: 8825: 8805: 8787: 8693: 8656: 8573: 8571: 8487: 8400: 8330: 8319: 8274: 8257: 8176: 8163: 8035: 7970: 7910: 7895: 7884: 7842: 7840: 7788: 7756: 7618: 7567: 7512: 7505: 7457: 7438: 7394: 7343: 7297: 7226: 7151: 7053: 6921: 6842: 6666: 6591: 6440: 6435: 6188: 6100: 6004: 5877: 5727: 5623: 5518: 5447:Helmholtz equation 5419: 5417: 5204: 5090: 5088: 5049: 5029: 4974: 4914: 4824: 4822: 4813: 4719: 4672: 4540: 4467: 4329:curvature of space 4325:General Relativity 3338:Partition function 3265:Topological charge 3185:General relativity 3180:Special relativity 3071: 2886: 2770: 2660: 2522: 2494:expectation values 2482: 2351: 2263: 2105: 2011:as an ensemble of 1997: 1869: 1819: 1801: 1658:Boltzmann constant 1626: 1545: 1522:special relativity 1428: 1331:general relativity 1246: 1183: 1072:symmetry of nature 1044:general relativity 1033:special relativity 945:quantum mechanical 583:Von Neumann–Wigner 563:Objective-collapse 362:Mach–Zehnder 352:Leggett inequality 347:Franck–Hertz 197:Old quantum theory 144: 63: 31425:Materials science 31382: 31381: 31268:Exciton-polariton 31153: 31152: 31125:Thermoelectricity 30794: 30793: 30656: 30655: 30190: 30189: 30149:Møller scattering 30119:Bhabha scattering 30088:Uehling potential 30037:Virtual particles 29928: 29927: 29902:Quantum mysticism 29880:Schrödinger's cat 29811:Quantum simulator 29781:Quantum metrology 29709:Quantum computing 29672:Quantum amplifier 29649:Quantum spacetime 29614:Quantum cosmology 29604:Quantum chemistry 29319:Scattering theory 29267:Zero-point energy 29262:Degenerate levels 29170:Quantum mechanics 29136: 29135: 29123:Physics education 29072:Materials science 29039:Interdisciplinary 28997:Quantum mechanics 28767:Nima Arkani-Hamed 28581:(11): 4234–4238. 28475:Annals of Physics 28227:(7373): 376–379. 28113:978-1-107-11166-0 27942:(5608): 862–863. 27877:978-0-521-43595-6 27850:978-1-57958-385-9 27767:978-0-7382-0055-2 27729:978-0-7382-0054-5 27691:978-0-7382-0053-8 27430:(6259): 420–423. 27391:978-0-7566-9841-6 27367:978-981-02-2095-2 27350:Reed, D. (1995). 27131:Physical Review B 27113:978-0-201-50397-5 27032:978-0-679-45443-4 26956:Peebles, P. J. E. 26788:Physics Letters A 26674:(5147): 612–613. 26551:978-1-107-60276-2 26516:978-0-19-502383-1 26476:978-3-642-55089-8 26400:978-0-691-09552-3 26354:Physical Review B 26236:Physical Review D 26031:Physical Review A 25998:978-0-387-13404-8 25956:(24): 1903–1906. 25932:978-0-691-03322-8 25903:978-0-691-03705-9 25827:978-0-444-10571-4 25645:978-0-671-47685-4 25602:10.1038/srep03949 25550:978-1-4478-3916-3 25438:(13): 2394–2397. 25314:10.1063/1.1761572 25294:Physics of Fluids 25268:978-0-19-923874-3 25216:(7622): 201–204. 25156:10.1109/13.485226 24867:on 2 August 2002. 24760:Popular Mechanics 24535:10.1063/1.3115561 24495:978-1-62410-115-1 24480:. p. 52236. 24368:10.1063/1.2169327 24186:Physical Review B 24143:Physical Review D 23995:superconductor". 23860:(13): 1446–1449. 23789:on 12 August 2011 23616:Physical Review D 23546:Physics Letters A 23397:(24): 1092–1093. 23286:10.1119/1.1969340 23078:978-3-540-70510-9 23026:10.1038/srep23408 22880:Physical Review A 22835:Physical Review E 22743:Physical Review A 22303:Physical Review B 22153:978-3-540-22495-2 22083:Physical Review B 22003:(26): 1863–1866. 21843:978-1-84816-724-7 21764:978-0-00-654115-8 21565:Lambek, Joachim. 21490:10.1063/1.1665933 21312:978-981-02-2095-2 21040:(16): 1593–1596. 20723:Physical Review D 20684:Annals of Physics 20596:978-94-010-2274-3 20501:(15): 3067–3070. 20134:Physical Review A 20007:(7226): 170–173. 19911:(21): 4549–4552. 19821:10.1063/1.1685696 19801:Rev. Sci. Instrum 19770:10.1038/2191120a0 19721:(6–10): 751–764. 19607:, pp. 42–43. 19516:, pp. 31–33. 19469:, pp. 73–74. 19433:978-0-684-86315-3 19339:Goy et al. (1983) 19267:(20): 2059–2062. 19212:(20): 2137–2140. 19017:(17): 1216–1219. 18846:978-0-124-65050-3 18837:Dispersion Forces 18446:978-1-461-29087-2 18225:978-0-8493-3786-4 18144:978-0-691-03327-3 18060:978-0-387-95262-8 17950:978-0-521-19118-0 17718:978-3-642-64900-4 17655:(2862): 349–350. 17320:See (Planck 16957:for press comment 16847:Topological order 16694:Evgeny Podkletnov 16650:Evgeny Podkletnov 16635:Miguel Alcubierre 16479:Single heat baths 16455:Garret Moddel at 16348:Topological order 16288:Beltrami vortices 16275:phase transitions 16250:conservation laws 16242:Lorentz force law 16126:Self-organization 16114: 16113: 15988:Quantum cosmology 15768:Quantum cosmology 15690:Strong CP problem 15650:Hierarchy problem 15507:. It is believed 15315:Lorentz violation 15261: 15187: 15133: 15035: 15022: 14976: 14906:electronic charge 14889:coupling constant 14792:notation) of the 14697:static friction. 14641:superconductivity 14299: 14262: 14229: 14196: 14164: 14107: 14070: 14057: 14009: 13952: 13917: 13842: 13757: 13610: 13498: 13475: 13396: 13328: 13280: 13054: 13041: 12993: 12499: 12454: 12452: 12351: 12350: 12316: 12315: 12263: 12248: 12246: 12188: 12001: 11857: 11844: 11796: 11665: 11636: 11536: 11525: 11382: 11309: 11307: 11112: 11111: 11069: 10999: 10965: 10928: 10827: 10771: 10770: 10547: 10405: 10394: 10393: 10310: 10180: 10179: 10137: 10120: 10079: 10010: 9993: 9967: 9900: 9889: 9823: 9812: 9777: 9757: 9741: 9717: 9706: 9692: 9673: 9629: 9588: 9577: 9549: 9535: 9514: 9472: 9435: 9412: 9389: 9350: 9322: 9298: 9250: 9222: 9154: 9129: 9106: 9046: 8925: 8895: 8871: 8780: 8654: 8547: 8486: 8450: 8399: 8372: 8329: 8305: 8303: 8248: 8246: 8195: 8167: 8149: 8013: 7965: 7894: 7870: 7504: 7424: 7094: 7093: 7039: 6981:and polarization 6760: 6759: 6534: 6533: 6364: 6363: 6063: 6062: 5956: 5384: 5299: 5048: 5028: 4973: 4913: 4812: 4718: 4671: 4657: 4498:. The fact that: 4225: 4224: 3318:Expectation value 3293:BRST quantization 3240:Poincaré symmetry 3195:Yang–Mills theory 3175:Quantum mechanics 3069: 3016:is a half-period 2989:According to the 2969:expectation value 2899:angular frequency 2803:angular frequency 2768: 2737: 2736: 2725: 2694: 2654: 2634: 2604: 2592: 2542: 2521: 2476: 2463: 2450: 2432: 2401: 2338: 2326: 2283: 2262: 2234: 2135:dielectric plates 1985:Werner Heisenberg 1961:Erwin Schrödinger 1913:Werner Heisenberg 1844:X-ray diffraction 1797: 1793: 1740: 1620: 1486:thermal radiation 1478:Otto von Guericke 1383:symmetry breaking 1223:quantum mechanics 1197:made up of their 1145:which deals with 1113:Nullpunktsenergie 1025:Lorentz invariant 933:Zero-point energy 930: 929: 637:Scattering theory 617:Quantum computing 390:Schrödinger's cat 322:Bell's inequality 130: 105: 74:Quantum mechanics 31457: 31440:Perpetual motion 31395:Energy (physics) 31370: 31369: 31358: 31346: 31345: 31334: 31333: 31273:Phonon polariton 31165:Amorphous magnet 31145:Electrostriction 31140:Flexoelectricity 31135:Ferroelectricity 31130:Piezoelectricity 30987:Josephson effect 30967:Spin Hall effect 30947: 30946: 30924:Phase transition 30906: 30889:Luttinger liquid 30836:States of matter 30821: 30814: 30807: 30798: 30797: 30786: 30785: 30703:Quantum dynamics 30376:Yang–Mills–Higgs 30331:Non-linear sigma 30321:Euler–Heisenberg 30306: 30305: 30217: 30210: 30203: 30194: 30193: 30179: 30178: 30154:Schwinger effect 29955: 29948: 29941: 29932: 29931: 29918: 29917: 29629:Quantum geometry 29624:Quantum dynamics 29481:Superdeterminism 29377:Matrix mechanics 29232:Bra–ket notation 29163: 29156: 29149: 29140: 29139: 29062:Chemical physics 29002:Particle physics 28928:Classical optics 28811: 28804: 28797: 28788: 28787: 28755: 28727: 28712: 28672: 28626: 28608: 28590: 28565: 28563: 28561: 28555: 28549:. Archived from 28539:10.1063/1.881491 28516: 28506: 28466:Journal articles 28460: 28443: 28427: 28426:on 6 March 2003. 28422:. Archived from 28410: 28393: 28376: 28359: 28324: 28307: 28290:(1–2): 101–107. 28278: 28236: 28215: 28213: 28211: 28205: 28193: 28183: 28181:2060/20170000277 28172:10.2514/1.B36120 28154: 28144: 28133: 28095: 28045: 28032: 27998: 27977: 27959: 27926: 27889: 27862: 27833: 27800:Brown, Harvey R. 27787: 27749: 27711: 27673: 27640:Brown, Harvey R. 27631: 27597: 27576: 27542: 27521: 27495: 27493:astro-ph/9805201 27471: 27421: 27411: 27371: 27346: 27318: 27290: 27268: 27222: 27186: 27170: 27154: 27125: 27105: 27094: 27068: 27066:astro-ph/9812133 27044: 27024: 27013: 26979: 26977:astro-ph/0207347 26951: 26915: 26891: 26851: 26811: 26779: 26761: 26743: 26725: 26715: 26658: 26618: 26584: 26563: 26536: 26496: 26459: 26433: 26412: 26385: 26348: 26306: 26285: 26251: 26230: 26193: 26175: 26157: 26155:astro-ph/9909339 26132: 26098: 26077: 26075: 26073: 26067: 26061:. Archived from 26028: 26018: 25973: 25944: 25915: 25886: 25852: 25831: 25804: 25775: 25773: 25771: 25765: 25759:. Archived from 25743:(767): 243–265. 25734: 25724: 25690: 25688:quant-ph/0306136 25665: 25637: 25628:Davies, P. C. W. 25623: 25613: 25587: 25562: 25533: 25491: 25481: 25447: 25445:astro-ph/9704263 25429: 25416: 25382: 25372: 25338: 25336:quant-ph/0203002 25317: 25288: 25251: 25233: 25231:10.1038/537S201a 25200: 25175:(8): 2037–2046. 25159: 25149: 25111: 25094: 25077: 25057: 25055: 25053: 25047: 25036: 25026: 25005: 24988: 24967: 24950: 24945:. Archived from 24933: 24910:Davies, P. C. W. 24905: 24900:. Archived from 24890:Davies, P. C. W. 24885: 24868: 24851: 24834: 24817: 24792: 24767: 24750: 24724: 24723: 24721: 24719: 24710:. Archived from 24696: 24690: 24684: 24678: 24677: 24675: 24673: 24658: 24652: 24651: 24649: 24647: 24632: 24626: 24620: 24614: 24613: 24603: 24585: 24553: 24547: 24546: 24506: 24500: 24499: 24473: 24467: 24461: 24455: 24454: 24452: 24450: 24445:on 1 August 2021 24435: 24429: 24426:Lucentini (2000) 24423: 24417: 24416: 24399:(2–3): 260–267. 24386: 24380: 24379: 24353: 24335: 24326: 24320: 24314: 24308: 24307: 24279: 24273: 24267: 24261: 24260: 24224: 24218: 24217: 24192:(9): 5489–5495. 24181: 24175: 24174: 24138: 24132: 24131: 24129: 24127: 24118: 24109: 24103: 24102: 24100: 24088: 24082: 24076: 24070: 24069: 24059: 24027: 24021: 24020: 24003:(3–4): 441–444. 23980: 23974: 23973: 23947: 23925: 23919: 23918: 23902: 23892: 23886: 23885: 23849: 23840: 23834: 23833: 23805: 23799: 23798: 23796: 23794: 23788: 23782:. Archived from 23777: 23768: 23762: 23761: 23750: 23744: 23743: 23707: 23701: 23700: 23666: 23646: 23640: 23639: 23622:(8): 2739–2740. 23611: 23605: 23604: 23576: 23570: 23569: 23541: 23535: 23534: 23498: 23492: 23490: 23484: 23478: 23476: 23470: 23464: 23462: 23456: 23450: 23449: 23432:(6): 1331–1335. 23421: 23415: 23414: 23386: 23380: 23379: 23377: 23375: 23360: 23354: 23353: 23351: 23339: 23333: 23332: 23296: 23290: 23289: 23263: 23254: 23248: 23247: 23245: 23233: 23227: 23226: 23224: 23222: 23207: 23201: 23200: 23198: 23174: 23168: 23167: 23141: 23121: 23112: 23106: 23105: 23089: 23083: 23082: 23054: 23048: 23047: 23037: 23011: 22987: 22981: 22980: 22954: 22934: 22928: 22922: 22916: 22910: 22904: 22903: 22875: 22869: 22868: 22858: 22826: 22820: 22819: 22793: 22773: 22767: 22766: 22738: 22732: 22726: 22720: 22714: 22708: 22707: 22679: 22673: 22672: 22652: 22642: 22636: 22635: 22607: 22601: 22600: 22572: 22566: 22565: 22537: 22531: 22530: 22514: 22508: 22507: 22505: 22503: 22497: 22491:. Archived from 22469:Physics Procedia 22466: 22457: 22451: 22449: 22443: 22437: 22436: 22434: 22432: 22417: 22411: 22410: 22400: 22390: 22366: 22360: 22359: 22357: 22345: 22339: 22333: 22327: 22326: 22298: 22292: 22286: 22277: 22276: 22255: 22249: 22248: 22236: 22230:Aiken, Amber M. 22227: 22216: 22215: 22213: 22211: 22191: 22185: 22184: 22164: 22158: 22157: 22131: 22125: 22124: 22098: 22078: 22072: 22062: 22056: 22055: 22027: 22021: 22020: 21992: 21986: 21985: 21983: 21981: 21975: 21960: 21951: 21945: 21944: 21942: 21940: 21934: 21919: 21910: 21904: 21903: 21901: 21899: 21893: 21878: 21869: 21863: 21862: 21854: 21848: 21847: 21827: 21821: 21820: 21800: 21794: 21793: 21775: 21769: 21768: 21750: 21744: 21743: 21725: 21719: 21718: 21716: 21714: 21698: 21692: 21691: 21671: 21665: 21655: 21649: 21648: 21636: 21630: 21629: 21611: 21605: 21604: 21602: 21592: 21586: 21580: 21574: 21573: 21571: 21562: 21556: 21555: 21543: 21537: 21536: 21500: 21494: 21493: 21465: 21459: 21453: 21447: 21446: 21444: 21442: 21436: 21430:. Archived from 21413: 21404: 21398: 21397: 21377: 21371: 21370: 21350: 21344: 21343: 21323: 21317: 21316: 21296: 21290: 21284: 21278: 21277: 21265: 21259: 21258: 21256: 21224: 21218: 21217: 21185: 21179: 21178: 21150: 21144: 21143: 21107: 21101: 21100: 21072: 21066: 21065: 21029: 21023: 21022: 20986: 20980: 20979: 20977: 20975: 20969: 20963:. Archived from 20946: 20937: 20931: 20930: 20904: 20882: 20876: 20875: 20865: 20835: 20829: 20823: 20817: 20816: 20796: 20790: 20784: 20778: 20772: 20766: 20765: 20739: 20717: 20708: 20707: 20679: 20673: 20672: 20636: 20630: 20615: 20609: 20608: 20574: 20568: 20564: 20558: 20551: 20545: 20544: 20510: 20508:astro-ph/9806099 20492: 20483: 20477: 20476: 20474: 20472: 20457: 20451: 20450: 20438: 20432: 20426: 20420: 20414: 20408: 20405:Battersby (2016) 20402: 20396: 20390: 20384: 20378: 20372: 20366: 20360: 20341: 20335: 20320: 20314: 20308: 20302: 20295: 20289: 20283: 20277: 20271: 20265: 20262:Weisskopf (1936) 20259: 20253: 20247: 20241: 20240: 20222: 20216: 20215: 20197: 20191: 20188:Le Bellac (2006) 20185: 20179: 20176:Le Bellac (2006) 20173: 20167: 20166: 20141:(3): 1824–1831. 20128: 20122: 20116: 20110: 20099: 20090: 20084: 20078: 20077: 20049: 20043: 20042: 20032: 19998: 19989: 19983: 19977: 19971: 19965: 19959: 19953: 19947: 19946: 19920: 19900: 19894: 19893: 19867: 19858: 19852: 19851: 19831: 19825: 19824: 19796: 19790: 19789: 19745: 19739: 19738: 19710: 19704: 19703: 19683: 19677: 19676: 19650: 19641: 19632: 19626: 19620: 19614: 19608: 19602: 19596: 19595: 19593: 19591: 19581: 19573: 19567: 19566: 19538: 19532: 19526: 19517: 19511: 19505: 19504: 19476: 19470: 19464: 19458: 19452: 19446: 19445: 19416:Touchstone Books 19407: 19401: 19400: 19360: 19354: 19348: 19342: 19336: 19330: 19329: 19301: 19295: 19294: 19284: 19252: 19246: 19245: 19203: 19194: 19188: 19182: 19176: 19170: 19164: 19163: 19143: 19137: 19131: 19122: 19121: 19087: 19085:cond-mat/0006404 19078:(9): 1799–1802. 19069: 19060: 19051: 19050: 19008: 18999: 18990: 18984: 18975: 18974: 18946: 18937: 18936: 18908: 18899: 18898: 18870: 18859: 18858: 18832: 18826: 18825: 18805: 18799: 18798: 18786: 18780: 18779: 18767: 18761: 18760: 18758: 18756: 18727: 18718: 18712: 18711: 18675: 18669: 18668: 18643:(7–8): 491–527. 18631: 18625: 18624: 18622: 18620: 18606: 18597: 18591: 18590: 18562: 18556: 18555: 18553: 18551: 18541: 18532: 18526: 18520: 18514: 18513: 18477: 18471: 18468:Weisskopf (1936) 18465: 18459: 18458: 18432: 18426: 18425: 18388: 18382: 18381: 18379: 18338: 18332: 18331: 18303: 18297: 18296: 18260: 18254: 18251:§1.5.2 pp. 22–23 18244: 18238: 18237: 18207: 18201: 18200: 18180: 18174: 18168: 18157: 18156: 18132: 18122: 18116: 18115: 18079: 18073: 18072: 18039: 18033: 18032: 18012: 18006: 18005: 17969: 17963: 17962: 17938: 17928: 17922: 17921: 17893: 17887: 17886: 17872: 17866: 17865: 17848:(155): 857–875. 17833: 17827: 17826: 17809:(153): 476–502. 17800: 17791: 17785: 17784: 17758: 17749: 17743: 17737: 17731: 17730: 17687: 17681: 17680: 17669:10.1038/114349a0 17644: 17638: 17637: 17601: 17595: 17594: 17584: 17582:10.1038/133529a0 17552: 17546: 17545: 17509: 17503: 17502: 17488: 17482: 17481: 17469: 17463: 17462: 17449: 17443: 17442: 17410: 17404: 17398: 17392: 17391: 17359: 17353: 17347: 17341: 17318: 17312: 17306: 17300: 17294: 17288: 17282: 17273: 17267: 17261: 17255: 17249: 17243: 17237: 17231: 17225: 17219: 17213: 17207: 17198: 17197: 17186: 17177: 17171: 17160: 17157:O'Carroll (2013) 17145: 17139: 17116: 17107: 17101: 17095: 17089: 17083: 17077: 17071: 17065: 17059: 17053: 17042: 17036: 17027: 17024:Battersby (2008) 17021: 17012: 17006: 17000: 16994: 16988: 16982: 16973: 16967: 16958: 16939: 16933: 16927: 16921: 16915: 16904: 16898: 16872:Virtual particle 16795: 16790: 16789: 16781: 16776: 16775: 16639:Alcubierre drive 16406:pseudoscientific 16371:cosmic inflation 16317: 16315: 16314: 16309: 16284:London equations 16218:Clifford algebra 16170:Josephson effect 16158:entangled states 16106: 16099: 16092: 15978:Quantum geometry 15933:Extra dimensions 15616: 15602: 15601: 15501:cosmic expansion 15497:Cosmic inflation 15483: 15470:Cosmic inflation 15429:Euclid telescope 15387: 15351:interstellar gas 15262: 15232: 15228: 15224: 15220: 15219: 15217: 15216: 15213: 15210: 15198: 15196: 15195: 15190: 15188: 15180: 15165: 15147: 15145: 15144: 15139: 15134: 15126: 15121: 15120: 15111: 15110: 15094: 15085: 15076: 15075: 15073: 15072: 15069: 15066: 15046: 15044: 15043: 15038: 15036: 15028: 15023: 15021: 15014: 15013: 14996: 14991: 14982: 14977: 14975: 14967: 14966: 14957: 14934: 14932: 14931: 14926: 14924: 14923: 14903: 14886: 14885: 14883: 14882: 14873: 14870: 14869: 14868: 14842: 14838: 14830: 14811: 14807: 14805: 14787: 14785: 14783: 14782: 14779: 14776: 14764: 14762: 14760: 14759: 14756: 14753: 14709: 14635: 14628: 14627: 14625: 14624: 14623: 14622: 14614: 14611: 14609: 14593: 14592: 14590: 14589: 14588: 14587: 14581: 14578: 14563: 14546: 14530: 14524: 14475:quark condensate 14471:gluon condensate 14462:non-perturbative 14439: 14437: 14436: 14431: 14419: 14413: 14409: 14396: 14390: 14389: 14387: 14386: 14385: 14384: 14373: 14370: 14358: 14347: 14331: 14329: 14328: 14323: 14312: 14311: 14306: 14300: 14292: 14287: 14281: 14276: 14264: 14263: 14255: 14251: 14246: 14231: 14230: 14222: 14212: 14210: 14209: 14204: 14202: 14198: 14197: 14189: 14185: 14180: 14165: 14163: 14149: 14146: 14135: 14129: 14124: 14109: 14108: 14100: 14096: 14083: 14082: 14077: 14071: 14063: 14058: 14056: 14042: 14034: 14028: 14023: 14011: 14010: 14002: 13998: 13987: 13985: 13984: 13979: 13977: 13961: 13957: 13953: 13951: 13949: 13944: 13928: 13922: 13918: 13916: 13915: 13914: 13895: 13893: 13888: 13879: 13878: 13859: 13847: 13843: 13841: 13839: 13834: 13825: 13824: 13812: 13811: 13801: 13793: 13790: 13785: 13772: 13767: 13758: 13756: 13755: 13754: 13735: 13734: 13733: 13714: 13705: 13701: 13691: 13690: 13650: 13637: 13625: 13623: 13622: 13617: 13615: 13611: 13609: 13608: 13607: 13598: 13597: 13585: 13584: 13579: 13575: 13573: 13568: 13556: 13555: 13539: 13538: 13537: 13520: 13517: 13512: 13503: 13499: 13494: 13486: 13480: 13476: 13474: 13473: 13472: 13460: 13459: 13446: 13445: 13444: 13428: 13416: 13412: 13408: 13398: 13397: 13389: 13357: 13353: 13349: 13339: 13338: 13329: 13321: 13296: 13292: 13282: 13281: 13273: 13241: 13230: 13228: 13227: 13222: 13220: 13219: 13218: 13198: 13193: 13192: 13191: 13171: 13167: 13156: 13151: 13150: 13142: 13141: 13115: 13114: 13110: 13086: 13084: 13083: 13078: 13067: 13066: 13061: 13055: 13047: 13042: 13040: 13026: 13018: 13012: 13007: 12995: 12994: 12986: 12974: 12972: 12971: 12966: 12964: 12948: 12923: 12922: 12901: 12885: 12881: 12871: 12870: 12832: 12831: 12816: 12812: 12808: 12786: 12785: 12767: 12766: 12721: 12720: 12699: 12695: 12685: 12684: 12640: 12639: 12637: 12636: 12633: 12630: 12608: 12594: 12580: 12569: 12567: 12566: 12561: 12559: 12546: 12545: 12528: 12523: 12508: 12504: 12500: 12495: 12494: 12493: 12474: 12467: 12463: 12453: 12451: 12440: 12432: 12428: 12424: 12413: 12408: 12407: 12399: 12398: 12375: 12374: 12370: 12352: 12346: 12345: 12344: 12343: 12320: 12319: 12317: 12311: 12310: 12309: 12290: 12289: 12286: 12285: 12277: 12276: 12261: 12257: 12247: 12245: 12234: 12225: 12221: 12210: 12205: 12200: 12189: 12187: 12176: 12163: 12148: 12146: 12145: 12140: 12126: 12112: 12111: 12106: 12100: 12074: 12073: 12068: 12052: 12041: 12037: 12035: 12034: 12029: 12020: 12019: 12010: 12002: 11999: 11997: 11983: 11971: 11964: 11957: 11935: 11917: 11916: 11914: 11913: 11907: 11904: 11889: 11887: 11886: 11881: 11870: 11869: 11864: 11858: 11850: 11845: 11843: 11829: 11821: 11815: 11810: 11798: 11797: 11789: 11779: 11777: 11776: 11771: 11760: 11748: 11729: 11713: 11706: 11692: 11690: 11689: 11684: 11676: 11671: 11667: 11666: 11664: 11663: 11662: 11649: 11648: 11639: 11637: 11635: 11634: 11622: 11617: 11616: 11594: 11580: 11564: 11548: 11546: 11545: 11540: 11538: 11537: 11529: 11526: 11524: 11523: 11522: 11509: 11501: 11487: 11486: 11478: 11465: 11455: 11453: 11452: 11447: 11445: 11441: 11434: 11421: 11420: 11405: 11401: 11400: 11396: 11384: 11383: 11375: 11369: 11368: 11364: 11349: 11337: 11332: 11322: 11318: 11308: 11303: 11292: 11275: 11274: 11266: 11253: 11251: 11250: 11245: 11243: 11242: 11238: 11228: 11224: 11223: 11222: 11218: 11217: 11190: 11185: 11181: 11168: 11167: 11163: 11162: 11133: 11132: 11128: 11113: 11107: 11106: 11105: 11086: 11085: 11082: 11078: 11053: 11052: 11047: 11034: 11032: 11031: 11026: 11015: 11014: 11006: 11000: 10992: 10978: 10977: 10972: 10966: 10958: 10953: 10947: 10942: 10930: 10929: 10921: 10912:may be written: 10911: 10897: 10895: 10894: 10889: 10887: 10886: 10885: 10881: 10874: 10861: 10860: 10840: 10829: 10828: 10820: 10814: 10813: 10809: 10798: 10786: 10781: 10772: 10769: 10765: 10764: 10751: 10743: 10742: 10731: 10730: 10726: 10725: 10696: 10695: 10691: 10669: 10668: 10664: 10644: 10642: 10641: 10636: 10598: 10597: 10573: 10572: 10549: 10548: 10540: 10524: 10513: 10482: 10471: 10464: 10450: 10444: 10435: 10433: 10432: 10427: 10425: 10424: 10420: 10407: 10406: 10398: 10395: 10392: 10388: 10387: 10374: 10366: 10365: 10354: 10353: 10349: 10339: 10338: 10323: 10322: 