Quantum field theory

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considers, such as Hilbert spaces (e.g. the space of square integrable real valued functions) or separable Banach spaces (e.g. the space of continuous real-valued functions on a compact interval, with the uniform convergence norm), have denumerable (i. e. countably infinite) dimension in the category of Banach spaces (though still their Euclidean vector space dimension is uncountable), so in these restricted contexts, the number of degrees of freedom (interpreted now as the vector space dimension of a dense subspace rather than the vector space dimension of the function space of interest itself) is denumerable.

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other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result... which was in agreement with Americans'.

10829: 8516:{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.} 5039: 22029: 20822: 2134:", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong interaction possible. 14487:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\pi ^{k}\right)\left(\partial ^{\mu }\pi ^{k}\right)+{\frac {1}{2}}\left(\partial _{\mu }\sigma \right)\left(\partial ^{\mu }\sigma \right)-{\frac {1}{2}}\left(2\mu ^{2}\right)\sigma ^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt {\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac {\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac {\lambda }{4}}\left(\pi ^{k}\pi ^{k}\right)^{2},} 9480:{\displaystyle {\begin{aligned}\langle 0|T\{\phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle &=\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle \langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle \\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle \langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle .\end{aligned}}} 9844: 3346: 7773: 10506: 22041: 19376: 6431: 5688:. However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, 1268: 15462:) can form a link configuration in the spacetime, which relates the braiding statistics of anyons in physics to the link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond. 4774: 13699: 1315:, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact". It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a numerical quantity (a 3108: 7323: 8861: 7414: 10824:{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda _{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta _{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac {1}{2}}\delta _{m}\phi _{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi _{r}^{4},} 1491:, where an electron spontaneously decreases in energy and emits a photon even without the action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an ordinary number while promoting spatial coordinates to 11539:. The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them. According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off 6102: 1669:; the reverse process, annihilation, could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the 1476:, were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred. It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations. 5655: 11776: 5034:{\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).} 15012: 11432: 1828:. The latter can be used to visually and intuitively organize and to help compute terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the 15209: 5843: 12663: 2977: 10392:

are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical

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The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite) number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this

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discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now

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It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal

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On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the

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photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon.

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In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general. Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with

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and the vacuum zero-point energy of the electron and photon fields, suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta. It was not until 20 years later that a systematic approach to remove such infinities was

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In fact, its number of degrees of freedom is uncountable, because the vector space dimension of the space of continuous (differentiable, real analytic) functions on even a finite dimensional Euclidean space is uncountable. On the other hand, subspaces (of these function spaces) that one typically

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Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as

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Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by gravity, and the perihelion precession of Mercury. The neglect of source theory by the physics community

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Since those parts of the modified mass and charge due to field reactions , it is impossible to calculate them by the theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the

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of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for

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carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole

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are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of

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The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of

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Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory. This makes the Feynman propagator one of the most important quantities in

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Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of

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coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of

15369: 3847: 13511: 3341:{\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),} 14791:

Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.

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approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory, can be given a sound mathematical interpretation from their finite-dimensional analogues.

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field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:

2284:. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems. 12165: 13470:

procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally. A more rigorous generalization of the Faddeev–Popov procedure is given by

11225: 7060: 7768:{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\left\langle 0\left|T\left\{\phi _{I}(x)\phi _{I}(y)\exp \left\right\}\right|0\right\rangle }{\left\langle 0\left|T\left\{\exp \left\right\}\right|0\right\rangle }},} 15607:, and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions, the three-dimensional scalar field theories with a quartic interaction, etc. 10354:

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The

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for relativistic Compton scattering. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.

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While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a

12939: 11938:. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories. 6426:{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .} 14129: 1278:. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated. 10983: 1616:(such as photons). Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the 3619: 14784:. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of 13201: 7400:

that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the

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can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.

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The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the

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is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods.

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that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.

3078: 15066: 9833: 5936: 5705: 1205:, a function of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by a 12485: 5418: 13982: 5302: 2757: 13307:

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called

11072:, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales. The way in which each parameter changes with scale is described by its 11041:

It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The

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Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory, but in 1951 he found a way around the problem of the infinities with a new method using

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external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the

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again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could not be generalized to other calculations.

15235: 11917: 13694:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\partial _{\mu }\phi ^{i}\right)\left(\partial ^{\mu }\phi ^{i}\right)+{\frac {1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda }{4}}\left(\phi ^{i}\phi ^{i}\right)^{2},} 11535:

particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable

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within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924,

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of the path integral may change. For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group

3455: 7927: 13892: 13807: 7318:{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \equiv D_{F}(x-y)=\lim _{\epsilon \to 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.} 5665:, which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called 1645:, but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom. 11980: 11105: 2679: 12759: 9505:

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a

8856:{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon )}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi (y)\exp \left}{\int {\mathcal {D}}\phi \,\exp \left}},} 6696:

In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.

1856:, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities. 15478:

participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a

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amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.

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Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.

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Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between

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quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular,

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implies that the observed elementary charge increases as the scale increases. The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.

9528: 1239:. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the 15632:

is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include

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in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state (the

4688: 3508: 10474: 10336:-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction. Lines whose end points are vertices can be thought of as the propagation of 10033: 5650:{\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .} 4760: 4199: 13439:

Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field

13392: 12274: 10175:, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space. 4530: 4362: 3933: 12978: 11771:{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },} 1708:. Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of 10146: 6055: 6013: 11479:

decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as

1890:, while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations. 15716: 6993: 4466: 15007:{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =\eta ^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,} 12389: 12230: 11427:{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda }}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi ^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right)+O{\mathord {\left(g^{5}\right)}},} 13359: 8888: 4650: 2735: 2706: 2250:

Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to

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These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the

15587:, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called 4030: 1874:, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the 17992:

Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (2017). "Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions".

15912: 7021: 5449: 5343: 4494: 4150: 3894: 7830: 1409:(the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles. 7947: 7398: 7362: 4015: 15204:{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu }A^{\mu }A^{\nu },\quad \partial _{\mu }\phi \partial ^{\mu }\phi =g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi ,} 2432:), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields. 1503:

Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.

1433:, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, 14533:

states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example,

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In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to

12339: 10090: 10063: 9984: 9952: 6965: 6938: 5838:{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},} 2992: 12658:{\displaystyle {\mathcal {L}}=i{\bar {\psi }}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac {1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar {\psi }}^{i}\psi ^{i},} 9697: 5857: 2972:{\displaystyle {\frac {\partial }{\partial t}}\left+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,} 10173: 9922: 9896: 9876: 6907: 6883: 6863: 5351: 2502: 13909: 5050: 15428:(TQFTs), all correlation functions are independent of continuous changes in the spacetime metric. QFTs in curved spacetime generally change according to the 1323:) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a mathematical trick. 3495:

The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a

10208: 21659: 4538: 20066: 15364:{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac {1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla _{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),} 6079:

small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let

2002:. In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of " 15487:

between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the

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is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.

2200: 11846: 3842:{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.} 1903:

as currents coupled to gauge fields. Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the

19462: 2417:. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many 1848:

is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the

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Notes on Recent Researches in Electricity and Magnetism: Intended as a Sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism'

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of radiation from atoms, where an electron emits a new photon under the action of an external electromagnetic field, but it was unable to explain

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Aharony, Ofer; Gur-Ari, Guy; Klinghoffer, Nizan (2015-05-19). "The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants".

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Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of

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The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and

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The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group. Examples of such theories include:

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supersymmetry, and so on. Supersymmetry can also be constructed in other dimensions, most notably in (1+1) dimensions for its application in

2182: 1366:, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the 14768:

of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the

12160:{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi (x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha (x)}\partial _{\mu }e^{i\alpha (x)},} 1835:

It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.

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Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing

5672:

The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields,

15401:. The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background. 11220:{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda }}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi ^{2}}}+O{\mathord {\left(e^{5}\right)}},} 21626: 19777: 19736: 17843: 15922: 14741: 13449:, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the 11506: 7040: 6706: 200: 1518:. With the exclusion of interactions, however, such a theory was yet incapable of making quantitative predictions about the real world. 1393:

with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as

18427: 2510: 2276:

Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first

2084:'s proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to 1620:

of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan,

20790: 11034:

such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit

3642:

is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label

22045: 20858: 10393:

quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off

6838:{\displaystyle \langle 0|T\{\phi (x)\phi (y)\}|0\rangle \quad {\text{or}}\quad \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle } 240: 12687: 11949:

Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an

21788: 20999: 20116: 19515: 15877: 15798: 15049: 11954:

Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.

5695:

The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a

1529:(QED), a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric 1298: 6622: 6584:{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp \left\{i\int _{0}^{T}dt\,L\right\},} 1817:. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities". 21652: 20802: 15541: 15471: 5689: 1538: 1210: 11794: 1904: 21505: 20964: 20486: 20059: 19540: 18528: 18485: 17941: 15833: 14757:

If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called

9681:{\displaystyle {\frac {-i\lambda }{4!}}\int d^{4}z\,\langle 0|T\{\phi (x)\phi (y)\phi (z)\phi (z)\phi (z)\phi (z)\}|0\rangle .} 1684:

showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron

17818:

Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008-12-11). "Undergraduate Lecture Notes in Topological Quantum Field Theory".

1716:

the energy of an atom) in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945,

21906: 20420: 19875: 19644: 19380: 19328: 19301: 19153: 19125: 19099: 19029: 19001: 18975: 18936: 18912: 18884: 18858: 18824: 18758: 18728: 18698: 18672: 18653: 18370: 18278: 18151: 18058: 17802: 17769: 17654: 17605: 17413: 17326: 17115: 16907: 16846: 16776: 16751: 16406: 16359: 16246: 16093: 15659: 14737: 14615:, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the 2424:

Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the

15563:

provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of

12934:{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c},} 1574:. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations. 20343: 19767: 19455: 19197: 12427:, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on 5454: 4693: 4243: 1510:

in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via

975: 21989: 19505: 19407: 15945: 15858: 15474:, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of 12193:

at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations

11555:

The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and

15496:

without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as

14124:{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots ,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda }}}+\sigma (x)\right),} 2442:

are two common formulations of QFT. To motivate the fundamentals of QFT, an overview of classical field theory follows.

21645: 19623: 19358: 19268: 19238: 19172: 19050: 18923: 18794: 18593: 15818: 15425: 14845: 1175: 1110: 10978:{\displaystyle \delta _{Z}=Z-1,\quad \delta _{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda }=\lambda Z^{2}-\lambda _{r}} 21813: 21300: 21273: 20089: 20052: 20035: 18957: 18623: 18337: 17871: 16726: 15808: 15793: 15588: 15497: 8908:-point correlation function in the free theory to a sum of products of two-point correlation functions. For example, 4390: 2166: 2051: 13233: 21994: 20989: 20886: 20199: 19965: 19772: 19715: 19535: 18816: 17155: 15935: 15611: 15410: 1907:

of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his

1773:

The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by

1385:'s study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit 1316: 1251:, to the point where some theorists called for the abandonment of the field theoretic approach. The development of 745: 15763: 4207: 3614:{\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},} 1665:. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called 20757: 20466: 20461: 20184: 19925: 19828: 19808: 19568: 19448: 17670:

Nath, P.; Arnowitt, R. (1975). "Generalized Super-Gauge Symmetry as a New Framework for Unified Gauge Theories".

15641: 15450:

is an example of TQFT and has been used to construct models of quantum gravity. Applications of TQFT include the

8526: 6086: 4021: 1222: 43: 5976:

of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the

21954: 21453: 21179: 21047: 20769: 20441: 19731: 19490: 19485: 18171: 15848: 15788: 15629: 15451: 12972:

are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:

11920: 11608: 4655: 2418: 1534: 1373:

Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in

11079:. Correlation functions, which underlie quantitative physical predictions, change with scale according to the 2149:, and its many predictions have been met with remarkable experimental confirmation in subsequent decades. The 2016:

developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964,

22077: 22067: 21845: 21793: 21062: 20974: 20936: 20851: 20747: 20524: 20446: 20229: 20099: 19399: 15927: 15583:

Since the 1950s, theoretical physicists and mathematicians have attempted to organize all QFTs into a set of

13489: 13484: 13424:

the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding

13196:{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi ^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\leftU^{\dagger }(x),} 10425: 2058: 1103: 308: 157: 117: 13410:, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the 4736: 4158: 2064:

By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking,

1864:, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the 22004: 21293: 21067: 20481: 20415: 20410: 20381: 20094: 20004: 19880: 19530: 19071: 15917: 15619: 15455: 14857: 14822: 14527:. The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken. 11080: 9989: 4499: 4331: 3902: 2269:

theory to elementary particles, while the concept of renormalization came out of the study of second-order

2170: 1911:

in 1966 then expanded the theory's applications to quantum electrodynamics in his three volume-set titled:

1704:

and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called

1629:

be possible to create material particles. Building on this idea, Fermi proposed in 1932 an explanation for

1311:"—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with 845: 775: 147: 19389: 18184:

Nguyen, Timothy (2016). "The perturbative approach to path integrals: A succinct mathematical treatment".

17544:

Brading, Katherine A. (March 2002). "Which symmetry? Noether, Weyl, and conservation of electric charge".

14810:(3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT 13367: 21590: 21145: 20946: 20549: 20456: 19999: 19793: 19394: 16116: 15940: 15600: 13453:. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing 3496: 2748: 1515: 1397:. This process of restricting energies to discrete values is called quantization. Building on this idea, 1394: 303: 245: 17546:

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

10095: 8890:

is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in

6025: 5983: 2024:

arrived at the same theory through a different path. This theory, nevertheless, was non-renormalizable.

22072: 21421: 21394: 21042: 20825: 20587: 20395: 20366: 19573: 19350: 19320: 19293: 19260: 19230: 19189: 19021: 18615: 18434: 15950: 15853: 12785: 11543:

from calculations in such a theory merely indicates that new physical phenomena appear at scales above

11450: 10996: 2414: 1339: 152: 19183: 17054: 15692: 12235: 10995:

follows. First select a regularization scheme (such as the cut-off regularization introduced above or

6970: 4437: 955: 21959: 21155: 21132: 21082: 21052: 20609: 20476: 20400: 20361: 20318: 20292: 20249: 20142: 19741: 19040: 17887:

Carqueville, Nils; Runkel, Ingo (2018). "Introductory lectures on topological quantum field theory".

17166: 15778: 15663: 13463:

To account for the gauge redundancy in the path integral formulation, one must perform the so-called

12355: 12196: 11607:

field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical

10202:

or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,

5969: 5963: 3084: 2481: 2439: 1821: 1803: 1793:

and can be applied to arbitrary order in perturbation theory. As Tomonaga said in his Nobel lecture:

1605: 1430: 1386: 1202: 293: 14677:. In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators 13812:

The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field

13315: 8869: 4626: 4111:{\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .} 2711: 2687: 1927:

The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger

1802:

By applying the renormalization procedure, calculations were finally made to explain the electron's

21964: 21527: 21209: 21163: 20844: 20676: 20656: 20646: 20636: 20592: 20167: 19977: 18690: 17566: 16935: 16423: 15828: 15644:, which give the necessary and sufficient conditions for a real time theory to be obtained from an 14818: 13312:. For instance, in the path integral formulation, despite the invariance of the Lagrangian density 11486: 10402: 6995:

is the ground state (vacuum state) of the interacting theory, different from the free ground state

4367: 3668: 2255: 1987: 1167: 780: 410: 255: 220: 17405: 7054:, can be found for the real scalar field by either canonical quantization or path integrals to be 770: 22033: 21878: 21746: 21736: 21199: 21009: 20941: 20901: 20371: 20297: 19849: 19705: 19674: 19649: 19588: 18985: 18585: 16238: 15872: 15637: 15513: 15447: 14834: 14666: 13492:

is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.

13464: 12185:) is invariant under a certain local transformation, then the transformation is referred to as a 11584: 11576:, as well as other types of interaction terms, including the electromagnetic interaction and the 11564:

of the real scalar field. A similar process can be done for other types of fields, including the

11073: 6613: 6093: 2316: 2142: 2069: 1971: 1865: 1526: 1473: 1236: 1055: 525: 333: 328: 190: 20785: 20333: 17532:

Methods in Field Theory, Proceedings of the Les Houches Summer School, Les Houches, France, 1975

14695: 8523:

This perturbation expansion expresses the interacting two-point function in terms of quantities

6998: 6606:

and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian

5426: 5320: 4471: 4127: 3870: 2157:, marking the complete verification of the existence of all constituents of the Standard Model. 1613: 1480: 21751: 21724: 21611: 21473: 20881: 20696: 20471: 20451: 20376: 20244: 20016: 19940: 19935: 19870: 19798: 19700: 19603: 19593: 19578: 19495: 19207: 17621:

de Wit, Bernard; Louis, Jan (1998-02-18). "Supersymmetry and Dualities in various dimensions".

17561: 15887: 15882: 15863: 15803: 15783: 15615: 15604: 15517: 14657: 14530: 12436: 11611:, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is: 11536: 11493:. They are insensitive to changes in the scale, as all their coupling constants have vanishing 11254: 10852:

are the experimentally measurable, renormalized, mass and coupling constant, respectively, and

7808: 3490: 2435: 2352: 2342: 2195:

also appeared in the same period. The first supersymmetric QFT in four dimensions was built by

2115: 1849: 1701: 1634: 1567: 1511: 1283: 1143: 925: 905: 405: 338: 250: 215: 20287: 17794: 15908:

Relation between Schrödinger's equation and the path integral formulation of quantum mechanics

7932: 7367: 7331: 3953: 3073:{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.} 21999: 21873: 21840: 21616: 21585: 21448: 21369: 21107: 21019: 20916: 20906: 20721: 20234: 20214: 19989: 19984: 19823: 19818: 19751: 19654: 19618: 19608: 19563: 15897: 15892: 15667: 15649: 15484: 15443: 13407: 13362: 12441: 12432: 11259: 11241: 11065: 11059: 5977: 3940: 3865: 2165:

The 1970s saw the development of non-perturbative methods in non-Abelian gauge theories. The

2153:, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at 1995: 1882:

With these difficulties looming, many theorists began to turn away from QFT. Some focused on

1542: 1363: 1343: 815: 675: 371: 235: 97: 18638: 18438: 17786: 17397: 15725:. Existence includes establishing axiomatic properties at least as strong as those cited in 14764:

Supersymmetry is a potential solution to many current problems in physics. For example, the

11518:

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off

9828:{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z).} 1468:

In the same year as his paper on the photoelectric effect, Einstein published his theory of

21949: 21888: 21883: 21857: 21575: 21399: 21359: 21140: 20681: 20626: 20356: 20179: 20137: 19930: 19664: 19500: 19079: 18993: 18776: 18537: 18494: 18258: 18203: 18088: 18013: 17970: 17950: 17724: 17679: 17553: 17469: 17354: 17212: 17000: 16953: 16887: 16686: 16564: 16497: 16435: 16319: 16135: 15843: 15592: 15415:

The correlation functions and physical predictions of a QFT depend on the spacetime metric

15398: 15053: 14769: 13417: 12678: 12279: 10068: 10041: 9962: 9937: 7798: 6943: 6916: 5973: 5931:{\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}} 5314: 3944: 2425: 1951: 1829: 1721: 1488: 1402: 1308: 1271: 1060: 840: 376: 230: 19412: 18398:(2012-01-06). "Survey of mathematical foundations of QFT and perturbative string theory". 17224: 12181:

is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the

5413:{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .} 1932: 1454: 142: 8: 21916: 21550: 21409: 21404: 21389: 21364: 21341: 21317: 20737: 20706: 20651: 20631: 20539: 20496: 20351: 20277: 20204: 20194: 20106: 19994: 19960: 19895: 19813: 19803: 19598: 18876: 17398: 16262: 15955: 15813: 15537: 15432:(local structure) of the spacetime background, while TQFTs are invariant under spacetime 14826: 13450: 13433: 12951: 12190: 11556: 7044: 5696: 5666: 3467: 3366: 2983: 2356: 2296: 2178: 2127: 1947: 1814: 1786: 1658: 1558: 1484: 1378: 1320: 1131: 1070: 400: 19420: 19083: 18898: 18541: 18498: 18307:

Summers, Stephen J. (2016-03-31). "A Perspective on Constructive Quantum Field Theory".

18262: 18207: 18092: 18017: 17954: 17728: 17683: 17557: 17473: 17358: 17216: 17004: 16957: 16891: 16690: 16568: 16501: 16439: 16323: 16139: 15442:(global structure) of spacetime. This means that all calculational results of TQFTs are 15229:. For a real scalar field, the Lagrangian density in a general spacetime background is 13977:{\displaystyle \phi _{0}^{i}=\left(0,\cdots ,0,{\frac {\mu }{\sqrt {\lambda }}}\right).} 5297:{\displaystyle \left=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left=\left=0,} 2080:. His theory was at first mostly ignored, until it was brought back to light in 1971 by 2035: 785: 515: 21944: 21714: 21606: 21483: 21478: 21431: 21194: 21122: 21077: 20797: 20666: 20564: 20272: 20219: 20111: 19628: 19424: 19114: 18843: 18838: 18738: 18608: 18553: 18510: 18468: 18399: 18308: 18284: 18248: 18219: 18193: 18166:, Mathematical Surveys and Monographs Volume 149, American Mathematical Society, 2008, 18146:, Mathematical Surveys and Monographs Volume 170, American Mathematical Society, 2011, 18029: 18003: 17974: 17914: 17896: 17849: 17819: 17740: 17714: 17622: 17511: 17485: 17459: 17430: 17393: 17370: 17228: 17202: 17135: 17131: 17078: 17016: 16913: 16877: 16869: 16654: 16619: 16595: 16552: 16528: 16485: 16212: 16151: 16125: 15902: 15838: 15755: 15576: 14730: 13403: 11577: 11490: 11480: 10158: 9907: 9881: 9861: 7051: 7023:. This expression represents the probability amplitude for the field to propagate from 6892: 6868: 6848: 2487: 2235: 2227: 2189:

by Polyakov and coauthors. These objects are inaccessible through perturbation theory.

2131: 2111: 2107: 2081: 1883: 1875: 1807: 1762: 1758: 1693: 1681: 1590: 1571: 1469: 1291: 1248: 1194: 1163: 1147: 1080: 1005: 935: 820: 805: 735: 700: 690: 650: 565: 455: 415: 205: 167: 122: 110: 87: 82: 17575: 21830: 21621: 21570: 21515: 21495: 21381: 21239: 21234: 21014: 20979: 20807: 20716: 20686: 20614: 20577: 20572: 20554: 20519: 20509: 20224: 20189: 20172: 20075: 19679: 19354: 19343: 19324: 19297: 19264: 19234: 19193: 19168: 19149: 19121: 19095: 19046: 19025: 18997: 18971: 18953: 18932: 18908: 18880: 18854: 18820: 18790: 18754: 18750: 18724: 18694: 18668: 18649: 18619: 18589: 18557: 18514: 18376: 18366: 18343: 18333: 18274: 18167: 18147: 18100: 18079: 18054: 18033: 17994: 17918: 17867: 17798: 17787: 17765: 17691: 17650: 17642: 17601: 17594: 17489: 17409: 17374: 17322: 17232: 17111: 17103: 17024: 16991: 16917: 16903: 16876:. Advanced Series on Directions in High Energy Physics. Vol. 26. pp. 1–27. 16842: 16772: 16747: 16722: 16659: 16641: 16600: 16582: 16533: 16515: 16451: 16402: 16355: 16345: 16242: 16089: 15749: 15572: 15568: 15533: 14806:

theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional

14765: 14645:

to fermions. Theorists have hypothesized the existence of a type of symmetry, called

13472: 13309: 12428: 11930: 11087: 11069: 8899: 5945: 4532:: and any state of the system can be expressed as a linear combination of the states 3897: 3099: 2360: 2057:

that the gauge symmetry in Yang–Mills theories could be broken by a mechanism called

1861: 1732: 1670: 1554: 1541:, he successfully explained the phenomenon of spontaneous emission. According to the 1446: 1434: 1374: 1287: 1243:

procedure. A second major barrier came with QFT's apparent inability to describe the

1151: 1035: 965: 850: 695: 630: 600: 570: 430: 425: 270: 195: 185: 77: 18223: 17978: 17744: 17046: 16155: 14653: 11968:

If the following transformation to the fields is performed at every spacetime point

10313:

In realistic applications, the scattering amplitude of a certain interaction or the

10287:{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots \phi (x_{n})\}|\Omega \rangle } 8572:

In the path integral formulation, the two-point correlation function can be written

755: 22009: 21756: 21741: 21682: 21560: 21500: 21426: 21057: 20534: 20529: 20386: 20282: 19950: 19087: 18904: 18545: 18502: 18288: 18266: 18211: 18118: 18096: 18021: 17958: 17906: 17859: 17736: 17732: 17687: 17571: 17477: 17429:

Fujita, Takehisa (2008-02-01). "Physics of Renormalization Group Equation in QED".

17362: 17220: 17042: 17008: 16961: 16895: 16694: 16649: 16631: 16590: 16572: 16523: 16505: 16443: 16327: 16305: 16230: 16143: 15475: 14838: 13718: 13425: 12182: 11974:(a local transformation), then the QED Lagrangian remains unchanged, or invariant: 10337: 4607:{\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .} 4468:. Any energy eigenstate state of a single harmonic oscillator can be obtained from 2300: 2270: 2261:

Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of

2251: 2103: 2003: 1887: 1853: 1825: 1774: 1766: 1442: 1426: 1355: 1327: 1244: 1198: 1155: 1075: 990: 970: 940: 890: 715: 710: 705: 590: 575: 505: 210: 137: 127: 62: 18808: 17012: 14814:

imposes no restriction on the number of dimensions nor the geometry of spacetime.

14750:, and superstring theory. In a supersymmetric theory, every fermion has a bosonic 10297:

is equal to the sum of (expressions corresponding to) all connected diagrams with

1753:. By ignoring the contribution of photons whose energy exceeds the electron mass, 21931: 21798: 21580: 21490: 21436: 21336: 21244: 20867: 20764: 20691: 20671: 20641: 20604: 20599: 20504: 20328: 19972: 19710: 19659: 19338: 19312: 19285: 19253: 19227:

Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories

19139: 19015: 18946: 18834: 18784: 18633: 18523: 18480: 18240: 18074: 18048: 17966: 17936: 17759: 17589: 17315: 16944: 16447: 16394: 16200: 16083: 15867: 15823: 15633: 15618:

are better supported mathematically — both can be classified in the framework of

15545: 15480: 14807: 14620: 14608: 11941: 11565: 11047: 10356: 10349: 9507: 9500: 7408:

In canonical quantization, the two-point correlation function can be written as:

3663: 2348: 2219: 2099: 2089: 2065: 2047: 2039: 2021: 2013: 1936: 1790: 1778: 1717: 1705: 1666: 1530: 1492: 1398: 1359: 1331: 1312: 1240: 1206: 1190: 1020: 1015: 985: 980: 950: 910: 880: 860: 765: 660: 655: 620: 605: 580: 560: 540: 470: 460: 381: 343: 260: 225: 57: 35: 30: 19211: 17481: 16349: 12439:, which describes the strong interaction, is a non-Abelian gauge theory with an 8896:, reducing the interacting two-point function to quantities in the free theory. 21969: 21668: 21510: 21465: 21443: 21331: 21214: 20742: 20711: 20701: 20323: 20313: 20147: 19915: 19890: 19669: 19145: 19135: 19109: 18894: 18868: 18804: 18742: 18603: 17389: 16986: 16931: 16899: 16079: 15769: 15645: 15556: 15521: 15488: 15433: 14612: 14604: 12186: 11788: 11043: 3358: 2396: 2378: 2325: 2266: 2262: 2138: 2093: 1999: 1998:(also known as Yang–Mills theories), which are based on more complicated local 1983: 1955: 1638: 1597: 1586: 1582: 1507: 1462: 1367: 1351: 1347: 1301:, despite the complete absence of the concept of fields from his 1687 treatise 1256: 1159: 1045: 995: 915: 900: 885: 865: 835: 830: 825: 810: 795: 760: 750: 680: 615: 585: 520: 485: 475: 450: 321: 288: 162: 19091: 18025: 11912:{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} 2231: 800: 22061: 21939: 21898: 21520: 21189: 21099: 20661: 20514: 20405: 20239: 20209: 20162: 20011: 19945: 19920: 19520: 18963: 18768: 18423: 18395: 18380: 18347: 17932: 17863: 17342: 17190: 16698: 16645: 16586: 16519: 16310: 16203:(1977). "The Search for Unity: Notes for a History of Quantum Field Theory". 15675: 15653: 14830: 14646: 14632: 13411: 12424: 11502: 11501:

functions does not imply conformal symmetry of the theory.) Examples include

7785: 7328:

In an interacting theory, where the Lagrangian or Hamiltonian contains terms

6910: 5685: 3091: 2312: 2288: 2277: 2223: 2207:. Supersymmetry only took off in the theoretical community after the work of 2192: 2123: 1782: 1621: 1617: 1601: 1578: 1182: 1171: 1050: 1040: 1030: 1025: 945: 930: 730: 725: 555: 550: 510: 480: 445: 435: 366: 361: 21285: 18270: 16636: 1757:

successfully estimated the numerical value of the Lamb shift. Subsequently,

21921: 21852: 21761: 21719: 21709: 21545: 21117: 21024: 20931: 20926: 20921: 20896: 20544: 20157: 20152: 19955: 19865: 19583: 19525: 19510: 18682: 18577: 18419: 17193:(2016-04-19). "Particle Physics and Condensed Matter: The Saga Continues". 17028: 16966: 16939: 16663: 16604: 16537: 16455: 15678: 15564: 15512:. Examples of QFTs that are completely solvable non-perturbatively include 14758: 14751: 14691: 13467: 11963: 11569: 7802: 5677: 4122: 2446: 2212: 2204: 2196: 2031: 1991: 1908: 1625: 1546: 1275: 1252: 1085: 1010: 895: 870: 855: 790: 670: 665: 545: 535: 530: 495: 490: 465: 420: 298: 265: 92: 72: 19434: 16771:(1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. 16746:(1. print ed.). Reading, Mass: Advanced Book Program, Perseus Books. 16577: 16510: 15670:

as set out by the above axioms. The full problem statement is as follows.

10397:, obtain expressions for the physical quantities, and then take the limit 3450:{\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.} 2203:

in 1970, but their result failed to garner widespread interest due to the

1437:, was formulated between 1925 and 1926, with important contributions from 1326:

Fields began to take on an existence of their own with the development of

21835: 21729: 21697: 21692: 21184: 20582: 19613: 19440: 19248: 19011: 18850: 18720: 18114: 17310: 16308:(November 1981). "The development of field theory in the last 50 years". 15681: 15596: 15501: 14785: 14777: 14773: 14698:. In general there can be more than one set of supersymmetry generators, 14687: 14616: 14553: 13496: 13446: 12777: 12414: 11573: 7922:{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}} 6886: 6616:. The initial and final conditions of the path integral are respectively 5673: 3095: 2450: 2292: 2208: 2150: 2119: 2077: 2043: 2027: 2017: 2007: 1975: 1967: 1728: 1685: 1065: 1000: 960: 920: 875: 740: 645: 640: 625: 610: 595: 18925:

Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation

16216: 14611:

at low temperatures. In the Standard Model of elementary particles, the

13887:{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac {\mu ^{2}}{\lambda }}.} 13802:{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad R\in \mathrm {O} (N).} 1297:

The earliest successful classical field theory is one that emerged from

21565: 21555: 21249: 21034: 20969: 20891: 19428: 18549: 18506: 18253: 17962: 17705:

Munoz, Carlos (2017-01-18). "Models of Supersymmetry for Dark Matter".

17627: 17516: 17435: 17366: 17074: 17020: 15932:

Theoretical and experimental justification for the Schrödinger equation

15057: 14694:. The symmetry group generated by all these generators is known as the 10314: 7036: 5662: 2141:

of elementary particles. The Standard Model successfully describes all

1754: 1750: 1709: 1662: 1650: 1630: 1563: 1458: 1438: 1417: 1413: 1390: 1382: 720: 635: 500: 440: 132: 67: 18215: 17134:(2012-01-04). "The Early History of String Theory and Supersymmetry". 16331: 16147: 10310:

interaction theory discussed above, every vertex must have four legs.

1720:

and Wheeler daringly suggested abandoning QFT altogether and proposed

21416: 21204: 21072: 20984: 20956: 20436: 20132: 19746: 18645: 16114:

Hobson, Art (2013). "There are no particles, there are only fields".

15640:. One way to construct theories satisfying Wightman axioms is to use 15623: 15509: 15505: 11950: 5699:

term could be introduced to the Lagrangian of the real scalar field:

2186: 2174: 2061:, through which originally massless gauge bosons could acquire mass. 1421: 1232: 20044: 18610:

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

18404: 18313: 17910: 17824: 11497:

function. (The converse is not true, however — the vanishing of all

11003:. Compute Feynman diagrams, in which divergent terms will depend on 6443:, the above product of integrals becomes the Feynman path integral: 3502:

The displacement of a classical harmonic oscillator is described by

22014: 21112: 20911: 19844: 18198: 18008: 17901: 17854: 17719: 17510:

supersymmetric Yang–Mills theory and the AdS/SCFT correspondence".

17464: 17207: 16882: 16839:

Climbing the mountain: the scientific biography of Julian Schwinger

16809:

Climbing the Mountain: The scientific biography of Julian Schwinger

16471:

Climbing the Mountain: The scientific biography of Julian Schwinger

15719: 15438: 14781: 14517: 11594: 10318: 2986:

for the field, which describe the way it varies in time and space:

1654: 1450: 685: 20836: 17140: 17083: 16130: 15458:. The world line trajectory of fractionalized particles (known as 11038:

is taken. In this way, meaningful finite quantities are obtained.

2299:, as well as the relation between frequency and voltage in the AC 21687: 21637: 18047:

Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997).

14642: 9843: 2674:{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left,} 2146: 1186: 17757: 10500:

is a constant to be determined. The Lagrangian density becomes:

10359:

procedure is a systematic process for removing such infinities.

9904:

connected with four edges and represented by dots (here labeled

1405:, that light is composed of individual packets of energy called 19375: 19076:

Relativistic Quantum Mechanics and Introduction to Field Theory

17758:

Morandi, G.; Sodano, P.; Tagliacozzo, A.; Tognetti, V. (2000).

15459: 11604: 2453:

at every point in space that changes in time. It is denoted as

2429: 2281: 2073: 1963: 1550: 1406: 12754:{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu }^{a}t^{a},} 10988:

are constants to be determined. The first three terms are the

10321:, which itself can be found using the Feynman diagram method. 9858:

connected with one edge and represented by dots (here labeled

2010:. Unlike photons, these gauge bosons themselves carry charge. 1523:

The quantum theory of the emission and absorption of radiation

19310: 19056: 17156:"Common Problems in Condensed Matter and High Energy Physics" 16940:"Conservation of Isotopic Spin and Isotopic Gauge Invariance" 15584: 15492: 14638: 14607:, spontaneous symmetry breaking can explain the alignment of 12472: 12457: 11923:. The parameters in this theory are the (bare) electron mass 11447: 2085: 1959: 1673:. QFT naturally incorporated antiparticles in its formalism. 1267: 1228: 18243:(2000). "Current Trends in Axiomatic Quantum Field Theory". 15465: 7929:, and the expansion of the two-point correlator in terms of 6686:{\displaystyle \phi (0)=\phi _{I},\quad \phi (T)=\phi _{F}.} 4617:

A similar procedure can be applied to the real scalar field

2747:

is a real parameter (the "mass" of the field). Applying the

1553:, there remains an oscillating electromagnetic field having 1294:. A brief overview of these theoretical precursors follows. 18247:. Lecture Notes in Physics. Vol. 558. pp. 43–64. 18046: 17761:

Field Theories for Low-Dimensional Condensed Matter Systems

15913:

Relationship between string theory and quantum field theory

13897:

Without loss of generality, let the ground state be in the

12393: 11526:

that the theory is no longer valid at energies higher than

9838:

This term can instead be obtained from the Feynman diagram

6913:

operator that shuffles its operands so the time-components

2154: 1307:. The force of gravity as described by Isaac Newton is an " 1259:

in the 1970s led to a renaissance of quantum field theory.