10318: 10312: 10311: 10303: 10292: 10276: 10274: 10273: 10268: 10266: 10262: 10261: 10257: 10247: 10243: 10232: 10227: 10223: 10201: 10200: 10196: 10181: 10175: 10174: 10173: 10154: 10153: 10150: 10146: 10130: 10126: 10121: 10113: 10104: 10098: 10093: 10081: 10080: 10072: 10058: 10047: 10045: 10044: 10039: 10033: 10032: 10027: 10012: 10011: 10003: 9994: 9992: 9984: 9983: 9974: 9969: 9968: 9960: 9950: 9948: 9947: 9942: 9940: 9936: 9930: 9925: 9910: 9902: 9901: 9893: 9890: 9882: 9877: 9863: 9859: 9853: 9848: 9833: 9825: 9824: 9816: 9813: 9805: 9800: 9796: 9795: 9790: 9786: 9779: 9778: 9770: 9759: 9758: 9750: 9742: 9734: 9723: 9719: 9718: 9710: 9707: 9699: 9694: 9693: 9685: 9675: 9674: 9666: 9649: 9647: 9646: 9641: 9639: 9635: 9631: 9630: 9622: 9618: 9613: 9598: 9590: 9589: 9581: 9578: 9570: 9565: 9551: 9550: 9542: 9536: 9528: 9520: 9516: 9515: 9507: 9503: 9498: 9483: 9479: 9478: 9473: 9465: 9446: 9442: 9441: 9436: 9428: 9423: 9413: 9405: 9400: 9396: 9395: 9390: 9382: 9372: 9368: 9361: 9357: 9356: 9351: 9343: 9338: 9323: 9315: 9304: 9300: 9299: 9291: 9287: 9282: 9267: 9266: 9261: 9257: 9256: 9251: 9243: 9238: 9224: 9215: 9206: 9193: 9185: 9184: 9156: 9155: 9147: 9140: 9136: 9135: 9130: 9122: 9117: 9107: 9099: 9085: 9077: 9076: 9048: 9047: 9039: 9025: 9024: 9022: 9021: 9016: 9013: 8989: 8980: 8978: 8977: 8972: 8970: 8969: 8957: 8956: 8951: 8944: 8939: 8927: 8918: 8912: 8911: 8906: 8902: 8901: 8896: 8888: 8883: 8872: 8870: 8859: 8834: 8832: 8831: 8826: 8824: 8820: 8818: 8813: 8800: 8795: 8781: 8779: 8778: 8777: 8768: 8767: 8751: 8737: 8736: 8720: 8719: 8718: 8708: 8707: 8706: 8685: 8676: 8665: 8663: 8662: 8657: 8655: 8653: 8652: 8651: 8642: 8641: 8628: 8627: 8626: 8613: 8599: 8598: 8582: 8580: 8579: 8574: 8572: 8568: 8567: 8548: 8546: 8545: 8544: 8535: 8534: 8518: 8510: 8506: 8502: 8501: 8500: 8488: 8479: 8471: 8470: 8451: 8449: 8448: 8447: 8434: 8426: 8418: 8414: 8413: 8401: 8392: 8386: 8385: 8373: 8371: 8370: 8369: 8353: 8344: 8343: 8331: 8322: 8318: 8314: 8304: 8296: 8283: 8281: 8280: 8275: 8270: 8269: 8256: 8247: 8245: 8244: 8243: 8227: 8219: 8218: 8206: 8205: 8200: 8196: 8194: 8183: 8175: 8162: 8158: 8141: 8140: 8138: 8137: 8134: 8131: 8105: 8093: 8065: 8058: 8054: 8048: 8044: 8042: 8041: 8036: 8027: 8026: 8014: 8012: 8011: 8002: 7988: 7979: 7977: 7976: 7971: 7966: 7964: 7963: 7954: 7947: 7946: 7930: 7919: 7917: 7916: 7911: 7909: 7908: 7896: 7887: 7883: 7879: 7862: 7851: 7849: 7848: 7843: 7841: 7831: 7827: 7816: 7811: 7810: 7802: 7798: 7774: 7769: 7765: 7743: 7739: 7729: 7728: 7727: 7719: 7715: 7692: 7691: 7687: 7668: 7667: 7666: 7646: 7641: 7640: 7636: 7627: 7610: 7606: 7595: 7590: 7589: 7581: 7577: 7554: 7553: 7549: 7521: 7519: 7518: 7513: 7511: 7507: 7506: 7497: 7491: 7490: 7486: 7475: 7470: 7466: 7451: 7450: 7437: 7433: 7420: 7419: 7403: 7401: 7400: 7395: 7393: 7392: 7391: 7371: 7366: 7365: 7361: 7352: 7336: 7327: 7322: 7321: 7313: 7309: 7302: 7290: 7282: 7281: 7277: 7271: 7262: 7261: 7252: 7251: 7235: 7233: 7232: 7227: 7225: 7224: 7220: 7210: 7206: 7205: 7204: 7203: 7195: 7169: 7164: 7160: 7147: 7146: 7145: 7137: 7115: 7114: 7110: 7095: 7092: 7088: 7087: 7077: 7076: 7075: 7056: 7055: 7052: 7048: 7029: 7021: 7009: 6984: 6980: 6976: 6975: 6974: 6957: 6943: 6930: 6928: 6927: 6922: 6920: 6916: 6915: 6914: 6910: 6902: 6891: 6890: 6860: 6855: 6851: 6838: 6837: 6833: 6825: 6814: 6813: 6781: 6780: 6776: 6761: 6758: 6754: 6753: 6743: 6742: 6741: 6722: 6721: 6707: 6699: 6698: 6694: 6688: 6675: 6673: 6672: 6667: 6665: 6664: 6660: 6650: 6646: 6645: 6644: 6643: 6635: 6609: 6604: 6600: 6587: 6586: 6585: 6577: 6555: 6554: 6550: 6535: 6532: 6528: 6527: 6517: 6516: 6515: 6496: 6495: 6481: 6473: 6472: 6468: 6462: 6449: 6447: 6446: 6441: 6439: 6438: 6403: 6402: 6401: 6393: 6380: 6379: 6375: 6365: 6359: 6355: 6347: 6339: 6338: 6334: 6328: 6315: 6313: 6312: 6298: 6297: 6282: 6259: 6247: 6236:, which we call 6235: 6224: 6208: 6197: 6195: 6194: 6189: 6181: 6180: 6175: 6171: 6167: 6159: 6158: 6157: 6151: 6136: 6135: 6126: 6125: 6109: 6107: 6106: 6101: 6099: 6098: 6097: 6089: 6076: 6075: 6074: 6064: 6058: 6054: 6046: 6038: 6037: 6036: 6030: 6017: 6013: 6011: 6010: 6005: 6003: 5999: 5998: 5997: 5985: 5984: 5972: 5971: 5957: 5952: 5944: 5939: 5935: 5934: 5933: 5921: 5920: 5908: 5907: 5887:or equivalently 5886: 5884: 5883: 5878: 5849: 5796: 5784: 5771: 5752: 5736: 5734: 5733: 5728: 5726: 5725: 5724: 5716: 5703: 5702: 5701: 5685: 5677: 5676: 5671: 5658: 5652: 5650: 5632: 5630: 5629: 5624: 5616: 5615: 5610: 5606: 5602: 5594: 5593: 5588: 5573: 5572: 5553: 5552: 5550: 5549: 5544: 5541: 5527: 5525: 5524: 5519: 5508: 5500: 5499: 5494: 5488: 5484: 5483: 5482: 5470: 5469: 5444: 5428: 5426: 5425: 5420: 5418: 5414: 5413: 5408: 5390: 5385: 5383: 5375: 5374: 5365: 5357: 5353: 5349: 5348: 5347: 5342: 5333: 5332: 5327: 5313: 5312: 5300: 5298: 5287: 5278: 5277: 5245: 5239: 5235: 5229: 5225: 5219: 5213: 5211: 5210: 5205: 5200: 5199: 5181: 5177: 5173: 5172: 5160: 5159: 5133: 5132: 5113: 5102:normally ordered 5099: 5097: 5096: 5091: 5089: 5082: 5081: 5060: 5050: 5041: 5035: 5031: 5030: 5021: 5012: 5011: 4985: 4975: 4966: 4960: 4956: 4952: 4951: 4939: 4938: 4915: 4906: 4896: 4892: 4888: 4883: 4882: 4873: 4857: 4856: 4833: 4831: 4830: 4825: 4823: 4819: 4815: 4814: 4805: 4796: 4795: 4769: 4765: 4761: 4757: 4756: 4744: 4743: 4720: 4711: 4702: 4698: 4697: 4692: 4686: 4685: 4673: 4664: 4658: 4656: 4648: 4647: 4638: 4629: 4628: 4605: 4604: 4602: 4601: 4598: 4595: 4585: 4579: 4572: 4568: 4562: 4558: 4549: 4547: 4546: 4541: 4533: 4529: 4525: 4524: 4497: 4488: 4486: 4476: 4474: 4473: 4468: 4460: 4456: 4455: 4454: 4427: 4421: 4417: 4410: 4355:Feynman diagrams 4303: 4302: 4300: 4299: 4296: 4293: 4259:zero-point field 4217: 4210: 4203: 3308:Effective action 3235:Lorentz symmetry 3160:Electromagnetism 3130: 3116: 3115: 3107: 3103: 3096: 3092: 3084: 3080: 3078: 3077: 3072: 3070: 3068: 3067: 3066: 3050: 3049: 3048: 3039: 3038: 3028: 2980:unitary operator 2966: 2965: 2963: 2962: 2959: 2956: 2942: 2941: 2939: 2938: 2935: 2932: 2922: 2918: 2914: 2904: 2896: 2883: 2882: 2880: 2879: 2876: 2873: 2841: 2840: 2838: 2837: 2834: 2831: 2800: 2799: 2798: 2779: 2777: 2776: 2771: 2769: 2764: 2756: 2751: 2750: 2738: 2729: 2728: 2726: 2718: 2713: 2712: 2700: 2696: 2695: 2687: 2669: 2667: 2666: 2661: 2655: 2647: 2645: 2644: 2639: 2635: 2627: 2617: 2613: 2612: 2611: 2606: 2605: 2597: 2593: 2591: 2580: 2573: 2569: 2568: 2567: 2562: 2558: 2557: 2556: 2544: 2543: 2535: 2523: 2514: 2491: 2489: 2488: 2483: 2477: 2469: 2464: 2462: 2458: 2457: 2452: 2451: 2443: 2435: 2433: 2431: 2427: 2426: 2421: 2417: 2416: 2415: 2403: 2402: 2394: 2381: 2369: 2360: 2358: 2357: 2352: 2346: 2345: 2340: 2339: 2331: 2327: 2325: 2314: 2309: 2308: 2303: 2299: 2298: 2297: 2285: 2284: 2276: 2264: 2255: 2249: 2248: 2236: 2235: 2227: 2082:Victor Weisskopf 2052: 2031:Victor Weisskopf 1995:Paul Dirac, 1933 1958: 1954: 1953: 1951: 1950: 1945: 1942: 1909:matrix mechanics 1880: 1867:Heisenberg, 1924 1810: 1808: 1807: 1802: 1795: 1794: 1792: 1785: 1784: 1771: 1754: 1746: 1741: 1736: 1728: 1712: 1708: 1704: 1703: 1701: 1700: 1697: 1694: 1684: 1664:is the absolute 1663: 1655: 1647: 1639: 1635: 1633: 1632: 1627: 1621: 1619: 1612: 1611: 1598: 1581: 1573: 1554: 1540:for his work on 1443: 1442: 1291: 1199:potential energy 1127:zero-point field 1115: 1109:from the German 1107:is a translation 922: 915: 908: 549:Superdeterminism 202:Bra–ket notation 153: 151: 150: 145: 137: 132: 131: 123: 111: 106: 104: 93: 65: 64: 31465: 31464: 31460: 31459: 31458: 31456: 31455: 31454: 31385: 31384: 31383: 31378: 31322: 31303:Granular matter 31298:Amorphous solid 31284: 31209: 31195:Antiferromagnet 31185:Superparamagnet 31158:Magnetic phases 31149: 31113: 31062: 31023:Bloch's theorem 30996: 30938: 30919:Order parameter 30912:Phase phenomena 30907: 30898: 30830: 30825: 30795: 30790: 30773: 30725:Quantum gravity 30652: 30611:Particle theory 30606: 30585: 30534: 30508: 30472: 30436: 30390:Low dimensional 30385: 30326:Ginzburg–Landau 30297: 30288:Topological QFT 30226: 30221: 30191: 30186: 30185: 30163: 30107: 30098:Vertex function 30083:Schwinger limit 30058:Furry's theorem 30041: 29995: 29981:Feynman diagram 29964: 29959: 29929: 29924: 29906: 29892:Wigner's friend 29868: 29859:Quantum gravity 29820: 29806:Quantum sensing 29786:Quantum network 29766:Quantum machine 29736:Quantum imaging 29699:Quantum circuit 29694:Quantum channel 29653: 29599:Quantum biology 29585: 29561:Elitzur–Vaidman 29536:Davisson–Germer 29519: 29471:Hidden-variable 29461:de Broglie–Bohm 29438:Interpretations 29432: 29396: 29350: 29237:Complementarity 29215: 29172: 29167: 29137: 29132: 29096: 29082:Medical physics 29033: 28992:Nuclear physics 28961: 28955:Non-equilibrium 28877: 28849: 28821: 28815: 28777:Steven Weinberg 28763: 28758: 28736: 28725: 28693: 28653: 28634: 28629: 28559: 28557: 28556:on 12 July 2018 28553: 28514: 28468: 28463: 28337: 28332: 28330:Further reading 28327: 28209: 28207: 28203: 28142: 28114: 28043: 27878: 27851: 27814: 27796:Saunders, Simon 27768: 27730: 27692: 27654: 27642:, eds. (1991). 27636:Saunders, Simon 27419: 27392: 27368: 27114: 27033: 26932: 26872: 26723: 26639: 26552: 26542:Quantum Physics 26517: 26477: 26401: 26211: 26096:physics/0605038 26071: 26069: 26068:on 31 July 2018 26065: 26026: 25999: 25933: 25904: 25843:(15): 1260004. 25828: 25769: 25767: 25763: 25737:Proc. R. Soc. A 25732: 25646: 25551: 25489: 25427: 25380: 25269: 25147:10.1.1.129.5792 25119: 25114: 25051: 25049: 25045: 25034: 24732: 24727: 24717: 24715: 24697: 24693: 24685: 24681: 24671: 24669: 24659: 24655: 24645: 24643: 24641:engineering.com 24633: 24629: 24621: 24617: 24554: 24550: 24507: 24503: 24496: 24474: 24470: 24462: 24458: 24448: 24446: 24437: 24436: 24432: 24424: 24420: 24387: 24383: 24333: 24327: 24323: 24315: 24311: 24280: 24276: 24268: 24264: 24225: 24221: 24182: 24178: 24139: 24135: 24125: 24123: 24116: 24110: 24106: 24098:gr-qc/0603033v1 24089: 24085: 24077: 24073: 24057:physics/9705043 24041: 24037: 24033: 24028: 24024: 23994: 23990: 23986: 23981: 23977: 23926: 23922: 23915: 23893: 23889: 23847: 23841: 23837: 23806: 23802: 23792: 23790: 23786: 23775: 23769: 23765: 23752: 23751: 23747: 23708: 23704: 23647: 23643: 23612: 23608: 23587:(35): 193–195. 23577: 23573: 23542: 23538: 23499: 23495: 23486: 23485: 23481: 23472: 23471: 23467: 23458: 23457: 23453: 23422: 23418: 23387: 23383: 23373: 23371: 23362: 23361: 23357: 23340: 23336: 23297: 23293: 23261: 23255: 23251: 23234: 23230: 23220: 23218: 23208: 23204: 23196:physics/0108005 23188: 23184: 23180: 23175: 23171: 23119: 23113: 23109: 23090: 23086: 23079: 23055: 23051: 22988: 22984: 22935: 22931: 22923: 22919: 22911: 22907: 22876: 22872: 22827: 22823: 22774: 22770: 22739: 22735: 22727: 22723: 22715: 22711: 22680: 22676: 22669: 22643: 22639: 22608: 22604: 22573: 22569: 22538: 22534: 22515: 22511: 22501: 22499: 22495: 22464: 22458: 22454: 22445: 22444: 22440: 22430: 22428: 22419: 22418: 22414: 22367: 22363: 22346: 22342: 22334: 22330: 22299: 22295: 22287: 22280: 22257: 22256: 22252: 22234: 22228: 22219: 22209: 22207: 22192: 22188: 22181: 22165: 22161: 22154: 22132: 22128: 22079: 22075: 22063: 22059: 22028: 22024: 21993: 21989: 21979: 21977: 21973: 21958: 21952: 21948: 21938: 21936: 21932: 21917: 21911: 21907: 21897: 21895: 21891: 21876: 21870: 21866: 21856: 21855: 21851: 21844: 21828: 21824: 21817: 21801: 21797: 21790: 21776: 21772: 21765: 21751: 21747: 21740: 21726: 21722: 21712: 21710: 21699: 21695: 21688: 21672: 21668: 21656: 21652: 21637: 21633: 21626: 21612: 21608: 21593: 21589: 21581: 21577: 21569: 21563: 21559: 21544: 21540: 21501: 21497: 21466: 21462: 21454: 21450: 21440: 21438: 21434: 21411: 21405: 21401: 21394: 21383:The Maxwellians 21378: 21374: 21351: 21347: 21340: 21329:The Maxwellians 21324: 21320: 21313: 21297: 21293: 21285: 21281: 21266: 21262: 21225: 21221: 21186: 21182: 21151: 21147: 21108: 21104: 21073: 21069: 21030: 21026: 20997:(1802): 45–57. 20987: 20983: 20973: 20971: 20967: 20944: 20938: 20934: 20888:Physical Review 20883: 20879: 20863:10.1.1.205.6343 20836: 20832: 20824: 20820: 20813: 20797: 20793: 20785: 20781: 20773: 20769: 20718: 20711: 20680: 20676: 20637: 20633: 20616: 20612: 20597: 20575: 20571: 20565: 20561: 20552: 20548: 20490: 20484: 20480: 20470: 20468: 20458: 20454: 20439: 20435: 20427: 20423: 20415: 20411: 20403: 20399: 20391: 20387: 20379: 20375: 20367: 20363: 20342: 20338: 20321: 20317: 20309: 20305: 20296: 20292: 20284: 20280: 20272: 20268: 20260: 20256: 20248: 20244: 20237: 20223: 20219: 20212: 20198: 20194: 20186: 20182: 20174: 20170: 20129: 20125: 20117: 20113: 20100: 20093: 20085: 20081: 20050: 20046: 19996: 19990: 19986: 19978: 19974: 19966: 19962: 19954: 19950: 19918:physics/9805038 19901: 19897: 19865: 19859: 19855: 19832: 19828: 19797: 19793: 19746: 19742: 19711: 19707: 19684: 19680: 19648: 19642: 19635: 19627: 19623: 19615: 19611: 19603: 19599: 19589: 19587: 19579: 19575: 19574: 19570: 19543:Physical Review 19539: 19535: 19527: 19520: 19512: 19508: 19481:Physical Review 19477: 19473: 19465: 19461: 19453: 19449: 19434: 19408: 19404: 19361: 19357: 19349: 19345: 19337: 19333: 19306:Physical Review 19302: 19298: 19253: 19249: 19201: 19195: 19191: 19185:Drexhage (1974) 19183: 19179: 19173:Drexhage (1970) 19171: 19167: 19144: 19140: 19132: 19125: 19067: 19061: 19054: 19006: 19000: 18993: 18985: 18978: 18951:Physical Review 18947: 18940: 18913:Physical Review 18909: 18902: 18875:Physical Review 18871: 18862: 18847: 18833: 18829: 18806: 18802: 18787: 18783: 18768: 18764: 18754: 18752: 18725: 18719: 18715: 18676: 18672: 18632: 18628: 18618: 18616: 18604: 18598: 18594: 18567:Physical Review 18563: 18559: 18549: 18547: 18539: 18533: 18529: 18521: 18517: 18482:Physical Review 18478: 18474: 18466: 18462: 18447: 18433: 18429: 18401:Physical Review 18389: 18385: 18355:Physical Review 18339: 18335: 18308:Physical Review 18304: 18300: 18271:(37): 631–637. 18261: 18257: 18245: 18241: 18226: 18208: 18204: 18181: 18177: 18169: 18160: 18145: 18123: 18119: 18080: 18076: 18061: 18040: 18036: 18013: 18009: 17970: 17966: 17951: 17929: 17925: 17904:(13): 361–376. 17894: 17890: 17873: 17869: 17834: 17830: 17798: 17792: 17788: 17756: 17750: 17746: 17738: 17734: 17719: 17688: 17684: 17645: 17641: 17602: 17598: 17553: 17549: 17510: 17506: 17489: 17485: 17470: 17466: 17450: 17446: 17411: 17407: 17401:Einstein (1993) 17399: 17395: 17360: 17356: 17348: 17344: 17319: 17315: 17307: 17303: 17295: 17291: 17283: 17276: 17268: 17264: 17256: 17252: 17244: 17240: 17232: 17228: 17220: 17216: 17208: 17201: 17196:on 31 May 2017. 17188: 17187: 17180: 17172: 17163: 17146: 17142: 17117: 17110: 17102: 17098: 17090: 17086: 17078: 17074: 17066: 17062: 17054: 17045: 17037: 17030: 17022: 17015: 17009:Einstein (1995) 17007: 17003: 16995: 16991: 16983: 16976: 16970:Weinberg (2015) 16968: 16961: 16943:Weinberg (1989) 16940: 16936: 16928: 16924: 16916: 16907: 16899: 16890: 16886: 16881: 16876: 16791: 16784: 16777: 16770: 16767: 16759:science fiction 16755: 16623:negative energy 16530:Meissner effect 16518: 16481: 16445:Physical Review 16419: 16379: 16303: 16300: 16299: 16280:superconductors 16246:Noether theorem 16128: 16110: 16081: 16080: 16036: 16028: 16027: 15953: 15951:Quantum gravity 15943: 15942: 15898: 15888: 15887: 15868:Massive gravity 15773:Brane cosmology 15708: 15700: 15699: 15645: 15630: 15600: 15572: 15557: 15494: 15493: 15488: 15485: 15478: 15472: 15464:renormalization 15402: 15401: 15396: 15389: 15382: 15376: 15371: 15323:RX J1856.5-3754 15310:quantum gravity 15299:RX J1856.5-3754 15280:Schwinger limit 15251: 15249: 15241:Main articles: 15239: 15230: 15226: 15222: 15214: 15211: 15208: 15207: 15205: 15200: 15179: 15171: 15168: 15167: 15161: 15156: 15125: 15116: 15112: 15106: 15102: 15100: 15097: 15096: 15093: 15087: 15084: 15078: 15070: 15067: 15062: 15061: 15059: 15054: 15027: 15009: 15005: 14998: 14992: 14987: 14981: 14968: 14962: 14958: 14956: 14948: 14945: 14944: 14919: 14915: 14913: 14910: 14909: 14901: 14896: 14881: 14874: 14871: 14867: 14862: 14861: 14860: 14856: 14855: 14853: 14848: 14840: 14836: 14833:Planck constant 14828: 14825: 14819: 14809: 14803: 14801: 14786: 14780: 14777: 14774: 14773: 14771: 14766: 14763: 14757: 14754: 14751: 14750: 14748: 14743: 14723: 14717: 14707: 14687:Hendrik Casimir 14669: 14663: 14654: 14630: 14618: 14616: 14615: 14612: 14605: 14603: 14602: 14600: 14595: 14585: 14583: 14582: 14579: 14574: 14573: 14571: 14565: 14559: 14544: 14526: 14520: 14513: 14511:Higgs mechanism 14507: 14457: 14451: 14425: 14422: 14421: 14415: 14411: 14405: 14392: 14383: 14380: 14379: 14378: 14374: 14371: 14368: 14367: 14365: 14360: 14349: 14341: 14335: 14307: 14302: 14301: 14291: 14283: 14277: 14272: 14254: 14253: 14247: 14242: 14221: 14220: 14218: 14215: 14214: 14200: 14199: 14188: 14187: 14181: 14176: 14153: 14148: 14145: 14131: 14125: 14120: 14099: 14098: 14094: 14093: 14078: 14073: 14072: 14062: 14046: 14041: 14030: 14024: 14019: 14001: 14000: 13995: 13993: 13990: 13989: 13975: 13974: 13959: 13958: 13945: 13940: 13932: 13927: 13923: 13910: 13906: 13896: 13889: 13884: 13874: 13870: 13860: 13858: 13854: 13845: 13844: 13835: 13830: 13820: 13816: 13807: 13803: 13802: 13794: 13792: 13786: 13778: 13768: 13763: 13750: 13746: 13736: 13729: 13725: 13715: 13713: 13706: 13686: 13682: 13666: 13662: 13658: 13656: 13653: 13652: 13649: 13639: 13635: 13629: 13613: 13612: 13603: 13599: 13593: 13589: 13580: 13569: 13564: 13551: 13547: 13546: 13542: 13541: 13540: 13533: 13529: 13521: 13519: 13513: 13508: 13487: 13485: 13481: 13468: 13464: 13455: 13451: 13447: 13440: 13436: 13429: 13427: 13423: 13414: 13413: 13388: 13387: 13368: 13364: 13355: 13354: 13334: 13330: 13320: 13304: 13300: 13272: 13271: 13252: 13248: 13238: 13236: 13233: 13232: 13211: 13207: 13203: 13194: 13184: 13180: 13179: 13152: 13143: 13134: 13133: 13132: 13106: 13105: 13101: 13100: 13096: 13094: 13091: 13090: 13062: 13057: 13056: 13046: 13030: 13025: 13014: 13008: 13003: 12985: 12984: 12982: 12979: 12978: 12962: 12961: 12946: 12945: 12918: 12914: 12899: 12898: 12866: 12862: 12846: 12842: 12827: 12823: 12814: 12813: 12781: 12777: 12762: 12758: 12716: 12712: 12711: 12707: 12700: 12680: 12676: 12660: 12656: 12652: 12650: 12647: 12646: 12634: 12631: 12628: 12623: 12622: 12620: 12619: 12610: 12607: 12596: 12593: 12582: 12578: 12573: 12557: 12556: 12541: 12537: 12524: 12519: 12506: 12505: 12489: 12485: 12475: 12473: 12469: 12459: 12458: 12444: 12439: 12430: 12429: 12409: 12400: 12391: 12390: 12389: 12366: 12365: 12361: 12360: 12356: 12336: 12335: 12331: 12321: 12318: 12305: 12301: 12291: 12288: 12278: 12269: 12268: 12267: 12253: 12252: 12238: 12233: 12226: 12206: 12201: 12196: 12194: 12190: 12180: 12175: 12171: 12169: 12166: 12165: 12161: 12150: 12122: 12107: 12102: 12101: 12096: 12069: 12064: 12063: 12058: 12055: 12054: 12049: 12043: 12039: 12015: 12011: 12006: 11998: 11993: 11979: 11977: 11974: 11973: 11966: 11959: 11951: 11940: 11929: 11923: 11908: 11905: 11899: 11898: 11896: 11891: 11865: 11860: 11859: 11849: 11833: 11828: 11817: 11811: 11806: 11788: 11787: 11785: 11782: 11781: 11756: 11754: 11751: 11750: 11742: 11734: 11723: 11715: 11708: 11700: 11694: 11672: 11658: 11654: 11650: 11644: 11640: 11638: 11630: 11626: 11621: 11612: 11608: 11607: 11603: 11601: 11598: 11597: 11588: 11582: 11574: 11566: 11558: 11552: 11528: 11527: 11518: 11514: 11510: 11502: 11500: 11479: 11474: 11473: 11471: 11468: 11467: 11463: 11427: 11426: 11422: 11416: 11412: 11389: 11385: 11374: 11373: 11360: 11359: 11355: 11354: 11350: 11342: 11333: 11328: 11314: 11313: 11293: 11291: 11267: 11262: 11261: 11259: 11256: 11255: 11234: 11233: 11229: 11213: 11209: 11205: 11201: 11186: 11177: 11176: 11158: 11154: 11147: 11143: 11124: 11123: 11119: 11118: 11114: 11101: 11097: 11087: 11084: 11074: 11073: 11048: 11043: 11042: 11040: 11037: 11036: 11007: 11002: 11001: 10991: 10973: 10968: 10967: 10957: 10949: 10943: 10938: 10920: 10919: 10917: 10914: 10913: 10910: 10899: 10867: 10866: 10862: 10856: 10852: 10848: 10844: 10833: 10819: 10818: 10805: 10804: 10800: 10791: 10782: 10777: 10760: 10756: 10752: 10744: 10741: 10721: 10717: 10710: 10706: 10687: 10686: 10682: 10660: 10659: 10655: 10653: 10650: 10649: 10590: 10586: 10568: 10564: 10539: 10538: 10530: 10527: 10526: 10515: 10484: 10473: 10466: 10463: 10452: 10446: 10440: 10416: 10415: 10411: 10397: 10396: 10383: 10379: 10375: 10367: 10364: 10345: 10344: 10340: 10334: 10330: 10314: 10313: 10302: 10301: 10300: 10298: 10295: 10294: 10291: 10280: 10264: 10263: 10253: 10252: 10248: 10228: 10219: 10218: 10192: 10191: 10187: 10186: 10182: 10169: 10165: 10155: 10152: 10142: 10141: 10128: 10127: 10122: 10112: 10105: 10100: 10094: 10089: 10071: 10070: 10066: 10064: 10061: 10060: 10051: 10028: 10023: 10022: 10002: 10001: 9985: 9979: 9975: 9973: 9959: 9958: 9956: 9953: 9952: 9938: 9937: 9932: 9926: 9921: 9906: 9892: 9891: 9881: 9873: 9861: 9860: 9855: 9849: 9844: 9829: 9815: 9814: 9804: 9791: 9769: 9768: 9767: 9763: 9749: 9748: 9747: 9743: 9733: 9721: 9720: 9709: 9708: 9698: 9684: 9683: 9676: 9665: 9664: 9657: 9655: 9652: 9651: 9637: 9636: 9633: 9632: 9621: 9620: 9614: 9609: 9594: 9580: 9579: 9569: 9561: 9541: 9540: 9527: 9518: 9517: 9506: 9505: 9499: 9494: 9474: 9464: 9460: 9456: 9437: 9427: 9419: 9418: 9414: 9404: 9391: 9381: 9377: 9373: 9352: 9342: 9334: 9333: 9329: 9328: 9324: 9314: 9302: 9301: 9290: 9289: 9283: 9278: 9262: 9252: 9242: 9234: 9233: 9229: 9228: 9213: 9203: 9189: 9177: 9173: 9157: 9146: 9145: 9142: 9141: 9131: 9121: 9113: 9112: 9108: 9098: 9081: 9069: 9065: 9049: 9038: 9037: 9033: 9031: 9028: 9027: 9017: 9014: 9006: 9005: 9003: 8991: 8985: 8965: 8961: 8952: 8947: 8946: 8940: 8935: 8916: 8907: 8897: 8887: 8879: 8878: 8874: 8873: 8863: 8858: 8850: 8847: 8846: 8814: 8809: 8796: 8791: 8786: 8782: 8773: 8769: 8763: 8759: 8755: 8750: 8732: 8728: 8714: 8710: 8709: 8702: 8698: 8697: 8691: 8688: 8687: 8684: 8678: 8675: 8669: 8647: 8643: 8637: 8633: 8629: 8622: 8618: 8614: 8612: 8594: 8590: 8588: 8585: 8584: 8570: 8569: 8563: 8559: 8540: 8536: 8530: 8526: 8522: 8517: 8508: 8507: 8496: 8492: 8477: 8476: 8472: 8466: 8462: 8443: 8439: 8435: 8427: 8425: 8416: 8415: 8409: 8405: 8390: 8381: 8377: 8365: 8361: 8357: 8352: 8345: 8339: 8335: 8320: 8310: 8309: 8295: 8291: 8289: 8286: 8285: 8265: 8261: 8252: 8239: 8235: 8231: 8226: 8214: 8210: 8201: 8187: 8182: 8178: 8177: 8171: 8154: 8153: 8147: 8144: 8143: 8135: 8132: 8129: 8124: 8123: 8121: 8120: 8111: 8095: 8091: 8080: 8077: 8063: 8056: 8050: 8046: 8022: 8018: 8007: 8003: 7989: 7987: 7985: 7982: 7981: 7959: 7955: 7942: 7938: 7931: 7929: 7927: 7924: 7923: 7904: 7900: 7885: 7875: 7874: 7868: 7865: 7864: 7860: 7855: 7839: 7838: 7812: 7803: 7794: 7793: 7792: 7770: 7761: 7760: 7755: 7751: 7744: 7720: 7711: 7710: 7709: 7705: 7683: 7682: 7678: 7677: 7673: 7670: 7669: 7659: 7652: 7648: 7642: 7632: 7631: 7623: 7622: 7611: 7591: 7582: 7573: 7572: 7571: 7545: 7544: 7540: 7539: 7535: 7531: 7529: 7526: 7525: 7495: 7482: 7481: 7477: 7471: 7462: 7461: 7456: 7452: 7446: 7442: 7429: 7428: 7415: 7411: 7409: 7406: 7405: 7384: 7377: 7373: 7367: 7357: 7356: 7348: 7347: 7332: 7323: 7314: 7305: 7304: 7303: 7298: 7286: 7273: 7272: 7267: 7266: 7257: 7253: 7247: 7243: 7241: 7238: 7237: 7216: 7215: 7211: 7199: 7191: 7184: 7180: 7165: 7156: 7155: 7141: 7133: 7129: 7125: 7106: 7105: 7101: 7100: 7096: 7083: 7079: 7078: 7071: 7067: 7057: 7054: 7044: 7043: 7025: 7017: 7015: 7012: 7011: 7006: 6999: 6992: 6986: 6982: 6978: 6973: 6965: 6964: 6963: 6959: 6956: 6945: 6937: 6932: 6906: 6898: 6886: 6882: 6875: 6871: 6856: 6847: 6846: 6829: 6821: 6809: 6805: 6795: 6791: 6772: 6771: 6767: 6766: 6762: 6749: 6745: 6744: 6737: 6733: 6723: 6720: 6703: 6690: 6689: 6684: 6683: 6681: 6678: 6677: 6656: 6655: 6651: 6639: 6631: 6624: 6620: 6605: 6596: 6595: 6581: 6573: 6569: 6565: 6546: 6545: 6541: 6540: 6536: 6523: 6519: 6518: 6511: 6507: 6497: 6494: 6477: 6464: 6463: 6458: 6457: 6455: 6452: 6451: 6434: 6433: 6427: 6426: 6416: 6415: 6397: 6389: 6385: 6381: 6371: 6370: 6366: 6354: 6343: 6330: 6329: 6324: 6323: 6321: 6318: 6317: 6311: 6305: 6304: 6303: 6296: 6290: 6289: 6288: 6284: 6280: 6270: 6261: 6258: 6249: 6246: 6237: 6234: 6226: 6222: 6210: 6207: 6199: 6176: 6163: 6153: 6152: 6147: 6146: 6145: 6141: 6140: 6131: 6127: 6121: 6117: 6115: 6112: 6111: 6093: 6085: 6081: 6077: 6070: 6069: 6065: 6053: 6042: 6032: 6031: 6026: 6025: 6023: 6020: 6019: 6015: 5993: 5989: 5980: 5976: 5967: 5963: 5962: 5958: 5945: 5943: 5929: 5925: 5916: 5912: 5903: 5899: 5898: 5894: 5892: 5889: 5888: 5845: 5792: 5790: 5787: 5786: 5776: 5754: 5750: 5738: 5720: 5712: 5708: 5704: 5697: 5696: 5692: 5681: 5672: 5667: 5666: 5664: 5661: 5660: 5654: 5644: 5638: 5636: 5611: 5598: 5589: 5584: 5583: 5582: 5578: 5577: 5568: 5564: 5559: 5556: 5555: 5545: 5542: 5537: 5536: 5534: 5529: 5504: 5495: 5490: 5489: 5478: 5474: 5465: 5461: 5460: 5456: 5454: 5451: 5450: 5438: 5432: 5416: 5415: 5409: 5404: 5403: 5386: 5376: 5370: 5366: 5364: 5355: 5354: 5343: 5338: 5337: 5328: 5323: 5322: 5321: 5317: 5308: 5304: 5291: 5286: 5279: 5273: 5269: 5265: 5263: 5260: 5259: 5256: 5241: 5237: 5231: 5227: 5221: 5217: 5195: 5191: 5168: 5164: 5155: 5151: 5147: 5143: 5128: 5124: 5119: 5116: 5115: 5111: 5105: 5087: 5086: 5077: 5073: 5058: 5057: 5039: 5019: 5007: 5003: 5002: 4998: 4983: 4982: 4964: 4947: 4943: 4934: 4930: 4926: 4922: 4904: 4897: 4884: 4878: 4874: 4869: 4865: 4861: 4852: 4848: 4844: 4842: 4839: 4838: 4821: 4820: 4803: 4791: 4787: 4786: 4782: 4767: 4766: 4752: 4748: 4739: 4735: 4731: 4727: 4709: 4700: 4699: 4693: 4688: 4687: 4681: 4677: 4662: 4649: 4643: 4639: 4637: 4630: 4621: 4617: 4613: 4611: 4608: 4607: 4599: 4596: 4591: 4590: 4588: 4587: 4581: 4577: 4570: 4564: 4560: 4551: 4520: 4516: 4509: 4505: 4503: 4500: 4499: 4490: 4482: 4480: 4450: 4446: 4439: 4435: 4433: 4430: 4429: 4423: 4419: 4412: 4408: 4405: 4384: 4378: 4353:and multi-loop 4306:renormalization 4297: 4294: 4289: 4288: 4286: 4281: 4239: 4227:Main articles: 4221: 4192: 4191: 4190: 4188: 3492: 3484: 3483: 3479:Quantum gravity 3454: 3446: 3445: 3441:Higgs mechanism 3421: 3411: 3410: 3396:Proca equations 3381: 3373: 3372: 3358:Renormalization 3323:Feynman diagram 3278: 3270: 3269: 3210: 3200: 3199: 3150: 3135: 3133:Feynman diagram 3114: 3105: 3098: 3094: 3090: 3087:Planck constant 3082: 3062: 3058: 3051: 3044: 3040: 3034: 3030: 3029: 3027: 3025: 3022: 3021: 2999:crystal lattice 2960: 2957: 2952: 2951: 2949: 2944: 2936: 2933: 2928: 2927: 2925: 2924: 2920: 2916: 2906: 2905:and defined by 2902: 2901:, denoted with 2894: 2877: 2874: 2869: 2868: 2866: 2861: 2854: 2848: 2835: 2832: 2827: 2826: 2824: 2822: 2815: 2809: 2790: 2788: 2783: 2757: 2755: 2746: 2742: 2727: 2717: 2708: 2704: 2686: 2685: 2681: 2679: 2676: 2675: 2646: 2640: 2626: 2622: 2621: 2607: 2596: 2595: 2594: 2584: 2579: 2578: 2574: 2563: 2552: 2548: 2534: 2533: 2532: 2528: 2527: 2512: 2511: 2507: 2505: 2502: 2501: 2468: 2453: 2442: 2441: 2440: 2436: 2434: 2422: 2411: 2407: 2393: 2392: 2391: 2387: 2386: 2382: 2380: 2378: 2375: 2374: 2368: 2362: 2341: 2330: 2329: 2328: 2318: 2313: 2304: 2293: 2289: 2275: 2274: 2273: 2269: 2268: 2253: 2244: 2240: 2226: 2225: 2223: 2220: 2219: 2193: 2187: 2109:Hendrik Casimir 2074:renormalisation 2070:magnetic moment 2046: 2044:Theodore Welton 1973: 1963:introduced the 1956: 1946: 1943: 1940: 1939: 1937: 1936: 1904:Robert Mulliken 1874: 1840:crystal lattice 1767: 1760: 1756: 1755: 1747: 1745: 1729: 1727: 1719: 1716: 1715: 1710: 1706: 1698: 1695: 1690: 1689: 1687: 1686: 1680: 1661: 1653: 1645: 1642:Planck constant 1637: 1594: 1587: 1583: 1582: 1574: 1572: 1564: 1561: 1560: 1557:energy radiator 1552: 1530: 1501:electrodynamics 1494:electromagnetic 1474:the renaissance 1438:the vacuum was 1420: 1415: 1292:, any point in 1281: 1278:Albert Einstein 1242:Planck constant 1211:Brownian motion 1175: 1100: 1084:broken symmetry 1074:; however, the 926: 897: 896: 895: 660: 652: 651: 597: 596:Advanced topics 589: 588: 587: 539:Hidden-variable 529:de Broglie–Bohm 508: 506:Interpretations 498: 497: 496: 466: 458: 457: 456: 414: 406: 405: 404: 371: 327:CHSH inequality 316: 308: 307: 306: 235:Complementarity 229: 221: 220: 219: 187: 158: 133: 122: 121: 107: 97: 92: 84: 81: 80: 38: 31: 24: 17: 12: 11: 5: 31463: 31453: 31452: 31450:Thermodynamics 31447: 31442: 31437: 31432: 31430:Quantum phases 31427: 31422: 31417: 31412: 31407: 31402: 31397: 31380: 31379: 31377: 31376: 31364: 31361:Physics Portal 31352: 31340: 31327: 31324: 31323: 31321: 31320: 31315: 31310: 31308:Liquid crystal 31305: 31300: 31294: 31292: 31286: 31285: 31283: 31282: 31277: 31276: 31275: 31270: 31260: 31255: 31250: 31245: 31240: 31235: 31230: 31225: 31219: 31217: 31215:Quasiparticles 31211: 31210: 31208: 31207: 31202: 31197: 31192: 31187: 31182: 31177: 31175:Superdiamagnet 31172: 31167: 31161: 31159: 31155: 31154: 31151: 31150: 31148: 31147: 31142: 31137: 31132: 31127: 31121: 31119: 31115: 31114: 31112: 31111: 31106: 31101: 31099:Superconductor 31096: 31091: 31086: 31081: 31079:Mott insulator 31076: 31070: 31068: 31064: 31063: 31061: 31060: 31055: 31050: 31045: 31040: 31035: 31030: 31025: 31020: 31015: 31010: 31004: 31002: 30998: 30997: 30995: 30994: 30989: 30984: 30979: 30974: 30969: 30964: 30959: 30953: 30951: 30944: 30940: 30939: 30937: 30936: 30931: 30926: 30921: 30915: 30913: 30909: 30908: 30901: 30899: 30897: 30896: 30891: 30886: 30881: 30876: 30871: 30866: 30861: 30856: 30851: 30846: 30840: 30838: 30832: 30831: 30824: 30823: 30816: 30809: 30801: 30792: 30791: 30778: 30775: 30774: 30772: 30771: 30766: 30761: 30760: 30759: 30749: 30744: 30739: 30734: 30733: 30732: 30722: 30721: 30720: 30710: 30705: 30700: 30695: 30690: 30685: 30680: 30675: 30670: 30668:Casimir effect 30664: 30662: 30658: 30657: 30654: 30653: 30651: 30650: 30645: 30643:Standard Model 30640: 30635: 30630: 30625: 30620: 30614: 30612: 30608: 30607: 30605: 30604: 30599: 30593: 30591: 30587: 30586: 30584: 30583: 30578: 30573: 30568: 30563: 30558: 30553: 30548: 30542: 30540: 30536: 30535: 30533: 30532: 30527: 30522: 30516: 30514: 30513:Superconformal 30510: 30509: 30507: 30506: 30501: 30496: 30494:Seiberg–Witten 30491: 30486: 30480: 30478: 30477:Supersymmetric 30474: 30473: 30471: 30470: 30465: 30460: 30455: 30450: 30444: 30442: 30438: 30437: 30435: 30434: 30429: 30424: 30419: 30414: 30409: 30404: 30399: 30393: 30391: 30387: 30386: 30384: 30383: 30378: 30373: 30368: 30363: 30358: 30353: 30348: 30343: 30338: 30333: 30328: 30323: 30318: 30312: 30310: 30303: 30299: 30298: 30296: 30295: 30290: 30285: 30280: 30275: 30270: 30265: 30260: 30255: 30250: 30245: 30240: 30234: 30232: 30228: 30227: 30220: 30219: 30212: 30205: 30197: 30188: 30187: 30184: 30183: 30169: 30168: 30165: 30164: 30162: 30161: 30156: 30151: 30146: 30141: 30136: 30131: 30129:Bremsstrahlung 30126: 30121: 30115: 30113: 30109: 30108: 30106: 30105: 30100: 30095: 30090: 30085: 30080: 30075: 30070: 30065: 30060: 30055: 30049: 30047: 30043: 30042: 30040: 30039: 30034: 30029: 30024: 30019: 30014: 30009: 30003: 30001: 29997: 29996: 29994: 29993: 29988: 29983: 29978: 29972: 29970: 29966: 29965: 29958: 29957: 29950: 29943: 29935: 29926: 29925: 29923: 29922: 29911: 29908: 29907: 29905: 29904: 29899: 29894: 29889: 29888: 29887: 29876: 29874: 29870: 29869: 29867: 29866: 29861: 29856: 29855: 29854: 29844: 29839: 29837:Casimir effect 29834: 29828: 29826: 29822: 29821: 29819: 29818: 29813: 29808: 29803: 29798: 29796:Quantum optics 29793: 29788: 29783: 29778: 29773: 29768: 29763: 29758: 29753: 29748: 29743: 29738: 29733: 29728: 29723: 29718: 29717: 29716: 29706: 29701: 29696: 29691: 29690: 29689: 29679: 29674: 29669: 29663: 29661: 29655: 29654: 29652: 29651: 29646: 29641: 29636: 29631: 29626: 29621: 29616: 29611: 29606: 29601: 29595: 29593: 29587: 29586: 29584: 29583: 29578: 29573: 29571:Quantum eraser 29568: 29563: 29558: 29553: 29548: 29543: 29538: 29533: 29527: 29525: 29521: 29520: 29518: 29517: 29512: 29507: 29502: 29497: 29492: 29487: 29486: 29485: 29484: 29483: 29468: 29463: 29458: 29453: 29448: 29442: 29440: 29434: 29433: 29431: 29430: 29425: 29420: 29415: 29410: 29404: 29402: 29398: 29397: 29395: 29394: 29389: 29384: 29379: 29374: 29369: 29364: 29358: 29356: 29352: 29351: 29349: 29348: 29347: 29346: 29341: 29331: 29326: 29321: 29316: 29311: 29306: 29301: 29296: 29291: 29286: 29281: 29276: 29271: 29270: 29269: 29264: 29259: 29254: 29244: 29242:Density matrix 29239: 29234: 29229: 29223: 29221: 29217: 29216: 29214: 29213: 29208: 29203: 29198: 29197: 29196: 29186: 29180: 29178: 29174: 29173: 29166: 29165: 29158: 29151: 29143: 29134: 29133: 29131: 29130: 29125: 29120: 29115: 29110: 29104: 29102: 29098: 29097: 29095: 29094: 29089: 29084: 29079: 29074: 29069: 29064: 29059: 29054: 29049: 29043: 29041: 29035: 29034: 29032: 29031: 29026: 29025: 29024: 29019: 29014: 29004: 28999: 28994: 28989: 28988: 28987: 28982: 28971: 28969: 28963: 28962: 28960: 28959: 28958: 28957: 28952: 28945:Thermodynamics 28942: 28941: 28940: 28935: 28925: 28920: 28915: 28914: 28913: 28908: 28903: 28898: 28887: 28885: 28879: 28878: 28876: 28875: 28874: 28873: 28863: 28857: 28855: 28851: 28850: 28848: 28847: 28846: 28845: 28835: 28829: 28827: 28823: 28822: 28814: 28813: 28806: 28799: 28791: 28785: 28784: 28774: 28762: 28761:External links 28759: 28757: 28756: 28735:978-3871448898 28734: 28713: 28692:978-0486645582 28691: 28673: 28652:978-0072448481 28651: 28635: 28633: 28630: 28628: 28627: 28566: 28507: 28481:(2): 474–503. 28469: 28467: 28464: 28462: 28461: 28444: 28428: 28411: 28394: 28377: 28360: 28338: 28336: 28335:Press articles 28333: 28331: 28328: 28326: 28325: 28308: 28279: 28216: 28194: 28166:(4): 830–841. 28155: 28134: 28112: 28096: 28033: 27978: 27927: 27901:(22). 220601. 27890: 27876: 27868:Quantum optics 27863: 27849: 27834: 27813:978-0198244493 27812: 27788: 27766: 27750: 27728: 27712: 27690: 27674: 27653:978-0198244493 27652: 27632: 27595:hep-th/0012253 27588:(4): 663–705. 27577: 27522: 27510:10.1086/300499 27486:(3): 1009–38. 27472: 27412: 27390: 27378:, ed. (2012). 27372: 27366: 27347: 27319: 27291: 27269: 27239:de Broglie, M. 27223: 27205:(4): 642–656. 27187: 27171: 27155: 27126: 27112: 27095: 27083:10.1086/307221 27049:Perlmutter, S. 27045: 27031: 27014: 26970:(2): 559–606. 26952: 26930: 26896:Milonni, P. W. 26892: 26871:978-0124980808 26870: 26856:Milonni, P. W. 26852: 26826:(4): 323–328. 26816:Milonni, P. W. 26812: 26794:(5): 225–226. 26784:Milonni, P. W. 26780: 26734:(1): 492–500. 26716: 26659: 26638:978-0198501770 26637: 26619: 26564: 26550: 26537: 26515: 26497: 26475: 26460: 26424:(3): 199–240. 26413: 26399: 26386: 26360:(1): 704–708. 26349: 26297:(20): 201601. 26286: 26231: 26210:978-0486445687 26209: 26194: 26148:(3): 555–564. 26133: 26078: 26037:(2): 678–694. 26019: 25997: 25974: 25945: 25931: 25916: 25902: 25887: 25832: 25826: 25805: 25787:(3): 108–119. 25776: 25725: 25666: 25644: 25624: 25578:(3949): 3949. 25563: 25549: 25534: 25482: 25421:Carroll, S. M. 25417: 25391:(2): 400–414. 25373: 25318: 25289: 25267: 25252: 25201: 25160: 25120: 25118: 25115: 25113: 25112: 25095: 25078: 25058: 25032:"To the Stars" 25027: 25006: 24989: 24972: 24951: 24934: 24906: 24886: 24869: 24852: 24835: 24818: 24793: 24768: 24751: 24733: 24731: 24728: 24726: 24725: 24691: 24679: 24653: 24627: 24615: 24568:(4): 989–994. 24548: 24521:(1): 524–531. 24501: 24494: 24468: 24456: 24430: 24418: 24381: 24338:AIP Conf. Proc 24321: 24309: 24290:(2): 201–208. 24274: 24262: 24235:(4): 371–383. 24219: 24176: 24149:(2): 457–459. 24133: 24104: 24083: 24071: 24039: 24035: 24031: 24022: 23992: 23988: 23984: 23975: 23938:(5): L73–L77. 23920: 23914:978-0521857147 23913: 23887: 23835: 23816:(3): 235–260. 23800: 23763: 23745: 23718:(6): 701–704. 23702: 23657:(2): 192–195. 23641: 23606: 23571: 23552:(6): 402–403. 23536: 23509:(2): 149–175. 23493: 23479: 23465: 23451: 23416: 23381: 23355: 23334: 23307:(5): 892–904. 23291: 23272:(3): 166–170. 23249: 23243:hep-th/9603077 23228: 23202: 23186: 23182: 23178: 23169: 23132:(5): 393–414. 23107: 23084: 23077: 23049: 22982: 22929: 22917: 22905: 22870: 22821: 22768: 22733: 22721: 22709: 22690:(7): 749–751. 22674: 22667: 22637: 22618:(8): 749–751. 22602: 22583:(7): 148–149. 22577:Electron. Lett 22567: 22548:(4): 409–416. 22532: 22509: 22452: 22438: 22412: 22361: 22355:hep-th/0104127 22340: 22328: 22293: 22278: 22250: 22217: 22186: 22180:978-0674009875 22179: 22159: 22152: 22126: 22089:(15): 155138. 22073: 22057: 22038:(1): 330–332. 22022: 21987: 21946: 21905: 21864: 21849: 21842: 21822: 21816:978-0674009875 21815: 21795: 21788: 21770: 21763: 21745: 21739:978-0471024019 21738: 21720: 21707:nobelprize.org 21693: 21686: 21666: 21650: 21631: 21625:978-0918388049 21624: 21606: 21587: 21575: 21557: 21538: 21511:(1): 138–162. 21495: 21460: 21458:, p. 201. 21456:Penrose (2004) 21448: 21399: 21392: 21372: 21345: 21338: 21318: 21311: 21291: 21287:Penrose (2004) 21279: 21260: 21239:(6): 494–497. 21219: 21200:(2): 251–254. 21194:Rev. Mod. Phys 21180: 21161:(7): 251–253. 21145: 21118:(8): 933–936. 21102: 21083:(5): 247–262. 21067: 21024: 20981: 20932: 20895:(3): 485–491. 20877: 20830: 20818: 20811: 20791: 20779: 20775:Milonni (1994) 20767: 20737:hep-th/0503158 20709: 20674: 20631: 20610: 20595: 20569: 20559: 20546: 20478: 20452: 20433: 20421: 20409: 20397: 20385: 20373: 20371:, p. 528. 20361: 20357:Bennett (2016) 20336: 20315: 20303: 20290: 20288:, p. 291. 20278: 20276:, p. 278. 20266: 20254: 20242: 20235: 20217: 20210: 20192: 20180: 20178:, p. 381. 20168: 20123: 20111: 20091: 20079: 20044: 19984: 19972: 19960: 19948: 19895: 19853: 19826: 19807:(4): 584–587. 19791: 19740: 19705: 19694:(3): 295–329. 19678: 19659:(5): 106–113. 19633: 19631:, p. 865. 19629:Milonni (2009) 19621: 19609: 19605:Milonni (1994) 19597: 19568: 19533: 19529:Milonni (1981) 19518: 19506: 19471: 19467:Milonni (1994) 19459: 19447: 19432: 19402: 19375:(3): 172–198. 19355: 19351:Milonni (1983) 19343: 19331: 19312:(11–12): 674. 19296: 19247: 19189: 19177: 19165: 19138: 19123: 19052: 18991: 18987:Milonni (1994) 18976: 18938: 18919:(1): 110–113. 18900: 18860: 18845: 18827: 18816:(3): 295–329. 18800: 18791:Soviet Physics 18781: 18762: 18713: 18670: 18626: 18592: 18557: 18544:nobelprize.org 18527: 18515: 18472: 18460: 18445: 18427: 18383: 18362:(3): 241–243. 18333: 18298: 18255: 18239: 18224: 18202: 18175: 18158: 18143: 18117: 18090:(3): 151–173. 18074: 18059: 18047:Rechenberg, H. 18034: 18007: 17980:(8): 557–615. 17964: 17949: 17923: 17888: 17867: 17828: 17786: 17744: 17742:, p. 162. 17732: 17717: 17682: 17639: 17596: 17547: 17520:(1): 183–199. 17504: 17483: 17464: 17444: 17405: 17393: 17374:(3): 551–560. 17354: 17352:, p. 235. 17342: 17313: 17309:Milonni (1994) 17301: 17297:Planck (1912a) 17289: 17274: 17262: 17250: 17238: 17236:, p. 225. 17226: 17214: 17199: 17178: 17161: 17140: 17108: 17096: 17084: 17072: 17060: 17058:, p. 111. 17056:Milonni (1994) 17043: 17041:, p. 111. 17028: 17013: 17001: 16999:, p. 104. 16989: 16987:, p. 138. 16974: 16972:, p. 376. 16959: 16934: 16922: 16918:Milonni (1994) 16905: 16903:, p. 137. 16887: 16885: 16882: 16880: 16877: 16875: 16874: 16869: 16864: 16859: 16854: 16849: 16844: 16839: 16834: 16829: 16824: 16819: 16814: 16809: 16804: 16802:Casimir effect 16798: 16797: 16796: 16782: 16779:Physics portal 16766: 16763: 16754: 16751: 16671:AC Gravity LLC 16666:spin alignment 16595:U.S. Air Force 16537:Robert Forward 16517: 16514: 16480: 16477: 16428:Robert Forward 16418: 16415: 16394:fringe science 16378: 16375: 16307: 16294:morphology as 16266:Ilya Prigogine 16238:Maxwell's laws 16136:Standard Model 16112: 16111: 16109: 16108: 16101: 16094: 16086: 16083: 16082: 16079: 16078: 16073: 16068: 16063: 16058: 16053: 16048: 16043: 16037: 16034: 16033: 16030: 16029: 16026: 16025: 16020: 16015: 16010: 16005: 16000: 15995: 15990: 15985: 15980: 15975: 15970: 15965: 15960: 15954: 15949: 15948: 15945: 15944: 15941: 15940: 15935: 15930: 15925: 15920: 15915: 15910: 15905: 15899: 15894: 15893: 15890: 15889: 15886: 15885: 15880: 15875: 15870: 15865: 15860: 15855: 15850: 15845: 15840: 15835: 15830: 15825: 15820: 15815: 15810: 15805: 15800: 15795: 15790: 15785: 15780: 15775: 15770: 15765: 15760: 15755: 15750: 15745: 15740: 15735: 15730: 15725: 15720: 15715: 15709: 15706: 15705: 15702: 15701: 15698: 15697: 15692: 15687: 15682: 15677: 15675:Dark radiation 15672: 15670:Phantom energy 15667: 15662: 15657: 15652: 15646: 15643: 15642: 15639: 15638: 15636:Standard Model 15632: 15631: 15617: 15609: 15608: 15599: 15596: 15592:unified theory 15571: 15568: 15560:Paul S. Wesson 15556: 15553: 15489: 15486: 15480: 15474:Main article: 15471: 15468: 15450:Standard Model 15397: 15390: 15384: 15378:Main article: 15375: 15372: 15370: 15367: 15288:Faraday effect 15238: 15235: 15225:. All in all, 15186: 15183: 15178: 15175: 15159: 15137: 15132: 15129: 15124: 15119: 15115: 15109: 15105: 15089: 15080: 15034: 15031: 15026: 15020: 15017: 15012: 15008: 15004: 15001: 14995: 14990: 14986: 14980: 14974: 14971: 14965: 14961: 14955: 14952: 14922: 14918: 14899: 14879: 14863: 14845:speed of light 14821:Main article: 14818: 14815: 14798:Dirac equation 14770: 14747: 14730:Fine structure 14719:Main article: 14716: 14713: 14667:Casimir effect 14665:Main article: 14662: 14661:Casimir effect 14659: 14653: 14650: 14555:gauge symmetry 14539:at the ground. 