18363:

Quantum Physics : a Functional Integral Point of View

17345:(1932-03-10). "Konfigurationsraum und zweite Quantelung". 11836:{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}} 4364:

increases the energy of the simple harmonic oscillator by

1692:

A series of papers was published between 1934 and 1938 by

18893: 17449: 13505:

real scalar fields, described by the Lagrangian density:

11257:, a non-Abelian gauge theory based on the symmetry group 10362:

Parameters appearing in the Lagrangian, such as the mass

1472:, built on Maxwell's electromagnetism. New rules, called 18582:

Inward Bound: Of Matter and Forces in the Physical World

18164:

Quantum Field Theory: A Tourist Guide for Mathematicians

17077:(2014-12-12). "The Standard Model of Particle Physics". 16794:

Proc of the 1967 Int. Conference on Particles and Fields

16351:

Physics and Philosophy: The Revolution in Modern Science

14516:

global symmetry is no longer manifest, leaving only the

13361:

under a certain local transformation of the fields, the

11086:

As an example, the coupling constant in QED, namely the

2110:

discovered in 1971 that certain phenomena involving the

16872:(2015-03-17). "The Evolution of Quantum Field Theory". 14576:. The number of broken symmetries is their difference, 11046:

of elementary particles is a renormalizable QFT, while

10405:, a class of methods to treat divergences in QFT, with 7797:

is the field operator under the free theory. Here, the

6845:

in the free or interacting theory, respectively. Here,

5972:

of QFT is concerned with the direct computation of the

1596:

is 2; it led to the correct Sommerfeld formula for the

13370: 9992: 7838: 2355:

of spatial and time coordinates. Examples include the

1370:. Action-at-a-distance was thus conclusively refuted. 18332:. Princeton, New Jersey: Princeton University Press. 16235:

The Oxford Companion to the History of Modern Science

15695: 15238: 15069: 14872: 14637:

All experimentally known symmetries in nature relate

14143: 14002: 13912: 13830: 13735: 13514: 13318: 13236: 12981: 12801: 12690: 12488: 12358: 12282: 12238: 12199: 11983: 11849: 11797: 11620: 11276: 11108: 10861: 10509: 10428: 10211: 10161: 10098: 10071: 10044: 9965: 9940: 9910: 9884: 9864: 9700: 9531: 8917: 8872: 8581: 8529: 8281: 8016: 7955: 7935: 7811: 7417: 7370: 7334: 7063: 7001: 6973: 6946: 6919: 6895: 6871: 6851: 6717: 6711:

In calculations, one often encounters expression like

6625: 6452: 6105: 6028: 5986: 5860: 5708: 5503: 5457: 5429: 5354: 5323: 5053: 4777: 4739: 4696: 4658: 4629: 4541: 4502: 4474: 4440: 4393: 4370: 4334: 4246: 4210: 4161: 4130: 4033: 3956: 3905: 3873: 3712: 3671: 3511: 3389: 3111: 2995: 2760: 2714: 2690: 2513: 2490: 2287:

Gauge theory is used to describe the quantization of

2114:

could also be explained by non-Abelian gauge theory.

1282:

Quantum field theory results from the combination of

1197:

of the particle is determined by minimization of the

1193:, which are more fundamental than the particles. The 15745: 15689:, a non-trivial quantum Yang–Mills theory exists on 7832:-theory, the interacting term of the Hamiltonian is 5487:{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }} 5423:

Any quantum state of the field can be obtained from

4726:{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }} 4321:{\displaystyle =\hbar \omega {\hat {a}}^{\dagger },} 4024:

of the simple harmonic oscillator can be written as

3630:

is a complex number (normalized by convention), and

1919:

no ‘distracting remarks’ about infinite quantities.

1749:

energy levels of the hydrogen atom, also called the

19165:

Finite Quantum Electrodynamics: The Causal Approach

18521: 18478: 17387: 16677:Schwinger, Julian (1966). "Particles and Sources". 16261: 15734: 15730: 15404: 11234:is the energy scale under which the measurement of 9691:After applying Wick's theorem, one of the terms is 5494:(or by a linear combination of such states), e.g. 2238:first proposed in 1974 that string theory could be 1585:. It had the following important consequences: the 19342: 19252: 19113: 18842: 18637: 18607: 18467: 17817: 17593: 17314: 16841:(Reprinted ed.). Oxford: Oxford Univ. Press. 16553:"On the Green's functions of quantized fields. II" 15710: 15363: 15203: 15006: 14486: 14123: 13976: 13886: 13801: 13693: 13386: 13353: 13296: 13195: 12933: 12753: 12657: 12383: 12333: 12268: 12224: 12159: 11911: 11835: 11770: 11426: 11219: 10977: 10823: 10468: 10286: 10167: 10140: 10084: 10057: 10027: 9978: 9946: 9916: 9890: 9870: 9827: 9680: 9516:term in the two-point correlation function in the 9479: 8882: 8855: 8557: 8515: 7941: 7921: 7824: 7767: 7392: 7356: 7317: 7015: 6987: 6959: 6932: 6901: 6877: 6857: 6837: 6685: 6583: 6425: 6049: 6007: 5930: 5837: 5649: 5486: 5443: 5412: 5337: 5296: 5033: 4754: 4725: 4682: 4644: 4606: 4524: 4488: 4460: 4426: 4379: 4356: 4320: 4232: 4193: 4152:, which is the lowest energy state, is defined by 4144: 4110: 4009: 3927: 3888: 3841: 3695: 3613: 3449: 3340: 3072: 2971: 2729: 2700: 2673: 2496: 1514:by treating the electromagnetic field as a set of 18737: 17886: 16989:(1979-12-14). "The 1979 Nobel Prize in Physics". 16486:"On the Green's functions of quantized fields. I" 16344: 15052:on the other hand, a general metric (such as the 9930:connecting the vertices and represented by lines. 7035:, and goes by multiple names, like the two-point 6700: 2315:are used in the following sections, in which the 1676: 22059: 19108: 18465: 18077:(1958). "A Soluble Relativistic Field Theory?". 16806: 16712: 16710: 16708: 16468: 16229: 15726: 13478: 10190:-th order, list all valid Feynman diagrams with 8641: 7477: 7151: 3460:Thus each normal mode corresponding to a single 3094:, so its solutions can be expressed as a sum of 1479:Two difficulties remained. Observationally, the 1235:, culminating in the first quantum field theory— 19134: 18584:(reprint ed.). Oxford, New York, Toronto: 17937:"Quantum Field Theory and the Jones Polynomial" 16624:Proceedings of the National Academy of Sciences 16557:Proceedings of the National Academy of Sciences 16490:Proceedings of the National Academy of Sciences 16078: 14841:to produce gauge theories in lower dimensions. 13495:To illustrate the mechanism, consider a linear 12445:gauge symmetry. It contains three Dirac fields 10419:theory, the field strength is first redefined: 7050:The free two-point function, also known as the 5980:for a system to evolve from some initial state 4496:by successively applying the creation operator 4427:{\displaystyle {\hat {a}}^{\dagger }|0\rangle } 1844:depression that lasted for almost two decades. 1832:of the interaction represented by the diagram. 1483:underlying quantum mechanics could explain the 1362:. Maxwell's equations implied the existence of 1346:, which described the relationship between the 21804:Mathematical formulation of the Standard Model 18687:Q is for Quantum: Particle Physics from A to Z 17848:. Cambridge University Press. pp. 27–29. 17098: 17096: 17094: 16826:. Reading, MA: Perseus Bookks. pp. 82–85. 16620:"The sources of Schwinger's Green's functions" 13297:{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.} 4623:, by promoting it to a quantum field operator 21653: 21315: 21301: 20852: 20060: 19456: 18945:Lancaster, T., & Blundell, S. J. (2014). 18833: 18803: 18393: 18330:The P(phi)_2 Euclidean (quantum) field theory 17991: 16705: 16338: 12479:gauge bosons. The QCD Lagrangian density is: 11530:, and all degrees of freedom above the scale 10151:Divide by the symmetry factor of the diagram. 2169:was discovered theoretically by 't Hooft and 1978:, which endow elementary particles with mass. 1111: 17669: 17663: 16868: 16300: 16298: 16296: 14552:continuous symmetries (the dimension of its 10281: 10270: 10226: 10212: 9672: 9661: 9586: 9572: 9467: 9456: 9415: 9401: 9398: 9387: 9346: 9332: 9319: 9308: 9267: 9253: 9250: 9239: 9198: 9184: 9171: 9160: 9119: 9105: 9102: 9091: 9050: 9036: 9026: 9015: 8936: 8922: 8634: 8623: 8596: 8582: 8552: 8530: 8504: 8493: 8421: 8407: 8277: 8266: 8156: 8142: 8008: 7997: 7970: 7956: 7470: 7459: 7432: 7418: 7116: 7105: 7078: 7064: 7010: 6982: 6832: 6821: 6794: 6780: 6770: 6759: 6732: 6718: 6505: 6453: 6417: 6357: 6354: 6294: 6288: 6222: 6158: 6106: 6044: 6002: 5641: 5451:by successively applying creation operators 5438: 5384: 5332: 4598: 4550: 4483: 4421: 4233:{\displaystyle {\frac {1}{2}}\hbar \omega .} 4182: 4139: 2226:, itself a type of two-dimensional QFT with 2130:showed that non-Abelian gauge theories are " 19042:Quantum Field Theory and the Standard Model 18948:Quantum field theory for the gifted amateur 18665:QED: The Strange Theory of Light and Matter 18524:"Axioms for Euclidean Green's functions II" 18235: 18233: 17785:Parker, Leonard E.; Toms, David J. (2009). 17649:. Vol. 2. Cambridge University Press. 17620: 17582: 17110:. Vol. 1. Cambridge University Press. 17091: 16837:Mehra, Jagdish; Milton, Kimball A. (2005). 16836: 16294: 16292: 16290: 16288: 16286: 16284: 16282: 16280: 16278: 16276: 16223: 16195: 16193: 16191: 16189: 16187: 16185: 15923:Static forces and virtual-particle exchange 8558:{\displaystyle \langle 0|\cdots |0\rangle } 6707:Correlation function (quantum field theory) 2245: 1724:as the mechanism of particle interactions. 1581:that described relativistic electrons: the 1506:Through the works of Born, Heisenberg, and 1304:Philosophiæ Naturalis Principia Mathematica 1142:) is a theoretical framework that combines 21660: 21646: 21308: 21294: 20859: 20845: 20067: 20053: 19470: 19463: 19449: 19317:General Principles of Quantum Field Theory 18360: 18302: 18300: 18298: 18144:Renormalization and Effective Field Theory 17784: 17641: 17185: 17183: 17102: 16981: 16979: 16977: 16791: 16721:. Reading, MA: Perseus Books. p. xi. 16255: 16183: 16181: 16179: 16177: 16175: 16173: 16171: 16169: 16167: 16165: 15561:Renormalization and Effective Field Theory 14833:is a type of (1+1)-dimensional QFT, while 13716:are real parameters. The theory admits an 10038:For every edge that connects two vertices 2428:are best explained by discrete particles ( 1498: 1118: 1104: 29: 19181: 18984: 18403: 18354: 18312: 18252: 18197: 18007: 17900: 17853: 17823: 17718: 17626: 17565: 17515: 17463: 17434: 17206: 17139: 17082: 16965: 16930: 16881: 16864: 16862: 16860: 16858: 16821: 16766: 16741: 16716: 16676: 16653: 16635: 16594: 16576: 16550: 16527: 16509: 16483: 16389: 16387: 16385: 16383: 16381: 16379: 16377: 16375: 16373: 16371: 16304: 16129: 16074: 16072: 16070: 16068: 16066: 16064: 16062: 16060: 16058: 16056: 16054: 16052: 16050: 16048: 16046: 16044: 16042: 16040: 16038: 16036: 16034: 16032: 16030: 16028: 16026: 16024: 16022: 16020: 16018: 16016: 16014: 16012: 16010: 16008: 16006: 16004: 16002: 16000: 15998: 15996: 15994: 15992: 15698: 15666:, concerns the well-defined existence of 15466:Perturbative and non-perturbative methods 13399:, in which all anomalies exactly cancel. 9746: 9571: 8834: 8788: 8760: 8690: 7877: 7717: 7612: 6569: 6533: 6221: 4683:{\displaystyle {\hat {a}}_{\mathbf {p} }} 3636:is the oscillator's frequency. Note that 3484: 2563: 2536: 1923:was a major disappointment for Schwinger: 1820:At the same time, Feynman introduced the 1262: 19337: 19224: 19070: 19038: 18921: 18782: 18767: 18602: 18481:"Axioms for Euclidean Green's functions" 18239: 18230: 18136: 18073: 17789:Quantum Field Theory in Curved Spacetime 17588: 17244: 17242: 16617: 16424:"Development of Quantum Electrodynamics" 16421: 16273: 16199: 16109: 16107: 16105: 15990: 15988: 15986: 15984: 15982: 15980: 15978: 15976: 15974: 15972: 15878:Quantum field theory in curved spacetime 15799:Common integrals in quantum field theory 14134:and the original Lagrangian density as: 11940: 11603:representing the electromagnetic field ( 11053: 2006:" interact via the exchange of massless 1946: 1838: 1401:proposed in 1905 an explanation for the 1266: 1211:perturbation theory in quantum mechanics 19141:An Introduction to Quantum Field Theory 18783:Frampton, Paul H. (22 September 2008). 18775:. Frontiers in Physics (2nd ed.). 18714: 18681: 18662: 18632: 18418: 18306: 18295: 17543: 17305: 17303: 17301: 17299: 17297: 17295: 17293: 17291: 17289: 17287: 17189: 17180: 17130: 16985: 16974: 16874:The Standard Theory of Particle Physics 16811:. Oxford University Press. p. 467. 16473:. Oxford University Press. p. 454. 16399:Advanced Topics in Quantum Field Theory 16393: 16162: 16085:An Introduction to Quantum Field Theory 13420:states that every continuous symmetry, 12276:is yet another symmetry transformation 12189:of the theory. Gauge symmetries form a 10469:{\displaystyle \phi =Z^{1/2}\phi _{r},} 10317:of a particle can be computed from the 2445:The simplest classical field is a real 2068:wrote down in 1967 a theory describing 22060: 21506:Atomic, molecular, and optical physics 19541:Two-dimensional conformal field theory 19167:(third ed.). Dover Publications. 19162: 18962: 18900:Critical Properties of φ-Theories 18529:Communications in Mathematical Physics 18522:Osterwalder, K.; Schrader, R. (1975). 18486:Communications in Mathematical Physics 18479:Osterwalder, K.; Schrader, R. (1973). 18183: 18177: 17942:Communications in Mathematical Physics 17931: 17838: 17793:. Cambridge University Press. p. 17502: 17428: 17404:. Cambridge University Press. p. 17285: 17283: 17281: 17279: 17277: 17275: 17273: 17271: 17269: 17267: 17073: 16855: 16792:C.R. Hagen; et al., eds. (1967). 16368: 16113: 15548:method, is fundamentally ill-defined. 15536:, there does not exist a well-defined 12460:fields as well as eight vector fields 10028:{\textstyle -i\lambda \int d^{4}z_{i}} 4755:{\displaystyle \omega _{\mathbf {p} }} 4194:{\displaystyle {\hat {a}}|0\rangle =0} 1735:measured the minute difference in the 21641: 21289: 20840: 20074: 20048: 19444: 19315:; Oksak, A.I.; Todorov, I.T. (1990). 19284: 19010: 18470:PCT, Spin and Statistics and all That 18327: 18321: 17704: 17239: 16102: 15969: 15527: 15424:. For a special class of QFTs called 14860:spacetime indices in the Lagrangian, 14738:Minimal Supersymmetric Standard Model 14656:, whose generators are the spacetime 13387:{\textstyle \int {\mathcal {D}}\phi } 12435:(also known as Yang–Mills theories). 9934:Every vertex corresponds to a single 5684:the electromagnetic field), and even 4525:{\displaystyle {\hat {a}}^{\dagger }} 4357:{\displaystyle {\hat {a}}^{\dagger }} 3928:{\displaystyle {\hat {a}}^{\dagger }} 3380:is the frequency of the normal mode: 2737:is the time-derivative of the field, 2160: 1604:; and it could be used to derive the 1299:Newton's law of universal gravitation 22040: 19206: 18897:; Schulte-Frohlinde, Verena (2001). 18867: 18576: 18361:Glimm, James; Jaffe, Arthur (1987). 18113: 17341: 17248: 16824:Particles, Sources and Fields vol. 1 16719:Particles, Sources and Fields vol. 1 14649:, that relates bosons and fermions. 5938:, etc. The summation over the index 1589:of an electron is 1/2; the electron 1189:levels) of their underlying quantum 20866: 19408:Stanford Encyclopedia of Philosophy 19247: 18466:Streater, R.; Wightman, A. (1964). 18365:. New York, NY: Springer New York. 18156: 17309: 17264: 15859:Noncommutative quantum field theory 15834:Green's function (many-body theory) 15595:, which has led to such results as 15544:of QFT, which underlies the entire 14795: 14715:, which generate the corresponding 11547:, where a new theory is necessary. 11489:(CFTs) are special QFTs that admit 10184:-point correlation function to the 3651:For a quantum harmonic oscillator, 2336: 13: 21667: 19255:Quantum Field Theory in a Nutshell 19212:"Lectures on Quantum Field Theory" 18873:Modern Elementary Particle Physics 18565: 17317:Quantum Field Theory in a Nutshell 17041: 16618:Schweber, Silvan S. (2005-05-31). 15819:Form factor (quantum field theory) 15491:carry the weak interaction, while 15426:topological quantum field theories 15308: 15295: 15241: 15186: 15173: 15144: 15131: 14989: 14976: 14947: 14934: 14266: 14243: 14200: 14170: 14146: 13783: 13571: 13541: 13517: 13376: 13336: 13321: 13157: 12852: 12824: 12705: 12491: 12123: 11890: 11867: 11662: 11623: 11316: 11297: 11148: 11129: 10718: 10692: 10560: 10534: 10512: 10343: 10278: 10215: 10141:{\displaystyle D_{F}(z_{i}-z_{j})} 9494: 8875: 8837: 8780: 8763: 8682: 8651: 8631: 8585: 8298: 8033: 8005: 7959: 7487: 7467: 7421: 6979: 6829: 6783: 6516: 6050:{\displaystyle |\phi _{F}\rangle } 6008:{\displaystyle |\phi _{I}\rangle } 5915: 5907: 5895: 5882: 5874: 5862: 5754: 5733: 5711: 4652:, while the annihilation operator 3034: 3014: 3004: 2951: 2944: 2939: 2910: 2899: 2893: 2886: 2881: 2858: 2854: 2812: 2801: 2795: 2788: 2783: 2767: 2763: 2693: 2614: 2539: 1178:is based on quantum field theory. 1176:standard model of particle physics 14: 22089: 21274:Template:Quantum mechanics topics 20036:Template:Quantum mechanics topics 19368: 18813:Gauge Theory of Weak Interactions 17845:Quantum Gravity in 2+1 Dimensions 16769:Particles, sources, and fields. 3 16744:Particles, sources, and fields. 2 15809:Constructive quantum field theory 15794:Introduction to quantum mechanics 15735:Osterwalder & Schrader (1975) 15731:Osterwalder & Schrader (1973) 15660:Yang–Mills existence and mass gap 15589:constructive quantum field theory 14776:—can be resolved by relating the 12401:gauge symmetry. The photon field 11957: 11550: 7047:or two-point function for short. 6967:increase from right to left, and 5957: 5044:Their commutation relations are: 4444: 4290: 4221: 4099: 4049: 1942: 22039: 22028: 22027: 21772: 21269: 21268: 20990:Topological quantum field theory 20821: 20820: 20031: 20030: 19374: 16796:. NY: Interscience. p. 128. 15936:Topological quantum field theory 15762: 15748: 15711:{\displaystyle \mathbb {R} ^{4}} 15612:topological quantum field theory 15411:Topological quantum field theory 15405:Topological quantum field theory 14823:(2+1)-dimensional electron gases 14748:supersymmetric Yang–Mills theory 14690:, which themselves transform as 14626: 12397:group, thus QED is said to have 12269:{\displaystyle e^{i\alpha '(x)}} 11513:supersymmetric Yang–Mills theory 9842: 6988:{\displaystyle |\Omega \rangle } 5606: 5568: 5525: 5473: 5403: 5370: 5269: 5240: 5206: 5182: 5151: 5143: 5098: 5074: 5017: 5009: 4993: 4968: 4944: 4936: 4920: 4897: 4868: 4794: 4746: 4712: 4674: 4461:{\displaystyle 3\hbar \omega /2} 3413: 3396: 3324: 3316: 3300: 3275: 3260: 3252: 3236: 3213: 3193: 3119: 1893: 1549:). Therefore, even in a perfect 1535:electromagnetic vector potential 21627:Timeline of physics discoveries 20975:Jackiw–Teitelboim gravity 18412: 18387: 18107: 18067: 18040: 17985: 17925: 17880: 17832: 17811: 17778: 17751: 17698: 17635: 17614: 17537: 17524: 17503:Kovacs, Stefano (1999-08-26). " 17496: 17443: 17422: 17381: 17335: 17254: 17225:10.1088/0031-8949/T168/1/014003 17148: 17124: 17067: 17035: 16924: 16830: 16815: 16800: 16785: 16760: 16735: 16670: 16611: 16551:Schwinger, Julian (July 1951). 16544: 16484:Schwinger, Julian (July 1951). 16477: 16462: 15946:Wheeler–Feynman absorber theory 15129: 14932: 13775: 13048: 12384:{\displaystyle e^{i\alpha (x)}} 12225:{\displaystyle e^{i\alpha (x)}} 12036: 10935: 10887: 10494:is the renormalized field, and 6779: 6773: 6654: 5944:has been omitted following the 5396: 5161: 3090:The Klein–Gordon equation is a 1913:Particles, Sources, and Fields. 1806:(the deviation of the electron 1223:History of quantum field theory 21955:Causal dynamical triangulation 21180:Causal dynamical triangulation 21048:Black hole information paradox 20770:Relativistic quantum mechanics 19045:. Cambridge University Press. 18667:. Princeton University Press. 17600:. Cambridge University Press. 17452:Journal of High Energy Physics 17321:. Princeton University Press. 16415: 16401:. Cambridge University Press. 