14509:Main article: 14506: 14503: 14453:Main article: 14450: 14447: 14429: 14381: 14339: 14321: 14318: 14315: 14310: 14305: 14298: 14295: 14290: 14286: 14280: 14275: 14271: 14267: 14261: 14258: 14250: 14245: 14241: 14237: 14234: 14228: 14225: 14195: 14192: 14184: 14179: 14175: 14171: 14168: 14162: 14159: 14156: 14152: 14147: 14144: 14141: 14138: 14134: 14128: 14123: 14119: 14115: 14112: 14106: 14103: 14097: 14095: 14092: 14089: 14086: 14081: 14076: 14069: 14066: 14061: 14055: 14052: 14049: 14045: 14040: 14037: 14033: 14027: 14022: 14018: 14014: 14008: 14005: 13999: 13997: 13973: 13970: 13967: 13964: 13962: 13960: 13956: 13948: 13943: 13939: 13935: 13931: 13926: 13921: 13913: 13909: 13905: 13902: 13899: 13892: 13887: 13883: 13877: 13873: 13869: 13866: 13863: 13857: 13853: 13850: 13848: 13846: 13838: 13833: 13829: 13823: 13819: 13815: 13810: 13806: 13800: 13797: 13789: 13784: 13781: 13777: 13771: 13766: 13762: 13753: 13749: 13745: 13742: 13739: 13732: 13728: 13724: 13721: 13718: 13712: 13709: 13707: 13704: 13700: 13697: 13694: 13689: 13685: 13681: 13678: 13675: 13672: 13669: 13665: 13661: 13660: 13647: 13633: 13606: 13602: 13596: 13592: 13588: 13583: 13578: 13572: 13567: 13563: 13559: 13554: 13550: 13545: 13536: 13532: 13527: 13524: 13516: 13511: 13507: 13502: 13497: 13493: 13490: 13484: 13479: 13471: 13467: 13463: 13458: 13454: 13450: 13443: 13439: 13435: 13432: 13426: 13422: 13419: 13417: 13415: 13411: 13407: 13404: 13401: 13395: 13392: 13386: 13383: 13380: 13377: 13374: 13371: 13367: 13363: 13360: 13358: 13356: 13352: 13348: 13345: 13342: 13337: 13333: 13327: 13324: 13319: 13316: 13313: 13310: 13307: 13303: 13299: 13295: 13291: 13288: 13285: 13279: 13276: 13270: 13267: 13264: 13261: 13258: 13255: 13251: 13247: 13244: 13242: 13240: 13217: 13214: 13210: 13206: 13202: 13197: 13190: 13187: 13183: 13178: 13174: 13170: 13166: 13163: 13160: 13155: 13149: 13146: 13140: 13137: 13131: 13127: 13124: 13121: 13118: 13113: 13109: 13104: 13099: 13076: 13073: 13070: 13065: 13060: 13053: 13050: 13045: 13039: 13036: 13033: 13029: 13024: 13021: 13017: 13011: 13006: 13002: 12998: 12992: 12989: 12960: 12957: 12954: 12951: 12949: 12947: 12944: 12941: 12938: 12935: 12932: 12929: 12926: 12921: 12917: 12913: 12910: 12907: 12904: 12902: 12900: 12897: 12894: 12891: 12888: 12884: 12880: 12877: 12874: 12869: 12865: 12861: 12858: 12855: 12852: 12849: 12845: 12841: 12838: 12835: 12830: 12826: 12822: 12819: 12817: 12815: 12811: 12807: 12804: 12801: 12798: 12795: 12792: 12789: 12784: 12780: 12776: 12773: 12770: 12765: 12761: 12757: 12754: 12751: 12748: 12745: 12742: 12739: 12736: 12733: 12730: 12727: 12724: 12719: 12715: 12710: 12706: 12703: 12701: 12698: 12694: 12691: 12688: 12683: 12679: 12675: 12672: 12669: 12666: 12663: 12659: 12655: 12654: 12626: 12612: 12600: 12586: 12576: 12555: 12552: 12549: 12544: 12540: 12535: 12532: 12527: 12522: 12518: 12514: 12511: 12509: 12507: 12503: 12498: 12492: 12488: 12484: 12481: 12478: 12472: 12466: 12462: 12457: 12450: 12447: 12443: 12438: 12435: 12433: 12431: 12427: 12423: 12420: 12417: 12412: 12406: 12403: 12397: 12394: 12388: 12384: 12381: 12378: 12373: 12369: 12364: 12359: 12355: 12349: 12342: 12339: 12334: 12330: 12327: 12324: 12314: 12308: 12304: 12300: 12297: 12294: 12284: 12281: 12275: 12272: 12266: 12260: 12256: 12251: 12244: 12241: 12237: 12232: 12229: 12227: 12224: 12220: 12217: 12214: 12209: 12204: 12199: 12193: 12186: 12183: 12179: 12174: 12173: 12154: 12138: 12135: 12132: 12129: 12125: 12121: 12118: 12115: 12110: 12105: 12099: 12095: 12092: 12089: 12086: 12083: 12080: 12077: 12072: 12067: 12062: 12047: 12027: 12023: 12018: 12014: 12009: 12005: 11996: 11992: 11989: 11986: 11982: 11944: 11927: 11879: 11876: 11873: 11868: 11863: 11856: 11853: 11848: 11842: 11839: 11836: 11832: 11827: 11824: 11820: 11814: 11809: 11805: 11801: 11795: 11792: 11769: 11766: 11763: 11759: 11738: 11719: 11698: 11682: 11679: 11675: 11670: 11661: 11657: 11653: 11647: 11643: 11633: 11629: 11625: 11620: 11615: 11611: 11606: 11586: 11570: 11556: 11535: 11532: 11521: 11517: 11513: 11508: 11505: 11499: 11496: 11493: 11490: 11485: 11482: 11477: 11444: 11440: 11437: 11433: 11430: 11425: 11419: 11415: 11411: 11408: 11404: 11399: 11395: 11392: 11388: 11381: 11378: 11372: 11367: 11363: 11358: 11353: 11348: 11345: 11341: 11336: 11331: 11327: 11321: 11317: 11312: 11306: 11302: 11299: 11296: 11290: 11287: 11284: 11281: 11278: 11273: 11270: 11265: 11241: 11237: 11232: 11227: 11221: 11216: 11212: 11208: 11204: 11200: 11197: 11194: 11189: 11184: 11180: 11175: 11171: 11166: 11161: 11157: 11153: 11150: 11146: 11142: 11139: 11136: 11131: 11127: 11122: 11117: 11110: 11104: 11100: 11096: 11093: 11090: 11081: 11077: 11072: 11068: 11065: 11062: 11059: 11056: 11051: 11046: 11024: 11021: 11018: 11013: 11010: 11005: 10998: 10995: 10990: 10987: 10984: 10981: 10976: 10971: 10964: 10961: 10956: 10952: 10946: 10941: 10937: 10933: 10927: 10924: 10903: 10884: 10880: 10877: 10873: 10870: 10865: 10859: 10855: 10851: 10847: 10843: 10839: 10836: 10832: 10826: 10823: 10817: 10812: 10808: 10803: 10797: 10794: 10790: 10785: 10780: 10776: 10768: 10763: 10759: 10755: 10750: 10747: 10740: 10737: 10734: 10729: 10724: 10720: 10716: 10713: 10709: 10705: 10702: 10699: 10694: 10690: 10685: 10681: 10678: 10675: 10672: 10667: 10663: 10658: 10634: 10630: 10627: 10624: 10621: 10618: 10615: 10611: 10607: 10604: 10601: 10596: 10593: 10589: 10585: 10582: 10579: 10576: 10571: 10567: 10562: 10558: 10555: 10552: 10546: 10543: 10537: 10534: 10456: 10423: 10419: 10414: 10410: 10404: 10401: 10391: 10386: 10382: 10378: 10373: 10370: 10363: 10360: 10357: 10352: 10348: 10343: 10337: 10333: 10329: 10326: 10321: 10317: 10309: 10306: 10284: 10260: 10256: 10251: 10246: 10242: 10239: 10236: 10231: 10226: 10222: 10217: 10213: 10210: 10207: 10204: 10199: 10195: 10190: 10185: 10178: 10172: 10168: 10164: 10161: 10158: 10149: 10145: 10140: 10136: 10133: 10131: 10129: 10125: 10119: 10116: 10111: 10108: 10106: 10103: 10097: 10092: 10088: 10084: 10078: 10075: 10069: 10068: 10037: 10031: 10026: 10021: 10018: 10015: 10009: 10006: 10000: 9997: 9991: 9988: 9982: 9978: 9972: 9966: 9963: 9935: 9929: 9924: 9920: 9916: 9913: 9909: 9905: 9899: 9896: 9888: 9885: 9880: 9876: 9872: 9869: 9866: 9864: 9862: 9858: 9852: 9847: 9843: 9839: 9836: 9832: 9828: 9822: 9819: 9811: 9808: 9803: 9799: 9794: 9789: 9785: 9782: 9776: 9773: 9766: 9762: 9756: 9753: 9746: 9740: 9737: 9732: 9729: 9726: 9724: 9722: 9716: 9713: 9705: 9702: 9697: 9691: 9688: 9682: 9679: 9677: 9672: 9669: 9663: 9660: 9659: 9628: 9625: 9617: 9612: 9608: 9604: 9601: 9597: 9593: 9587: 9584: 9576: 9573: 9568: 9564: 9560: 9557: 9554: 9548: 9545: 9539: 9534: 9531: 9526: 9523: 9521: 9519: 9513: 9510: 9502: 9497: 9493: 9489: 9486: 9482: 9477: 9471: 9468: 9463: 9459: 9455: 9452: 9449: 9445: 9440: 9434: 9431: 9426: 9422: 9417: 9411: 9408: 9403: 9399: 9394: 9388: 9385: 9380: 9376: 9371: 9367: 9364: 9360: 9355: 9349: 9346: 9341: 9337: 9332: 9327: 9321: 9318: 9313: 9310: 9307: 9305: 9303: 9297: 9294: 9286: 9281: 9277: 9273: 9270: 9265: 9260: 9255: 9249: 9246: 9241: 9237: 9232: 9227: 9221: 9218: 9212: 9209: 9207: 9205: 9202: 9199: 9196: 9192: 9188: 9183: 9180: 9176: 9172: 9169: 9166: 9163: 9160: 9158: 9153: 9150: 9144: 9143: 9139: 9134: 9128: 9125: 9120: 9116: 9111: 9105: 9102: 9097: 9094: 9091: 9088: 9084: 9080: 9075: 9072: 9068: 9064: 9061: 9058: 9055: 9052: 9050: 9045: 9042: 9036: 9035: 8968: 8964: 8960: 8955: 8950: 8943: 8938: 8934: 8930: 8924: 8921: 8915: 8910: 8905: 8900: 8894: 8891: 8886: 8882: 8877: 8869: 8866: 8862: 8857: 8854: 8823: 8817: 8812: 8808: 8804: 8799: 8794: 8790: 8785: 8776: 8772: 8766: 8762: 8758: 8754: 8749: 8746: 8743: 8740: 8735: 8731: 8727: 8724: 8717: 8713: 8705: 8701: 8696: 8686:is therefore: 8682: 8673: 8650: 8646: 8640: 8636: 8632: 8625: 8621: 8617: 8611: 8608: 8605: 8602: 8597: 8593: 8566: 8562: 8557: 8554: 8551: 8543: 8539: 8533: 8529: 8525: 8521: 8516: 8513: 8511: 8509: 8505: 8499: 8495: 8491: 8485: 8482: 8475: 8469: 8465: 8460: 8457: 8454: 8446: 8442: 8438: 8433: 8430: 8424: 8421: 8419: 8417: 8412: 8408: 8404: 8398: 8395: 8389: 8384: 8380: 8376: 8368: 8364: 8360: 8356: 8351: 8348: 8346: 8342: 8338: 8334: 8328: 8325: 8317: 8313: 8308: 8302: 8299: 8294: 8293: 8273: 8268: 8264: 8260: 8255: 8251: 8242: 8238: 8234: 8230: 8225: 8222: 8217: 8213: 8209: 8204: 8199: 8193: 8190: 8186: 8181: 8174: 8170: 8166: 8161: 8157: 8152: 8127: 8113: 8094:for all modes 8084: 8076: 8073: 8034: 8031: 8025: 8021: 8017: 8010: 8006: 8001: 7998: 7995: 7992: 7969: 7962: 7958: 7953: 7950: 7945: 7941: 7937: 7934: 7907: 7903: 7899: 7893: 7890: 7882: 7878: 7873: 7858: 7837: 7834: 7830: 7826: 7823: 7820: 7815: 7809: 7806: 7801: 7797: 7791: 7787: 7784: 7781: 7778: 7773: 7768: 7764: 7759: 7754: 7750: 7747: 7745: 7742: 7738: 7735: 7732: 7726: 7723: 7718: 7714: 7708: 7704: 7701: 7698: 7695: 7690: 7686: 7681: 7676: 7672: 7671: 7665: 7662: 7658: 7655: 7651: 7645: 7639: 7635: 7630: 7626: 7621: 7617: 7614: 7612: 7609: 7605: 7602: 7599: 7594: 7588: 7585: 7580: 7576: 7570: 7566: 7563: 7560: 7557: 7552: 7548: 7543: 7538: 7534: 7533: 7510: 7503: 7500: 7494: 7489: 7485: 7480: 7474: 7469: 7465: 7460: 7455: 7449: 7445: 7441: 7436: 7432: 7427: 7423: 7418: 7414: 7390: 7387: 7383: 7380: 7376: 7370: 7364: 7360: 7355: 7351: 7346: 7342: 7339: 7335: 7331: 7326: 7320: 7317: 7312: 7308: 7301: 7296: 7293: 7289: 7285: 7280: 7276: 7270: 7265: 7260: 7256: 7250: 7246: 7223: 7219: 7214: 7209: 7202: 7198: 7194: 7190: 7187: 7183: 7179: 7176: 7173: 7168: 7163: 7159: 7154: 7150: 7144: 7140: 7136: 7132: 7128: 7124: 7121: 7118: 7113: 7109: 7104: 7099: 7091: 7086: 7082: 7074: 7070: 7066: 7063: 7060: 7051: 7047: 7042: 7038: 7035: 7032: 7028: 7024: 7020: 7004: 6997: 6990: 6966: 6949: 6935: 6919: 6913: 6909: 6905: 6901: 6897: 6894: 6889: 6885: 6881: 6878: 6874: 6870: 6867: 6864: 6859: 6854: 6850: 6845: 6841: 6836: 6832: 6828: 6824: 6820: 6817: 6812: 6808: 6804: 6801: 6798: 6794: 6790: 6787: 6784: 6779: 6775: 6770: 6765: 6757: 6752: 6748: 6740: 6736: 6732: 6729: 6726: 6719: 6716: 6713: 6710: 6706: 6702: 6697: 6693: 6687: 6663: 6659: 6654: 6649: 6642: 6638: 6634: 6630: 6627: 6623: 6619: 6616: 6613: 6608: 6603: 6599: 6594: 6590: 6584: 6580: 6576: 6572: 6568: 6564: 6561: 6558: 6553: 6549: 6544: 6539: 6531: 6526: 6522: 6514: 6510: 6506: 6503: 6500: 6493: 6490: 6487: 6484: 6480: 6476: 6471: 6467: 6461: 6437: 6432: 6429: 6428: 6425: 6422: 6421: 6419: 6414: 6411: 6407: 6400: 6396: 6392: 6388: 6384: 6378: 6374: 6369: 6362: 6358: 6353: 6350: 6346: 6342: 6337: 6333: 6327: 6306: 6291: 6275: 6265: 6253: 6241: 6230: 6218: 6203: 6187: 6184: 6179: 6174: 6170: 6166: 6162: 6156: 6150: 6144: 6139: 6134: 6130: 6124: 6120: 6096: 6092: 6088: 6084: 6080: 6073: 6068: 6061: 6057: 6052: 6049: 6045: 6041: 6035: 6029: 6002: 5996: 5992: 5988: 5983: 5979: 5975: 5970: 5966: 5961: 5955: 5951: 5948: 5942: 5938: 5932: 5928: 5924: 5919: 5915: 5911: 5906: 5902: 5897: 5876: 5873: 5870: 5867: 5864: 5861: 5858: 5855: 5852: 5848: 5844: 5841: 5838: 5835: 5832: 5829: 5826: 5823: 5820: 5817: 5814: 5811: 5808: 5805: 5802: 5799: 5795: 5746: 5723: 5719: 5715: 5711: 5707: 5700: 5695: 5691: 5688: 5684: 5680: 5675: 5670: 5642: 5622: 5619: 5614: 5609: 5605: 5601: 5597: 5592: 5587: 5581: 5576: 5571: 5567: 5563: 5517: 5514: 5511: 5507: 5503: 5498: 5493: 5487: 5481: 5477: 5473: 5468: 5464: 5459: 5436: 5412: 5407: 5402: 5399: 5396: 5393: 5389: 5382: 5379: 5373: 5369: 5363: 5360: 5358: 5356: 5352: 5346: 5341: 5336: 5331: 5326: 5320: 5316: 5311: 5307: 5303: 5297: 5294: 5290: 5285: 5282: 5280: 5276: 5272: 5268: 5267: 5255: 5252: 5203: 5198: 5194: 5190: 5187: 5184: 5180: 5176: 5171: 5167: 5163: 5158: 5154: 5150: 5146: 5142: 5139: 5136: 5131: 5127: 5123: 5109: 5085: 5080: 5076: 5072: 5069: 5066: 5063: 5061: 5059: 5056: 5053: 5047: 5044: 5038: 5034: 5027: 5024: 5018: 5015: 5010: 5006: 5001: 4997: 4994: 4991: 4988: 4986: 4984: 4981: 4978: 4972: 4969: 4963: 4959: 4955: 4950: 4946: 4942: 4937: 4933: 4929: 4925: 4921: 4918: 4912: 4909: 4903: 4900: 4898: 4895: 4891: 4887: 4881: 4877: 4872: 4868: 4864: 4860: 4855: 4851: 4847: 4846: 4818: 4811: 4808: 4802: 4799: 4794: 4790: 4785: 4781: 4778: 4775: 4772: 4770: 4768: 4764: 4760: 4755: 4751: 4747: 4742: 4738: 4734: 4730: 4726: 4723: 4717: 4714: 4708: 4705: 4703: 4701: 4696: 4691: 4684: 4680: 4676: 4670: 4667: 4661: 4655: 4652: 4646: 4642: 4636: 4633: 4631: 4627: 4624: 4620: 4616: 4615: 4539: 4536: 4532: 4528: 4523: 4519: 4515: 4512: 4508: 4466: 4463: 4459: 4453: 4449: 4445: 4442: 4438: 4428:that satisfy: 4404: 4401: 4380:Main article: 4377: 4374: 4337:virtual energy 4275:quantum vacuum 4223: 4222: 4220: 4219: 4212: 4205: 4197: 4194: 4193: 4186: 4185: 4180: 4175: 4170: 4165: 4160: 4155: 4150: 4145: 4140: 4135: 4130: 4125: 4120: 4115: 4110: 4105: 4100: 4095: 4090: 4085: 4080: 4075: 4070: 4065: 4060: 4055: 4050: 4045: 4040: 4035: 4030: 4025: 4020: 4015: 4010: 4005: 4000: 3995: 3990: 3985: 3980: 3975: 3970: 3965: 3960: 3955: 3950: 3945: 3940: 3935: 3930: 3925: 3920: 3915: 3910: 3905: 3900: 3895: 3890: 3885: 3880: 3875: 3870: 3865: 3860: 3855: 3850: 3845: 3840: 3835: 3830: 3825: 3820: 3815: 3810: 3805: 3800: 3795: 3790: 3785: 3780: 3775: 3770: 3765: 3760: 3755: 3750: 3745: 3740: 3735: 3730: 3725: 3720: 3715: 3710: 3705: 3700: 3695: 3690: 3685: 3680: 3675: 3670: 3665: 3660: 3655: 3650: 3645: 3640: 3635: 3630: 3625: 3620: 3615: 3610: 3605: 3600: 3595: 3590: 3585: 3580: 3575: 3570: 3565: 3560: 3555: 3550: 3545: 3540: 3535: 3530: 3525: 3520: 3515: 3510: 3505: 3500: 3494: 3493: 3490: 3489: 3486: 3485: 3482: 3481: 3476: 3471: 3466: 3461: 3455: 3452: 3451: 3448: 3447: 3444: 3443: 3438: 3433: 3428: 3422: 3419:Standard Model 3417: 3416: 3413: 3412: 3409: 3408: 3403: 3398: 3393: 3388: 3386:Dirac equation 3382: 3379: 3378: 3375: 3374: 3371: 3370: 3368:Wick's theorem 3365: 3360: 3355: 3353:Regularization 3350: 3345: 3340: 3335: 3330: 3325: 3320: 3315: 3310: 3305: 3300: 3295: 3290: 3285: 3279: 3276: 3275: 3272: 3271: 3268: 3267: 3262: 3260:Noether charge 3257: 3252: 3247: 3245:Gauge symmetry 3242: 3237: 3232: 3227: 3222: 3217: 3211: 3206: 3205: 3202: 3201: 3198: 3197: 3192: 3187: 3182: 3177: 3172: 3167: 3162: 3157: 3151: 3148: 3147: 3144: 3143: 3137: 3136: 3131: 3123: 3122: 3113: 3110: 3065: 3061: 3057: 3054: 3047: 3043: 3037: 3033: 2850:Main article: 2847: 2846:Atomic physics 2844: 2820: 2813: 2767: 2763: 2760: 2754: 2749: 2745: 2741: 2735: 2732: 2724: 2721: 2716: 2711: 2707: 2703: 2699: 2693: 2690: 2684: 2659: 2653: 2650: 2643: 2638: 2633: 2630: 2625: 2620: 2616: 2610: 2603: 2600: 2590: 2587: 2583: 2577: 2572: 2566: 2561: 2555: 2551: 2547: 2541: 2538: 2531: 2526: 2520: 2517: 2510: 2481: 2475: 2472: 2467: 2461: 2456: 2449: 2446: 2439: 2430: 2425: 2420: 2414: 2410: 2406: 2400: 2397: 2390: 2385: 2366: 2350: 2344: 2337: 2334: 2324: 2321: 2317: 2312: 2307: 2302: 2296: 2292: 2288: 2282: 2279: 2272: 2267: 2261: 2258: 2252: 2247: 2243: 2239: 2233: 2230: 2204:potential well 2189:Main article: 2186: 2183: 2179:Purcell effect 2171:Rabi frequency 2166:two-level atom 2142:Herbert Callen 2090:Wolfgang Pauli 1977:Pascual Jordan 1972: 1969: 1887:Walther Nernst 1872:Kurt Bennewitz 1848:Walther Nernst 1832:Paul Ehrenfest 1800: 1791: 1788: 1783: 1780: 1777: 1774: 1770: 1766: 1763: 1759: 1753: 1750: 1744: 1739: 1735: 1732: 1726: 1723: 1625: 1618: 1615: 1610: 1607: 1604: 1601: 1597: 1593: 1590: 1586: 1580: 1577: 1571: 1568: 1542:quantum theory 1529: 1526: 1419: 1416: 1414: 1411: 1403:speed of light 1323:quantum vacuum 1267:potential well 1203:kinetic energy 1174: 1171: 1099: 1096: 1070:were an exact 1056:Casimir effect 987:fields, whose 928: 927: 925: 924: 917: 910: 902: 899: 898: 894: 893: 888: 883: 878: 873: 868: 863: 858: 853: 848: 843: 838: 833: 828: 823: 818: 813: 808: 803: 798: 793: 788: 783: 778: 773: 768: 763: 758: 753: 748: 743: 738: 733: 728: 723: 718: 713: 708: 703: 698: 693: 688: 683: 678: 673: 668: 662: 661: 658: 657: 654: 653: 650: 649: 644: 639: 634: 632:Density matrix 629: 624: 619: 614: 609: 604: 598: 595: 594: 591: 590: 586: 585: 580: 575: 570: 565: 560: 555: 554: 553: 552: 551: 536: 531: 526: 521: 516: 510: 509: 504: 503: 500: 499: 495: 494: 489: 484: 479: 474: 468: 467: 464: 463: 460: 459: 455: 454: 449: 444: 439: 434: 429: 423: 422: 421: 415: 412: 411: 408: 407: 403: 402: 397: 392: 386: 385: 384: 383: 382: 380:Delayed-choice 375:Quantum eraser 370: 369: 364: 359: 354: 349: 344: 339: 334: 329: 324: 318: 317: 314: 313: 310: 309: 305: 304: 303: 302: 292: 287: 282: 277: 272: 267: 265:Quantum number 262: 257: 252: 247: 242: 237: 231: 230: 227: 226: 223: 222: 218: 217: 212: 206: 205: 204: 199: 194: 188: 185: 184: 181: 180: 179: 178: 173: 168: 160: 159: 154: 143: 140: 136: 129: 126: 120: 117: 114: 110: 103: 100: 96: 91: 88: 77: 76: 70: 69: 51:kinetic energy 15: 9: 6: 4: 3: 2: 31462: 31451: 31448: 31446: 31443: 31441: 31438: 31436: 31433: 31431: 31428: 31426: 31423: 31421: 31418: 31416: 31413: 31411: 31408: 31406: 31403: 31401: 31398: 31396: 31393: 31392: 31390: 31375: 31374: 31365: 31363: 31362: 31357: 31353: 31351: 31350: 31341: 31339: 31338: 31329: 31328: 31325: 31319: 31316: 31314: 31311: 31309: 31306: 31304: 31301: 31299: 31296: 31295: 31293: 31291: 31287: 31281: 31278: 31274: 31271: 31269: 31266: 31265: 31264: 31261: 31259: 31256: 31254: 31251: 31249: 31246: 31244: 31241: 31239: 31236: 31234: 31231: 31229: 31226: 31224: 31221: 31220: 31218: 31216: 31212: 31206: 31203: 31201: 31198: 31196: 31193: 31191: 31188: 31186: 31183: 31181: 31178: 31176: 31173: 31171: 31168: 31166: 31163: 31162: 31160: 31156: 31146: 31143: 31141: 31138: 31136: 31133: 31131: 31128: 31126: 31123: 31122: 31120: 31116: 31110: 31107: 31105: 31102: 31100: 31097: 31095: 31092: 31090: 31087: 31085: 31084:Semiconductor 31082: 31080: 31077: 31075: 31072: 31071: 31069: 31065: 31059: 31056: 31054: 31053:Hubbard model 31051: 31049: 31046: 31044: 31041: 31039: 31036: 31034: 31031: 31029: 31026: 31024: 31021: 31019: 31016: 31014: 31011: 31009: 31006: 31005: 31003: 30999: 30993: 30990: 30988: 30985: 30983: 30980: 30978: 30975: 30973: 30970: 30968: 30965: 30963: 30960: 30958: 30955: 30954: 30952: 30948: 30945: 30941: 30935: 30932: 30930: 30927: 30925: 30922: 30920: 30917: 30916: 30914: 30910: 30905: 30895: 30892: 30890: 30887: 30885: 30882: 30880: 30877: 30875: 30872: 30870: 30867: 30865: 30862: 30860: 30857: 30855: 30852: 30850: 30847: 30845: 30842: 30841: 30839: 30837: 30833: 30829: 30822: 30817: 30815: 30810: 30808: 30803: 30802: 30799: 30789: 30781: 30776: 30770: 30767: 30765: 30764:Quantum logic 30762: 30758: 30755: 30754: 30753: 30750: 30748: 30745: 30743: 30740: 30738: 30735: 30731: 30728: 30727: 30726: 30723: 30719: 30716: 30715: 30714: 30711: 30709: 30706: 30704: 30701: 30699: 30698:Quantum chaos 30696: 30694: 30691: 30689: 30686: 30684: 30681: 30679: 30676: 30674: 30673:Cosmic string 30671: 30669: 30666: 30665: 30663: 30659: 30649: 30646: 30644: 30641: 30639: 30636: 30634: 30631: 30629: 30626: 30624: 30621: 30619: 30616: 30615: 30613: 30609: 30603: 30600: 30598: 30595: 30594: 30592: 30588: 30582: 30579: 30577: 30574: 30572: 30569: 30567: 30564: 30562: 30559: 30557: 30554: 30552: 30549: 30547: 30546:Pure 4D N = 1 30544: 30543: 30541: 30537: 30531: 30528: 30526: 30523: 30521: 30518: 30517: 30515: 30511: 30505: 30502: 30500: 30497: 30495: 30492: 30490: 30487: 30485: 30482: 30481: 30479: 30475: 30469: 30466: 30464: 30461: 30459: 30456: 30454: 30451: 30449: 30446: 30445: 30443: 30439: 30433: 30430: 30428: 30427:Thirring–Wess 30425: 30423: 30420: 30418: 30415: 30413: 30410: 30408: 30405: 30403: 30402:Bullough–Dodd 30400: 30398: 30397:2D Yang–Mills 30395: 30394: 30392: 30388: 30382: 30379: 30377: 30374: 30372: 30369: 30367: 30364: 30362: 30359: 30357: 30354: 30352: 30349: 30347: 30344: 30342: 30339: 30337: 30334: 30332: 30329: 30327: 30324: 30322: 30319: 30317: 30314: 30313: 30311: 30307: 30304: 30300: 30294: 30291: 30289: 30286: 30284: 30281: 30279: 30276: 30274: 30273:String theory 30271: 30269: 30266: 30264: 30261: 30259: 30256: 30254: 30251: 30249: 30246: 30244: 30243:Axiomatic QFT 30241: 30239: 30238:Algebraic QFT 30236: 30235: 30233: 30229: 30225: 30218: 30213: 30211: 30206: 30204: 30199: 30198: 30195: 30182: 30174: 30171: 30170: 30166: 30160: 30157: 30155: 30152: 30150: 30147: 30145: 30142: 30140: 30137: 30135: 30132: 30130: 30127: 30125: 30122: 30120: 30117: 30116: 30114: 30110: 30104: 30101: 30099: 30096: 30094: 30091: 30089: 30086: 30084: 30081: 30079: 30076: 30074: 30071: 30069: 30066: 30064: 30061: 30059: 30056: 30054: 30051: 30050: 30048: 30044: 30038: 30035: 30033: 30030: 30028: 30025: 30023: 30020: 30018: 30015: 30013: 30010: 30008: 30005: 30004: 30002: 29998: 29992: 29989: 29987: 29984: 29982: 29979: 29977: 29974: 29973: 29971: 29967: 29963: 29956: 29951: 29949: 29944: 29942: 29937: 29936: 29933: 29921: 29913: 29912: 29909: 29903: 29900: 29898: 29895: 29893: 29890: 29886: 29883: 29882: 29881: 29878: 29877: 29875: 29871: 29865: 29862: 29860: 29857: 29853: 29850: 29849: 29848: 29845: 29843: 29840: 29838: 29835: 29833: 29830: 29829: 29827: 29823: 29817: 29814: 29812: 29809: 29807: 29804: 29802: 29799: 29797: 29794: 29792: 29789: 29787: 29784: 29782: 29779: 29777: 29774: 29772: 29769: 29767: 29764: 29762: 29759: 29757: 29756:Quantum logic 29754: 29752: 29749: 29747: 29744: 29742: 29739: 29737: 29734: 29732: 29729: 29727: 29724: 29722: 29719: 29715: 29712: 29711: 29710: 29707: 29705: 29702: 29700: 29697: 29695: 29692: 29688: 29685: 29684: 29683: 29680: 29678: 29675: 29673: 29670: 29668: 29665: 29664: 29662: 29660: 29656: 29650: 29647: 29645: 29642: 29640: 29637: 29635: 29632: 29630: 29627: 29625: 29622: 29620: 29617: 29615: 29612: 29610: 29609:Quantum chaos 29607: 29605: 29602: 29600: 29597: 29596: 29594: 29592: 29588: 29582: 29579: 29577: 29576:Stern–Gerlach 29574: 29572: 29569: 29567: 29564: 29562: 29559: 29557: 29554: 29552: 29549: 29547: 29544: 29542: 29539: 29537: 29534: 29532: 29529: 29528: 29526: 29522: 29516: 29513: 29511: 29510:Transactional 29508: 29506: 29503: 29501: 29500:Quantum logic 29498: 29496: 29493: 29491: 29488: 29482: 29479: 29478: 29477: 29474: 29473: 29472: 29469: 29467: 29464: 29462: 29459: 29457: 29454: 29452: 29449: 29447: 29444: 29443: 29441: 29439: 29435: 29429: 29426: 29424: 29421: 29419: 29416: 29414: 29411: 29409: 29406: 29405: 29403: 29399: 29393: 29390: 29388: 29385: 29383: 29380: 29378: 29375: 29373: 29370: 29368: 29365: 29363: 29360: 29359: 29357: 29353: 29345: 29342: 29340: 29337: 29336: 29335: 29334:Wave function 29332: 29330: 29327: 29325: 29322: 29320: 29317: 29315: 29312: 29310: 29309:Superposition 29307: 29305: 29304:Quantum state 29302: 29300: 29297: 29295: 29292: 29290: 29287: 29285: 29282: 29280: 29277: 29275: 29272: 29268: 29265: 29263: 29260: 29258: 29257:Excited state 29255: 29253: 29250: 29249: 29248: 29245: 29243: 29240: 29238: 29235: 29233: 29230: 29228: 29225: 29224: 29222: 29218: 29212: 29209: 29207: 29204: 29202: 29199: 29195: 29192: 29191: 29190: 29187: 29185: 29182: 29181: 29179: 29175: 29171: 29164: 29159: 29157: 29152: 29150: 29145: 29144: 29141: 29129: 29126: 29124: 29121: 29119: 29116: 29114: 29111: 29109: 29106: 29105: 29103: 29099: 29093: 29090: 29088: 29087:Ocean physics 29085: 29083: 29080: 29078: 29075: 29073: 29070: 29068: 29065: 29063: 29060: 29058: 29055: 29053: 29050: 29048: 29045: 29044: 29042: 29040: 29036: 29030: 29027: 29023: 29022:Modern optics 29020: 29018: 29015: 29013: 29010: 29009: 29008: 29005: 29003: 29000: 28998: 28995: 28993: 28990: 28986: 28983: 28981: 28978: 28977: 28976: 28973: 28972: 28970: 28968: 28964: 28956: 28953: 28951: 28948: 28947: 28946: 28943: 28939: 28936: 28934: 28931: 28930: 28929: 28926: 28924: 28921: 28919: 28916: 28912: 28909: 28907: 28904: 28902: 28899: 28897: 28894: 28893: 28892: 28889: 28888: 28886: 28884: 28880: 28872: 28871:Computational 28869: 28868: 28867: 28864: 28862: 28859: 28858: 28856: 28852: 28844: 28841: 28840: 28839: 28836: 28834: 28831: 28830: 28828: 28824: 28820: 28812: 28807: 28805: 28800: 28798: 28793: 28792: 28789: 28782: 28778: 28775: 28772: 28768: 28765: 28764: 28753: 28749: 28745: 28741: 28737: 28731: 28724: 28723: 28718: 28714: 28710: 28706: 28702: 28698: 28694: 28688: 28684: 28683: 28678: 28674: 28670: 28666: 28662: 28658: 28654: 28648: 28644: 28643: 28637: 28636: 28624: 28620: 28616: 28612: 28607: 28602: 28598: 28594: 28589: 28584: 28580: 28576: 28572: 28567: 28552: 28548: 28544: 28540: 28536: 28532: 28528: 28524: 28520: 28519:Physics Today 28513: 28508: 28504: 28500: 28496: 28492: 28488: 28484: 28480: 28476: 28471: 28470: 28458: 28454: 28450: 28445: 28441: 28440:New Scientist 28437: 28433: 28432:Sciama, D. W. 28429: 28425: 28421: 28417: 28412: 28408: 28404: 28400: 28395: 28391: 28387: 28386:Physics World 28383: 28378: 28374: 28370: 28369:New Scientist 28366: 28361: 28357: 28353: 28349: 28345: 28340: 28339: 28322: 28318: 28314: 28309: 28305: 28301: 28297: 28293: 28289: 28285: 28280: 28276: 28272: 28268: 28264: 28260: 28256: 28252: 28248: 28244: 28240: 28235: 28230: 28226: 28222: 28217: 28202: 28201: 28195: 28191: 28187: 28182: 28177: 28173: 28169: 28165: 28161: 28156: 28152: 28148: 28140: 28135: 28131: 28127: 28123: 28119: 28115: 28109: 28105: 28101: 28097: 28093: 28089: 28085: 28081: 28077: 28073: 28069: 28065: 28061: 28057: 28053: 28049: 28042: 28038: 28034: 28030: 28026: 28022: 28018: 28014: 28010: 28006: 28002: 27997: 27992: 27988: 27984: 27979: 27975: 27971: 27967: 27963: 27958: 27953: 27949: 27945: 27941: 27937: 27933: 27928: 27924: 27920: 27916: 27912: 27908: 27904: 27900: 27896: 27891: 27887: 27883: 27879: 27873: 27869: 27864: 27860: 27856: 27852: 27846: 27843:. Routledge. 