15849:List of quantum field theories 15789:Axiomatic quantum field theory 15727:Streater & Wightman (1964) 15630:Algebraic quantum field theory 15452:fractional quantum Hall effect 15260: 15252: 14110: 14104: 14080: 14074: 14046: 14040: 14019: 14013: 13793: 13787: 13746: 13402:The theoretical foundation of 13354:{\displaystyle {\mathcal {L}}} 13348: 13326: 13270: 13263: 13246: 13240: 13187: 13181: 13124: 13118: 13095: 13089: 13083: 13070: 13064: 13042: 13036: 13023: 13017: 13001: 12998: 12992: 12630: 12545: 12531: 12509: 12376: 12370: 12326: 12323: 12317: 12303: 12297: 12291: 12261: 12255: 12217: 12211: 12149: 12143: 12117: 12111: 12075: 12069: 12056: 12053: 12047: 12030: 12024: 12016: 12010: 11996: 11993: 11987: 11921:electromagnetic field strength 11804: 11736: 11637: 11609:electromagnetic four-potential 11286: 11280: 11118: 11112: 10737: 10714: 10711: 10688: 10579: 10556: 10553: 10530: 10274: 10267: 10254: 10245: 10232: 10219: 10135: 10109: 9819: 9807: 9794: 9782: 9769: 9757: 9665: 9658: 9652: 9646: 9640: 9634: 9628: 9622: 9616: 9610: 9604: 9598: 9592: 9579: 9460: 9453: 9440: 9434: 9421: 9408: 9391: 9384: 9371: 9365: 9352: 9339: 9312: 9305: 9292: 9286: 9273: 9260: 9243: 9236: 9223: 9217: 9204: 9191: 9164: 9157: 9144: 9138: 9125: 9112: 9095: 9088: 9075: 9069: 9056: 9043: 9019: 9012: 8999: 8993: 8980: 8974: 8961: 8955: 8942: 8929: 8883:{\displaystyle {\mathcal {L}}} 8712: 8706: 8700: 8694: 8669: 8654: 8648: 8627: 8620: 8614: 8608: 8602: 8589: 8545: 8537: 8497: 8484: 8470: 8448: 8434: 8414: 8340: 8330: 8319: 8306: 8270: 8257: 8243: 8221: 8207: 8194: 8188: 8175: 8169: 8149: 8075: 8065: 8054: 8041: 8001: 7994: 7988: 7982: 7976: 7963: 7910: 7903: 7855: 7849: 7734: 7728: 7629: 7623: 7568: 7562: 7549: 7543: 7505: 7490: 7484: 7463: 7456: 7450: 7444: 7438: 7425: 7387: 7381: 7351: 7345: 7307: 7281: 7197: 7187: 7158: 7144: 7132: 7109: 7102: 7096: 7090: 7084: 7071: 7003: 6975: 6825: 6818: 6812: 6806: 6800: 6787: 6763: 6756: 6750: 6744: 6738: 6725: 6701:Two-point correlation function 6664: 6658: 6635: 6629: 6530: 6524: 6491: 6467: 6403: 6371: 6340: 6308: 6268: 6236: 6144: 6120: 6030: 5988: 5745: 5729: 5634: 5597: 5559: 5516: 5465: 5431: 5377: 5362: 5325: 5261: 5232: 5198: 5174: 5155: 5139: 5127: 5117: 5090: 5066: 4960: 4889: 4841: 4831: 4804: 4790: 4784: 4704: 4666: 4645:{\displaystyle {\hat {\phi }}} 4636: 4591: 4568: 4543: 4510: 4476: 4414: 4401: 4342: 4303: 4284: 4272: 4256: 4247: 4175: 4168: 4132: 4080: 4062: 4040: 3984: 3968: 3913: 3880: 3808: 3758: 3731: 3725: 3719: 3690: 3684: 3678: 3521: 3515: 3419: 3408: 3166: 3156: 3129: 3115: 2923: 2896: 2818: 2798: 2741:is the gradient operator, and 2730:{\displaystyle {\dot {\phi }}} 2701:{\displaystyle {\mathcal {L}}} 2621: 2611: 1677:Infinities and renormalization 1377:, nor for the distribution of 1: 21794:Spontaneous symmetry breaking 21146:Wheeler–DeWitt equation 20748:Quantum statistical mechanics 20525:Quantum differential calculus 20447:Delayed-choice quantum eraser 20230:Symmetry in quantum mechanics 18640:The Character of Physical Law 17576:10.1016/S1355-2198(01)00033-8 17013:10.1126/science.206.4424.1290 16422:Tomonaga, Shinichiro (1966). 15962: 15928:Symmetry in quantum mechanics 15456:topological quantum computers 15446:of the underlying spacetime. 14844:In Minkowski space, the flat 13490:Spontaneous symmetry breaking 13485:Spontaneous symmetry breaking 13479:Spontaneous symmetry-breaking 10374:, have no physical meaning — 4380:{\displaystyle \hbar \omega } 3696:{\displaystyle {\hat {x}}(t)} 2306: 2059:spontaneous symmetry breaking 1700:Faced with these infinities, 1420:of atomic structure, wherein 158:Spontaneous symmetry breaking 118:Symmetry in quantum mechanics 21068:Cosmic censorship hypothesis 20887:Batalin–Vilkovisky formalism 19345:The Quantum Theory of Fields 18101:10.1016/0003-4916(58)90015-0 17737:10.1051/epjconf/201713601002 17692:10.1016/0370-2693(75)90297-x 17596:The Quantum Theory of Fields 16448:10.1126/science.154.3751.864 15540:for QFT, which implies that 13993:fields can be rewritten as: 7801:should be understood as its 6600:is the Lagrangian involving 3361:(normalized by convention), 2474:is the position vector, and 1577:In 1928, Dirac wrote down a 1516:quantum harmonic oscillators 1395:quantum harmonic oscillators 1335:fields remains to this day. 7: 21591:Quantum information science 20550:Quantum stochastic calculus 20540:Quantum measurement problem 20462:Mach–Zehnder interferometer 20000:Quantum information science 19395:Encyclopedia of Mathematics 18970:. Oxford University Press. 18968:The Quantum Theory of Light 18428:"Quantum Yang–Mills Theory" 18119:"On Quantum Field Theories" 16117:American Journal of Physics 15741: 15642:Osterwalder–Schrader axioms 14780:and its super-partner, the 14583:, which corresponds to the 7805:expansion. For example, in 6094:generator of time evolution 4434:is an eigenstate of energy 3497:quantum harmonic oscillator 3466:can be seen as a classical 2708:is the Lagrangian density, 2242:quantum theory of gravity. 1935:" between J. Schwinger and 1653:. Indeed, the evidence for 1429:proposed the hypothesis of 1342:was completed in 1864 with 10: 22094: 21422:Classical electromagnetism 21164:Hartle–Hawking state 21063:Bousso's holographic bound 21043:Black hole complementarity 19351:Cambridge University Press 19321:Kluwer Academic Publishers 19294:Cambridge University Press 19261:Princeton University Press 19231:Cambridge University Press 19190:Cambridge University Press 19078:(1st ed.). Springer. 19022:Cambridge University Press 18616:Princeton University Press 18435:Clay Mathematics Institute 17889:Banach Center Publications 17530:Veltman, M. J. G. (1976). 17400:String Theory and M-Theory 16900:10.1142/9789814733519_0001 16822:Schwinger, Julian (1998). 16767:Schwinger, Julian (1998). 16742:Schwinger, Julian (1998). 16717:Schwinger, Julian (1998). 15854:Local quantum field theory 15610:Compared to ordinary QFT, 15498:'t Hooft–Polyakov monopole 15408: 14821:, QFT is used to describe 14630: 13482: 12786:fundamental representation 12770:is the coupling constant, 12433:non-Abelian gauge theories 12410:may be referred to as the 11961: 11057: 10997:dimensional regularization 10368:and the coupling constant 10347: 9959:For every internal vertex 9498: 8565:that are evaluated in the 7016:{\displaystyle |0\rangle } 6704: 5961: 5444:{\displaystyle |0\rangle } 5338:{\displaystyle |0\rangle } 4733:and the angular frequency 4489:{\displaystyle |0\rangle } 4240:One can easily check that 4145:{\displaystyle |0\rangle } 3889:{\displaystyle {\hat {a}}} 3488: 2415:classical electromagnetism 2340: 2222:led to the development of 2167:'t Hooft–Polyakov monopole 2096:, marking its completion. 1996:non-Abelian gauge theories 1657:was discovered in 1932 by 1521:In his seminal 1927 paper 1381:in different wavelengths. 1340:classical electromagnetism 1255:and the completion of the 1220: 1216: 153:Explicit symmetry breaking 22023: 21982: 21960:Canonical quantum gravity 21930: 21897: 21866: 21823: 21812: 21781: 21770: 21675: 21599: 21536: 21464: 21380: 21352: 21324: 21260: 21227: 21172: 21156:Euclidean quantum gravity 21154: 21133:Canonical quantum gravity 21131: 21098: 21091: 21083:Gravitational singularity 21053:Black-hole thermodynamics 21033: 20998: 20955: 20937:Ryu–Takayanagi conjecture 20874: 20816: 20778: 20730: 20610:Quantum complexity theory 20588:Quantum cellular automata 20563: 20495: 20429: 20342: 20306: 20293:Path integral formulation 20260: 20125: 20082: 20025: 19908: 19858: 19837: 19786: 19760: 19724: 19688: 19637: 19556: 19549: 19478: 19092:10.1007/978-3-642-61057-8 18789:. John Wiley & Sons. 18691:Weidenfeld & Nicolson 18026:10.1016/j.aop.2017.06.019 17167:U.S. Department of Energy 16807:Mehra and Milton (2000). 16469:Mehra and Milton (2000). 15664:Millennium Prize Problems 15436:but are sensitive to the 14652:The Standard Model obeys 13221:at every spacetime point 11597:field and a vector field 11473:), the coupling constant 10386:, and the field strength 10330:loops are referred to as 10155:With the symmetry factor 7825:{\displaystyle \phi ^{4}} 5970:path integral formulation 5964:Path integral formulation 4762:are now for a particular 4387:. For example, the state 2480:is the time. Suppose the 2118:(QCD) was born. In 1973, 2050:proposed in their famous 1824:of quantum mechanics and 1822:path integral formulation 1804:anomalous magnetic moment 1641:do not contain electrons 1624:, Heisenberg, Pauli, and 1387:electromagnetic radiation 309:Bargmann–Wigner equations 21965:Superfluid vacuum theory 21528:Condensed matter physics 21210:Superfluid vacuum theory 20965:2+1D topological gravity 20677:Quantum machine learning 20657:Quantum key distribution 20647:Quantum image processing 20637:Quantum error correction 20487:Wheeler's delayed choice 19163:Scharf, Günter (2014) . 19138:; Schroeder, D. (1995). 19112:; Reinhardt, J. (1996). 17864:10.1017/CBO9780511564192 16699:10.1103/PhysRev.152.1219 16082:; Schroeder, D. (1995). 15918:Schwinger–Dyson equation 15050:QFTs in curved spacetime 14819:condensed matter physics 13432:symmetry of QED implies 11487:Conformal field theories 11463:(the Standard Model has 11081:Callan–Symanzik equation 10401:. This is an example of 10178:In order to compute the 9851:The diagram consists of 7942:{\displaystyle \lambda } 7393:{\displaystyle H_{I}(t)} 7357:{\displaystyle L_{I}(t)} 4690:, the creation operator 4010:{\displaystyle \left=1.} 2291:in superconductors, the 2256:condensed matter physics 2246:Condensed-matter-physics 2143:fundamental interactions 2070:electroweak interactions 1972:fundamental interactions 1525:, Dirac coined the term 1181:QFT treats particles as 1168:condensed matter physics 21747:Quantum electrodynamics 21737:Electroweak interaction 21200:Noncommutative geometry 20947:Weinberg–Witten theorem 20942:Trans-Planckian problem 20593:Quantum finite automata 19696:2D free massless scalar 19589:Quantum electrodynamics 19516:QFT in curved spacetime 19415:", by Meinard Kuhlmann. 19225:Williams, A.G. (2022). 19039:Schwartz, M.D. (2014). 18663:Feynman, R.P. (2006) . 18586:Oxford University Press 18271:10.1007/3-540-44482-3_4 17482:10.1007/JHEP05(2015)031 17055:Encyclopædia Britannica 16637:10.1073/pnas.0405167101 16239:Oxford University Press 15951:Wigner's classification 15941:Ward–Takahashi identity 15873:Quantum electrodynamics 15601:spin–statistics theorem 14667:Lorentz transformations 14619:, a process called the 11587:contains a Dirac field 11585:quantum electrodynamics 11050:is non-renormalizable. 6614:Legendre transformation 2749:Euler–Lagrange equation 2317:reduced Planck constant 2076:and the effects of the 2053:Physical Review Letters 1866:fine-structure constant 1527:quantum electrodynamics 1499:Quantum electrodynamics 1474:Lorentz transformations 1237:quantum electrodynamics 1170:to construct models of 334:Electroweak interaction 329:Quantum electrodynamics 304:Wheeler–DeWitt equation 191:Background field method 21725:Quantum chromodynamics 21612:Nobel Prize in Physics 21474:Relativistic mechanics 20882:AdS/CFT correspondence 20697:Quantum neural network 20017:Quantum thermodynamics 19941:On shell and off shell 19936:Loop quantum cosmology 19778:N = 4 super Yang–Mills 19737:N = 1 super Yang–Mills 19604:Scalar electrodynamics 19594:Quantum chromodynamics 19496:Conformal field theory 19472:Quantum field theories 19418:Siegel, Warren, 2005. 19390:"Quantum field theory" 19182:Srednicki, M. (2007). 18050:Conformal Field Theory 17707:EPJ Web of Conferences 17347:Zeitschrift für Physik 16967:10.1103/PhysRev.96.191 15888:Quantum flavordynamics 15883:Quantum chromodynamics 15804:Conformal field theory 15784:AdS/CFT correspondence 15739: 15712: 15616:conformal field theory 15518:conformal field theory 15444:topological invariants 15365: 15205: 15008: 14488: 14125: 13978: 13888: 13803: 13695: 13388: 13355: 13298: 13197: 12935: 12755: 12659: 12475:fields, which are the 12437:Quantum chromodynamics 12385: 12335: 12270: 12226: 12161: 11945: 11913: 11837: 11772: 11537:effective field theory 11428: 11255:quantum chromodynamics 11247:The coupling constant 11221: 10999:); call the regulator 10979: 10825: 10470: 10288: 10169: 10142: 10092:, write down a factor 10086: 10059: 10029: 9986:, write down a factor 9980: 9948: 9918: 9892: 9872: 9829: 9682: 9491:quantum field theory. 9481: 8884: 8857: 8559: 8517: 8302: 8037: 7943: 7923: 7826: 7769: 7394: 7358: 7319: 7017: 6989: 6961: 6934: 6903: 6879: 6859: 6839: 6687: 6585: 6427: 6051: 6009: 5932: 5854:is a spacetime index, 5839: 5651: 5488: 5445: 5414: 5339: 5298: 5035: 4756: 4727: 4684: 4646: 4608: 4526: 4490: 4462: 4428: 4381: 4358: 4322: 4234: 4195: 4146: 4112: 4011: 3935:, respectively, where 3929: 3890: 3843: 3697: 3615: 3491:Canonical quantization 3485:Canonical quantization 3451: 3342: 3074: 2973: 2851: 2731: 2702: 2675: 2498: 2436:Canonical quantization 2343:Classical field theory 2116:Quantum chromodynamics 1979: 1929: 1800: 1702:John Archibald Wheeler 1568:resonance fluorescence 1512:canonical quantization 1284:classical field theory 1279: 1263:Theoretical background 1144:classical field theory 339:Quantum chromodynamics 216:Effective field theory 21841:Cosmological constant 21617:Philosophy of physics 21108:Bosonic string theory 21020:Semiclassical gravity 20917:Holographic principle 20907:Gravitational anomaly 20722:Quantum teleportation 20250:Wave–particle duality 19990:Quantum hydrodynamics 19985:Quantum hadrodynamics 19609:Scalar chromodynamics 18994:John Wiley & Sons 18922:Kleinert, H. (2008). 18328:Simon, Barry (1974). 17165:. Office of Science, 16578:10.1073/pnas.37.7.455 16511:10.1073/pnas.37.7.452 15898:Quantum hydrodynamics 15893:Quantum hadrodynamics 15779:Abraham–Lorentz force 15713: 15672: 15650:analytic continuation 15485:electromagnetic force 15366: 15206: 15009: 14489: 14126: 13979: 13889: 13804: 13696: 13408:equivalence principle 13389: 13356: 13299: 13198: 12936: 12756: 12660: 12391:is an element of the 12386: 12336: 12334:{\displaystyle e^{i}} 12271: 12227: 12162: 11944: 11914: 11838: 11773: 11429: 11242:differential equation 11222: 11066:renormalization group 11060:Renormalization group 11054:Renormalization group 10980: 10826: 10471: 10409:being the regulator. 10289: 10170: 10143: 10087: 10085:{\displaystyle z_{j}} 10060: 10058:{\displaystyle z_{i}} 10030: 9981: 9979:{\displaystyle z_{i}} 9949: 9947:{\displaystyle \phi } 9919: 9893: 9873: 9830: 9683: 9482: 8885: 8858: 8560: 8518: 8282: 8017: 7944: 7924: 7827: 7770: 7395: 7359: 7320: 7018: 6990: 6962: 6960:{\displaystyle y^{0}} 6935: 6933:{\displaystyle x^{0}} 6904: 6880: 6860: 6840: 6688: 6586: 6428: 6052: 6010: 5978:probability amplitude 5933: 5840: 5652: 5489: 5446: 5415: 5340: 5299: 5036: 4757: 4728: 4685: 4647: 4609: 4527: 4491: 4463: 4429: 4382: 4359: 4323: 4235: 4196: 4147: 4113: 4012: 3941:Hermitian conjugation 3930: 3891: 3866:annihilation operator 3844: 3698: 3616: 3452: 3343: 3085:Klein–Gordon equation 3083:This is known as the 3075: 2974: 2831: 2732: 2703: 2676: 2499: 2333:are both set to one. 1950: 1925: 1839:Non-renormalizability 1795: 1606:Klein–Nishina formula 1570:and non-relativistic 1543:uncertainty principle 1431:wave–particle duality 1364:electromagnetic waves 1330:in the 19th century. 1270: 294:Klein–Gordon equation 236:LSZ reduction formula 16:Theoretical framework 22078:Mathematical physics 22068:Quantum field theory 21950:Loop quantum gravity 21889:Theory of everything 21884:Grand Unified Theory 21858:Neutrino oscillation 21705:Quantum field theory 21576:Mathematical physics 21141:Loop quantum gravity 21000:Quantum field theory 20753:Quantum field theory 20682:Quantum metamaterial 20627:Quantum cryptography 20357:Consistent histories 19961:Quantum fluctuations 19931:Loop quantum gravity 19501:Lattice field theory 19435:Quantum Field Theory 19413:Quantum Field Theory 19383:at Wikimedia Commons 19381:Quantum field theory 19290:Quantum Field Theory 19185:Quantum Field Theory 19017:Quantum Field Theory 18990:Quantum Field Theory 18931:. World Scientific. 18845:Quantum Field Theory 18773:Gauge Field Theories 18717:Quantum Field Theory 18715:McMahon, D. (2008). 18245:Quantum Field Theory 16354:. Prometheus Books. 16233:(14 February 2003). 15844:Lattice field theory 15829:Green–Kubo relations 15693: 15593:mathematical physics 15575:, together with the 15399:covariant derivative 15236: 15067: 15054:Schwarzschild metric 14870: 14696:super-Poincaré group 14141: 14000: 13910: 13828: 13733: 13512: 13397:SU(3) × SU(2) × U(1) 13368: 13316: 13234: 12979: 12799: 12688: 12679:covariant derivative 12486: 12356: 12280: 12236: 12197: 11981: 11847: 11795: 11618: 11453:. In the case where 11274: 11263:, has the following 11106: 11095:, has the following 10859: 10507: 10426: 10209: 10196:external points and 10159: 10096: 10069: 10042: 9990: 9963: 9938: 9908: 9882: 9862: 9698: 9529: 8915: 8870: 8579: 8527: 7953: 7933: 7836: 7809: 7415: 7405:two-point function. 7368: 7332: 7061: 7041:correlation function 6999: 6971: 6944: 6917: 6893: 6869: 6849: 6715: 6623: 6450: 6103: 6026: 6022:to some final state 5984: 5974:scattering amplitude 5858: 5706: 5692:would be necessary. 