27842: 27841: 27835: 27831: 27827: 27823: 27819: 27815: 27809: 27805: 27801: 27797: 27793: 27792:Sciama, D. W. 27789: 27785: 27781: 27777: 27773: 27769: 27763: 27759: 27755: 27754:Schwinger, J. 27751: 27747: 27743: 27739: 27735: 27731: 27725: 27721: 27717: 27716:Schwinger, J. 27713: 27709: 27705: 27701: 27697: 27693: 27687: 27683: 27679: 27678:Schwinger, J. 27675: 27671: 27667: 27663: 27659: 27655: 27649: 27645: 27641: 27637: 27633: 27629: 27625: 27621: 27617: 27613: 27609: 27605: 27601: 27596: 27591: 27587: 27583: 27578: 27574: 27570: 27566: 27562: 27558: 27554: 27550: 27546: 27541: 27536: 27533:(3): 030602. 27532: 27528: 27523: 27519: 27515: 27511: 27507: 27503: 27499: 27494: 27489: 27485: 27481: 27477: 27473: 27469: 27465: 27461: 27457: 27453: 27449: 27445: 27441: 27437: 27433: 27429: 27425: 27418: 27413: 27409: 27405: 27401: 27397: 27393: 27387: 27383: 27382: 27377: 27373: 27369: 27363: 27359: 27358: 27353: 27348: 27344: 27340: 27336: 27332: 27328: 27324: 27320: 27316: 27312: 27308: 27304: 27300: 27296: 27292: 27288: 27284: 27280: 27279: 27274: 27270: 27266: 27262: 27258: 27254: 27250: 27246: 27245: 27240: 27236: 27232: 27228: 27224: 27220: 27216: 27212: 27208: 27204: 27200: 27196: 27192: 27188: 27184: 27180: 27176: 27172: 27168: 27164: 27160: 27156: 27152: 27148: 27144: 27140: 27137:(21): 14740. 27136: 27132: 27127: 27123: 27119: 27115: 27109: 27104: 27103: 27096: 27092: 27088: 27084: 27080: 27076: 27072: 27067: 27062: 27059:(2): 565–86. 27058: 27054: 27050: 27046: 27042: 27038: 27034: 27028: 27023: 27022: 27015: 27011: 27007: 27003: 26999: 26995: 26991: 26987: 26983: 26978: 26973: 26969: 26965: 26961: 26960:Ratra, Bharat 26957: 26953: 26949: 26945: 26941: 26937: 26933: 26931:9783540706229 26927: 26923: 26919: 26914: 26909: 26905: 26901: 26897: 26893: 26889: 26885: 26881: 26877: 26873: 26867: 26863: 26862: 26857: 26853: 26849: 26845: 26841: 26837: 26833: 26829: 26825: 26821: 26817: 26813: 26809: 26805: 26801: 26797: 26793: 26789: 26785: 26781: 26777: 26773: 26769: 26765: 26760: 26755: 26751: 26747: 26742: 26737: 26733: 26729: 26722: 26717: 26713: 26709: 26705: 26701: 26697: 26693: 26689: 26685: 26681: 26677: 26673: 26669: 26665: 26660: 26656: 26652: 26648: 26644: 26640: 26634: 26630: 26629: 26624: 26620: 26616: 26612: 26608: 26604: 26600: 26596: 26592: 26588: 26583: 26578: 26574: 26570: 26565: 26561: 26557: 26553: 26547: 26543: 26538: 26534: 26530: 26526: 26522: 26518: 26512: 26508: 26507: 26502: 26498: 26494: 26490: 26486: 26482: 26478: 26472: 26468: 26467: 26461: 26457: 26453: 26449: 26445: 26441: 26437: 26432: 26427: 26423: 26419: 26414: 26410: 26406: 26402: 26396: 26392: 26387: 26383: 26379: 26375: 26371: 26367: 26363: 26359: 26355: 26350: 26346: 26342: 26338: 26334: 26330: 26326: 26322: 26318: 26314: 26310: 26305: 26300: 26296: 26292: 26287: 26283: 26279: 26275: 26271: 26267: 26263: 26259: 26255: 26250: 26245: 26242:(1): 015020. 26241: 26237: 26232: 26228: 26224: 26220: 26216: 26212: 26206: 26202: 26201: 26195: 26191: 26187: 26183: 26179: 26174: 26169: 26165: 26161: 26156: 26151: 26147: 26143: 26139: 26134: 26130: 26126: 26122: 26118: 26114: 26110: 26106: 26102: 26097: 26092: 26088: 26084: 26079: 26064: 26060: 26056: 26052: 26048: 26044: 26040: 26036: 26032: 26025: 26020: 26016: 26012: 26008: 26004: 26000: 25994: 25990: 25986: 25982: 25981: 25975: 25971: 25967: 25963: 25959: 25955: 25951: 25946: 25942: 25938: 25934: 25928: 25924: 25923: 25917: 25913: 25909: 25905: 25899: 25895: 25894: 25888: 25884: 25880: 25876: 25872: 25868: 25864: 25860: 25856: 25851: 25846: 25842: 25838: 25833: 25829: 25823: 25819: 25815: 25811: 25806: 25802: 25798: 25794: 25790: 25786: 25782: 25777: 25762: 25758: 25754: 25750: 25746: 25742: 25738: 25731: 25726: 25722: 25718: 25714: 25710: 25706: 25702: 25698: 25694: 25689: 25684: 25681:(5): 050402. 25680: 25676: 25672: 25667: 25663: 25659: 25655: 25651: 25647: 25641: 25636: 25635: 25629: 25625: 25621: 25617: 25612: 25607: 25603: 25599: 25595: 25591: 25586: 25581: 25577: 25573: 25569: 25564: 25560: 25556: 25552: 25546: 25542: 25541: 25535: 25531: 25527: 25523: 25519: 25515: 25511: 25507: 25503: 25499: 25495: 25488: 25483: 25479: 25475: 25471: 25467: 25463: 25459: 25455: 25451: 25446: 25441: 25437: 25433: 25426: 25422: 25418: 25414: 25410: 25406: 25402: 25398: 25394: 25390: 25386: 25379: 25374: 25370: 25366: 25362: 25358: 25354: 25350: 25346: 25342: 25337: 25332: 25329:(4): 041804. 25328: 25324: 25319: 25315: 25311: 25307: 25303: 25299: 25295: 25290: 25286: 25282: 25278: 25274: 25270: 25264: 25260: 25259: 25253: 25249: 25245: 25241: 25237: 25232: 25227: 25223: 25219: 25215: 25211: 25207: 25202: 25198: 25194: 25190: 25186: 25182: 25178: 25174: 25170: 25166: 25161: 25157: 25153: 25148: 25143: 25139: 25135: 25131: 25127: 25122: 25121: 25109: 25105: 25101: 25096: 25092: 25088: 25087:New Scientist 25084: 25079: 25075: 25071: 25067: 25063: 25059: 25044: 25040: 25033: 25028: 25024: 25020: 25016: 25012: 25007: 25003: 24999: 24995: 24990: 24986: 24982: 24978: 24973: 24970: 24965: 24961: 24960:New Scientist 24957: 24952: 24948: 24944: 24940: 24935: 24931: 24927: 24923: 24919: 24915: 24914:New Scientist 24911: 24907: 24903: 24899: 24895: 24891: 24887: 24883: 24879: 24878:New Scientist 24875: 24870: 24866: 24862: 24858: 24853: 24849: 24845: 24844:New Scientist 24841: 24836: 24832: 24828: 24824: 24819: 24815: 24811: 24807: 24803: 24799: 24794: 24790: 24786: 24782: 24778: 24774: 24769: 24765: 24761: 24757: 24752: 24748: 24744: 24743:New Scientist 24740: 24735: 24734: 24713: 24709: 24705: 24701: 24695: 24688: 24683: 24668: 24664: 24657: 24642: 24638: 24631: 24624: 24619: 24611: 24607: 24602: 24597: 24593: 24589: 24584: 24579: 24575: 24571: 24567: 24563: 24559: 24552: 24544: 24540: 24536: 24532: 24528: 24524: 24520: 24516: 24512: 24505: 24497: 24491: 24487: 24483: 24479: 24472: 24465: 24460: 24444: 24440: 24434: 24427: 24422: 24414: 24410: 24406: 24402: 24398: 24394: 24393: 24385: 24377: 24373: 24369: 24365: 24361: 24357: 24352: 24351:gr-qc/0607086 24347: 24344:: 1415–1420. 24343: 24339: 24332: 24325: 24318: 24313: 24305: 24301: 24297: 24293: 24289: 24285: 24278: 24271: 24270:Kowitt (1994) 24266: 24258: 24254: 24250: 24246: 24242: 24238: 24234: 24230: 24223: 24215: 24211: 24207: 24203: 24199: 24195: 24191: 24187: 24180: 24172: 24168: 24164: 24160: 24156: 24152: 24148: 24144: 24137: 24122: 24115: 24108: 24099: 24094: 24087: 24080: 24075: 24067: 24063: 24058: 24053: 24049: 24045: 24026: 24018: 24014: 24010: 24006: 24002: 23998: 23979: 23971: 23967: 23963: 23959: 23955: 23951: 23946: 23945:gr-qc/0009013 23941: 23937: 23933: 23932: 23924: 23916: 23910: 23906: 23901: 23900: 23891: 23883: 23879: 23875: 23871: 23867: 23863: 23859: 23855: 23854: 23846: 23839: 23831: 23827: 23823: 23819: 23815: 23811: 23804: 23785: 23781: 23774: 23767: 23759: 23758:New Scientist 23755: 23749: 23741: 23737: 23733: 23729: 23725: 23721: 23717: 23713: 23706: 23698: 23694: 23690: 23686: 23682: 23678: 23674: 23670: 23665: 23664:gr-qc/0012102 23660: 23656: 23652: 23645: 23637: 23633: 23629: 23625: 23621: 23617: 23610: 23602: 23598: 23594: 23590: 23586: 23582: 23575: 23567: 23563: 23559: 23555: 23551: 23547: 23540: 23532: 23528: 23524: 23520: 23516: 23512: 23508: 23504: 23497: 23489: 23483: 23475: 23469: 23461: 23455: 23447: 23443: 23439: 23435: 23431: 23427: 23420: 23412: 23408: 23404: 23400: 23396: 23392: 23385: 23370:. 6 July 2010 23369: 23365: 23359: 23350: 23345: 23338: 23330: 23326: 23322: 23318: 23314: 23310: 23306: 23302: 23295: 23287: 23283: 23279: 23275: 23271: 23267: 23260: 23253: 23244: 23239: 23232: 23217: 23216:New Scientist 23213: 23206: 23197: 23192: 23173: 23165: 23161: 23157: 23153: 23149: 23145: 23140: 23139:gr-qc/0009036 23135: 23131: 23127: 23126: 23118: 23111: 23103: 23099: 23095: 23088: 23080: 23074: 23070: 23066: 23062: 23061: 23053: 23045: 23041: 23036: 23031: 23027: 23023: 23019: 23015: 23010: 23005: 23001: 22997: 22993: 22986: 22978: 22974: 22970: 22966: 22962: 22958: 22953: 22948: 22944: 22940: 22933: 22926: 22921: 22914: 22909: 22901: 22897: 22893: 22889: 22886:(6): 063845. 22885: 22881: 22874: 22866: 22862: 22857: 22852: 22848: 22844: 22841:(5): 051105. 22840: 22836: 22832: 22825: 22817: 22813: 22809: 22805: 22801: 22797: 22792: 22787: 22783: 22779: 22772: 22764: 22760: 22756: 22752: 22749:(5): 055804. 22748: 22744: 22737: 22730: 22729:Scully (2001) 22725: 22718: 22713: 22705: 22701: 22697: 22693: 22689: 22685: 22678: 22670: 22668:9780444869920 22664: 22660: 22656: 22651: 22650: 22641: 22633: 22629: 22625: 22621: 22617: 22613: 22606: 22598: 22594: 22590: 22586: 22582: 22578: 22571: 22563: 22559: 22555: 22551: 22547: 22543: 22536: 22528: 22524: 22520: 22513: 22498:on 7 May 2021 22494: 22490: 22486: 22482: 22478: 22474: 22470: 22463: 22456: 22448: 22442: 22426: 22422: 22416: 22408: 22404: 22399: 22394: 22389: 22384: 22380: 22376: 22372: 22365: 22356: 22351: 22344: 22337: 22332: 22324: 22320: 22316: 22312: 22308: 22304: 22297: 22290: 22285: 22283: 22274: 22270: 22266: 22265: 22260: 22254: 22247: 22243: 22240: 22233: 22226: 22224: 22222: 22205: 22201: 22197: 22190: 22182: 22176: 22172: 22171: 22163: 22155: 22149: 22145: 22141: 22137: 22130: 22122: 22118: 22114: 22110: 22106: 22102: 22097: 22092: 22088: 22084: 22077: 22070: 22066: 22061: 22053: 22049: 22045: 22041: 22037: 22033: 22026: 22018: 22014: 22010: 22006: 22002: 21998: 21991: 21972: 21968: 21964: 21957: 21950: 21931: 21927: 21923: 21916: 21909: 21890: 21886: 21882: 21875: 21868: 21860: 21853: 21845: 21839: 21835: 21834: 21826: 21818: 21812: 21808: 21807: 21799: 21791: 21789:9780749386061 21785: 21781: 21774: 21766: 21760: 21756: 21749: 21741: 21735: 21731: 21724: 21708: 21704: 21697: 21689: 21687:9783540304319 21683: 21679: 21678: 21670: 21663: 21659: 21654: 21646: 21642: 21635: 21627: 21621: 21617: 21610: 21601: 21600: 21591: 21584: 21579: 21568: 21561: 21554:(2): 213–219. 21553: 21549: 21542: 21534: 21530: 21526: 21522: 21518: 21514: 21510: 21506: 21499: 21491: 21487: 21483: 21479: 21475: 21471: 21470:J. Math. Phys 21464: 21457: 21452: 21433: 21429: 21425: 21421: 21417: 21410: 21403: 21395: 21393:9780801482342 21389: 21385: 21384: 21376: 21368: 21364: 21360: 21356: 21349: 21341: 21339:9780801482342 21335: 21331: 21330: 21322: 21314: 21308: 21304: 21303: 21295: 21288: 21283: 21275: 21271: 21264: 21255: 21250: 21246: 21242: 21238: 21234: 21230: 21223: 21215: 21211: 21207: 21203: 21199: 21195: 21191: 21184: 21176: 21172: 21168: 21164: 21160: 21156: 21149: 21141: 21137: 21133: 21129: 21125: 21121: 21117: 21113: 21106: 21098: 21094: 21090: 21086: 21082: 21078: 21071: 21063: 21059: 21055: 21051: 21047: 21043: 21039: 21035: 21028: 21020: 21016: 21012: 21008: 21004: 21000: 20996: 20992: 20985: 20966: 20962: 20958: 20954: 20950: 20943: 20936: 20928: 20924: 20920: 20916: 20912: 20908: 20903: 20898: 20894: 20890: 20889: 20881: 20873: 20869: 20864: 20859: 20855: 20851: 20847: 20843: 20842: 20834: 20828:, p. 23. 20827: 20822: 20814: 20812:9789812779977 20808: 20804: 20803: 20795: 20789:, p. 20. 20788: 20783: 20777:, p. 48. 20776: 20771: 20763: 20759: 20755: 20751: 20747: 20743: 20738: 20733: 20730:(2): 021301. 20729: 20725: 20724: 20716: 20714: 20705: 20701: 20697: 20693: 20689: 20685: 20678: 20670: 20666: 20662: 20658: 20654: 20650: 20646: 20642: 20635: 20628: 20624: 20620: 20614: 20606: 20602: 20598: 20592: 20588: 20584: 20580: 20573: 20563: 20556: 20550: 20542: 20538: 20534: 20530: 20526: 20522: 20518: 20514: 20509: 20504: 20500: 20496: 20489: 20482: 20467: 20463: 20456: 20448: 20447:New Scientist 20444: 20437: 20430: 20425: 20418: 20413: 20406: 20401: 20394: 20389: 20382: 20377: 20370: 20365: 20358: 20354: 20350: 20346: 20340: 20333: 20329: 20325: 20319: 20312: 20307: 20300: 20294: 20287: 20282: 20275: 20270: 20263: 20258: 20251: 20246: 20238: 20236:9789814545143 20232: 20228: 20221: 20213: 20211:9781466512993 20207: 20203: 20196: 20190:, p. 33. 20189: 20184: 20177: 20172: 20164: 20160: 20156: 20152: 20148: 20144: 20140: 20136: 20135: 20127: 20121:, p. 80. 20120: 20115: 20108: 20104: 20098: 20096: 20088: 20083: 20075: 20071: 20067: 20063: 20059: 20055: 20048: 20040: 20036: 20031: 20026: 20022: 20018: 20014: 20010: 20006: 20002: 19995: 19988: 19981: 19976: 19969: 19964: 19957: 19952: 19944: 19940: 19936: 19932: 19928: 19924: 19919: 19914: 19910: 19906: 19899: 19891: 19887: 19883: 19879: 19875: 19871: 19864: 19857: 19849: 19845: 19842:: 2637–2651. 19841: 19837: 19830: 19822: 19818: 19814: 19810: 19806: 19802: 19795: 19787: 19783: 19779: 19775: 19771: 19767: 19763: 19759: 19755: 19751: 19744: 19736: 19732: 19728: 19724: 19720: 19716: 19709: 19701: 19697: 19693: 19689: 19682: 19674: 19670: 19666: 19662: 19658: 19654: 19647: 19640: 19638: 19630: 19625: 19619:, p. 22. 19618: 19613: 19606: 19601: 19585: 19578: 19572: 19564: 19560: 19556: 19552: 19548: 19544: 19537: 19530: 19525: 19523: 19515: 19510: 19502: 19498: 19494: 19490: 19486: 19482: 19475: 19468: 19463: 19456: 19451: 19443: 19439: 19435: 19429: 19425: 19421: 19417: 19413: 19406: 19398: 19394: 19390: 19386: 19382: 19378: 19374: 19371:(in German). 19370: 19366: 19359: 19352: 19347: 19340: 19335: 19327: 19323: 19319: 19315: 19311: 19307: 19300: 19292: 19288: 19283: 19278: 19274: 19270: 19266: 19262: 19258: 19251: 19243: 19239: 19235: 19231: 19227: 19223: 19219: 19215: 19211: 19207: 19200: 19193: 19186: 19181: 19174: 19169: 19161: 19157: 19154:(1): 89–109. 19153: 19149: 19142: 19135: 19130: 19128: 19119: 19115: 19111: 19107: 19103: 19099: 19095: 19091: 19086: 19081: 19077: 19073: 19066: 19059: 19057: 19048: 19044: 19040: 19036: 19032: 19028: 19024: 19020: 19016: 19012: 19005: 18998: 18996: 18989:, p. 54. 18988: 18983: 18981: 18972: 18968: 18964: 18960: 18957:(1): 97–109. 18956: 18952: 18945: 18943: 18934: 18930: 18926: 18922: 18918: 18914: 18907: 18905: 18896: 18892: 18888: 18884: 18880: 18876: 18869: 18867: 18865: 18856: 18852: 18848: 18842: 18838: 18831: 18823: 18819: 18815: 18811: 18804: 18796: 18792: 18785: 18777: 18773: 18766: 18751: 18747: 18743: 18739: 18735: 18731: 18730:Physics World 18724: 18717: 18709: 18705: 18701: 18697: 18693: 18689: 18685: 18682:(in German). 18681: 18674: 18666: 18662: 18658: 18654: 18650: 18646: 18642: 18639:(in German). 18638: 18630: 18614: 18610: 18603: 18596: 18588: 18584: 18580: 18576: 18572: 18568: 18561: 18545: 18538: 18531: 18525:, p. 35. 18524: 18519: 18511: 18507: 18503: 18499: 18495: 18491: 18487: 18483: 18476: 18469: 18464: 18456: 18452: 18448: 18442: 18438: 18431: 18423: 18419: 18415: 18411: 18407: 18403: 18402: 18397: 18393: 18387: 18378: 18373: 18369: 18365: 18361: 18357: 18356: 18351: 18347: 18343: 18337: 18329: 18325: 18321: 18317: 18313: 18309: 18302: 18294: 18290: 18286: 18282: 18278: 18274: 18270: 18267:(in German). 18266: 18259: 18252: 18248: 18243: 18235: 18231: 18227: 18221: 18217: 18213: 18206: 18198: 18194: 18190: 18186: 18179: 18172: 18167: 18165: 18163: 18154: 18150: 18146: 18140: 18136: 18131: 18130: 18121: 18113: 18109: 18105: 18101: 18097: 18093: 18089: 18086:(in German). 18085: 18078: 18070: 18066: 18062: 18056: 18052: 18048: 18044: 18038: 18030: 18026: 18022: 18018: 18011: 18003: 17999: 17995: 17991: 17987: 17983: 17979: 17976:(in German). 17975: 17968: 17960: 17956: 17952: 17946: 17942: 17937: 17936: 17927: 17919: 17915: 17911: 17907: 17903: 17900:(in German). 17899: 17892: 17884: 17880: 17879: 17871: 17863: 17859: 17855: 17851: 17847: 17843: 17839: 17832: 17824: 17820: 17816: 17812: 17808: 17804: 17797: 17790: 17782: 17778: 17774: 17770: 17767:(151): 1–24. 17766: 17762: 17755: 17748: 17741: 17736: 17728: 17724: 17720: 17714: 17710: 17706: 17702: 17698: 17694: 17686: 17678: 17674: 17670: 17666: 17662: 17658: 17654: 17650: 17643: 17635: 17631: 17627: 17623: 17619: 17615: 17612:(1134): 291. 17611: 17607: 17600: 17592: 17588: 17583: 17578: 17574: 17570: 17567:(3362): 529. 17566: 17562: 17558: 17551: 17543: 17539: 17535: 17531: 17527: 17523: 17519: 17516:(in German). 17515: 17508: 17500: 17496: 17495: 17487: 17479: 17475: 17468: 17460: 17457:(in German). 17456: 17448: 17440: 17436: 17432: 17428: 17424: 17421:(in German). 17420: 17416: 17409: 17402: 17397: 17389: 17385: 17381: 17377: 17373: 17370:(in German). 17369: 17365: 17358: 17351: 17346: 17339: 17338:Planck (1958) 17335: 17331: 17327: 17323: 17317: 17311:, p. 10. 17310: 17305: 17298: 17293: 17286: 17281: 17279: 17271: 17270:Loudon (2000) 17266: 17259: 17258:Planck (1900) 17254: 17247: 17242: 17235: 17234:Conlon (2011) 17230: 17223: 17218: 17211: 17206: 17204: 17195: 17191: 17185: 17183: 17175: 17170: 17168: 17166: 17158: 17154: 17150: 17144: 17137: 17136:Davies (1994) 17133: 17132:Powell (1994) 17129: 17125: 17121: 17115: 17113: 17105: 17100: 17093: 17088: 17081: 17076: 17070:, p. 12. 17069: 17064: 17057: 17052: 17050: 17048: 17040: 17035: 17033: 17025: 17020: 17018: 17010: 17005: 16998: 16997:Davies (1985) 16993: 16986: 16985:Sciama (1991) 16981: 16979: 16971: 16966: 16964: 16956: 16955:Siegel (2016) 16952: 16948: 16944: 16938: 16931: 16930:Davies (2011) 16926: 16920:, p. 35. 16919: 16914: 16912: 16910: 16902: 16901:Sciama (1991) 16897: 16895: 16893: 16888: 16873: 16870: 16868: 16865: 16863: 16860: 16858: 16857:Vacuum energy 16855: 16853: 16850: 16848: 16845: 16843: 16840: 16838: 16835: 16833: 16830: 16828: 16825: 16823: 16820: 16818: 16815: 16813: 16810: 16808: 16805: 16803: 16800: 16799: 16794: 16788: 16783: 16780: 16774: 16769: 16762: 16760: 16750: 16747: 16742: 16738: 16734: 16729: 16727: 16723: 16719: 16715: 16714:Linus Pauling 16711: 16706: 16704: 16698: 16695: 16692:, approached 16691: 16687: 16683: 16682:Phantom Works 16678: 16676: 16672: 16667: 16663: 16659: 16655: 16651: 16646: 16644: 16640: 16636: 16632: 16628: 16624: 16620: 16619:exotic matter 16616: 16612: 16607: 16605: 16600: 16596: 16591: 16588: 16584: 16579: 16578:New Scientist 16574: 16570: 16565: 16563: 16559: 16554: 16550: 16546: 16542: 16538: 16533: 16531: 16527: 16523: 16513: 16509: 16505: 16503: 16498: 16497:Johnson noise 16494: 16490: 16486: 16476: 16472: 16469: 16464: 16462: 16458: 16453: 16451: 16447: 16446: 16441: 16437: 16432: 16429: 16424: 16423:Casimir force 16414: 16412: 16407: 16403: 16399: 16395: 16390: 16388: 16384: 16374: 16372: 16364: 16360: 16355: 16351: 16349: 16345: 16341: 16337: 16333: 16329: 16325: 16321: 16305: 16297: 16293: 16289: 16285: 16281: 16276: 16270: 16267: 16263: 16259: 16255: 16251: 16247: 16243: 16239: 16235: 16231: 16226: 16223: 16219: 16215: 16211: 16207: 16203: 16199: 16195: 16190: 16188: 16183: 16182:Sagnac effect 16179: 16175: 16171: 16167: 16163: 16162:Aharonov–Bohm 16159: 16155: 16151: 16146: 16141: 16138:all view the 16137: 16133: 16127: 16123: 16119: 16107: 16102: 16100: 16095: 16093: 16088: 16087: 16085: 16084: 16077: 16074: 16072: 16069: 16067: 16064: 16062: 16059: 16057: 16054: 16052: 16049: 16047: 16044: 16042: 16039: 16038: 16032: 16031: 16024: 16021: 16019: 16016: 16014: 16011: 16009: 16006: 16004: 16001: 15999: 15996: 15994: 15991: 15989: 15986: 15984: 15981: 15979: 15976: 15974: 15971: 15969: 15966: 15964: 15963:String theory 15961: 15959: 15956: 15955: 15952: 15947: 15946: 15939: 15936: 15934: 15931: 15929: 15926: 15924: 15921: 15919: 15916: 15914: 15911: 15909: 15906: 15904: 15901: 15900: 15897: 15896:Supersymmetry 15892: 15891: 15884: 15881: 15879: 15876: 15874: 15871: 15869: 15866: 15864: 15861: 15859: 15856: 15854: 15851: 15849: 15846: 15844: 15841: 15839: 15836: 15834: 15831: 15829: 15826: 15824: 15821: 15819: 15816: 15814: 15811: 15809: 15806: 15804: 15801: 15799: 15798:Mirror matter 15796: 15794: 15791: 15789: 15786: 15784: 15781: 15779: 15778:String theory 15776: 15774: 15771: 15769: 15766: 15764: 15761: 15759: 15756: 15754: 15751: 15749: 15746: 15744: 15741: 15739: 15736: 15734: 15731: 15729: 15726: 15724: 15721: 15719: 15716: 15714: 15711: 15710: 15704: 15703: 15696: 15693: 15691: 15688: 15686: 15683: 15681: 15678: 15676: 15673: 15671: 15668: 15666: 15663: 15661: 15658: 15656: 15653: 15651: 15648: 15647: 15641: 15640: 15637: 15634: 15633: 15628: 15624: 15621: 15615: 15611: 15610: 15607: 15604: 15603: 15595: 15593: 15588: 15584: 15582: 15578: 15567: 15565: 15561: 15552: 15550: 15546: 15542: 15538: 15532: 15530: 15526: 15522: 15518: 15514: 15510: 15506: 15502: 15498: 15492: 15477: 15467: 15465: 15461: 15457: 15456:string theory 15453: 15451: 15446: 15442: 15438: 15434: 15430: 15427: 15422: 15420: 15414: 15412: 15407: 15400: 15394: 15381: 15366: 15364: 15360: 15356: 15352: 15347: 15343: 15339: 15334: 15332: 15328: 15324: 15320: 15316: 15311: 15307: 15300: 15295: 15291: 15289: 15285: 15281: 15276: 15267: 15248: 15244: 15234: 15203: 15184: 15181: 15176: 15173: 15163: 15155:rest energy, 15154: 15148: 15130: 15127: 15122: 15117: 15107: 15092: 15083: 15065: 15057: 15051: 15047: 15032: 15029: 15024: 15018: 15010: 15006: 15002: 14999: 14993: 14988: 14984: 14978: 14972: 14969: 14963: 14959: 14953: 14950: 14942: 14938: 14920: 14916: 14907: 14902: 14894: 14890: 14878: 14866: 14859: 14851: 14846: 14834: 14824: 14814: 14799: 14795: 14794:hydrogen atom 14791: 14769: 14746: 14742: 14741:energy levels 14735: 14731: 14727: 14722: 14712: 14710: 14703: 14698: 14694: 14690: 14688: 14685: 14682: 14673: 14668: 14658: 14649: 14647: 14642: 14637: 14634: 14631:246 GeV/ 14621: 14608: 14598: 14577: 14569: 14562: 14556: 14553: 14550: 14538: 14534: 14529: 14523: 14517: 14512: 14502: 14500: 14495: 14491: 14488:that mediate 14487: 14484: 14480: 14476: 14472: 14468: 14464: 14463: 14456: 14446: 14442: 14427: 14418: 14408: 14402: 14398: 14395: 14377: 14363: 14356: 14352: 14345: 14338: 14332: 14316: 14308: 14296: 14293: 14288: 14278: 14273: 14269: 14265: 14248: 14243: 14239: 14235: 14232: 14182: 14177: 14173: 14169: 14166: 14139: 14126: 14121: 14117: 14113: 14110: 14087: 14079: 14067: 14064: 14059: 14038: 14035: 14025: 14020: 14016: 14012: 13968: 13965: 13963: 13954: 13946: 13941: 13937: 13933: 13929: 13924: 13919: 13911: 13907: 13903: 13900: 13897: 13890: 13885: 13881: 13875: 13871: 13864: 13861: 13855: 13851: 13849: 13836: 13831: 13827: 13821: 13817: 13813: 13808: 13804: 13798: 13795: 13779: 13775: 13769: 13764: 13760: 13751: 13747: 13743: 13740: 13737: 13730: 13726: 13719: 13716: 13710: 13708: 13702: 13695: 13687: 13683: 13679: 13673: 13667: 13663: 13646: 13642: 13632: 13626: 13604: 13600: 13594: 13590: 13586: 13581: 13576: 13570: 13565: 13561: 13557: 13552: 13548: 13543: 13534: 13530: 13525: 13522: 13509: 13505: 13500: 13495: 13491: 13488: 13482: 13477: 13469: 13465: 13461: 13456: 13452: 13448: 13441: 13437: 13430: 13424: 13420: 13418: 13409: 13402: 13393: 13390: 13384: 13381: 13375: 13369: 13365: 13361: 13359: 13350: 13343: 13335: 13331: 13325: 13322: 13317: 13311: 13305: 13301: 13297: 13293: 13286: 13277: 13274: 13268: 13265: 13259: 13253: 13249: 13245: 13243: 13215: 13212: 13208: 13204: 13200: 13195: 13188: 13176: 13172: 13168: 13161: 13153: 13147: 13144: 13138: 13129: 13125: 13119: 13111: 13102: 13097: 13087: 13071: 13063: 13051: 13048: 13043: 13022: 13019: 13009: 13004: 13000: 12996: 12975: 12955: 12952: 12950: 12939: 12933: 12927: 12919: 12915: 12908: 12905: 12903: 12892: 12886: 12882: 12875: 12867: 12863: 12859: 12853: 12847: 12843: 12836: 12828: 12824: 12820: 12818: 12809: 12802: 12796: 12790: 12782: 12778: 12771: 12763: 12759: 12755: 12749: 12743: 12737: 12731: 12725: 12717: 12713: 12708: 12704: 12702: 12696: 12689: 12681: 12677: 12673: 12667: 12661: 12657: 12642: 12629: 12618: 12615: 12606: 12603: 12599: 12592: 12589: 12585: 12579: 12570: 12550: 12542: 12538: 12533: 12530: 12520: 12516: 12512: 12510: 12501: 12496: 12490: 12486: 12479: 12476: 12470: 12464: 12455: 12448: 12445: 12441: 12436: 12434: 12425: 12418: 12410: 12404: 12401: 12395: 12386: 12379: 12371: 12362: 12357: 12353: 12347: 12340: 12337: 12332: 12325: 12322: 12312: 12306: 12302: 12295: 12292: 12282: 12279: 12273: 12264: 12258: 12249: 12242: 12239: 12235: 12230: 12228: 12222: 12215: 12207: 12202: 12191: 12184: 12181: 12177: 12160: 12157: 12153: 12136: 12133: 12116: 12108: 12087: 12078: 12070: 12050: 12025: 12016: 12012: 11990: 11969: 11962: 11955: 11950: 11947: 11943: 11937: 11933: 11926: 11922:vacuum field 11919: 11912: 11903: 11894: 11874: 11866: 11854: 11851: 11846: 11825: 11822: 11812: 11807: 11803: 11799: 11764: 11746: 11741: 11737: 11731: 11727: 11722: 11718: 11711: 11704: 11697: 11680: 11677: 11668: 11659: 11655: 11645: 11631: 11627: 11623: 11618: 11613: 11604: 11592: 11585: 11578: 11573: 11569: 11562: 11555: 11549: 11519: 11515: 11511: 11506: 11503: 11497: 11491: 11483: 11480: 11461: 11456: 11442: 11438: 11435: 11431: 11428: 11423: 11417: 11413: 11409: 11406: 11402: 11397: 11393: 11390: 11386: 11370: 11365: 11356: 11351: 11346: 11343: 11339: 11334: 11329: 11325: 11319: 11310: 11304: 11300: 11297: 11294: 11288: 11285: 11279: 11271: 11268: 11239: 11230: 11225: 11219: 11214: 11210: 11206: 11202: 11195: 11187: 11182: 11173: 11169: 11164: 11159: 11155: 11151: 11148: 11144: 11137: 11129: 11120: 11115: 11108: 11102: 11098: 11091: 11088: 11079: 11070: 11066: 11063: 11057: 11049: 11019: 11011: 11008: 10996: 10993: 10988: 10982: 10974: 10962: 10959: 10954: 10944: 10939: 10935: 10931: 10909: 10906: 10902: 10882: 10878: 10875: 10871: 10868: 10863: 10857: 10853: 10849: 10845: 10837: 10834: 10815: 10810: 10801: 10795: 10792: 10788: 10783: 10778: 10774: 10766: 10761: 10757: 10753: 10748: 10745: 10738: 10735: 10732: 10727: 10722: 10718: 10714: 10711: 10707: 10700: 10692: 10683: 10679: 10673: 10665: 10656: 10648:equation is: 10645: 10632: 10628: 10625: 10619: 10613: 10609: 10602: 10594: 10591: 10587: 10583: 10577: 10569: 10565: 10560: 10556: 10553: 10550: 10544: 10541: 10532: 10522: 10518: 10511: 10507: 10503: 10499: 10495: 10491: 10487: 10480: 10476: 10469: 10462: 10459: 10455: 10449: 10443: 10437: 10421: 10412: 10408: 10389: 10384: 10380: 10376: 10371: 10368: 10361: 10358: 10355: 10350: 10341: 10335: 10331: 10327: 10324: 10319: 10307: 10304: 10290: 10287: 10283: 10277: 10258: 10249: 10244: 10237: 10229: 10224: 10215: 10211: 10205: 10197: 10188: 10183: 10176: 10170: 10166: 10159: 10156: 10147: 10138: 10134: 10132: 10117: 10114: 10109: 10107: 10095: 10090: 10086: 10082: 10057: 10054: 10048: 10035: 10029: 10013: 9995: 9989: 9970: 9927: 9922: 9918: 9914: 9911: 9903: 9886: 9883: 9878: 9870: 9867: 9865: 9850: 9845: 9841: 9837: 9834: 9826: 9809: 9806: 9801: 9797: 9787: 9780: 9764: 9760: 9744: 9738: 9735: 9730: 9727: 9725: 9703: 9700: 9695: 9680: 9678: 9661: 9615: 9610: 9606: 9602: 9599: 9591: 9574: 9571: 9566: 9552: 9532: 9529: 9524: 9522: 9500: 9495: 9491: 9487: 9484: 9480: 9469: 9466: 9461: 9457: 9453: 9447: 9443: 9432: 9429: 9424: 9415: 9409: 9406: 9401: 9397: 9386: 9383: 9378: 9374: 9369: 9362: 9358: 9347: 9344: 9339: 9330: 9325: 9319: 9316: 9311: 9308: 9306: 9284: 9279: 9275: 9271: 9268: 9263: 9258: 9247: 9244: 9239: 9230: 9219: 9216: 9210: 9208: 9197: 9194: 9181: 9178: 9167: 9161: 9159: 9137: 9126: 9123: 9118: 9109: 9103: 9100: 9095: 9089: 9086: 9073: 9070: 9059: 9053: 9051: 9020: 9012: 9009: 9001: 8998: 8994: 8988: 8981: 8966: 8962: 8958: 8953: 8941: 8936: 8932: 8928: 8922: 8919: 8913: 8908: 8903: 8892: 8889: 8884: 8875: 8867: 8864: 8860: 8855: 8852: 8842: 8840: 8835: 8821: 8815: 8810: 8806: 8802: 8797: 8792: 8788: 8783: 8774: 8770: 8764: 8760: 8756: 8752: 8747: 8741: 8733: 8729: 8725: 8722: 8715: 8711: 8703: 8699: 8694: 8681: 8672: 8666: 8648: 8644: 8638: 8634: 8630: 8623: 8619: 8615: 8609: 8603: 8595: 8591: 8564: 8560: 8555: 8552: 8549: 8541: 8537: 8531: 8527: 8523: 8519: 8514: 8512: 8503: 8497: 8493: 8483: 8480: 8473: 8467: 8463: 8458: 8455: 8452: 8444: 8440: 8436: 8431: 8428: 8422: 8420: 8410: 8406: 8396: 8393: 8387: 8382: 8378: 8374: 8366: 8362: 8358: 8354: 8349: 8347: 8340: 8336: 8326: 8323: 8315: 8306: 8300: 8297: 8271: 8266: 8262: 8258: 8253: 8249: 8240: 8236: 8232: 8228: 8223: 8220: 8215: 8211: 8207: 8202: 8197: 8191: 8188: 8184: 8179: 8172: 8168: 8159: 8150: 8130: 8119: 8116: 8107: 8103: 8099: 8090: 8087: 8083: 8072: 8068: 8060: 8053: 8032: 8029: 8023: 8019: 8015: 8008: 8004: 7999: 7996: 7993: 7990: 7967: 7960: 7956: 7951: 7948: 7943: 7939: 7935: 7932: 7920: 7905: 7901: 7891: 7888: 7880: 7871: 7861: 7852: 7835: 7832: 7828: 7821: 7813: 7807: 7804: 7799: 7789: 7785: 7779: 7771: 7766: 7757: 7752: 7748: 7746: 7740: 7733: 7724: 7721: 7716: 7706: 7702: 7696: 7688: 7679: 7674: 7663: 7660: 7656: 7653: 7649: 7643: 7637: 7628: 7619: 7615: 7613: 7607: 7600: 7592: 7586: 7583: 7578: 7568: 7564: 7558: 7550: 7541: 7536: 7522: 7508: 7501: 7498: 7492: 7487: 7478: 7472: 7467: 7458: 7453: 7447: 7443: 7434: 7425: 7421: 7416: 7412: 7388: 7385: 7381: 7378: 7374: 7368: 7362: 7353: 7344: 7340: 7324: 7318: 7315: 7310: 7294: 7278: 7263: 7258: 7254: 7248: 7244: 7221: 7212: 7207: 7196: 7188: 7185: 7181: 7174: 7166: 7161: 7152: 7148: 7138: 7130: 7126: 7119: 7111: 7102: 7097: 7089: 7084: 7080: 7072: 7068: 7061: 7058: 7049: 7040: 7036: 7030: 7007: 7000: 6993: 6972: 6969: 6962: 6955: 6952: 6948: 6942: 6938: 6917: 6903: 6895: 6892: 6887: 6883: 6876: 6872: 6865: 6857: 6852: 6843: 6839: 6826: 6818: 6815: 6810: 6806: 6799: 6796: 6792: 6785: 6777: 6768: 6763: 6755: 6750: 6746: 6738: 6734: 6727: 6724: 6717: 6711: 6708: 6695: 6661: 6652: 6647: 6636: 6628: 6625: 6621: 6614: 6606: 6601: 6592: 6588: 6578: 6570: 6566: 6559: 6551: 6542: 6537: 6529: 6524: 6520: 6512: 6508: 6501: 6498: 6491: 6485: 6482: 6469: 6430: 6423: 6417: 6412: 6409: 6405: 6394: 6386: 6382: 6376: 6367: 6360: 6356: 6351: 6335: 6309: 6302: 6294: 6287: 6278: 6274: 6268: 6264: 6256: 6252: 6244: 6240: 6233: 6229: 6221: 6217: 6213: 6206: 6202: 6185: 6182: 6177: 6172: 6142: 6137: 6132: 6128: 6122: 6118: 6090: 6082: 6078: 6066: 6059: 6055: 6050: 6000: 5994: 5990: 5986: 5981: 5977: 5973: 5968: 5964: 5959: 5953: 5949: 5946: 5940: 5936: 5930: 5926: 5922: 5917: 5913: 5909: 5904: 5900: 5895: 5871: 5868: 5865: 5862: 5859: 5856: 5853: 5842: 5836: 5833: 5830: 5827: 5824: 5821: 5818: 5815: 5812: 5809: 5806: 5803: 5800: 5783: 5779: 5773: 5769: 5765: 5761: 5757: 5749: 5745: 5741: 5717: 5709: 5705: 5693: 5689: 5673: 5657: 5648: 5641: 5633: 5620: 5617: 5612: 5607: 5590: 5579: 5574: 5569: 5565: 5561: 5548: 5540: 5532: 5515: 5512: 5496: 5485: 5479: 5475: 5471: 5466: 5457: 5448: 5442: 5435: 5429: 5410: 5397: 5391: 5380: 5377: 5371: 5367: 5361: 5359: 5350: 5344: 5334: 5329: 5318: 5314: 5309: 5305: 5301: 5295: 5292: 5288: 5283: 5281: 5274: 5270: 5251: 5247: 5244: 5234: 5224: 5214: 5201: 5196: 5192: 5188: 5182: 5178: 5174: 5169: 5165: 5161: 5156: 5152: 5148: 5144: 5140: 5134: 5129: 5125: 5121: 5112: 5103: 5083: 5078: 5074: 5070: 5064: 5062: 5054: 5045: 5042: 5036: 5032: 5025: 5022: 5016: 5013: 5008: 5004: 4999: 4995: 4989: 4987: 4979: 4970: 4967: 4961: 4957: 4953: 4948: 4944: 4940: 4935: 4931: 4927: 4923: 4919: 4910: 4907: 4901: 4899: 4893: 4889: 4879: 4875: 4866: 4862: 4858: 4853: 4849: 4834: 4816: 4809: 4806: 4800: 4797: 4792: 4788: 4783: 4779: 4773: 4771: 4762: 4758: 4753: 4749: 4745: 4740: 4736: 4732: 4728: 4724: 4715: 4712: 4706: 4704: 4694: 4689: 4682: 4678: 4674: 4668: 4665: 4659: 4653: 4650: 4644: 4640: 4634: 4632: 4625: 4622: 4618: 4594: 4584: 4574: 4567: 4557: 4554: 4537: 4534: 4530: 4526: 4521: 4517: 4513: 4510: 4506: 4496: 4493: 4485: 4477: 4464: 4461: 4457: 4451: 4447: 4443: 4440: 4436: 4426: 4415: 4400: 4398: 4393: 4389: 4383: 4373: 4371: 4367: 4363: 4360: 4356: 4352: 4348: 4343: 4341: 4338: 4334: 4330: 4326: 4322: 4321:sea of energy 4318: 4313: 4311: 4307: 4292: 4284: 4278: 4276: 4272: 4268: 4262: 4260: 4256: 4252: 4248: 4244: 4238: 4234: 4233:Vacuum energy 4230: 4218: 4213: 4211: 4206: 4204: 4199: 4198: 4196: 4195: 4189: 4184: 4181: 4179: 4176: 4174: 4171: 4169: 4166: 4164: 4161: 4159: 4158:Zamolodchikov 4156: 4154: 4153:Zamolodchikov 4151: 4149: 4146: 4144: 4141: 4139: 4136: 4134: 4131: 4129: 4126: 4124: 4121: 4119: 4116: 4114: 4111: 4109: 4106: 4104: 4101: 4099: 4096: 4094: 4091: 4089: 4086: 4084: 4081: 4079: 4076: 4074: 4071: 4069: 4066: 4064: 4061: 4059: 4056: 4054: 4051: 4049: 4046: 4044: 4041: 4039: 4036: 4034: 4031: 4029: 4026: 4024: 4021: 4019: 4016: 4014: 4011: 4009: 4006: 4004: 4001: 3999: 3996: 3994: 3991: 3989: 3986: 3984: 3981: 3979: 3976: 3974: 3971: 3969: 3966: 3964: 3961: 3959: 3956: 3954: 3951: 3949: 3946: 3944: 3941: 3939: 3936: 3934: 3931: 3929: 3926: 3924: 3921: 3919: 3916: 3914: 3911: 3909: 3906: 3904: 3901: 3899: 3896: 3894: 3891: 3889: 3886: 3884: 3881: 3879: 3876: 3874: 3871: 3869: 3866: 3864: 3861: 3859: 3856: 3854: 3851: 3849: 3846: 3844: 3841: 3839: 3836: 3834: 3831: 3829: 3826: 3824: 3821: 3819: 3816: 3814: 3811: 3809: 3806: 3804: 3801: 3799: 3796: 3794: 3791: 3789: 3786: 3784: 3781: 3779: 3776: 3774: 3771: 3769: 3766: 3764: 3761: 3759: 3756: 3754: 3751: 3749: 3746: 3744: 3741: 3739: 3736: 3734: 3731: 3729: 3726: 3724: 3721: 3719: 3716: 3714: 3711: 3709: 3706: 3704: 3701: 3699: 3696: 3694: 3691: 3689: 3686: 3684: 3681: 3679: 3676: 3674: 3671: 3669: 3666: 3664: 3661: 3659: 3656: 3654: 3651: 3649: 3646: 3644: 3641: 3639: 3636: 3634: 3631: 3629: 3626: 3624: 3621: 3619: 3616: 3614: 3611: 3609: 3606: 3604: 3601: 3599: 3596: 3594: 3591: 3589: 3586: 3584: 3581: 3579: 3576: 3574: 3571: 3569: 3566: 3564: 3561: 3559: 3556: 3554: 3551: 3549: 3546: 3544: 3541: 3539: 3536: 3534: 3531: 3529: 3526: 3524: 3521: 3519: 3516: 3514: 3511: 3509: 3506: 3504: 3501: 3499: 3496: 3495: 3488: 3487: 3480: 3477: 3475: 3472: 3470: 3467: 3465: 3464:Supersymmetry 3462: 3460: 3459:String theory 3457: 3456: 3450: 3449: 3442: 3439: 3437: 3434: 3432: 3429: 3427: 3424: 3423: 3420: 3415: 3414: 3407: 3404: 3402: 3399: 3397: 3394: 3392: 3389: 3387: 3384: 3383: 3377: 3376: 3369: 3366: 3364: 3361: 3359: 3356: 3354: 3351: 3349: 3346: 3344: 3341: 3339: 3336: 3334: 3331: 3329: 3326: 3324: 3321: 3319: 3316: 3314: 3311: 3309: 3306: 3304: 3301: 3299: 3296: 3294: 3291: 3289: 3286: 3284: 3281: 3280: 3274: 3273: 3266: 3263: 3261: 3258: 3256: 3253: 3251: 3248: 3246: 3243: 3241: 3238: 3236: 3233: 3231: 3228: 3226: 3223: 3221: 3218: 3216: 3213: 3212: 3209: 3204: 3203: 3196: 3193: 3191: 3188: 3186: 3183: 3181: 3178: 3176: 3173: 3171: 3168: 3166: 3163: 3161: 3158: 3156: 3153: 3152: 3146: 3145: 3142: 3139: 3138: 3134: 3129: 3125: 3124: 3121: 3118: 3117: 3109: 3101: 3088: 3063: 3059: 3055: 3052: 3045: 3041: 3035: 3031: 3019: 3015: 3011: 3010:wave function 3006: 3004: 3000: 2996: 2992: 2987: 2985: 2981: 2977: 2972: 2970: 2955: 2947: 2931: 2913: 2909: 2900: 2897:above, using 2892: 2872: 2864: 2858: 2853: 2843: 2830: 2819: 2812: 2806: 2804: 2797: 2793: 2786: 2780: 2765: 2761: 2758: 2752: 2747: 2743: 2739: 2733: 2730: 2722: 2719: 2714: 2709: 2705: 2701: 2697: 2688: 2682: 2673: 2670: 2657: 2651: 2648: 2641: 2636: 2631: 2628: 2623: 2618: 2614: 2608: 2598: 2588: 2585: 2581: 2575: 2570: 2564: 2559: 2553: 2549: 2545: 2536: 2529: 2524: 2518: 2515: 2508: 2499: 2495: 2479: 2473: 2470: 2465: 2459: 2454: 2444: 2437: 2428: 2423: 2418: 2412: 2408: 2404: 2395: 2388: 2383: 2371: 2365: 2348: 2342: 2332: 2322: 2319: 2315: 2310: 2305: 2300: 2294: 2290: 2286: 2277: 2270: 2265: 2259: 2256: 2250: 2245: 2241: 2237: 2228: 2217: 2213: 2209: 2205: 2200: 2198: 2192: 2182: 2180: 2176: 2172: 2167: 2163: 2158: 2155: 2154:Johnson noise 2151: 2147: 2143: 2138: 2136: 2131: 2127: 2123: 2119: 2115: 2110: 2101: 2097: 2095: 2094:Nobel lecture 2091: 2087: 2083: 2079: 2075: 2071: 2067: 2066:hydrogen atom 2063: 2058: 2056: 2050: 2045: 2039: 2034: 2032: 2026: 2021: 2018: 2014: 2010: 2006: 2002: 1993: 1989: 1986: 1982: 1978: 1968: 1966: 1962: 1949: 1934: 1929: 1925: 1920: 1918: 1914: 1910: 1905: 1900: 1896: 1892: 1888: 1884: 1883:Francis Simon 1878: 1873: 1865: 1860: 1855: 1853: 1849: 1845: 1841: 1837: 1833: 1828: 1827:specific heat 1824: 1815: 1811: 1798: 1789: 1786: 1778: 1775: 1768: 1764: 1761: 1757: 1751: 1748: 1742: 1737: 1733: 1730: 1724: 1721: 1693: 1683: 1676: 1674: 1669: 1667: 1659: 1651: 1643: 1623: 1616: 1613: 1605: 1602: 1595: 1591: 1588: 1584: 1578: 1575: 1569: 1566: 1558: 1550: 1543: 1539: 1534: 1525: 1523: 1519: 1513: 1508: 1506: 1502: 1499: 1495: 1491: 1487: 1482: 1479: 1475: 1471: 1467: 1463: 1459: 1455: 1451: 1447: 1437: 1433: 1424: 1410: 1408: 1404: 1400: 1396: 1392: 1388: 1384: 1380: 1376: 1372: 1368: 1364: 1363:Casimir force 1360: 1355: 1353: 1349: 1344: 1340: 1336: 1332: 1326: 1324: 1320: 1316: 1312: 1308: 1304: 1300: 1295: 1290: 1289: 1285: 1279: 1274: 1272: 1271:liquid helium 1268: 1264: 1260: 1256: 1255:wave function 1252: 1243: 1239: 1235: 1230: 1226: 1224: 1220: 1216: 1215:absolute zero 1212: 1208: 1204: 1200: 1196: 1192: 1188: 1179: 1170: 1168: 1164: 1160: 1159:vacuum energy 1156: 1152: 1148: 1144: 1140: 1136: 1132: 1128: 1124: 1120: 1116: 1114: 1108: 1105: 1095: 1093: 1089: 1085: 1081: 1077: 1073: 1069: 1068:supersymmetry 1065: 1061: 1060:fermion field 1057: 1053: 1049: 1045: 1041: 1036: 1034: 1030: 1026: 1022: 1018: 1014: 1010: 1006: 1002: 998: 994: 990: 986: 982: 978: 974: 970: 966: 962: 961:absolute zero 958: 954: 950: 946: 942: 938: 934: 923: 918: 916: 911: 909: 904: 903: 901: 900: 892: 889: 887: 884: 882: 879: 877: 874: 872: 869: 867: 864: 862: 859: 857: 854: 852: 849: 847: 844: 842: 839: 837: 834: 832: 829: 827: 824: 822: 819: 817: 814: 812: 809: 807: 804: 802: 799: 797: 794: 792: 789: 787: 784: 782: 779: 777: 774: 772: 769: 767: 764: 762: 759: 757: 754: 752: 749: 747: 744: 742: 739: 737: 734: 732: 729: 727: 724: 722: 719: 717: 714: 712: 709: 707: 704: 702: 699: 697: 694: 692: 689: 687: 684: 682: 679: 677: 674: 672: 669: 667: 664: 663: 656: 655: 648: 645: 643: 640: 638: 635: 633: 630: 628: 625: 623: 622:Quantum chaos 620: 618: 615: 613: 610: 608: 605: 603: 600: 599: 593: 592: 584: 581: 579: 578:Transactional 576: 574: 571: 569: 568:Quantum logic 566: 564: 561: 559: 556: 550: 547: 546: 545: 542: 541: 540: 537: 535: 532: 530: 527: 525: 522: 520: 517: 515: 512: 511: 507: 502: 501: 493: 490: 488: 485: 483: 480: 478: 475: 473: 470: 469: 462: 461: 453: 450: 448: 445: 443: 440: 438: 435: 433: 430: 428: 425: 424: 420: 417: 416: 410: 409: 401: 398: 396: 393: 391: 388: 387: 381: 378: 377: 376: 373: 372: 368: 365: 363: 360: 358: 355: 353: 350: 348: 345: 343: 340: 338: 335: 333: 330: 328: 325: 323: 320: 319: 312: 311: 301: 298: 297: 296: 295:Wave function 293: 291: 288: 286: 283: 281: 278: 276: 275:Superposition 273: 271: 268: 266: 263: 261: 258: 256: 253: 251: 248: 246: 243: 241: 238: 236: 233: 232: 225: 224: 216: 213: 211: 208: 207: 203: 200: 198: 195: 193: 190: 189: 183: 182: 177: 174: 172: 169: 167: 164: 163: 162: 161: 157: 124: 118: 101: 98: 94: 86: 79: 78: 75: 72: 71: 67: 66: 60: 59:superfluidity 56: 52: 48: 47:Liquid helium 44: 40: 36: 29: 22: 31371: 31359: 31347: 31335: 31253:Pines' demon 30992:Kondo effect 30894:Time crystal 30779: 30708:Quantum foam 30648:Stueckelberg 30602:Chern–Simons 30539:Supergravity 30278:Supergravity 30263:Gauge theory 30172: 29639:Quantum mind 29551:Franck–Hertz 29413:Klein–Gordon 29362:Formulations 29355:Formulations 29284:Interference 29274:Entanglement 29266: 29252:Ground state 29247:Energy level 29220:Fundamentals 29184:Introduction 29047:Astrophysics 28861:Experimental 28721: 28717:Rafelski, J. 28681: 28641: 28578: 28574: 28558:. Retrieved 28551:the original 28525:(3): 39–45. 28522: 28518: 28478: 28474: 28452: 28439: 28424:the original 28419: 28402: 28385: 28368: 28347: 28312: 28287: 28283: 28224: 28220: 28208:. Retrieved 28199: 28163: 28159: 28150: 28146: 28103: 28100:Weinberg, S. 28051: 28047: 28037:Weinberg, S. 27986: 27982: 27939: 27935: 27898: 27894: 27867: 27839: 27803: 27757: 27719: 27681: 27643: 27585: 27581: 27530: 27526: 27483: 27479: 27476:Riess, A. G. 27427: 27423: 27380: 27376:Rees, Martin 27356: 27326: 27323:Power, E. A. 27298: 27277: 27243: 27231:Langevin, P. 27202: 27198: 27182: 27178: 27166: 27162: 27134: 27130: 27101: 27056: 27052: 27020: 26967: 26963: 26903: 26860: 26823: 26819: 26791: 26787: 26731: 26727: 26671: 26667: 26627: 26572: 26568: 26541: 26505: 26465: 26421: 26417: 26390: 26357: 26353: 26294: 26290: 26239: 26235: 26199: 26145: 26141: 26086: 26082: 26070:. Retrieved 26063:the original 26034: 26030: 25983:. Springer. 