5501: 5455: 5427: 5352: 5321: 5315:Dirac delta function 5051: 4775: 4737: 4694: 4656: 4627: 4539: 4500: 4472: 4438: 4391: 4368: 4332: 4244: 4208: 4159: 4128: 4031: 3954: 3945:commutation relation 3903: 3871: 3864:are replaced by the 3710: 3669: 3648:of a quantum field. 3509: 3387: 3109: 2993: 2758: 2712: 2688: 2511: 2488: 2426:photoelectric effect 2088:in 1970 by Glashow, 1952:Elementary particles 1830:scattering amplitude 1722:action-at-a-distance 1649:of the existence of 1537:. Using first-order 1489:spontaneous emission 1481:Schrödinger equation 1403:photoelectric effect 1309:action at a distance 1272:Magnetic field lines 1136:quantum field theory 377:Theory of everything 231:Lattice field theory 201:Correlation function 23:Quantum field theory 21917:Split supersymmetry 21879:Kaluza–Klein theory 21752:Fermi's interaction 21551:Atmospheric physics 21390:Classical mechanics 21318:branches of physics 21010:Bunch–Davies vacuum 21002:in curved spacetime 20902:Faddeev–Popov ghost 20738:Quantum fluctuation 20707:Quantum programming 20667:Quantum logic gates 20652:Quantum information 20632:Quantum electronics 20107:Classical mechanics 19995:Quantum information 19599:Quartic interaction 19084:1996rqmi.book.....Y 18988:; Shaw, G. (1993). 18542:1975CMaPh..42..281O 18499:1973CMaPh..31...83O 18263:2000LNP...558...43B 18208:2016JMP....57i2301N 18162:Gerald B. Folland, 18093:1958AnPhy...3...91T 18018:2017AnPhy.384..254P 17955:1989CMaPh.121..351W 17729:2017EPJWC.13601002M 17684:1975PhLB...56..177N 17558:2002SHPMP..33....3B 17474:2015JHEP...05..031A 17359:1932ZPhy...75..622F 17217:2016PhST..168a4003W 17005:1979Sci...206.1290C 16999:(4424): 1290–1292. 16958:1954PhRv...96..191Y 16892:2016stpp.conf....1T 16691:1966PhRv..152.1219S 16569:1951PNAS...37..455S 16502:1951PNAS...37..452S 16440:1966Sci...154..864T 16324:1981PhT....34k..69W 16263:Joseph John Thomson 16140:2013AmJPh..81..211H 15674:Prove that for any 15668:Yang–Mills theories 15638:Haag–Kastler axioms 15605:Goldstone's theorem 15542:perturbation theory 15538:interaction picture 15472:perturbation theory 15448:Chern–Simons theory 14835:Kaluza–Klein theory 14827:high-energy physics 14770:grand unified scale 14531:Goldstone's theorem 13927: 13860: 13845: 13434:charge conservation 13428:. For example, the 13117: 13063: 12958:. Repeated indices 12952:structure constants 12927: 12912: 12875: 12847: 12819: 12737: 12597: 11240:is performed. This 10931: 10817: 10777: 10664: 10624: 10609: 10485:is the bare field, 9510:. For example, the 8902:further reduce any 8820: 8746: 7710: 7605: 6562: 5948:. If the parameter 5697:quartic interaction 5690:perturbation theory 5667:second quantization 5622: 5584: 5541: 5483: 5317:. The vacuum state 5279: 5250: 5108: 4978: 4722: 4328:which implies that 3947:between the two is 3468:harmonic oscillator 3367:complex conjugation 3285: 2984:equations of motion 2751:on the Lagrangian: 2357:gravitational field 2297:quantum Hall effect 2179:Holger Bech Nielsen 2132:asymptotically free 2128:Hugh David Politzer 1994:of QED, leading to 1815:vacuum polarization 1787:Shinichiro Tomonaga 1659:Carl David Anderson 1635:Fermi's interaction 1559:quantum fluctuation 1539:perturbation theory 1485:stimulated emission 1379:blackbody radiation 1344:Maxwell's equations 1321:gravitational field 1249:strong interactions 1164:subatomic particles 1132:theoretical physics 356:Incomplete theories 21945:Superstring theory 21715:Strong interaction 21607:History of physics 21195:Group field theory 21123:Superstring theory 21078:Firewall (physics) 20791:in popular culture 20573:Quantum algorithms 20421:Von Neumann–Wigner 20401:Objective collapse 20112:Old quantum theory 19881:Nambu–Jona-Lasinio 19809:Higher dimensional 19716:Wess–Zumino–Witten 19506:Noncommutative QFT 19116:Field Quantization 18709:Introductory texts 18550:10.1007/BF01608978 18507:10.1007/BF01645738 17963:10.1007/BF01217730 17643:Polchinski, Joseph 17367:10.1007/BF01344458 17163:science.energy.gov 17104:Polchinski, Joseph 16088:. Westview Press. 15903:Quantum triviality 15839:Group field theory 15756:Mathematics portal 15708: 15577:Batalin-Vilkovisky 15528:Mathematical rigor 15361: 15220:is the inverse of 15201: 15023:is the inverse of 15004: 14731:superstring theory 14484: 14121: 13974: 13913: 13884: 13846: 13831: 13799: 13691: 13404:general relativity 13384: 13351: 13294: 13193: 13103: 13049: 12931: 12913: 12898: 12861: 12833: 12802: 12751: 12723: 12655: 12580: 12429:non-Abelian groups 12381: 12331: 12266: 12222: 12157: 11946: 11909: 11833: 11768: 11578:Yukawa interaction 11568:scalar field, the 11491:conformal symmetry 11481:asymptotic freedom 11424: 11217: 10975: 10917: 10821: 10803: 10763: 10650: 10610: 10595: 10466: 10284: 10165: 10138: 10082: 10055: 10025: 9976: 9944: 9914: 9888: 9868: 9825: 9678: 9477: 9475: 8880: 8853: 8803: 8729: 8673: 8555: 8513: 8507: 8280: 7939: 7919: 7822: 7765: 7693: 7588: 7509: 7390: 7354: 7315: 7165: 7052:Feynman propagator 7013: 6985: 6957: 6930: 6899: 6875: 6855: 6835: 6683: 6581: 6548: 6423: 6047: 6005: 5928: 5835: 5647: 5590: 5552: 5509: 5484: 5458: 5441: 5410: 5335: 5294: 5254: 5225: 5083: 5031: 4953: 4752: 4723: 4697: 4680: 4642: 4604: 4522: 4486: 4458: 4424: 4377: 4354: 4318: 4230: 4191: 4142: 4108: 4007: 3925: 3886: 3839: 3693: 3611: 3447: 3338: 3269: 3070: 2969: 2727: 2698: 2671: 2494: 2419:degrees of freedom 2265:'s application of 2228:conformal symmetry 2171:Alexander Polyakov 2161:Other developments 2112:strong interaction 2108:Heinrich Leutwyler 1980: 1933:the shoes incident 1876:strong interaction 1763:James Bruce French 1759:Norman Myles Kroll 1694:Ernst Stueckelberg 1682:Robert Oppenheimer 1572:Compton scattering 1470:special relativity 1292:special relativity 1280: 1195:equation of motion 1148:special relativity 241:Partition function 168:Topological charge 88:General relativity 83:Special relativity 22073:Quantum mechanics 22055: 22054: 21978: 21977: 21853:Strong CP problem 21831:Hierarchy problem 21635: 21634: 21622:Physics education 21571:Materials science 21538:Interdisciplinary 21496:Quantum mechanics 21283: 21282: 21240:Eternal inflation 21235:Quantum cosmology 21223: 21222: 21015:Hawking radiation 20980:Liouville gravity 20834: 20833: 20808:Quantum mysticism 20786:Schrödinger's cat 20717:Quantum simulator 20687:Quantum metrology 20615:Quantum computing 20578:Quantum amplifier 20555:Quantum spacetime 20520:Quantum cosmology 20510:Quantum chemistry 20225:Scattering theory 20173:Zero-point energy 20168:Degenerate levels 20076:Quantum mechanics 20042: 20041: 19904: 19903: 19379:Media related to 19330:978-0-7923-0540-8 19303:978-0-521-46946-3 19155:978-0-201-50397-5 19127:978-3-540-59179-5 19101:978-3-540-60453-2 19031:978-0-521-33859-2 19003:978-0-471-94186-6 18977:978-0-19-851155-7 18938:978-981-279-170-2 18914:978-981-02-4658-7 18886:978-0-201-11749-3 18860:978-0-07-032071-0 18826:978-3-540-67672-0 18786:2008, 3rd edition 18760:978-0-8053-0983-6 18751:Benjamin Cummings 18730:978-0-07-154382-8 18700:978-0-297-81752-9 18674:978-0-691-12575-6 18655:978-0-262-56003-0 18474:. W. A. Benjamin. 18372:978-1-4612-4728-9 18280:978-3-540-67972-1 18216:10.1063/1.4962800 18152:978-0-8218-5288-0 18080:Annals of Physics 18060:978-1-4612-7475-9 17995:Annals of Physics 17804:978-0-521-87787-9 17771:978-3-662-04273-1 17672:Physics Letters B 17656:978-0-521-67228-3 17607:978-0-521-55001-7 17415:978-0-521-86069-7 17353:(9–10): 622–647. 17328:978-0-691-01019-9 17117:978-0-521-67227-6 17075:Kibble, Tom W. B. 17043:Sutton, Christine 16909:978-981-4733-50-2 16848:978-0-19-852745-9 16778:978-0-7382-0055-2 16753:978-0-7382-0054-5 16630:(22): 7783–7788. 16434:(3751): 864–868. 16408:978-0-521-19084-8 16361:978-1-57392-694-2 16346:Werner Heisenberg 16332:10.1063/1.2914365 16306:Weisskopf, Victor 16248:978-0-19-974376-6 16148:10.1119/1.4789885 16095:978-0-201-50397-5 15476:virtual particles 15331: 15279: 15264: 14766:hierarchy problem 14654:Poincaré symmetry 14442: 14399: 14360: 14337: 14294: 14235: 14162: 14096: 14095: 13964: 13963: 13879: 13726:global symmetry: 13649: 13606: 13533: 13473:BRST quantization 13418:Noether's theorem 13217:is an element of 12633: 12578: 12512: 12431:, giving rise to 11931:elementary charge 11807: 11739: 11696: 11640: 11593:representing the 11446:is the number of 11376: 11352: 11320: 11300: 11184: 11152: 11132: 11088:elementary charge 10801: 10751: 10676: 10648: 10593: 10528: 10338:virtual particles 10168:{\displaystyle 2} 9917:{\displaystyle z} 9902:internal vertices 9891:{\displaystyle y} 9871:{\displaystyle x} 9856:external vertices 9728: 9553: 8848: 8640: 8508: 8356: 8091: 7891: 7760: 7476: 7258: 7207: 7150: 6902:{\displaystyle T} 6878:{\displaystyle y} 6858:{\displaystyle x} 6777: 6436:Taking the limit 6067:, the total time 5946:Einstein notation 5893: 5820: 5782: 5727: 5600: 5562: 5519: 5468: 5400: 5365: 5264: 5235: 5201: 5177: 5093: 5069: 4963: 4892: 4875: 4874: 4851: 4787: 4707: 4669: 4639: 4571: 4513: 4404: 4345: 4306: 4275: 4259: 4219: 4171: 4097: 4083: 4065: 4043: 3987: 3971: 3916: 3898:creation operator 3883: 3811: 3799: 3798: 3761: 3750: 3749: 3722: 3681: 3662:is promoted to a 3580: 3579: 3540: 3539: 3442: 3200: 3199: 3176: 3100:Fourier transform 3028: 2958: 2927: 2872: 2822: 2774: 2724: 2641: 2609: 2589: 2577: 2497:{\displaystyle L} 2361:Newtonian gravity 2271:phase transitions 2252:many-body systems 1974:, as well as the 1905:classical process 1888:conservation laws 1862:coupling constant 1733:Robert Retherford 1671:Dirac hole theory 1555:zero-point energy 1455:Erwin Schrödinger 1447:Werner Heisenberg 1435:quantum mechanics 1288:quantum mechanics 1274:visualized using 1201:computed for the 1154:. QFT is used in 1152:quantum mechanics 1128: 1127: 221:Expectation value 196:BRST quantization 143:Poincaré symmetry 98:Yang–Mills theory 78:Quantum mechanics 22085: 22043: 22042: 22031: 22030: 21821: 21820: 21776: 21775: 21757:Weak hypercharge 21742:Weak interaction 21683:Particle physics 21662: 21655: 21648: 21639: 21638: 21561:Chemical physics 21501:Particle physics 21427:Classical optics 21310: 21303: 21296: 21287: 21286: 21272: 21271: 21096: 21095: 21058:Bekenstein bound 20875:Central concepts 20861: 20854: 20847: 20838: 20837: 20824: 20823: 20535:Quantum geometry 20530:Quantum dynamics 20387:Superdeterminism 20283:Matrix mechanics 20138:Bra–ket notation 20069: 20062: 20055: 20046: 20045: 20034: 20033: 19951:Quantum dynamics 19624:Yang–Mills–Higgs 19579:Non-linear sigma 19569:Euler–Heisenberg 19554: 19553: 19465: 19458: 19451: 19442: 19441: 19437:by P. J. Mulders 19403: 19378: 19364: 19348: 19334: 19311:Bogoliubov, N.; 19307: 19286:Brown, Lowell S. 19274: 19259:(2nd ed.). 19258: 19244: 19221: 19219: 19218: 19203: 19199:978-0521-8644-97 19178: 19159: 19131: 19119: 19105: 19067: 19065: 19064: 19055:. Archived from 19035: 19007: 18981: 18942: 18930: 18918: 18905:World Scientific 18890: 18864: 18848: 18830: 18800: 18780: 18764: 18734: 18704: 18678: 18659: 18643: 18629: 18613: 18599: 18561: 18518: 18475: 18473: 18453: 18452: 18450: 18449: 18443: 18437:. Archived from 18432: 18416: 18410: 18409: 18407: 18391: 18385: 18384: 18358: 18352: 18351: 18325: 18319: 18318: 18316: 18304: 18293: 18292: 18256: 18241:Buchholz, Detlev 18237: 18228: 18227: 18201: 18181: 18175: 18160: 18154: 18142:Kevin Costello, 18140: 18134: 18133: 18126:Dan Mat Fys Medd 18123: 18111: 18105: 18104: 18071: 18065: 18064: 18044: 18038: 18037: 18011: 17989: 17983: 17982: 17929: 17923: 17922: 17904: 17884: 17878: 17877: 17857: 17836: 17830: 17829: 17827: 17815: 17809: 17808: 17792: 17782: 17776: 17775: 17755: 17749: 17748: 17722: 17702: 17696: 17695: 17667: 17661: 17660: 17639: 17633: 17632: 17630: 17618: 17612: 17611: 17599: 17590:Weinberg, Steven 17586: 17580: 17579: 17569: 17541: 17535: 17528: 17522: 17521: 17519: 17509: 17500: 17494: 17493: 17467: 17447: 17441: 17440: 17438: 17426: 17420: 17419: 17403: 17394:Schwarz, John H. 17388:Becker, Katrin; 17385: 17379: 17378: 17339: 17333: 17332: 17320: 17307: 17262: 17258: 17252: 17246: 17237: 17236: 17210: 17201:(T168): 014003. 17187: 17178: 17177: 17175: 17174: 17160: 17152: 17146: 17145: 17143: 17132:Schwarz, John H. 17128: 17122: 17121: 17100: 17089: 17088: 17086: 17071: 17065: 17064: 17062: 17061: 17047:"Standard model" 17039: 17033: 17032: 16983: 16972: 16971: 16969: 16928: 16922: 16921: 16885: 16870:'t Hooft, Gerard 16866: 16853: 16852: 16834: 16828: 16827: 16819: 16813: 16812: 16804: 16798: 16797: 16789: 16783: 16782: 16764: 16758: 16757: 16739: 16733: 16732: 16714: 16703: 16702: 16674: 16668: 16667: 16657: 16639: 16615: 16609: 16608: 16598: 16580: 16548: 16542: 16541: 16531: 16513: 16481: 16475: 16474: 16466: 16460: 16459: 16419: 16413: 16412: 16391: 16366: 16365: 16342: 16336: 16335: 16302: 16271: 16270: 16259: 16253: 16252: 16231:John L. Heilbron 16227: 16221: 16220: 16201:Weinberg, Steven 16197: 16160: 16159: 16133: 16124:(211): 211–223. 16111: 16100: 16099: 16076: 15956:Wigner's theorem 15814:Dirac's equation 15772: 15767: 15766: 15758: 15753: 15752: 15724: 15717: 15715: 15714: 15709: 15707: 15706: 15701: 15688: 15591:, a subfield of 15423: 15396: 15387: 15370: 15368: 15367: 15362: 15357: 15353: 15352: 15351: 15342: 15341: 15332: 15324: 15316: 15315: 15303: 15302: 15293: 15292: 15280: 15272: 15265: 15263: 15255: 15250: 15245: 15244: 15228: 15219: 15210: 15208: 15207: 15202: 15194: 15193: 15181: 15180: 15171: 15170: 15152: 15151: 15139: 15138: 15125: 15124: 15115: 15114: 15105: 15104: 15089: 15088: 15079: 15078: 15047: 15031: 15022: 15013: 15011: 15010: 15005: 14997: 14996: 14984: 14983: 14974: 14973: 14955: 14954: 14942: 14941: 14928: 14927: 14918: 14917: 14908: 14907: 14892: 14891: 14882: 14881: 14855: 14839:extra dimensions 14837:uses gravity in 14805: 14796:Other spacetimes 14754:and vice versa. 14747: 14728: 14721: 14714: 14685: 14676: 14664: 14609:magnetic dipoles 14595: 14590:massless fields 14589: 14582: 14575: 14563: 14551: 14540: 14526: 14515: 14507: 14493: 14491: 14490: 14485: 14480: 14479: 14474: 14470: 14469: 14468: 14459: 14458: 14443: 14435: 14430: 14429: 14420: 14419: 14410: 14409: 14400: 14392: 14384: 14383: 14374: 14373: 14361: 14356: 14351: 14350: 14338: 14333: 14328: 14327: 14318: 14314: 14313: 14312: 14295: 14287: 14282: 14278: 14274: 14273: 14259: 14255: 14251: 14250: 14236: 14228: 14223: 14219: 14218: 14217: 14208: 14207: 14193: 14189: 14188: 14187: 14178: 14177: 14163: 14155: 14150: 14149: 14130: 14128: 14127: 14122: 14117: 14113: 14097: 14091: 14087: 14073: 14072: 14039: 14038: 14012: 14011: 13992: 13983: 13981: 13980: 13975: 13970: 13966: 13965: 13959: 13955: 13926: 13921: 13902: 13893: 13891: 13890: 13885: 13880: 13875: 13874: 13865: 13859: 13854: 13844: 13839: 13820: 13808: 13806: 13805: 13800: 13786: 13771: 13770: 13761: 13760: 13745: 13744: 13725: 13715: 13709: 13700: 13698: 13697: 13692: 13687: 13686: 13681: 13677: 13676: 13675: 13666: 13665: 13650: 13642: 13637: 13636: 13627: 13626: 13617: 13616: 13607: 13599: 13594: 13590: 13589: 13588: 13579: 13578: 13564: 13560: 13559: 13558: 13549: 13548: 13534: 13526: 13521: 13520: 13504: 13458: 13444: 13431: 13426:conservation law 13398: 13393: 13391: 13390: 13385: 13380: 13379: 13360: 13358: 13357: 13352: 13344: 13343: 13325: 13324: 13303: 13301: 13300: 13295: 13290: 13289: 13288: 13287: 13278: 13277: 13226: 13220: 13216: 13202: 13200: 13199: 13194: 13180: 13179: 13170: 13166: 13165: 13164: 13155: 13154: 13136: 13135: 13116: 13111: 13082: 13081: 13062: 13057: 13035: 13034: 13016: 13015: 12991: 12990: 12971: 12957: 12949: 12940: 12938: 12937: 12932: 12926: 12921: 12911: 12906: 12897: 12896: 12874: 12869: 12860: 12859: 12846: 12841: 12832: 12831: 12818: 12813: 12791: 12783: 12775: 12769: 12760: 12758: 12757: 12752: 12747: 12746: 12736: 12731: 12713: 12712: 12700: 12699: 12676: 12664: 12662: 12661: 12656: 12651: 12650: 12641: 12640: 12635: 12634: 12626: 12616: 12615: 12596: 12591: 12579: 12571: 12566: 12565: 12556: 12555: 12543: 12542: 12530: 12529: 12520: 12519: 12514: 12513: 12505: 12495: 12494: 12478: 12470: 12455: 12444: 12422: 12413: 12409: 12400: 12396: 12390: 12388: 12387: 12382: 12380: 12379: 12351: 12340: 12338: 12337: 12332: 12330: 12329: 12316: 12275: 12273: 12272: 12267: 12265: 12264: 12254: 12231: 12229: 12228: 12223: 12221: 12220: 12180: 12166: 12164: 12163: 12158: 12153: 12152: 12131: 12130: 12121: 12120: 12096: 12095: 12068: 12067: 12046: 12045: 12020: 12019: 11973: 11937: 11928: 11918: 11916: 11915: 11910: 11908: 11907: 11898: 11897: 11885: 11884: 11875: 11874: 11862: 11861: 11842: 11840: 11839: 11834: 11832: 11831: 11822: 11821: 11809: 11808: 11800: 11786: 11777: 11775: 11774: 11769: 11764: 11763: 11751: 11750: 11741: 11740: 11732: 11723: 11722: 11710: 11709: 11697: 11689: 11681: 11677: 11670: 11669: 11660: 11659: 11642: 11641: 11633: 11627: 11626: 11602: 11592: 11561: 11546: 11542: 11533: 11529: 11521: 11512: 11478: 11472: 11462: 11445: 11433: 11431: 11430: 11425: 11420: 11419: 11418: 11414: 11413: 11392: 11388: 11387: 11386: 11377: 11369: 11353: 11351: 11350: 11349: 11336: 11335: 11326: 11321: 11319: 11311: 11303: 11301: 11293: 11262: 11252: 11239: 11233: 11226: 11224: 11223: 11218: 11213: 11212: 11211: 11207: 11206: 11185: 11183: 11182: 11181: 11168: 11167: 11158: 11153: 11151: 11143: 11135: 11133: 11125: 11094: 11037: 11033: 11024: 11015: 11006: 11002: 10993: 10984: 10982: 10981: 10976: 10974: 10973: 10961: 10960: 10945: 10944: 10930: 10925: 10910: 10909: 10897: 10896: 10871: 10870: 10851: 10842: 10830: 10828: 10827: 10822: 10816: 10811: 10802: 10800: 10792: 10791: 10782: 10776: 10771: 10762: 10761: 10752: 10744: 10736: 10735: 10726: 10725: 10710: 10709: 10700: 10699: 10687: 10686: 10677: 10669: 10663: 10658: 10649: 10647: 10639: 10638: 10629: 10623: 10618: 10608: 10603: 10594: 10586: 10578: 10577: 10568: 10567: 10552: 10551: 10542: 10541: 10529: 10521: 10516: 10515: 10499: 10493: 10484: 10475: 10473: 10472: 10467: 10462: 10461: 10452: 10451: 10447: 10418: 10408: 10400: 10396: 10391: 10385: 10379: 10373: 10367: 10335: 10329: 10309: 10302: 10293: 10291: 10290: 10285: 10277: 10266: 10265: 10244: 10243: 10222: 10201: 10195: 10189: 10183: 10174: 10172: 10171: 10166: 10147: 10145: 10144: 10139: 10134: 10133: 10121: 10120: 10108: 10107: 10091: 10089: 10088: 10083: 10081: 10080: 10064: 10062: 10061: 10056: 10054: 10053: 10034: 10032: 10031: 10026: 10024: 10023: 10014: 10013: 9985: 9983: 9982: 9977: 9975: 9974: 9953: 9951: 9950: 9945: 9923: 9921: 9920: 9915: 9897: 9895: 9894: 9889: 9877: 9875: 9874: 9869: 9846: 9834: 9832: 9831: 9826: 9806: 9805: 9781: 9780: 9756: 9755: 9742: 9741: 9729: 9727: 9719: 9708: 9687: 9685: 9684: 9679: 9668: 9582: 9567: 9566: 9554: 9552: 9544: 9533: 9521: 9515: 9486: 9484: 9483: 9478: 9476: 9463: 9452: 9451: 9433: 9432: 9411: 9394: 9383: 9382: 9364: 9363: 9342: 9325: 9315: 9304: 9303: 9285: 9284: 9263: 9246: 9235: 9234: 9216: 9215: 9194: 9177: 9167: 9156: 9155: 9137: 9136: 9115: 9098: 9087: 9086: 9068: 9067: 9046: 9022: 9011: 9010: 8992: 8991: 8973: 8972: 8954: 8953: 8932: 8907: 8895: 8889: 8887: 8886: 8881: 8879: 8878: 8862: 8860: 8859: 8854: 8849: 8847: 8846: 8842: 8841: 8840: 8830: 8829: 8819: 8814: 8784: 8783: 8773: 8772: 8768: 8767: 8766: 8756: 8755: 8745: 8740: 8686: 8685: 8675: 8672: 8630: 8592: 8564: 8562: 8561: 8556: 8548: 8540: 8522: 8520: 8519: 8514: 8509: 8500: 8492: 8491: 8482: 8481: 8469: 8468: 8456: 8455: 8446: 8445: 8433: 8432: 8417: 8406: 8405: 8396: 8395: 8380: 8379: 8370: 8369: 8357: 8355: 8348: 8347: 8328: 8327: 8326: 8304: 8301: 8296: 8273: 8265: 8264: 8255: 8254: 8242: 8241: 8229: 8228: 8219: 8218: 8206: 8205: 8187: 8186: 8168: 8167: 8152: 8141: 8140: 8131: 8130: 8115: 8114: 8105: 8104: 8092: 8090: 8083: 8082: 8063: 8062: 8061: 8039: 8036: 8031: 8015: 8004: 7966: 7948: 7946: 7945: 7940: 7928: 7926: 7925: 7920: 7918: 7917: 7902: 7901: 7892: 7890: 7879: 7873: 7872: 7848: 7847: 7831: 7829: 7828: 7823: 7821: 7820: 7796: 7783: 7774: 7772: 7771: 7766: 7761: 7759: 7755: 7751: 7747: 7746: 7742: 7741: 7737: 7727: 7726: 7709: 7704: 7654: 7650: 7646: 7642: 7641: 7637: 7636: 7632: 7622: 7621: 7604: 7599: 7561: 7560: 7542: 7541: 7511: 7508: 7466: 7428: 7399: 7397: 7396: 7391: 7380: 7379: 7363: 7361: 7360: 7355: 7344: 7343: 7324: 7322: 7321: 7316: 7311: 7310: 7306: 7305: 7293: 7292: 7280: 7279: 7259: 7257: 7247: 7246: 7234: 7233: 7224: 7223: 7210: 7208: 7206: 7205: 7204: 7185: 7181: 7180: 7170: 7164: 7131: 7130: 7112: 7074: 7045:Green's function 7034: 7028: 7022: 7020: 7019: 7014: 7006: 6994: 6992: 6991: 6986: 6978: 6966: 6964: 6963: 6958: 6956: 6955: 6939: 6937: 6936: 6931: 6929: 6928: 6908: 6906: 6905: 6900: 6884: 6882: 6881: 6876: 6864: 6862: 6861: 6856: 6844: 6842: 6841: 6836: 6828: 6790: 6778: 6775: 6766: 6728: 6692: 6690: 6689: 6684: 6679: 6678: 6650: 6649: 6611: 6605: 6599: 6590: 6588: 6587: 6582: 6577: 6573: 6561: 6556: 6520: 6519: 6504: 6503: 6494: 6489: 6488: 6470: 6465: 6464: 6442: 6432: 6430: 6429: 6424: 6416: 6415: 6406: 6401: 6400: 6396: 6374: 6369: 6368: 6353: 6352: 6343: 6338: 6337: 6333: 6311: 6306: 6305: 6287: 6286: 6271: 6266: 6265: 6261: 6239: 6234: 6233: 6220: 6219: 6195: 6194: 6179: 6178: 6157: 6156: 6147: 6142: 6141: 6123: 6118: 6117: 6084: 6078: 6073:is divided into 6072: 6066: 6056: 6054: 6053: 6048: 6043: 6042: 6033: 6021: 6014: 6012: 6011: 6006: 6001: 6000: 5991: 5953: 5943: 5937: 5935: 5934: 5929: 5927: 5926: 5914: 5903: 5902: 5891: 5881: 5870: 5869: 5853: 5844: 5842: 5841: 5836: 5831: 5830: 5821: 5819: 5808: 5803: 5802: 5793: 5792: 5783: 5775: 5770: 5766: 5762: 5761: 5741: 5740: 5728: 5720: 5715: 5714: 5656: 5654: 5653: 5648: 5637: 5632: 5631: 5626: 5621: 5616: 5615: 5614: 5609: 5602: 5601: 5593: 5583: 5578: 5577: 5576: 5571: 5564: 5563: 5555: 5551: 5550: 5545: 5540: 5535: 5534: 5533: 5528: 5521: 5520: 5512: 5493: 5491: 5490: 5485: 5482: 5477: 5476: 5470: 5469: 5461: 5450: 5448: 5447: 5442: 5434: 5419: 5417: 5416: 5411: 5406: 5401: 5398: 5380: 5375: 5374: 5373: 5367: 5366: 5358: 5344: 5342: 5341: 5336: 5328: 5312: 5303: 5301: 5300: 5295: 5284: 5280: 5278: 5273: 5272: 5266: 5265: 5257: 5249: 5244: 5243: 5237: 5236: 5228: 5216: 5212: 5211: 5210: 5209: 5203: 5202: 5194: 5187: 5186: 5185: 5179: 5178: 5170: 5154: 5146: 5135: 5134: 5113: 5109: 5107: 5102: 5101: 5095: 5094: 5086: 5079: 5078: 5077: 5071: 5070: 5062: 5040: 5038: 5037: 5032: 5027: 5023: 5022: 5021: 5020: 5012: 4998: 4997: 4996: 4977: 4972: 4971: 4965: 4964: 4956: 4949: 4948: 4947: 4939: 4925: 4924: 4923: 4902: 4901: 4900: 4894: 4893: 4885: 4876: 4873: 4872: 4871: 4858: 4854: 4852: 4850: 4849: 4848: 4829: 4825: 4824: 4814: 4797: 4789: 4788: 4780: 4767: 4761: 4759: 4758: 4753: 4751: 4750: 4749: 4732: 4730: 4729: 4724: 4721: 4716: 4715: 4709: 4708: 4700: 4689: 4687: 4686: 4681: 4679: 4678: 4677: 4671: 4670: 4662: 4651: 4649: 4648: 4643: 4641: 4640: 4632: 4622: 4613: 4611: 4610: 4605: 4594: 4589: 4588: 4583: 4579: 4578: 4573: 4572: 4564: 4546: 4531: 4529: 4528: 4523: 4521: 4520: 4515: 4514: 4506: 4495: 4493: 4492: 4487: 4479: 4467: 4465: 4464: 4459: 4454: 4433: 4431: 4430: 4425: 4417: 4412: 4411: 4406: 4405: 4397: 4386: 4384: 4383: 4378: 4363: 4361: 4360: 4355: 4353: 4352: 4347: 4346: 4338: 4327: 4325: 4324: 4319: 4314: 4313: 4308: 4307: 4299: 4283: 4282: 4277: 4276: 4268: 4261: 4260: 4252: 4239: 4237: 4236: 4231: 4220: 4212: 4200: 4198: 4197: 4192: 4178: 4173: 4172: 4164: 4151: 4149: 4148: 4143: 4135: 4117: 4115: 4114: 4109: 4098: 4090: 4085: 4084: 4076: 4073: 4072: 4067: 4066: 4058: 4045: 4044: 4036: 4016: 4014: 4013: 4008: 4000: 3996: 3995: 3994: 3989: 3988: 3980: 3973: 3972: 3964: 3938: 3934: 3932: 3931: 3926: 3924: 3923: 3918: 3917: 3909: 3895: 3893: 3892: 3887: 3885: 3884: 3876: 3863: 3857: 3852:Complex numbers 3848: 3846: 3845: 3840: 3835: 3834: 3819: 3818: 3813: 3812: 3804: 3800: 3791: 3787: 3782: 3781: 3763: 3762: 3754: 3751: 3742: 3738: 3724: 3723: 3715: 3702: 3700: 3699: 3694: 3683: 3682: 3674: 3661: 3647: 3641: 3635: 3629: 3620: 3618: 3617: 3612: 3607: 3606: 3591: 3590: 3581: 3572: 3568: 3563: 3562: 3541: 3532: 3528: 3480: 3465: 3456: 3454: 3453: 3448: 3443: 3441: 3440: 3428: 3427: 3422: 3416: 3411: 3406: 3401: 3400: 3399: 3379: 3364: 3356: 3347: 3345: 3344: 3339: 3334: 3330: 3329: 3328: 3327: 3319: 3305: 3304: 3303: 3284: 3279: 3278: 3265: 3264: 3263: 3255: 3241: 3240: 3239: 3218: 3217: 3216: 3201: 3198: 3197: 3196: 3183: 3179: 3177: 3175: 3174: 3173: 3154: 3150: 3149: 3139: 3122: 3079: 3077: 3076: 3071: 3060: 3056: 3055: 3054: 3042: 3041: 3029: 3027: 3026: 3025: 3012: 3011: 3002: 2978: 2976: 2975: 2970: 2959: 2957: 2949: 2948: 2947: 2937: 2932: 2928: 2926: 2922: 2921: 2909: 2891: 2890: 2889: 2879: 2873: 2871: 2870: 2869: 2853: 2850: 2845: 2827: 2823: 2821: 2811: 2793: 2792: 2791: 2781: 2775: 2773: 2762: 2746: 2740: 2736: 2734: 2733: 2728: 2726: 2725: 2717: 2707: 2705: 2704: 2699: 2697: 2696: 2680: 2678: 2677: 2672: 2667: 2663: 2662: 2661: 2652: 2651: 2642: 2634: 2629: 2628: 2610: 2602: 2597: 2596: 2591: 2590: 2582: 2578: 2570: 2559: 2558: 2543: 2542: 2532: 2531: 2503: 2501: 2500: 2495: 2479: 2473: 2467: 2412: 2394: 2376: 2337:Classical fields 2332: 2323: 2311:For simplicity, 2301:Josephson effect 2104:Murray Gell-Mann 2036:François Englert 1990:generalized the 1966:, four types of 1901:external sources 1873: 1854:weak interaction 1826:Feynman diagrams 1775:Julian Schwinger 1767:Victor Weisskopf 1493:linear operators 1443:Louis de Broglie 1427:Louis de Broglie 1356:electric current 1328:electromagnetism 1156:particle physics 1120: 1113: 1106: 211:Effective action 138:Lorentz symmetry 63:Electromagnetism 33: 19: 18: 22093: 22092: 22088: 22087: 22086: 22084: 22083: 22082: 22058: 22057: 22056: 22051: 22019: 21974: 21932:Quantum gravity 21926: 21893: 21862: 21815: 21808: 21799:Higgs mechanism 21777: 21773: 21768: 21671: 21666: 21636: 21631: 21595: 21581:Medical physics 21532: 21491:Nuclear physics 21460: 21454:Non-equilibrium 21376: 21348: 21320: 21314: 21284: 21279: 21278: 21256: 21245:FRW/CFT duality 21219: 21168: 21150: 21127: 21087: 21029: 21001: 20994: 20951: 20870: 20868:Quantum gravity 20865: 20835: 20830: 20812: 20798:Wigner's friend 20774: 20765:Quantum gravity 20726: 20712:Quantum sensing 20692:Quantum network 20672:Quantum machine 20642:Quantum imaging 20605:Quantum circuit 20600:Quantum channel 20559: 20505:Quantum biology 20491: 20467:Elitzur–Vaidman 20442:Davisson–Germer 20425: 20377:Hidden-variable 20367:de Broglie–Bohm 20344:Interpretations 20338: 20302: 20256: 20143:Complementarity 20121: 20078: 20073: 20043: 20038: 20021: 19973:Quantum gravity 19900: 19859:Particle theory 19854: 19833: 19782: 19756: 19720: 19684: 19638:Low dimensional 19633: 19574:Ginzburg–Landau 19545: 19536:Topological QFT 19474: 19469: 19388: 19371: 19361: 19349:. Vol. 1. 19331: 19304: 19271: 19241: 19216: 19214: 19200: 19175: 19156: 19128: 19102: 19062: 19060: 19053: 19032: 19004: 18978: 18939: 18928: 18915: 18887: 18861: 18827: 18797: 18761: 18731: 18701: 18675: 18656: 18626: 18604:Schweber, S. S. 18596: 18571:General readers 18568: 18566:Further reading 18457: 18456: 18447: 18445: 18441: 18430: 18417: 18413: 18392: 18388: 18373: 18359: 18355: 18340: 18326: 18322: 18305: 18296: 18281: 18238: 18231: 18182: 18178: 18161: 18157: 18141: 18137: 18121: 18112: 18108: 18072: 18068: 18061: 18045: 18041: 17990: 17986: 17930: 17926: 17911:10.4064/bc114-1 17885: 17881: 17874: 17837: 17833: 17816: 17812: 17805: 17783: 17779: 17772: 17756: 17752: 17703: 17699: 17668: 17664: 17657: 17640: 17636: 17619: 17615: 17608: 17587: 17583: 17542: 17538: 17529: 17525: 17504: 17501: 17497: 17448: 17444: 17427: 17423: 17416: 17390:Becker, Melanie 17386: 17382: 17340: 17336: 17329: 17308: 17265: 17259: 17255: 17247: 17240: 17195:Physica Scripta 17188: 17181: 17172: 17170: 17158: 17154: 17153: 17149: 17129: 17125: 17118: 17101: 17092: 17072: 17068: 17059: 17057: 17040: 17036: 16987:Coleman, Sidney 16984: 16975: 16945:Physical Review 16929: 16925: 16910: 16867: 16856: 16849: 16835: 16831: 16820: 16816: 16805: 16801: 16790: 16786: 16779: 16765: 16761: 16754: 16740: 16736: 16729: 16715: 16706: 16675: 16671: 16616: 16612: 16549: 16545: 16482: 16478: 16467: 16463: 16420: 16416: 16409: 16392: 16369: 16362: 16343: 16339: 16303: 16274: 16260: 16256: 16249: 16228: 16224: 16198: 16163: 16112: 16103: 16096: 16077: 15970: 15965: 15960: 15824:Feynman diagram 15768: 15761: 15754: 15747: 15744: 15722: 15702: 15697: 15696: 15694: 15691: 15690: 15684: 15634:Wightman axioms 15620:representations 15546:Feynman diagram 15530: 15481:Feynman diagram 15468: 15434:diffeomorphisms 15421: 15416: 15413: 15407: 15395: 15389: 15384: 15375: 15347: 15343: 15337: 15333: 15323: 15311: 15307: 15298: 15294: 15285: 15281: 15271: 15270: 15266: 15259: 15251: 15249: 15240: 15239: 15237: 15234: 15233: 15226: 15221: 15215: 15189: 15185: 15176: 15172: 15163: 15159: 15147: 15143: 15134: 15130: 15120: 15116: 15110: 15106: 15097: 15093: 15084: 15080: 15074: 15070: 15068: 15065: 15064: 15045: 15038: 15033: 15029: 15024: 15018: 14992: 14988: 14979: 14975: 14966: 14962: 14950: 14946: 14937: 14933: 14923: 14919: 14913: 14909: 14900: 14896: 14887: 14883: 14877: 14873: 14871: 14868: 14867: 14858:raise and lower 14853: 14848: 14808:Minkowski space 14801: 14798: 14742: 14723: 14722:supersymmetry, 14716: 14704: 14699: 14683: 14678: 14674: 14669: 14660: 14635: 14629: 14621:Higgs mechanism 14591: 14584: 14577: 14565: 14557: 14542: 14534: 14520: 14509: 14508:. The original 14498: 14475: 14464: 14460: 14454: 14450: 14449: 14445: 14444: 14434: 14425: 14421: 14415: 14411: 14405: 14401: 14391: 14379: 14375: 14369: 14365: 14355: 14346: 14342: 14332: 14323: 14319: 14308: 14304: 14300: 14296: 14286: 14269: 14265: 14264: 14260: 14246: 14242: 14241: 14237: 14227: 14213: 14209: 14203: 14199: 14198: 14194: 14183: 14179: 14173: 14169: 14168: 14164: 14154: 14145: 14144: 14142: 14139: 14138: 14086: 14062: 14058: 14034: 14030: 14029: 14025: 14007: 14003: 14001: 13998: 13997: 13988: 13954: 13935: 13931: 13922: 13917: 13911: 13908: 13907: 13903:-th direction: 13898: 13870: 13866: 13864: 13855: 13850: 13840: 13835: 13829: 13826: 13825: 13819: 13813: 13782: 13766: 13762: 13753: 13749: 13740: 13736: 13734: 13731: 13730: 13717: 13711: 13705: 13682: 13671: 13667: 13661: 13657: 13656: 13652: 13651: 13641: 13632: 13628: 13622: 13618: 13612: 13608: 13598: 13584: 13580: 13574: 13570: 13569: 13565: 13554: 13550: 13544: 13540: 13539: 13535: 13525: 13516: 13515: 13513: 13510: 13509: 13500: 13487: 13481: 13454: 13440: 13429: 13396: 13375: 13374: 13369: 13366: 13365: 13339: 13335: 13320: 13319: 13317: 13314: 13313: 13283: 13279: 13273: 13269: 13256: 13252: 13235: 13232: 13231: 13222: 13218: 13207: 13175: 13171: 13160: 13156: 13147: 13143: 13131: 13127: 13112: 13107: 13102: 13098: 13077: 13073: 13058: 13053: 13030: 13026: 13008: 13004: 12986: 12982: 12980: 12977: 12976: 12959: 12955: 12945: 12922: 12917: 12907: 12902: 12886: 12882: 12870: 12865: 12855: 12851: 12842: 12837: 12827: 12823: 12814: 12806: 12800: 12797: 12796: 12789: 12781: 12771: 12765: 12742: 12738: 12732: 12727: 12708: 12704: 12695: 12691: 12689: 12686: 12685: 12674: 12669: 12646: 12642: 12636: 12625: 12624: 12623: 12602: 12598: 12592: 12584: 12570: 12561: 12557: 12548: 12544: 12538: 12534: 12525: 12521: 12515: 12504: 12503: 12502: 12490: 12489: 12487: 12484: 12483: 12476: 12461: 12446: 12440: 12420: 12411: 12407: 12402: 12398: 12392: 12363: 12359: 12357: 12354: 12353: 12342: 12309: 12287: 12283: 12281: 12278: 12277: 12247: 12243: 12239: 12237: 12234: 12233: 12204: 12200: 12198: 12195: 12194: 12171: 12136: 12132: 12126: 12122: 12101: 12097: 12088: 12084: 12063: 12059: 12041: 12037: 12003: 11999: 11982: 11979: 11978: 11969: 11966: 11960: 11948: 11933: 11929:and the (bare) 11924: 11903: 11899: 11893: 11889: 11880: 11876: 11870: 11866: 11854: 11850: 11848: 11845: 11844: 11827: 11823: 11817: 11813: 11799: 11798: 11796: 11793: 11792: 11782: 11759: 11755: 11746: 11742: 11731: 11730: 11715: 11711: 11702: 11698: 11688: 11665: 11661: 11655: 11651: 11647: 11643: 11632: 11631: 11622: 11621: 11619: 11616: 11615: 11598: 11588: 11583:As an example, 11557: 11553: 11544: 11540: 11531: 11527: 11519: 11507: 11474: 11469: 11464: 11459: 11454: 11443: 11438: 11409: 11405: 11401: 11400: 11399: 11382: 11378: 11368: 11358: 11354: 11345: 11341: 11337: 11331: 11327: 11325: 11312: 11304: 11302: 11292: 11275: 11272: 11271: 11258: 11248: 11235: 11231: 11202: 11198: 11194: 11193: 11192: 11177: 11173: 11169: 11163: 11159: 11157: 11144: 11136: 11134: 11124: 11107: 11104: 11103: 11090: 11068:, developed by 11062: 11056: 11048:quantum gravity 11035: 11031: 11026: 11022: 11017: 11013: 11008: 11007:. Then, define 11004: 11000: 10989: 10969: 10965: 10956: 10952: 10940: 10936: 10926: 10921: 10905: 10901: 10892: 10888: 10866: 10862: 10860: 10857: 10856: 10849: 10844: 10840: 10835: 10812: 10807: 10793: 10787: 10783: 10781: 10772: 10767: 10757: 10753: 10743: 10731: 10727: 10721: 10717: 10705: 10701: 10695: 10691: 10682: 10678: 10668: 10659: 10654: 10640: 10634: 10630: 10628: 10619: 10614: 10604: 10599: 10585: 10573: 10569: 10563: 10559: 10547: 10543: 10537: 10533: 10520: 10511: 10510: 10508: 10505: 10504: 10495: 10491: 10486: 10480: 10457: 10453: 10443: 10439: 10435: 10427: 10424: 10423: 10414: 10406: 10398: 10394: 10387: 10381: 10375: 10369: 10363: 10357:renormalisation 10352: 10350:Renormalization 10346: 10344:Renormalization 10331: 10325: 10305: 10298: 10273: 10261: 10257: 10239: 10235: 10218: 