25979: 25953: 25949: 25921: 25892: 25840: 25836: 25809: 25784: 25780: 25768:. Retrieved 25761:the original 25740: 25736: 25678: 25674: 25633: 25575: 25571: 25539: 25497: 25493: 25435: 25431: 25388: 25384: 25326: 25322: 25300:(10): 2078. 25297: 25293: 25257: 25213: 25209: 25172: 25168: 25164: 25129: 25125: 25117:Bibliography 25103: 25086: 25069: 25050:. Retrieved 25043:the original 25038: 25017:(5): 30–31. 25014: 25010: 24998:The Guardian 24997: 24980: 24959: 24947:the original 24942: 24913: 24902:the original 24898:The Guardian 24897: 24877: 24865:the original 24860: 24843: 24826: 24801: 24776: 24759: 24742: 24716:. Retrieved 24712:the original 24707: 24700:Drake, Nadia 24694: 24682: 24670:. Retrieved 24666: 24656: 24644:. Retrieved 24640: 24630: 24618: 24565: 24561: 24551: 24518: 24514: 24504: 24477: 24471: 24459: 24447:. 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Retrieved 20465: 20455: 20446: 20436: 20424: 20412: 20400: 20388: 20381:Crane (2016) 20376: 20364: 20359:for comment. 20353:Crane (2016) 20339: 20318: 20313:, p. 1. 20306: 20299:Dunne (2012) 20293: 20281: 20269: 20264:, p. 3. 20257: 20245: 20226: 20220: 20201: 20195: 20183: 20171: 20138: 20132: 20126: 20114: 20107:Chown (1990) 20082: 20057: 20053: 20047: 20004: 20000: 19987: 19975: 19963: 19951: 19908: 19904: 19898: 19873: 19869: 19856: 19839: 19835: 19829: 19804: 19800: 19794: 19753: 19749: 19743: 19718: 19714: 19708: 19691: 19687: 19681: 19656: 19652: 19624: 19612: 19600: 19588:. Retrieved 19583: 19571: 19546: 19542: 19536: 19514:Power (1964) 19509: 19484: 19480: 19474: 19462: 19450: 19411: 19405: 19372: 19368: 19358: 19346: 19334: 19309: 19305: 19299: 19264: 19260: 19250: 19209: 19205: 19192: 19180: 19168: 19151: 19147: 19141: 19075: 19071: 19014: 19010: 18954: 18950: 18916: 18912: 18881:(1): 34–40. 18878: 18874: 18836: 18830: 18813: 18809: 18803: 18794: 18790: 18784: 18775: 18771: 18765: 18753:. Retrieved 18733: 18729: 18716: 18686:(3–4): 245. 18683: 18679: 18673: 18640: 18636: 18629: 18617:. Retrieved 18612: 18608: 18595: 18570: 18566: 18560: 18548:. Retrieved 18543: 18530: 18523:Power (1964) 18518: 18485: 18481: 18475: 18470:, p. 6. 18463: 18436: 18430: 18405: 18399: 18386: 18359: 18353: 18342:Lamb, Willis 18336: 18311: 18307: 18301: 18268: 18264: 18258: 18242: 18211: 18205: 18191:(4): 17–35. 18188: 18184: 18178: 18171:Dirac (1927) 18128: 18120: 18087: 18083: 18077: 18050: 18037: 18020: 18016: 18010: 17977: 17973: 17967: 17934: 17926: 17901: 17897: 17891: 17877: 17870: 17845: 17841: 17831: 17806: 17802: 17789: 17764: 17760: 17747: 17740:Kragh (2002) 17735: 17700: 17696: 17685: 17652: 17648: 17642: 17609: 17605: 17599: 17564: 17560: 17550: 17517: 17513: 17507: 17493: 17486: 17477: 17473: 17467: 17458: 17454: 17447: 17425:(1): 49–92. 17422: 17418: 17408: 17396: 17371: 17367: 17357: 17345: 17340:for reprints 17316: 17304: 17292: 17287:, p. 7. 17285:Kragh (2012) 17272:, p. 9. 17265: 17253: 17248:, p. 7. 17241: 17229: 17224:, p. 1. 17217: 17194:the original 17159:for comment. 17143: 17138:for comment. 17099: 17087: 17082:, p. 4. 17075: 17063: 17004: 16992: 16951:Shiga (2005) 16937: 16925: 16867:Vacuum state 16852:Unruh effect 16842:Time crystal 16837:Scalar field 16832:Quantum foam 16807:Ground state 16756: 16730: 16707: 16699: 16679: 16670: 16647: 16642: 16608: 16593:In 1986 the 16592: 16569:GE Aerospace 16566: 16534: 16519: 16510: 16506: 16482: 16473: 16465: 16454: 16443: 16433: 16420: 16398:Nikola Tesla 16391: 16380: 16368: 16331: 16271: 16252:such as the 16227: 16191: 16129: 16118:Chaos theory 15973:Quantum foam 15958:False vacuum 15923:Supergravity 15883:CPT symmetry 15665:Quintessence 15585: 15573: 15558: 15537:baryogenesis 15533: 15495: 15448: 15437:scalar field 15433:quintessence 15423: 15415: 15403: 15335: 15327:neutron star 15303: 15271: 15266:neutron star 15201: 15157: 15149: 15090: 15081: 15063: 15055: 15052: 15048: 14897: 14876: 14864: 14857: 14849: 14826: 14767: 14744: 14738: 14706:faster than 14699: 14695: 14691: 14678: 14655: 14638: 14632: 14619: 14606: 14596: 14575: 14567: 14560: 14542: 14536: 14532: 14527: 14521: 14486:gauge bosons 14479:quark matter 14469:such as the 14460: 14458: 14443: 14416: 14406: 14403: 14399: 14393: 14375: 14361: 14354: 14350: 14343: 14336: 14333: 13644: 13640: 13630: 13627: 13088: 12976: 12643: 12624: 12616: 12613: 12604: 12601: 12597: 12590: 12587: 12583: 12574: 12571: 12162:(0)|vac⟩ = 0 12158: 12155: 12151: 12045: 11967: 11960: 11953: 11948: 11945: 11941: 11938: 11931: 11924: 11920: 11910: 11901: 11892: 11744: 11739: 11735: 11732: 11725: 11720: 11716: 11709: 11702: 11695: 11590: 11583: 11576: 11571: 11567: 11560: 11553: 11550: 11457: 10907: 10904: 10900: 10646: 10520: 10516: 10509: 10505: 10501: 10497: 10493: 10489: 10485: 10478: 10474: 10467: 10460: 10457: 10453: 10447: 10441: 10438: 10288: 10285: 10281: 10278: 10055: 10052: 10049: 9018: 9010: 9007: 8999: 8996: 8992: 8986: 8982: 8843: 8836: 8679: 8670: 8667: 8125: 8117: 8114: 8108: 8101: 8097: 8088: 8085: 8081: 8078: 8069: 8061: 8051: 7921: 7856: 7853: 7523: 7002: 6995: 6988: 6970: 6967: 6960: 6953: 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single 1546: 1515: 1510: 1504: 1483: 1446:the elements 1429: 1356: 1327: 1287: 1283: 1275: 1247: 1184: 1166: 1151:color charge 1130: 1126: 1122: 1118: 1110: 1103: 1101: 1037: 1005:force fields 936: 932: 931: 477:Klein–Gordon 413:Formulations 250:Energy level 245:Entanglement 228:Fundamentals 215:Interference 166:Introduction 55:Lambda point 39: 31290:Soft matter 31190:Ferromagnet 31008:Drude model 30977:Berry phase 30957:Hall effect 30590:Topological 30504:Wess–Zumino 30417:Sine-Gordon 30407:Gross–Neveu 30316:Born–Infeld 30283:Thermal QFT 30078:Self-energy 30068:Landau pole 30032:Positronium 30007:Dual photon 29897:EPR paradox 29677:Quantum bus 29546:Double-slit 29524:Experiments 29490:Many-worlds 29428:Schrödinger 29392:Phase space 29382:Schrödinger 29372:Interaction 29329:Uncertainty 29299:Nonlocality 29294:Measurement 29289:Decoherence 29279:Hamiltonian 28950:Statistical 28866:Theoretical 28843:Engineering 28677:Heitler, W. 28054:(1): 1–23. 27257:unk84021539 25132:(1): 1–13. 24718:22 November 24464:Cook (2002) 22431:22 February 22309:(4): 1700. 22065:Reed (1995) 21928:(1): 9–52. 20848:(1): 8–21. 20690:(1): 1–23. 20369:Rees (2012) 19187:, p. . 18797:(1): 73–83. 18314:(9): 1157. 18023:: 185–193. 17350:Kuhn (1978) 17210:Wall (2014) 17104:Choi (2013) 16643:warp bubble 16604:BAE Systems 16035:Experiments 16008:Causal sets 15853:Graviscalar 15848:Graviphoton 15748:Technicolor 15723:Fifth force 15680:Dark photon 15660:Dark energy 15655:Dark matter 15627:Higgs boson 15523:), why the 15380:Dark energy 15374:Dark energy 15346:astronomers 15319:astronomers 14835:divided by 14808:is roughly 14790:term symbol 14552:electroweak 14533:Mexican-hat 14531:. It has a 14505:Higgs field 14467:condensates 4362:propagators 4345:In quantum 4255:Higgs boson 4173:Zinn-Justin 4023:Sommerfield 3948:Pomeranchuk 3918:Osterwalder 3913:Oppenheimer 3843:Łopuszański 3668:Fredenhagen 3469:Technicolor 2492:making the 2208:Hamiltonian 2126:Dirk Polder 2047: [ 1875: [ 1836:Peter Debye 1666:temperature 1348:dark energy 1251:Hamiltonian 1207:Temperature 1125:. The term 1064:boson field 1052:dark energy 866:von Neumann 851:Schrödinger 627:EPR paradox 558:Many-worlds 492:Schrödinger 447:Schrödinger 442:Phase-space 432:Interaction 337:Double-slit 315:Experiments 290:Uncertainty 260:Nonlocality 255:Measurement 240:Decoherence 210:Hamiltonian 31389:Categories 31205:Spin glass 31200:Metamagnet 31180:Paramagnet 31067:Conduction 31043:BCS theory 30884:Superfluid 30879:Supersolid 30371:Yang–Mills 30144:Lamb shift 30073:QED vacuum 29825:Extensions 29659:Technology 29505:Relational 29456:Copenhagen 29367:Heisenberg 29314:Tunnelling 29177:Background 29067:Geophysics 29057:Biophysics 28901:Analytical 28854:Approaches 28661:2001044743 28210:24 October 28153:(6): 3–39. 28122:2015021123 28076:2152/61094 27400:2011277855 27295:Planck, M. 27273:Planck, M. 27265:5894537227 27235:Solvay, E. 27227:Planck, M. 27191:Planck, M. 27185:: 138–148. 27175:Planck, M. 27169:: 237–245. 27159:Planck, M. 26940:2008942038 26741:1610.08323 26647:2001265846 26623:Loudon, R. 26485:2014938218 26219:2005053026 26072:17 October 25770:17 October 25277:2009279136 25062:Siegel, E. 25052:25 October 24672:24 October 24646:24 October 24126:26 October 23793:26 October 23374:28 October 23221:26 October 23009:1511.06256 22502:1 November 22210:24 October 22067:, p. 21713:3 November 21660:, p. 21647:: 235–257. 21441:3 November 21155:Phys. Lett 20974:3 November 20902:1911.10555 20393:Cho (2016) 20349:Cho (2016) 20060:(2): 154. 19876:(1): 5–8. 19549:(2): 670. 19487:(2): 511. 19234:1911/79433 18755:24 October 18619:19 October 18573:(4): 360. 18550:20 October 18488:(4): 339. 18455:1015092892 18408:(3): 412. 17727:7331244990 17092:Cho (2015) 16879:References 16822:QCD vacuum 16817:QED vacuum 16812:Lamb shift 16746:pilot-wave 16737:Eagleworks 16718:dineutrons 16611:Kip Thorne 16214:quaternion 16198:FitzGerald 16116:See also: 16046:Gran Sasso 15818:Dark fluid 15618:Simulated 15462:) occurs: 15406:supernovae 14806:10 eV 14734:Bohr model 14721:Lamb shift 14715:Lamb shift 14455:QCD vacuum 8055:for large 5183::≡ 5135::≡ 4382:QED vacuum 4317:Paul Dirac 4168:Zimmermann 4063:Vainshtein 3808:Kontsevich 3753:Iliopoulos 3728:Heisenberg 3553:Bogoliubov 3491:Scientists 3343:Propagator 3230:T-symmetry 3225:P-symmetry 3220:C-symmetry 3208:Symmetries 3165:Weak force 3149:Background 2976:degenerate 2086:Hans Bethe 2001:Paul Dirac 1928:Bohr model 1924:Niels Bohr 1823:Otto Stern 1673:Max Planck 1549:Max Planck 1470:philosophy 1367:Lamb shift 1238:oscillator 1167:condensate 1143:QCD vacuum 1135:QED vacuum 1031:theory of 1029:Einstein's 973:the vacuum 861:Sommerfeld 776:Heisenberg 771:Gutzwiller 711:de Broglie 659:Scientists 573:Relational 524:Copenhagen 427:Heisenberg 285:Tunnelling 186:Background 31263:Polariton 31170:Diamagnet 31118:Couplings 31094:Conductor 31089:Semimetal 31074:Insulator 30950:Phenomena 30874:Fermi gas 30780:See also: 30499:Super QCD 30453:Liouville 30441:Conformal 30412:Schwinger 30173:See also: 30112:Processes 30000:Particles 29969:Formalism 29531:Bell test 29401:Equations 29227:Born rule 29017:Molecular 28918:Acoustics 28911:Continuum 28906:Celestial 28896:Newtonian 28883:Classical 28826:Divisions 28752:946050522 28709:924845769 28615:0027-8424 28588:1111.5608 28547:0031-9228 28503:0003-4916 28259:0028-0836 28234:1105.4714 28190:126303009 28130:910664598 28092:122259372 28084:0034-6861 28021:1434-6060 27996:1302.6165 27989:(3): 58. 27974:120884236 27886:444869786 27859:937249213 27830:774073198 27756:(1998c). 27718:(1998b). 27680:(1998a). 27670:774073198 27620:1355-2198 27540:1308.5935 27452:0036-8075 27408:851193468 27343:490279969 27315:603096370 27287:924400975 27193:(1912a). 27122:635667163 27091:118910636 27041:474890537 27010:118961123 27002:0034-6861 26948:297803628 26913:0811.2516 26888:422797902 26848:119991060 26768:0035-8711 26696:0036-8075 26615:118643015 26607:1434-6060 26582:1301.3923 26575:(3): 57. 26560:957316740 26533:803538583 26493:884863929 26456:118593162 26448:0003-9519 26431:1111.4623 26409:248763258 26329:0031-9007 26304:1301.5367 26282:119241509 26274:1550-7998 26249:0905.0031 26182:0035-8711 26129:120354480 26121:1434-6001 26015:317097176 25941:921496342 25912:929349643 25883:119258601 25875:0217-751X 25850:1202.1557 25585:1308.4174 25559:840927124 25470:0031-9007 25285:319209483 25197:0305-4470 25142:CiteSeerX 25104:SPACE.com 24943:SPACE.com 24592:0027-8424 24543:0094-243X 24392:Physica C 24304:115204136 24257:122075917 23830:110771631 23531:121728055 23349:1006.5754 23002:: 23408. 22977:118468331 22952:1303.6558 22816:119262651 22791:0803.4067 22388:0910.5893 22381:(2). 51. 22239:U.S. Army 22096:1004.3835 21969:(1): 68. 21859:The Times 21533:123315936 21428:0182-4295 21361:(9): 70. 21097:118184376 20927:121421318 20858:CiteSeerX 20669:126297065 20605:118779716 20533:0031-9007 20471:23 August 19590:15 August 19442:869069919 19397:122763326 19047:119728862 18855:925046024 18778:: 94–110. 18750:0953-8585 18708:123122363 18665:125644826 18615:: 793–795 18510:120434909 18396:Kusch, P. 18392:Foley, H. 18234:832589969 18153:439849774 18112:120536476 18069:722601833 18043:Mehra, J. 18002:186237037 17959:638472161 17542:121049183 17461:: 83–116. 16724:put in a 16621:that has 16615:wormholes 16466:In 2008, 16306:μ 16292:flux-tube 16194:Heaviside 16187:emergence 16122:Emergence 15968:Spin foam 15863:Gravitino 15577:Schwinger 15555:Cosmology 15541:asymmetry 15521:isotropic 15359:telescope 15177:≈ 15174:α 15136:ℏ 15123:≥ 15114:Δ 15104:Δ 15025:≈ 15016:ℏ 15007:ε 15003:π 14970:ℏ 14951:α 14917:ε 14684:physicist 14428:ω 14289:≈ 14270:ω 14260:˙ 14240:ω 14236:τ 14227:¨ 14194:˙ 14174:ω 14170:− 14167:≈ 14118:ω 14114:− 14111:≈ 14105:¨ 14039:τ 14036:− 14017:ω 14007:¨ 13972:ℏ 13938:ω 13934:τ 13930:π 13901:π 13882:ω 13868:ℏ 13828:ω 13818:τ 13788:∞ 13783:∞ 13780:− 13776:∫ 13761:ω 13741:π 13723:ℏ 13711:≈ 13601:ω 13591:τ 13562:ω 13558:− 13549:ω 13531:ω 13526:ω 13515:∞ 13506:∫ 13492:π 13453:π 13434:ℏ 13394:˙ 13278:˙ 13213:λ 13209:λ 13205:δ 13177:δ 13154:† 13145:λ 13112:λ 13023:τ 13020:− 13001:ω 12991:¨ 12959:ℏ 12920:† 12912:ℏ 12829:† 12764:† 12718:† 12551:ω 12539:ρ 12526:∞ 12517:∫ 12487:ω 12483:ℏ 12480:π 12465:λ 12456:∑ 12449:π 12411:† 12402:λ 12372:λ 12354:× 12333:ω 12329:ℏ 12326:π 12303:ω 12299:ℏ 12296:π 12280:λ 12265:∑ 12259:λ 12250:∑ 12243:π 12185:π 12131:⟩ 12128:Ψ 12094:Ψ 12091:⟨ 12085:⟩ 12061:⟨ 12022:⟩ 12013:ψ 12004:⟩ 11988:⟩ 11985:Ψ 11826:τ 11823:− 11804:ω 11794:¨ 11652:∂ 11642:∂ 11619:− 11610:∇ 11534:¨ 11436:− 11414:ω 11410:⁡ 11380:˙ 11371:⋅ 11366:λ 11326:∫ 11320:λ 11311:∑ 11298:π 11289:− 11240:λ 11211:ω 11188:† 11183:λ 11170:− 11156:ω 11149:− 11130:λ 11099:ω 11095:ℏ 11092:π 11080:λ 11071:∑ 10936:ω 10926:¨ 10876:− 10854:ω 10825:˙ 10816:⋅ 10811:λ 10775:∫ 10758:ω 10754:ℏ 10749:π 10719:ω 10712:− 10693:λ 10666:λ 10592:− 10570:† 10545:˙ 10536:ℏ 10422:λ 10409:⋅ 10403:˙ 10381:ω 10377:ℏ 10372:π 10351:λ 10332:ω 10320:λ 10308:˙ 10259:λ 10230:† 10225:λ 10198:λ 10167:ω 10163:ℏ 10160:π 10148:λ 10139:∑ 10135:≈ 10110:≈ 10087:ω 10077:¨ 10017:∇ 10014:⋅ 10008:˙ 9987:∂ 9977:∂ 9965:˙ 9919:ω 9912:− 9904:× 9898:˙ 9842:ω 9835:− 9827:× 9821:˙ 9784:∇ 9781:⋅ 9775:˙ 9761:− 9755:˙ 9731:− 9715:˙ 9696:− 9690:˙ 9671:¨ 9627:˙ 9607:ω 9600:− 9592:× 9586:˙ 9556:∇ 9553:⋅ 9547:˙ 9512:˙ 9492:ω 9485:− 9462:− 9454:× 9451:∇ 9448:× 9425:− 9402:− 9379:− 9366:∇ 9363:⋅ 9340:− 9312:− 9296:˙ 9276:ω 9269:− 9240:− 9226:∇ 9179:− 9171:ℏ 9152:˙ 9119:− 9071:− 9063:ℏ 9044:˙ 8933:ω 8885:− 8807:ω 8803:− 8789:ω 8761:π 8753:ℏ 8742:ω 8730:ρ 8726:ω 8712:ω 8700:ω 8695:∫ 8635:π 8620:ω 8616:ℏ 8604:ω 8592:ρ 8561:ω 8556:ω 8550:∫ 8528:π 8520:ℏ 8494:ω 8490:ℏ 8453:∫ 8441:π 8432:π 8407:ω 8403:ℏ 8375:∫ 8363:π 8337:ω 8333:ℏ 8316:λ 8307:∑ 8259:∫ 8254:λ 8250:∑ 8237:π 8208:∫ 8192:π 8173:λ 8169:∑ 8165:⟶ 8160:λ 8151:∑ 8016:∫ 7994:π 7936:π 7902:ω 7898:ℏ 7881:λ 7872:∑ 7814:† 7805:λ 7772:† 7767:λ 7722:λ 7689:λ 7661:λ 7654:λ 7650:δ 7620:δ 7593:† 7584:λ 7551:λ 7488:λ 7473:† 7468:λ 7444:ω 7440:ℏ 7435:λ 7426:∑ 7386:λ 7379:λ 7375:δ 7345:δ 7325:∗ 7316:λ 7295:⋅ 7279:λ 7245:∫ 7222:λ 7197:⋅ 7186:− 7167:† 7162:λ 7139:⋅ 7112:λ 7081:ω 7065:ℏ 7062:π 7050:λ 7041:∑ 6904:⋅ 6896:− 6884:ω 6858:† 6853:λ 6827:⋅ 6819:− 6807:ω 6797:− 6778:λ 6747:ω 6731:ℏ 6728:π 6696:λ 6662:λ 6637:⋅ 6626:− 6607:† 6602:λ 6579:⋅ 6552:λ 6521:ω 6505:ℏ 6502:π 6470:λ 6410:λ 6395:⋅ 6377:λ 6336:λ 6119:∫ 6091:⋅ 5950:π 5718:⋅ 5562:∫ 5463:∇ 5392:α 5381:π 5302:∫ 5296:π 5197:† 5189:ω 5186:ℏ 5170:† 5157:† 5141:ω 5138:ℏ 5079:† 5071:ω 5068:ℏ 5055:ω 5052:ℏ 5037:− 5009:† 4996:ω 4993:ℏ 4980:ω 4977:ℏ 4962:− 4949:† 4936:† 4920:ω 4917:ℏ 4859:− 4793:† 4780:ω 4777:ℏ 4754:† 4741:† 4725:ω 4722:ℏ 4679:ω 4535:≠ 4522:† 4452:† 4340:potential 4333:astronomy 4310:cosmology 4251:electrons 4103:Wetterich 4088:Weisskopf 4038:Sudarshan 3988:Schwinger 3903:Nishijima 3868:Maldacena 3833:Leutwyler 3798:Kinoshita 3698:Goldstone 3688:Gell-Mann 3603:Doplicher 3380:Equations 3018:sine wave 2762:ω 2759:ℏ 2720:ℏ 2702:≥ 2692:^ 2629:ℏ 2619:≥ 2602:^ 2546:− 2540:^ 2498:potential 2471:ℏ 2466:≥ 2448:^ 2405:− 2399:^ 2336:^ 2287:− 2281:^ 2232:^ 2080:and also 2062:microwave 1787:− 1765:ν 1752:ν 1734:ν 1722:ε 1650:frequency 1614:− 1592:ν 1579:ν 1567:ε 1547:In 1900, 1498:Maxwell's 1436:Aristotle 1391:emergence 1191:particles 1102:The term 1040:cosmology 969:molecules 953:fluctuate 891:Zeilinger 736:Ehrenfest 465:Equations 142:⟩ 139:Ψ 128:^ 116:⟩ 113:Ψ 90:ℏ 31337:Category 31318:Colloids 30576:Type IIB 30571:Type IIA 30556:4D N = 8 30551:4D N = 1 30520:6D (2,0) 30484:4D N = 1 30463:Polyakov 30422:Thirring 30231:Theories 30046:Concepts 30027:Positron 30012:Electron 29920:Category 29714:Timeline 29466:Ensemble 29446:Bayesian 29339:Collapse 29211:Glossary 29194:Timeline 28744:86175968 28701:83005201 28679:(1984). 28669:48965418 28623:10972781 28457:Archived 28453:Phys.org 28407:Archived 28390:Archived 28373:Archived 28267:22094697 28102:(2015). 28039:(1989). 28029:15753833 27966:12511655 27923:11736390 27822:90048906 27802:(eds.). 27784:40544377 27776:98087896 27746:40544377 27738:98087896 27708:40544377 27700:98087896 27662:90048906 27565:24484127 27518:15640044 27468:40368170 27460:26429882 27381:Universe 27335:65020006 27325:(1964). 27307:59047616 27297:(1958). 27275:(1913). 27241:(eds.). 26898:(2009). 26880:93029780 26858:(1994). 26776:38736536 26712:17747645 26655:44602993 26625:(2000). 26525:77019022 26503:(1978). 26501:Kuhn, T. 26382:10009347 26337:25167393 26227:61200849 26190:11717019 26007:84026824 25721:20243276 25713:12906584 25662:12397205 25654:84005473 25630:(1985). 25620:24492860 25530:17072357 25522:11239149 25478:13943605 25413:32996610 25369:43354557 25361:11801108 25240:27681049 25108:Archived 25091:Archived 25074:Archived 25002:Archived 24985:Archived 24964:Archived 24882:Archived 24848:Archived 24831:Archived 24814:Archived 24789:Archived 24764:Archived 24747:Archived 24667:Wired UK 24610:16578621 24376:24997124 24214:10004334 24171:10013404 23882:10038800 23740:10054182 23697:35509153 23689:11177789 23329:51650940 23164:15382105 23044:27003686 22865:23214736 22475:: 8–17. 22407:17095906 22273:Archived 22121:14593420 21971:Archived 21930:Archived 21889:Archived 21140:10034203 21062:10034484 21019:46623507 20762:13171179 20567:466-470. 20541:14539052 20039:19129843 19943:56132451 19291:10034639 19242:10032058 19118:32579381 19110:10970617 18348:(1947). 18197:20024506 18185:Daedalus 18049:(2002). 17634:98061516 16765:See also 16731:In 2014 16722:dumbbell 16703:toroidal 16680:In 2002 16648:In 1992 16609:In 1988 16573:Dewitt's 16549:toroidal 16535:In 1963 16483:In 1951 16210:Hamilton 16154:solitons 16071:Tevatron 15918:M-theory 15858:Graviton 15788:M-theory 15728:F-theory 15707:Theories 15644:Evidence 14594:, where 14473:and the 13216:′ 13189:′ 13148:′ 13139:′ 12426:⟩ 12405:′ 12396:′ 12358:⟨ 12341:′ 12283:′ 12274:′ 12223:⟩ 12192:⟨ 11432:′ 11394:′ 11347:′ 10872:′ 10838:′ 10796:′ 10514:, where 7808:′ 7800:′ 7725:′ 7717:′ 7664:′ 7638:′ 7587:′ 7579:′ 7389:′ 7363:′ 7319:′ 7311:′ 5114:, i.e.: 4894:⟩ 4863:⟨ 4351:one-loop 4118:Wightman 4083:Weinberg 4073:Virasoro 4053:Tomonaga 4048:Thirring 4043:Symanzik 4003:Semenoff 3978:Schrader 3943:Polyakov 3863:Majorana 3803:Klebanov 3758:Ivanenko 3748:'t Hooft 3718:Guralnik 3663:Fröhlich 3658:Fritzsch 3653:Frampton 3568:Buchholz 3513:Bargmann 3503:Anderson 3303:Crossing 2984:commutes 2698:⟩ 2683:⟨ 2615:⟩ 2576:⟨ 2571:⟩ 2509:⟨ 2460:⟩ 2438:⟨ 2429:⟩ 2384:⟨ 2212:operator 2197:momentum 2140:In 1951 2107:In 1948 2092:'s 1945 1981:Max Born 1975:In 1926 1922:In 1913 1891:hydrogen 1466:atomists 1441:τὸ κενόν 1343:galaxies 1339:Big Bang 1303:fermions 1234:electron 1173:Overview 1054:and the 993:fermions 816:Millikan 741:Einstein 726:Davisson 681:Blackett 666:Aharonov 534:Ensemble 514:Bayesian 419:Overview 300:Collapse 280:Symmetry 171:Glossary 49:retains 31349:Commons 31313:Polymer 31280:Polaron 31258:Plasmon 31238:Exciton 30678:History 30661:Related 30458:Minimal 30309:Regular 29873:Related 29852:History 29591:Science 29423:Rydberg 29189:History 29101:Related 28985:General 28980:Special 28838:Applied 28593:Bibcode 28527:Bibcode 28483:Bibcode 28292:Bibcode 28239:Bibcode 28056:Bibcode 28001:Bibcode 27944:Bibcode 27936:Science 27903:Bibcode 27628:9007190 27600:Bibcode 27573:1826585 27545:Bibcode 27498:Bibcode 27432:Bibcode 27424:Science 27207:Bibcode 27139:Bibcode 27071:Bibcode 26982:Bibcode 26828:Bibcode 26796:Bibcode 26746:Bibcode 26704:2883066 26676:Bibcode 26668:Science 26587:Bibcode 26362:Bibcode 26345:8579347 26309:Bibcode 26254:Bibcode 26160:Bibcode 26101:Bibcode 26059:9910287 26039:Bibcode 25958:Bibcode 25855:Bibcode 25789:Bibcode 25745:Bibcode 25693:Bibcode 25611:3912482 25590:Bibcode 25502:Bibcode 25494:Science 25450:Bibcode 25393:Bibcode 25341:Bibcode 25302:Bibcode 25248:4398296 25218:Bibcode 25177:Bibcode 25134:Bibcode 24918:Bibcode 24802:Science 24777:Science 24570:Bibcode 24523:Bibcode 24449:6 March 24401:Bibcode 24356:Bibcode 24237:Bibcode 24194:Bibcode 24151:Bibcode 24062:Bibcode 24005:Bibcode 23970:4797900 23950:Bibcode 23862:Bibcode 23720:Bibcode 23669:Bibcode 23624:Bibcode 23589:Bibcode 23554:Bibcode 23511:Bibcode 23434:Bibcode 23399:Bibcode 23309:Bibcode 23274:Bibcode 23144:Bibcode 23035:4802220 23014:Bibcode 22957:Bibcode 22888:Bibcode 22843:Bibcode 22796:Bibcode 22751:Bibcode 22692:Bibcode 22620:Bibcode 22585:Bibcode 22550:Bibcode 22542:Physica 22477:Bibcode 22311:Bibcode 22101:Bibcode 22040:Bibcode 22005:Bibcode 21513:Bibcode 21478:Bibcode 21276:: 1106. 21241:Bibcode 21202:Bibcode 21163:Bibcode 21120:Bibcode 21042:Bibcode 20999:Bibcode 20957:Bibcode 20955:: 101. 20907:Bibcode 20850:Bibcode 20742:Bibcode 20692:Bibcode 20649:Bibcode 20513:Bibcode 20163:9909797 20143:Bibcode 20062:Bibcode 20030:4169270 20009:Bibcode 19923:Bibcode 19878:Bibcode 19809:Bibcode 19786:4258508 19778:5675624 19758:Bibcode 19723:Bibcode 19715:Physica 19661:Bibcode 19584:PDGLive 19551:Bibcode 19489:Bibcode 19420:Bibcode 19377:Bibcode 19314:Bibcode 19269:Bibcode 19214:Bibcode 19090:Bibcode 19039:1136482 19019:Bibcode 18959:Bibcode 18921:Bibcode 18883:Bibcode 18688:Bibcode 18645:Bibcode 18575:Bibcode 18490:Bibcode 18410:Bibcode 18364:Bibcode 18316:Bibcode 18293:6780937 18273:Bibcode 18092:Bibcode 18025:Bibcode 17982:Bibcode 17906:Bibcode 17850:Bibcode 17811:Bibcode 17769:Bibcode 17677:4121118 17657:Bibcode 17614:Bibcode 17591:4130047 17569:Bibcode 17522:Bibcode 17427:Bibcode 17376:Bibcode 16690:Seattle 16654:Ning Li 16493:Nyquist 16222:Tesla's 16066:Super-K 15587:Milonni 15545:baryons 15218:⁠ 15206:⁠ 15153:Z boson 15074:⁠ 15060:⁠ 14935:is the 14895:(where 14884:⁠ 14854:⁠ 14847:), and 14827:Taking 14784:⁠ 14772:⁠ 14761:⁠ 14749:⁠ 14626:⁠ 14617:√ 14601:⁠ 14591:⁠ 14584:√ 14572:⁠ 14388:⁠ 14366:⁠ 12638:⁠ 12621:⁠ 11915:⁠ 11897:⁠ 9023:⁠ 9004:⁠ 8139:⁠ 8122:⁠ 5551:⁠ 5535:⁠ 4603:⁠ 4589:⁠ 4301:⁠ 4287:⁠ 4128:Wilczek 4093:Wentzel 4068:Veltman 4013:Shirkov 4008:Shifman 3998:Seiberg 3983:Schwarz 3963:Rubakov 3888:Naimark 3838:Lipatov 3828:Lehmann 3793:Kendall 3683:Gelfand 3678:Glashow 3638:Feynman 3618:Faddeev 3613:Englert 3583:Coleman 3573:Cachazo 3558:Brodsky 3543:Bjorken 3533:Berezin 3523:Belavin 3283:Anomaly 3141:History 3085:is the 2995:entropy 2964:⁠ 2950:⁠ 2940:⁠ 2926:⁠ 2881:⁠ 2867:⁠ 2839:⁠ 2825:⁠ 2801:is the 2789:√ 2150:Nyquist 1952:⁠ 1938:⁠ 1899:mercury 1702:⁠ 1688:⁠ 1656:is the 1648:is the 1640:is the 1476:, that 1413:History 1315:photons 1307:leptons 1149:(e.g., 1013:photons 1011:(e.g., 1003:), and 997:leptons 995:(i.e., 943:that a 856:Simmons 846:Rydberg 811:Moseley 791:Kramers 781:Hilbert 766:Glauber 761:Feynman 746:Everett 716:Compton 487:Rydberg 176:History 31248:Phonon 31243:Magnon 31001:Theory 30859:Plasma 30849:Liquid 30618:Chiral 30566:Type I 30381:Yukawa 30302:Models 30022:Photon 29566:Popper 29012:Atomic 28967:Modern 28817:Major 28750: 28742: 28732: 28707: 28699: 28689: 28667: 28659: 28649: 28621: 28613: 28560:31 May 28545: 28501: 28348:Nature 28275:219735 28273: 28265: 28257: 28221:Nature 28188: 28128: 28120: 28110: 28090: 28082: 28027: 28019: 27972: 27964: 27921: 27884: 27874: 27857: 27847: 27828: 27820: 27810: 27782: 27774: 27764: 27744: 27736: 27726: 27706: 27698: 27688: 27668: 27660: 27650: 27626: 27618: 27571: 27563: 27516: 27466: 27458: 27450: 27406: 27398: 27388: 27364: 27341: 27333: 27313: 27305: 27285: 27263: 27255: 27251:–114. 27120: 27110: 27089: 27039: 27029: 27008: 27000: 26946: 26938: 26928: 26886: 26878: 26868: 26846: 26774: 26766: 26710: 26702: 26694: 26653: 26645: 26635: 26613: 26605: 26558: 26548: 26531: 26523: 26513: 26491: 26483: 26473: 26454: 26446: 26407: 26397: 26380: 26343: 26335: 26327: 26280: 26272: 26225: 26217: 26207: 26188: 26180: 26127: 26119: 26057: 26013: 26005: 25995: 25939: 25929: 25910: 25900: 25881: 25873: 25824: 25719: 25711: 25660: 25652: 25642: 25618: 25608: 25557: 25547: 25528: 25520: 25476: 25468: 25411: 25367: 25359: 25283: 25275: 25265: 25246: 25238: 25210:Nature 25195: 25144: 25070:Forbes 24608: 24601:219778 24598: 24590: 24541: 24492: 24374: 24302: 24255: 24212: 24169: 23968: 23911: 23880: 23828: 23738: 23695: 23687: 23581:Nature 23529: 23327: 23162: 23075: 23042: 23032: 22975: 22863: 22814: 22665: 22405: 22177: 22150: 22119: 21980:22 May 21939:22 May 21898:22 May 21840: 21813: 21786: 21761: 21736: 21684: 21622: 21531: 21426: 21390: 21336: 21309: 21138: 21095: 21060: 21017: 20925: 20860: 20809: 20760: 20667: 20603: 20593: 20539: 20531: 20233: 20208: 20161: 20037: 20027: 20001:Nature 19941: 19784: 19776: 19750:Nature 19440: 19430: 19395: 19289: 19240: 19116: 19108: 19045: 19037: 18853: 18843: 18748: 18706: 18663: 18508: 18453: 18443: 18291: 18232: 18222: 18195: 18151: 18141: 18137:–112. 18110: 18067: 18057: 18000: 17957: 17947: 17725: 17715: 17675: 17649:Nature 17632: 17589: 17561:Nature 17540: 17336:) and 16686:Boeing 16558:Dewitt 16485:Callen 16402:aether 16324:Type I 16180:, the 16176:, the 16150:plasma 16124:, and 15393:vacuum 15077:, and 14610:| 14604:| 14483:vector 12149:since 12038:where 11890:where 11035:where 10451:, and 6931:where 6260:where 6198:where 6014:where 5737:where 5651:| 5637:| 5528:where 4487:| 4481:| 4370:masses 4269:(e.g. 4247:quanta 4243:fields 4235:, and 4183:Zumino 4148:Yukawa 4138:Witten 4133:Wilson 4123:Wigner 4058:Tyutin 4018:Skyrme 3968:Ruelle 3938:Plefka 3933:Peskin 3923:Parisi 3883:Møller 3873:Migdal 3858:Maiani 3853:Lüders 3818:Landau 3813:Kuraev 3788:Källén 3778:Jordan 3763:Jackiw 3703:Gribov 3593:DeWitt 3588:Dashen 3578:Callan 3548:Bleuer 3518:Becchi 3508:Anselm 3081:where 2782:where 2361:where 2206:, the 1796: 1660:, and 1636:where 1505:ad hoc 1490:aether 1460:, and 1432:vacuum 1319:gluons 1313:(e.g. 1311:bosons 1305:(i.e. 1195:energy 1021:aether 1017:gluons 1009:bosons 1001:quarks 989:quanta 985:matter 981:fields 941:energy 886:Zeeman 881:Wigner 831:Planck 801:Landau 786:Jordan 437:Matrix 367:Popper 31223:Anyon 30844:Solid 30757:links 30730:links 30718:links 30638:NMSSM 30623:Fermi 30366:Soler 30336:Proca 29476:Local 29418:Pauli 29408:Dirac 28726:(PDF) 28632:Books 28619:S2CID 28583:arXiv 28554:(PDF) 28515:(PDF) 28271:S2CID 28229:arXiv 28204:(PDF) 28186:S2CID 28143:(PDF) 28088:S2CID 28044:(PDF) 28025:S2CID 27991:arXiv 27970:S2CID 27624:S2CID 27590:arXiv 27569:S2CID 27535:arXiv 27514:S2CID 27488:arXiv 27464:S2CID 27420:(PDF) 27087:S2CID 27061:arXiv 27006:S2CID 26972:arXiv 26908:arXiv 26844:S2CID 26772:S2CID 26736:arXiv 26724:(PDF) 26700:JSTOR 26611:S2CID 26577:arXiv 26452:S2CID 26426:arXiv 26341:S2CID 26299:arXiv 26278:S2CID 26244:arXiv 26186:S2CID 26150:arXiv 26125:S2CID 26091:arXiv 26066:(PDF) 26027:(PDF) 25879:S2CID 25845:arXiv 25764:(PDF) 25733:(PDF) 25717:S2CID 25683:arXiv 25580:arXiv 25526:S2CID 25490:(PDF) 25474:S2CID 25440:arXiv 25428:(PDF) 25409:S2CID 25381:(PDF) 25365:S2CID 25331:arXiv 25244:S2CID 25046:(PDF) 25035:(PDF) 24372:S2CID 24346:arXiv 24334:(PDF) 24300:S2CID 24253:S2CID 24117:(PDF) 24093:arXiv 24052:arXiv 23966:S2CID 23940:arXiv 23848:(PDF) 23826:S2CID 23787:(PDF) 23776:(PDF) 23693:S2CID 23659:arXiv 23527:S2CID 23344:arXiv 23325:S2CID 23262:(PDF) 23238:arXiv 23191:arXiv 23160:S2CID 23134:arXiv 23120:(PDF) 23004:arXiv 22973:S2CID 22947:arXiv 22812:S2CID 22786:arXiv 22496:(PDF) 22465:(PDF) 22403:S2CID 22383:arXiv 22375:Atoms 22350:arXiv 22235:(PDF) 22117:S2CID 22091:arXiv 21974:(PDF) 21959:(PDF) 21933:(PDF) 21918:(PDF) 21892:(PDF) 21877:(PDF) 21570:(PDF) 21529:S2CID 21435:(PDF) 21412:(PDF) 21093:S2CID 21015:S2CID 20968:(PDF) 20945:(PDF) 20923:S2CID 20897:arXiv 20758:S2CID 20732:arXiv 20665:S2CID 20627:1998c 20623:1998b 20619:1998a 20601:S2CID 20537:S2CID 20503:arXiv 20491:(PDF) 19997:(PDF) 19939:S2CID 19913:arXiv 19866:(PDF) 19782:S2CID 19649:(PDF) 19580:(PDF) 19393:S2CID 19202:(PDF) 19114:S2CID 19080:arXiv 19068:(PDF) 19043:S2CID 19007:(PDF) 18726:(PDF) 18704:S2CID 18661:S2CID 18605:(PDF) 18540:(PDF) 18506:S2CID 18289:S2CID 18193:JSTOR 18108:S2CID 17998:S2CID 17799:(PDF) 17757:(PDF) 17699:[ 17673:S2CID 17630:S2CID 17587:S2CID 17538:S2CID 17476:[ 17330:1912b 17326:1912a 16884:Notes 16660:in a 16468:DARPA 16206:Hertz 16202:Lodge 16166:Berry 16041:ANNIE 15908:NMSSM 15581:Jaffe 15355:X-ray 15342:Milan 15331:Earth 14843:(the 14831:(the 14681:Dutch 13651:and: 12040:|vac⟩ 8841:/cm. 4178:Zuber 4028:Stora 3993:Segal 3973:Salam 3958:Proca 3953:Popov 3928:Pauli 3908:Oehme 3898:Neveu 3893:Nambu 3878:Mills 3768:Jaffe 3743:Hagen 3738:Higgs 3713:Gupta 3708:Gross 3693:Glimm 3673:Furry 3643:Fierz 3633:Fermi 3628:Fayet 3623:Fadin 3608:Dyson 3598:Dirac 3563:Brout 3538:Bethe 3498:Adler 3277:Tools 2051:] 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15030:1 15019:c 15011:0 15000:4 14994:2 14989:e 14985:q 14979:= 14973:c 14964:2 14960:e 14954:= 14921:0 14900:e 14898:q 14880:0 14877:ε 14872:/ 14865:e 14858:q 14850:e 14841:c 14829:ħ 14804:× 14781:2 14778:/ 14775:1 14768:P 14758:2 14755:/ 14752:1 14745:S 14708:c 14633:c 14620:λ 14613:/ 14607:μ 14597:v 14586:2 14580:/ 14576:v 14568:ϕ 14566:⟨ 14561:ϕ 14545:ϕ 14528:ϕ 14522:ϕ 14417:ẋ 14412:ω 14407:x 14382:0 14372:/ 14369:1 14362:t 14357:) 14355:t 14353:( 14351:x 14346:) 14344:t 14342:( 14340:0 14337:E 14320:) 14317:t 14314:( 14309:0 14304:E 14297:m 14294:e 14285:x 14279:2 14274:0 14266:+ 14257:x 14249:2 14244:0 14233:+ 14224:x 14191:x 14183:2 14178:0 14161:. 14158:. 14155:. 14151:x 14143:) 14140:t 14137:( 14133:x 14127:2 14122:0 14102:x 14091:) 14088:t 14085:( 14080:0 14075:E 14068:m 14065:e 14060:= 14054:. 14051:. 14048:. 14044:x 14032:x 14026:2 14021:0 14013:+ 14004:x 13969:i 13966:= 13955:) 13947:3 13942:0 13925:( 13920:) 13912:3 13908:c 13904:m 13898:3 13891:3 13886:0 13876:2 13872:e 13865:i 13862:2 13856:( 13852:= 13837:6 13832:0 13822:2 13814:+ 13809:2 13805:x 13799:x 13796:d 13770:3 13765:0 13752:3 13748:c 13744:m 13738:3 13731:2 13727:e 13720:i 13717:2 13703:] 13699:) 13696:t 13693:( 13688:z 13684:p 13680:, 13677:) 13674:t 13671:( 13668:z 13664:[ 13648:0 13645:ω 13641:ω 13634:0 13605:6 13595:2 13587:+ 13582:2 13577:) 13571:2 13566:0 13553:2 13544:( 13535:4 13523:d 13510:0 13501:) 13496:3 13489:8 13483:( 13478:) 13470:3 13466:c 13462:m 13457:2 13449:2 13442:2 13438:e 13431:i 13425:( 13421:= 13410:] 13406:) 13403:t 13400:( 13391:z 13385:m 13382:, 13379:) 13376:t 13373:( 13370:z 13366:[ 13362:= 13351:] 13347:) 13344:t 13341:( 13336:z 13332:A 13326:c 13323:e 13318:, 13315:) 13312:t 13309:( 13306:z 13302:[ 13298:+ 13294:] 13290:) 13287:t 13284:( 13275:z 13269:m 13266:, 13263:) 13260:t 13257:( 13254:z 13250:[ 13246:= 13201:, 13196:3 13186:k 13182:k 13173:= 13169:] 13165:) 13162:0 13159:( 13136:k 13130:a 13126:, 13123:) 13120:0 13117:( 13108:k 13103:a 13098:[ 13075:) 13072:t 13069:( 13064:0 13059:E 13052:m 13049:e 13044:= 13038:. 13035:. 13032:. 13028:x 13016:x 13010:2 13005:0 12997:+ 12988:x 12956:i 12953:= 12943:) 12940:t 12937:( 12934:U 12931:) 12928:t 12925:( 12916:U 12909:i 12906:= 12896:) 12893:t 12890:( 12887:U 12883:] 12879:) 12876:0 12873:( 12868:z 12864:p 12860:, 12857:) 12854:0 12851:( 12848:z 12844:[ 12840:) 12837:t 12834:( 12825:U 12821:= 12810:] 12806:) 12803:t 12800:( 12797:U 12794:) 12791:0 12788:( 12783:z 12779:p 12775:) 12772:t 12769:( 12760:U 12756:, 12753:) 12750:t 12747:( 12744:U 12741:) 12738:0 12735:( 12732:z 12729:) 12726:t 12723:( 12714:U 12709:[ 12705:= 12697:] 12693:) 12690:t 12687:( 12682:z 12678:p 12674:, 12671:) 12668:t 12665:( 12662:z 12658:[ 12635:2 12632:/ 12627:k 12617:λ 12614:k 12611:Σ 12605:λ 12602:k 12598:a 12591:λ 12588:k 12584:a 12577:F 12575:H 12554:) 12548:( 12543:0 12534:w 12531:d 12521:0 12513:= 12502:) 12497:V 12491:k 12477:2 12471:( 12461:k 12446:4 12442:1 12437:= 12422:) 12419:0 12416:( 12393:k 12387:a 12383:) 12380:0 12377:( 12368:k 12363:a 12348:V 12338:k 12323:2 12313:V 12307:k 12293:2 12271:k 12255:k 12240:4 12236:1 12231:= 12219:) 12216:t 12213:( 12208:2 12203:0 12198:E 12182:4 12178:1 12159:λ 12156:k 12152:a 12137:0 12134:= 12124:| 12120:) 12117:t 12114:( 12109:0 12104:E 12098:| 12088:= 12082:) 12079:t 12076:( 12071:0 12066:E 12051:⟩ 12048:D 12046:ψ 12044:| 12026:, 12017:D 12008:| 11995:| 11991:= 11981:| 11968:t 11961:t 11956:) 11954:t 11952:( 11949:λ 11946:k 11942:a 11934:) 11932:t 11930:( 11928:0 11925:E 11909:3 11906:/ 11902:e 11900:2 11893:τ 11878:) 11875:t 11872:( 11867:0 11862:E 11855:m 11852:e 11847:= 11841:. 11838:. 11835:. 11831:x 11819:x 11813:2 11808:0 11800:+ 11791:x 11768:) 11765:t 11762:( 11758:x 11747:) 11745:t 11743:( 11736:E 11728:) 11726:t 11724:( 11717:E 11710:t 11705:) 11703:t 11701:( 11699:0 11696:E 11681:0 11678:= 11674:E 11669:] 11660:2 11656:t 11646:2 11632:2 11628:c 11624:1 11614:2 11605:[ 11593:) 11591:t 11589:( 11587:0 11584:E 11579:) 11577:t 11575:( 11568:E 11563:) 11561:t 11559:( 11557:0 11554:E 11531:x 11520:3 11516:c 11512:3 11507:e 11504:2 11498:= 11495:) 11492:t 11489:( 11484:R 11481:R 11476:E 11464:m 11443:) 11439:t 11429:t 11424:( 11418:k 11403:] 11398:) 11391:t 11387:( 11377:x 11362:k 11357:e 11352:[ 11344:t 11340:d 11335:t 11330:0 11316:k 11305:V 11301:e 11295:4 11286:= 11283:) 11280:t 11277:( 11272:R 11269:R 11264:E 11236:k 11231:e 11226:] 11220:t 11215:k 11207:i 11203:e 11199:) 11196:0 11193:( 11179:k 11174:a 11165:t 11160:k 11152:i 11145:e 11141:) 11138:0 11135:( 11126:k 11121:a 11116:[ 11109:V 11103:k 11089:2 11076:k 11067:i 11064:= 11061:) 11058:t 11055:( 11050:0 11045:E 11023:) 11020:t 11017:( 11012:R 11009:R 11004:E 10997:m 10994:e 10989:+ 10986:) 10983:t 10980:( 10975:0 10970:E 10963:m 10960:e 10955:= 10951:x 10945:2 10940:0 10932:+ 10923:x 10908:λ 10905:k 10901:ȧ 10883:) 10879:t 10869:t 10864:( 10858:k 10850:i 10846:e 10842:) 10835:t 10831:( 10822:x 10807:k 10802:e 10793:t 10789:d 10784:t 10779:0 10767:V 10762:k 10746:2 10739:e 10736:i 10733:+ 10728:t 10723:k 10715:i 10708:e 10704:) 10701:0 10698:( 10689:k 10684:a 10680:= 10677:) 10674:t 10671:( 10662:k 10657:a 10633:. 10629:1 10626:= 10623:) 10620:0 10617:( 10614:U 10610:, 10606:) 10603:t 10600:( 10595:1 10588:U 10584:= 10581:) 10578:t 10575:( 10566:U 10561:, 10557:U 10554:H 10551:= 10542:U 10533:i 10523:) 10521:t 10519:( 10517:U 10512:) 10510:t 10508:( 10506:U 10502:A 10500:) 10498:t 10496:( 10494:U 10490:t 10488:( 10486:A 10481:) 10479:t 10477:( 10475:A 10468:t 10461:λ 10458:k 10454:a 10448:p 10442:x 10418:k 10413:e 10400:x 10390:V 10385:k 10369:2 10362:e 10359:i 10356:+ 10347:k 10342:a 10336:k 10328:i 10325:= 10316:k 10305:a 10289:λ 10286:k 10282:a 10255:k 10250:e 10245:] 10241:) 10238:t 10235:( 10221:k 10216:a 10212:+ 10209:) 10206:t 10203:( 10194:k 10189:a 10184:[ 10177:V 10171:k 10157:2 10144:k 10124:E 10118:m 10115:e 10102:x 10096:2 10091:0 10083:+ 10074:x 10056:ẍ 10053:m 10036:. 10030:3 10025:A 10020:) 10005:x 9999:( 9996:+ 9990:t 9981:A 9971:= 9962:A 9934:x 9928:2 9923:0 9915:m 9908:B 9895:x 9887:c 9884:e 9879:+ 9875:E 9871:e 9868:= 9857:x 9851:2 9846:0 9838:m 9831:B 9818:x 9810:c 9807:e 9802:+ 9798:] 9793:A 9788:) 9772:x 9765:( 9752:A 9745:[ 9739:c 9736:e 9728:= 9712:A 9704:c 9701:e 9687:p 9681:= 9668:x 9662:m 9624:x 9616:2 9611:0 9603:m 9596:B 9583:x 9575:c 9572:e 9567:+ 9563:A 9559:) 9544:x 9538:( 9533:c 9530:e 9525:= 9509:x 9501:2 9496:0 9488:m 9481:] 9476:A 9470:c 9467:e 9458:[ 9444:) 9439:A 9433:c 9430:e 9421:p 9416:( 9410:m 9407:1 9398:] 9393:A 9387:c 9384:e 9375:[ 9370:] 9359:) 9354:A 9348:c 9345:e 9336:p 9331:( 9326:[ 9320:m 9317:1 9309:= 9293:x 9285:2 9280:0 9272:m 9264:2 9259:) 9254:A 9248:c 9245:e 9236:p 9231:( 9220:2 9217:1 9211:= 9201:] 9198:H 9195:. 9191:p 9187:[ 9182:1 9175:) 9168:i 9165:( 9162:= 9149:p 9138:) 9133:A 9127:c 9124:e 9115:p 9110:( 9104:m 9101:1 9096:= 9093:] 9090:H 9087:. 9083:x 9079:[ 9074:1 9067:) 9060:i 9057:( 9054:= 9041:x 9019:c 9015:/ 9011:A 9008:e 9002:+ 9000:ẋ 8997:m 8993:p 8987:x 8967:F 8963:H 8959:+ 8954:2 8949:x 8942:2 8937:0 8929:m 8923:2 8920:1 8914:+ 8909:2 8904:) 8899:A 8893:c 8890:e 8881:p 8876:( 8868:m 8865:2 8861:1 8856:= 8853:H 8822:) 8816:4 8811:1 8798:4 8793:2 8784:( 8775:3 8771:c 8765:2 8757:8 8748:= 8745:) 8739:( 8734:0 8723:d 8716:2 8704:1 8683:2 8680:ω 8674:1 8671:ω 8649:3 8645:c 8639:2 8631:2 8624:3 8610:= 8607:) 8601:( 8596:0 8565:3 8553:d 8542:3 8538:c 8532:2 8524:2 8515:= 8504:) 8498:k 8484:2 8481:1 8474:( 8468:2 8464:k 8459:k 8456:d 8445:3 8437:4 8429:4 8423:= 8411:k 8397:2 8394:1 8388:k 8383:3 8379:d 8367:3 8359:8 8355:2 8350:= 8341:k 8327:2 8324:1 8312:k 8301:V 8298:1 8272:k 8267:3 8263:d 8241:3 8233:8 8229:V 8224:= 8221:k 8216:3 8212:d 8203:3 8198:) 8189:2 8185:1 8180:( 8156:k 8136:2 8133:/ 8128:k 8118:λ 8115:k 8112:Σ 8104:) 8102:λ 8098:k 8096:( 8089:λ 8086:k 8082:n 8064:V 8057:v 8052:v 8047:v 8033:v 8030:d 8024:3 8020:v 8009:3 8005:c 8000:V 7997:h 7991:4 7968:V 7961:3 7957:c 7952:v 7949:d 7944:2 7940:v 7933:8 7906:k 7892:2 7889:1 7877:k 7859:F 7857:H 7836:0 7833:= 7829:] 7825:) 7822:t 7819:( 7796:k 7790:a 7786:, 7783:) 7780:t 7777:( 7763:k 7758:a 7753:[ 7749:= 7741:] 7737:) 7734:t 7731:( 7713:k 7707:a 7703:, 7700:) 7697:t 7694:( 7685:k 7680:a 7675:[ 7657:, 7644:3 7634:k 7629:, 7625:k 7616:= 7608:] 7604:) 7601:t 7598:( 7575:k 7569:a 7565:, 7562:) 7559:t 7556:( 7547:k 7542:a 7537:[ 7509:) 7502:2 7499:1 7493:+ 7484:k 7479:a 7464:k 7459:a 7454:( 7448:k 7431:k 7422:= 7417:F 7413:H 7382:, 7369:3 7359:k 7354:, 7350:k 7341:= 7338:) 7334:r 7330:( 7307:k 7300:A 7292:) 7288:r 7284:( 7275:k 7269:A 7264:r 7259:3 7255:d 7249:V 7218:k 7213:e 7208:] 7201:r 7193:k 7189:i 7182:e 7178:) 7175:0 7172:( 7158:k 7153:a 7149:+ 7143:r 7135:k 7131:i 7127:e 7123:) 7120:0 7117:( 7108:k 7103:a 7098:[ 7090:V 7085:k 7073:2 7069:c 7059:2 7046:k 7037:= 7034:) 7031:t 7027:r 7023:( 7019:A 7008:) 7005:z 7003:k 6998:y 6996:k 6991:x 6989:k 6987:( 6983:λ 6979:k 6971:λ 6968:k 6961:a 6954:λ 6951:k 6947:a 6936:k 6934:ω 6918:] 6912:) 6908:r 6900:k 6893:t 6888:k 6880:( 6877:i 6873:e 6869:) 6866:0 6863:( 6849:k 6844:a 6840:+ 6835:) 6831:r 6823:k 6816:t 6811:k 6803:( 6800:i 6793:e 6789:) 6786:0 6783:( 6774:k 6769:a 6764:[ 6756:V 6751:k 6739:2 6735:c 6725:2 6718:= 6715:) 6712:t 6709:, 6705:r 6701:( 6692:k 6686:A 6658:k 6653:e 6648:] 6641:r 6633:k 6629:i 6622:e 6618:) 6615:0 6612:( 6598:k 6593:a 6589:+ 6583:r 6575:k 6571:i 6567:e 6563:) 6560:0 6557:( 6548:k 6543:a 6538:[ 6530:V 6525:k 6513:2 6509:c 6499:2 6492:= 6489:) 6486:t 6483:, 6479:r 6475:( 6466:k 6460:A 6431:2 6424:1 6418:{ 6413:= 6406:, 6399:r 6391:k 6387:i 6383:e 6373:k 6368:e 6361:V 6357:1 6352:= 6349:) 6345:r 6341:( 6332:k 6326:A 6310:2 6308:k 6301:e 6295:1 6293:k 6286:e 6279:2 6277:k 6273:e 6269:1 6267:k 6263:e 6257:2 6255:k 6251:e 6245:1 6243:k 6239:e 6232:k 6228:e 6220:k 6216:e 6212:k 6205:k 6201:e 6186:1 6183:= 6178:2 6173:| 6169:) 6165:r 6161:( 6155:k 6149:A 6143:| 6138:r 6133:3 6129:d 6123:V 6095:r 6087:k 6083:i 6079:e 6072:k 6067:e 6060:V 6056:1 6051:= 6048:) 6044:r 6040:( 6034:k 6028:A 6016:n 6001:) 5995:z 5991:n 5987:, 5982:y 5978:n 5974:, 5969:x 5965:n 5960:( 5954:L 5947:2 5941:= 5937:) 5931:z 5927:k 5923:, 5918:y 5914:k 5910:, 5905:x 5901:k 5896:( 5875:) 5872:t 5869:, 5866:z 5863:, 5860:y 5857:, 5854:x 5851:( 5847:A 5843:= 5840:) 5837:t 5834:, 5831:L 5828:+ 5825:z 5822:, 5819:L 5816:+ 5813:y 5810:, 5807:L 5804:+ 5801:x 5798:( 5794:A 5782:L 5778:V 5770:) 5768:t 5766:, 5764:r 5762:( 5760:A 5756:∇ 5748:k 5744:e 5740:k 5722:r 5714:k 5710:i 5706:e 5699:k 5694:e 5690:= 5687:) 5683:r 5679:( 5674:0 5669:A 5656:r 5649:) 5647:r 5645:( 5643:0 5640:A 5621:1 5618:= 5613:2 5608:| 5604:) 5600:r 5596:( 5591:0 5586:A 5580:| 5575:r 5570:3 5566:d 5547:c 5543:/ 5539:ω 5531:k 5516:0 5513:= 5510:) 5506:r 5502:( 5497:0 5492:A 5486:) 5480:2 5476:k 5472:+ 5467:2 5458:( 5443:) 5441:r 5439:( 5437:0 5434:A 5411:2 5406:| 5401:) 5398:t 5395:( 5388:| 5378:2 5372:2 5368:k 5362:= 5351:) 5345:2 5340:B 5335:+ 5330:2 5325:E 5319:( 5315:r 5310:3 5306:d 5293:8 5289:1 5284:= 5275:F 5271:H 5243:a 5238:a 5233:a 5228:a 5223:a 5218:a 5202:a 5193:a 5179:) 5175:a 5166:a 5162:+ 5153:a 5149:a 5145:( 5130:F 5126:H 5122:: 5110:F 5108:H 5106:: 5084:a 5075:a 5065:= 5046:2 5043:1 5033:) 5026:2 5023:1 5017:+ 5014:a 5005:a 5000:( 4990:= 4971:2 4968:1 4958:) 4954:a 4945:a 4941:+ 4932:a 4928:a 4924:( 4911:2 4908:1 4902:= 4890:0 4886:| 4880:F 4876:H 4871:| 4867:0 4854:F 4850:H 4817:) 4810:2 4807:1 4801:+ 4798:a 4789:a 4784:( 4774:= 4763:) 4759:a 4750:a 4746:+ 4737:a 4733:a 4729:( 4716:2 4713:1 4707:= 4695:2 4690:q 4683:2 4675:m 4669:2 4666:1 4660:+ 4654:m 4651:2 4645:2 4641:p 4635:= 4626:l 4623:c 4619:H 4600:2 4597:/ 4583:a 4578:a 4571:a 4566:a 4561:a 4556:a 4553:a 4538:1 4531:] 4527:a 4518:a 4514:, 4511:a 4507:[ 4495:a 4492:a 4484:α 4465:1 4462:= 4458:] 4448:a 4444:, 4441:a 4437:[ 4425:a 4420:a 4416:* 4414:α 4409:α 4298:2 4295:/ 4283:E 4216:e 4209:t 4202:v 3106:L 3100:n 3095:n 3091:m 3083:h 3064:2 3060:L 3056:m 3053:8 3046:2 3042:n 3036:2 3032:h 2961:2 2958:/ 2946:E 2934:/ 2930:h 2921:ħ 2917:h 2908:ω 2903:ω 2895:ν 2878:2 2875:/ 2863:E 2836:2 2833:/ 2821:0 2818:V 2814:0 2811:E 2796:m 2794:/ 2792:k 2785:ω 2766:2 2753:+ 2748:0 2744:V 2740:= 2734:m 2731:k 2723:2 2715:+ 2710:0 2706:V 2689:H 2658:. 2652:m 2649:k 2642:2 2637:) 2632:4 2624:( 2609:2 2599:p 2589:m 2586:2 2582:1 2565:2 2560:) 2554:0 2550:x 2537:x 2530:( 2525:k 2519:2 2516:1 2480:, 2474:2 2455:2 2445:p 2424:2 2419:) 2413:0 2409:x 2396:x 2389:( 2367:0 2364:V 2349:, 2343:2 2333:p 2323:m 2320:2 2316:1 2311:+ 2306:2 2301:) 2295:0 2291:x 2278:x 2271:( 2266:k 2260:2 2257:1 2251:+ 2246:0 2242:V 2238:= 2229:H 2169:( 1957:r 1948:r 1944:/ 1941:1 1799:. 1790:1 1782:) 1779:T 1776:k 1773:( 1769:/ 1762:h 1758:e 1749:h 1743:+ 1738:2 1731:h 1725:= 1711:h 1707:ν 1699:2 1696:/ 1662:T 1654:k 1646:ν 1638:h 1624:, 1617:1 1609:) 1606:T 1603:k 1600:( 1596:/ 1589:h 1585:e 1576:h 1570:= 1553:ε 1284:E 1244:. 1129:( 935:( 921:e 914:t 907:v 135:| 125:H 119:= 109:| 102:t 99:d 95:d 87:i 61:. 37:. 30:. 23:.

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