10210: 10207: 10206: 10197: 10191: 10185: 10179: 10160: 10157: 10156: 10129: 10125: 10116: 10112: 10103: 10099: 10097: 10094: 10093: 10076: 10072: 10070: 10067: 10066: 10049: 10045: 10043: 10040: 10039: 10019: 10015: 10009: 10005: 9991: 9988: 9987: 9970: 9966: 9964: 9961: 9960: 9939: 9936: 9935: 9909: 9906: 9905: 9883: 9880: 9879: 9863: 9860: 9859: 9801: 9797: 9776: 9772: 9751: 9747: 9737: 9733: 9720: 9709: 9707: 9699: 9696: 9695: 9664: 9578: 9562: 9558: 9545: 9534: 9532: 9530: 9527: 9526: 9517: 9511: 9508:Feynman diagram 9503: 9501:Feynman diagram 9497: 9495:Feynman diagram 9474: 9473: 9459: 9447: 9443: 9428: 9424: 9407: 9390: 9378: 9374: 9359: 9355: 9338: 9323: 9322: 9311: 9299: 9295: 9280: 9276: 9259: 9242: 9230: 9226: 9211: 9207: 9190: 9175: 9174: 9163: 9151: 9147: 9132: 9128: 9111: 9094: 9082: 9078: 9063: 9059: 9042: 9029: 9018: 9006: 9002: 8987: 8983: 8968: 8964: 8949: 8945: 8928: 8918: 8916: 8913: 8912: 8903: 8891: 8874: 8873: 8871: 8868: 8867: 8836: 8835: 8825: 8821: 8815: 8807: 8799: 8795: 8779: 8778: 8774: 8762: 8761: 8751: 8747: 8741: 8733: 8725: 8721: 8681: 8680: 8676: 8674: 8644: 8626: 8588: 8580: 8577: 8576: 8544: 8536: 8528: 8525: 8524: 8496: 8487: 8483: 8477: 8473: 8464: 8460: 8451: 8447: 8441: 8437: 8428: 8424: 8413: 8401: 8397: 8391: 8387: 8375: 8371: 8365: 8361: 8343: 8339: 8329: 8322: 8318: 8305: 8303: 8297: 8286: 8269: 8260: 8256: 8250: 8246: 8237: 8233: 8224: 8220: 8214: 8210: 8201: 8197: 8182: 8178: 8163: 8159: 8148: 8136: 8132: 8126: 8122: 8110: 8106: 8100: 8096: 8078: 8074: 8064: 8057: 8053: 8040: 8038: 8032: 8021: 8014: 8000: 7962: 7954: 7951: 7950: 7934: 7931: 7930: 7913: 7909: 7897: 7893: 7883: 7878: 7868: 7864: 7843: 7839: 7837: 7834: 7833: 7816: 7812: 7810: 7807: 7806: 7794: 7789: 7779: 7722: 7718: 7705: 7697: 7686: 7682: 7675: 7671: 7667: 7663: 7659: 7655: 7617: 7613: 7600: 7592: 7581: 7577: 7556: 7552: 7537: 7533: 7532: 7528: 7524: 7520: 7516: 7512: 7510: 7480: 7462: 7424: 7416: 7413: 7412: 7375: 7371: 7369: 7366: 7365: 7339: 7335: 7333: 7330: 7329: 7301: 7297: 7288: 7284: 7275: 7271: 7264: 7260: 7242: 7238: 7229: 7225: 7219: 7215: 7214: 7209: 7200: 7196: 7186: 7176: 7172: 7171: 7169: 7154: 7126: 7122: 7108: 7070: 7062: 7059: 7058: 7030: 7024: 7002: 7000: 6997: 6996: 6974: 6972: 6969: 6968: 6951: 6947: 6945: 6942: 6941: 6924: 6920: 6918: 6915: 6914: 6894: 6891: 6890: 6870: 6867: 6866: 6850: 6847: 6846: 6824: 6786: 6774: 6762: 6724: 6716: 6713: 6712: 6709: 6703: 6674: 6670: 6645: 6641: 6624: 6621: 6620: 6607: 6601: 6595: 6557: 6552: 6544: 6540: 6515: 6514: 6499: 6495: 6490: 6475: 6471: 6466: 6460: 6456: 6451: 6448: 6447: 6437: 6411: 6407: 6402: 6392: 6379: 6375: 6370: 6364: 6360: 6348: 6344: 6339: 6329: 6316: 6312: 6307: 6301: 6297: 6276: 6272: 6267: 6257: 6244: 6240: 6235: 6229: 6225: 6209: 6205: 6190: 6186: 6174: 6170: 6152: 6148: 6143: 6128: 6124: 6119: 6113: 6109: 6104: 6101: 6100: 6080: 6074: 6068: 6058: 6038: 6034: 6029: 6027: 6024: 6023: 6016: 5996: 5992: 5987: 5985: 5982: 5981: 5966: 5960: 5949: 5939: 5922: 5918: 5910: 5898: 5894: 5877: 5865: 5861: 5859: 5856: 5855: 5849: 5826: 5822: 5812: 5807: 5798: 5794: 5788: 5784: 5774: 5757: 5753: 5752: 5748: 5736: 5732: 5719: 5710: 5709: 5707: 5704: 5703: 5633: 5627: 5617: 5610: 5605: 5604: 5603: 5592: 5591: 5586: 5585: 5579: 5572: 5567: 5566: 5565: 5554: 5553: 5546: 5536: 5529: 5524: 5523: 5522: 5511: 5510: 5505: 5504: 5502: 5499: 5498: 5478: 5472: 5471: 5460: 5459: 5456: 5453: 5452: 5430: 5428: 5425: 5424: 5402: 5397: 5376: 5369: 5368: 5357: 5356: 5355: 5353: 5350: 5349: 5324: 5322: 5319: 5318: 5308: 5274: 5268: 5267: 5256: 5255: 5245: 5239: 5238: 5227: 5226: 5224: 5220: 5205: 5204: 5193: 5192: 5191: 5181: 5180: 5169: 5168: 5167: 5166: 5162: 5150: 5142: 5130: 5126: 5103: 5097: 5096: 5085: 5084: 5073: 5072: 5061: 5060: 5059: 5058: 5054: 5052: 5049: 5048: 5016: 5008: 4992: 4991: 4987: 4983: 4979: 4973: 4967: 4966: 4955: 4954: 4943: 4935: 4919: 4918: 4914: 4907: 4903: 4896: 4895: 4884: 4883: 4882: 4881: 4877: 4867: 4866: 4862: 4853: 4844: 4840: 4830: 4820: 4816: 4815: 4813: 4793: 4779: 4778: 4776: 4773: 4772: 4763: 4745: 4744: 4740: 4738: 4735: 4734: 4717: 4711: 4710: 4699: 4698: 4695: 4692: 4691: 4673: 4672: 4661: 4660: 4659: 4657: 4654: 4653: 4631: 4630: 4628: 4625: 4624: 4618: 4590: 4584: 4574: 4563: 4562: 4561: 4557: 4556: 4542: 4540: 4537: 4536: 4516: 4505: 4504: 4503: 4501: 4498: 4497: 4475: 4473: 4470: 4469: 4450: 4439: 4436: 4435: 4413: 4407: 4396: 4395: 4394: 4392: 4389: 4388: 4369: 4366: 4365: 4348: 4337: 4336: 4335: 4333: 4330: 4329: 4309: 4298: 4297: 4296: 4278: 4267: 4266: 4265: 4251: 4250: 4245: 4242: 4241: 4211: 4209: 4206: 4205: 4204:and has energy 4174: 4163: 4162: 4160: 4157: 4156: 4131: 4129: 4126: 4125: 4089: 4075: 4074: 4068: 4057: 4056: 4055: 4035: 4034: 4032: 4029: 4028: 3990: 3979: 3978: 3977: 3963: 3962: 3961: 3957: 3955: 3952: 3951: 3936: 3919: 3908: 3907: 3906: 3904: 3901: 3900: 3875: 3874: 3872: 3869: 3868: 3859: 3853: 3824: 3820: 3814: 3803: 3802: 3801: 3786: 3768: 3764: 3753: 3752: 3737: 3714: 3713: 3711: 3708: 3707: 3673: 3672: 3670: 3667: 3666: 3664:linear operator 3652: 3643: 3637: 3631: 3625: 3596: 3592: 3586: 3582: 3567: 3549: 3545: 3527: 3510: 3507: 3506: 3493: 3487: 3479: 3471: 3470:with frequency 3461: 3436: 3432: 3423: 3418: 3417: 3412: 3407: 3405: 3395: 3394: 3390: 3388: 3385: 3384: 3378: 3370: 3362: 3352: 3323: 3315: 3299: 3298: 3294: 3290: 3286: 3280: 3274: 3273: 3259: 3251: 3235: 3234: 3230: 3223: 3219: 3212: 3211: 3207: 3206: 3202: 3192: 3191: 3187: 3178: 3169: 3165: 3155: 3145: 3141: 3140: 3138: 3118: 3110: 3107: 3106: 3050: 3046: 3037: 3033: 3021: 3017: 3013: 3007: 3003: 3001: 3000: 2996: 2994: 2991: 2990: 2950: 2943: 2942: 2938: 2936: 2917: 2913: 2905: 2892: 2885: 2884: 2880: 2878: 2874: 2865: 2861: 2857: 2852: 2846: 2835: 2807: 2794: 2787: 2786: 2782: 2780: 2776: 2766: 2761: 2759: 2756: 2755: 2742: 2738: 2716: 2715: 2713: 2710: 2709: 2692: 2691: 2689: 2686: 2685: 2657: 2653: 2647: 2643: 2633: 2624: 2620: 2601: 2592: 2581: 2580: 2579: 2569: 2568: 2564: 2554: 2550: 2538: 2537: 2527: 2523: 2512: 2509: 2508: 2489: 2486: 2485: 2475: 2469: 2454: 2399: 2381: 2363: 2345: 2339: 2328: 2319: 2309: 2248: 2220:quantum gravity 2201:Evgeny Likhtman 2163: 2100:Harald Fritzsch 2090:John Iliopoulos 2082:Gerard 't Hooft 2066:Steven Weinberg 2040:Gerald Guralnik 2022:John Clive Ward 2014:Sheldon Glashow 2000:symmetry groups 1962:, six types of 1958:: six types of 1945: 1896: 1886:principles and 1868: 1841: 1791:renormalization 1779:Richard Feynman 1748: 1741: 1718:Richard Feynman 1706:S-matrix theory 1679: 1667:pair production 1531:current density 1501: 1416:introduced the 1399:Albert Einstein 1360:electric charge 1332:Michael Faraday 1319:in the case of 1313:Richard Bentley 1265: 1241:renormalization 1225: 1219: 1207:Feynman diagram 1160:physical models 1124: 1095: 1094: 1093: 1091: 395: 387: 386: 382:Quantum gravity 357: 349: 348: 344:Higgs mechanism 324: 314: 313: 299:Proca equations 284: 276: 275: 261:Renormalization 226:Feynman diagram 181: 173: 172: 113: 103: 102: 53: 38: 36:Feynman diagram 17: 12: 11: 5: 22091: 22081: 22080: 22075: 22070: 22053: 22052: 22050: 22049: 22037: 22024: 22021: 22020: 22018: 22017: 22012: 22007: 22002: 21997: 21992: 21986: 21984: 21980: 21979: 21976: 21975: 21973: 21972: 21970:Twistor theory 21967: 21962: 21957: 21952: 21947: 21942: 21936: 21934: 21928: 21927: 21925: 21924: 21919: 21914: 21909: 21903: 21901: 21895: 21894: 21892: 21891: 21886: 21881: 21876: 21870: 21868: 21864: 21863: 21861: 21860: 21855: 21850: 21849: 21848: 21838: 21833: 21827: 21825: 21818: 21816:Standard Model 21810: 21809: 21807: 21806: 21801: 21796: 21791: 21785: 21783: 21779: 21778: 21771: 21769: 21767: 21766: 21765: 21764: 21759: 21754: 21749: 21744: 21734: 21733: 21732: 21727: 21722: 21712: 21707: 21702: 21701: 21700: 21695: 21690: 21679: 21677: 21673: 21672: 21669:Standard Model 21665: 21664: 21657: 21650: 21642: 21633: 21632: 21630: 21629: 21624: 21619: 21614: 21609: 21603: 21601: 21597: 21596: 21594: 21593: 21588: 21583: 21578: 21573: 21568: 21563: 21558: 21553: 21548: 21542: 21540: 21534: 21533: 21531: 21530: 21525: 21524: 21523: 21518: 21513: 21503: 21498: 21493: 21488: 21487: 21486: 21481: 21470: 21468: 21462: 21461: 21459: 21458: 21457: 21456: 21451: 21444:Thermodynamics 21441: 21440: 21439: 21434: 21424: 21419: 21414: 21413: 21412: 21407: 21402: 21397: 21386: 21384: 21378: 21377: 21375: 21374: 21373: 21372: 21362: 21356: 21354: 21350: 21349: 21347: 21346: 21345: 21344: 21334: 21328: 21326: 21322: 21321: 21313: 21312: 21305: 21298: 21290: 21281: 21280: 21277: 21276: 21262: 21261: 21258: 21257: 21255: 21254: 21253: 21252: 21247: 21242: 21231: 21229: 21225: 21224: 21221: 21220: 21218: 21217: 21215:Twistor theory 21212: 21207: 21202: 21197: 21192: 21187: 21182: 21176: 21174: 21170: 21169: 21167: 21166: 21160: 21158: 21152: 21151: 21149: 21148: 21143: 21137: 21135: 21129: 21128: 21126: 21125: 21120: 21115: 21110: 21104: 21102: 21093: 21089: 21088: 21086: 21085: 21080: 21075: 21070: 21065: 21060: 21055: 21050: 21045: 21039: 21037: 21031: 21030: 21028: 21027: 21022: 21017: 21012: 21006: 21004: 20996: 20995: 20993: 20992: 20987: 20982: 20977: 20972: 20967: 20961: 20959: 20953: 20952: 20950: 20949: 20944: 20939: 20934: 20929: 20924: 20919: 20914: 20909: 20904: 20899: 20894: 20889: 20884: 20878: 20876: 20872: 20871: 20864: 20863: 20856: 20849: 20841: 20832: 20831: 20829: 20828: 20817: 20814: 20813: 20811: 20810: 20805: 20800: 20795: 20794: 20793: 20782: 20780: 20776: 20775: 20773: 20772: 20767: 20762: 20761: 20760: 20750: 20745: 20743:Casimir effect 20740: 20734: 20732: 20728: 20727: 20725: 20724: 20719: 20714: 20709: 20704: 20702:Quantum optics 20699: 20694: 20689: 20684: 20679: 20674: 20669: 20664: 20659: 20654: 20649: 20644: 20639: 20634: 20629: 20624: 20623: 20622: 20612: 20607: 20602: 20597: 20596: 20595: 20585: 20580: 20575: 20569: 20567: 20561: 20560: 20558: 20557: 20552: 20547: 20542: 20537: 20532: 20527: 20522: 20517: 20512: 20507: 20501: 20499: 20493: 20492: 20490: 20489: 20484: 20479: 20477:Quantum eraser 20474: 20469: 20464: 20459: 20454: 20449: 20444: 20439: 20433: 20431: 20427: 20426: 20424: 20423: 20418: 20413: 20408: 20403: 20398: 20393: 20392: 20391: 20390: 20389: 20374: 20369: 20364: 20359: 20354: 20348: 20346: 20340: 20339: 20337: 20336: 20331: 20326: 20321: 20316: 20310: 20308: 20304: 20303: 20301: 20300: 20295: 20290: 20285: 20280: 20275: 20270: 20264: 20262: 20258: 20257: 20255: 20254: 20253: 20252: 20247: 20237: 20232: 20227: 20222: 20217: 20212: 20207: 20202: 20197: 20192: 20187: 20182: 20177: 20176: 20175: 20170: 20165: 20160: 20150: 20148:Density matrix 20145: 20140: 20135: 20129: 20127: 20123: 20122: 20120: 20119: 20114: 20109: 20104: 20103: 20102: 20092: 20086: 20084: 20080: 20079: 20072: 20071: 20064: 20057: 20049: 20040: 20039: 20026: 20023: 20022: 20020: 20019: 20014: 20009: 20008: 20007: 19997: 19992: 19987: 19982: 19981: 19980: 19970: 19969: 19968: 19958: 19953: 19948: 19943: 19938: 19933: 19928: 19923: 19918: 19916:Casimir effect 19912: 19910: 19906: 19905: 19902: 19901: 19899: 19898: 19893: 19891:Standard Model 19888: 19883: 19878: 19873: 19868: 19862: 19860: 19856: 19855: 19853: 19852: 19847: 19841: 19839: 19835: 19834: 19832: 19831: 19826: 19821: 19816: 19811: 19806: 19801: 19796: 19790: 19788: 19784: 19783: 19781: 19780: 19775: 19770: 19764: 19762: 19761:Superconformal 19758: 19757: 19755: 19754: 19749: 19744: 19742:Seiberg–Witten 19739: 19734: 19728: 19726: 19725:Supersymmetric 19722: 19721: 19719: 19718: 19713: 19708: 19703: 19698: 19692: 19690: 19686: 19685: 19683: 19682: 19677: 19672: 19667: 19662: 19657: 19652: 19647: 19641: 19639: 19635: 19634: 19632: 19631: 19626: 19621: 19616: 19611: 19606: 19601: 19596: 19591: 19586: 19581: 19576: 19571: 19566: 19560: 19558: 19551: 19547: 19546: 19544: 19543: 19538: 19533: 19528: 19523: 19518: 19513: 19508: 19503: 19498: 19493: 19488: 19482: 19480: 19476: 19475: 19468: 19467: 19460: 19453: 19445: 19439: 19438: 19432: 19429:hep-th/9912205 19416: 19404: 19385: 19384: 19370: 19369:External links 19367: 19366: 19365: 19360:978-0521550017 19359: 19335: 19329: 19308: 19302: 19281: 19280: 19279:Advanced texts 19276: 19275: 19270:978-0691140346 19269: 19245: 19240:978-1108470902 19239: 19222: 19204: 19198: 19179: 19174:978-0486492735 19173: 19160: 19154: 19146:Westview Press 19132: 19126: 19106: 19100: 19072:Ynduráin, F.J. 19068: 19052:978-1107034730 19051: 19036: 19030: 19008: 19002: 18982: 18976: 18960: 18943: 18937: 18919: 18913: 18891: 18885: 18865: 18859: 18831: 18825: 18801: 18796:978-3527408351 18795: 18769:Frampton, P.H. 18765: 18759: 18747:Quantum Fields 18739:Bogolyubov, N. 18735: 18729: 18711: 18710: 18706: 18705: 18699: 18679: 18673: 18660: 18654: 18630: 18624: 18600: 18595:978-0198519973 18594: 18573: 18572: 18567: 18564: 18563: 18562: 18536:(3): 281–305. 18519: 18476: 18462: 18461: 18455: 18454: 18424:Witten, Edward 18411: 18396:Schreiber, Urs 18394:Sati, Hisham; 18386: 18371: 18353: 18338: 18320: 18294: 18279: 18254:hep-th/9811233 18229: 18176: 18155: 18135: 18106: 18066: 18059: 18039: 18002:(C): 254–287. 17984: 17949:(3): 351–399. 17933:Witten, Edward 17924: 17879: 17872: 17840:Carlip, Steven 17831: 17810: 17803: 17777: 17770: 17750: 17697: 17662: 17655: 17634: 17628:hep-th/9801132 17613: 17606: 17581: 17567:10.1.1.569.106 17536: 17523: 17517:hep-th/9908171 17495: 17442: 17436:hep-th/0606101 17421: 17414: 17380: 17334: 17327: 17263: 17253: 17238: 17191:Wilczek, Frank 17179: 17147: 17123: 17116: 17090: 17066: 17051:britannica.com 17034: 16973: 16952:(1): 191–195. 16938:(1954-10-01). 16923: 16908: 16854: 16847: 16829: 16814: 16799: 16784: 16777: 16759: 16752: 16734: 16727: 16704: 16669: 16610: 16563:(7): 455–459. 16543: 16496:(7): 452–455. 16476: 16461: 16414: 16407: 16367: 16360: 16337: 16272: 16254: 16247: 16222: 16161: 16101: 16094: 15967: 15966: 15964: 15961: 15959: 15958: 15953: 15948: 15943: 15938: 15933: 15930: 15925: 15920: 15915: 15910: 15905: 15900: 15895: 15890: 15885: 15880: 15875: 15870: 15861: 15856: 15851: 15846: 15841: 15836: 15831: 15826: 15821: 15816: 15811: 15806: 15801: 15796: 15791: 15786: 15781: 15775: 15774: 15773: 15770:Physics portal 15759: 15743: 15740: 15705: 15700: 15646:imaginary time 15557:Kevin Costello 15534:Haag's theorem 15529: 15526: 15522:Thirring model 15514:minimal models 15489:W and Z bosons 15467: 15464: 15419: 15409:Main article: 15406: 15403: 15391: 15382: 15372: 15371: 15360: 15356: 15350: 15346: 15340: 15336: 15330: 15327: 15322: 15319: 15314: 15310: 15306: 15301: 15297: 15291: 15288: 15284: 15278: 15275: 15269: 15262: 15258: 15254: 15248: 15243: 15224: 15212: 15211: 15200: 15197: 15192: 15188: 15184: 15179: 15175: 15169: 15166: 15162: 15158: 15155: 15150: 15146: 15142: 15137: 15133: 15128: 15123: 15119: 15113: 15109: 15103: 15100: 15096: 15092: 15087: 15083: 15077: 15073: 15043: 15036: 15027: 15015: 15014: 15003: 15000: 14995: 14991: 14987: 14982: 14978: 14972: 14969: 14965: 14961: 14958: 14953: 14949: 14945: 14940: 14936: 14931: 14926: 14922: 14916: 14912: 14906: 14903: 14899: 14895: 14890: 14886: 14880: 14876: 14851: 14797: 14794: 14702: 14681: 14672: 14641:to bosons and 14631:Main article: 14628: 14625: 14613:W and Z bosons 14605:ferromagnetism 14603:In the QFT of 14495: 14494: 14483: 14478: 14473: 14467: 14463: 14457: 14453: 14448: 14441: 14438: 14433: 14428: 14424: 14418: 14414: 14408: 14404: 14398: 14395: 14390: 14387: 14382: 14378: 14372: 14368: 14364: 14359: 14354: 14349: 14345: 14341: 14336: 14331: 14326: 14322: 14317: 14311: 14307: 14303: 14299: 14293: 14290: 14285: 14281: 14277: 14272: 14268: 14263: 14258: 14254: 14249: 14245: 14240: 14234: 14231: 14226: 14222: 14216: 14212: 14206: 14202: 14197: 14192: 14186: 14182: 14176: 14172: 14167: 14161: 14158: 14153: 14148: 14132: 14131: 14120: 14116: 14112: 14109: 14106: 14103: 14100: 14094: 14090: 14085: 14082: 14079: 14076: 14071: 14068: 14065: 14061: 14057: 14054: 14051: 14048: 14045: 14042: 14037: 14033: 14028: 14024: 14021: 14018: 14015: 14010: 14006: 13985: 13984: 13973: 13969: 13962: 13958: 13953: 13950: 13947: 13944: 13941: 13938: 13934: 13930: 13925: 13920: 13916: 13895: 13894: 13883: 13878: 13873: 13869: 13863: 13858: 13853: 13849: 13843: 13838: 13834: 13817: 13810: 13809: 13798: 13795: 13792: 13789: 13785: 13781: 13778: 13774: 13769: 13765: 13759: 13756: 13752: 13748: 13743: 13739: 13702: 13701: 13690: 13685: 13680: 13674: 13670: 13664: 13660: 13655: 13648: 13645: 13640: 13635: 13631: 13625: 13621: 13615: 13611: 13605: 13602: 13597: 13593: 13587: 13583: 13577: 13573: 13568: 13563: 13557: 13553: 13547: 13543: 13538: 13532: 13529: 13524: 13519: 13483:Main article: 13480: 13477: 13383: 13378: 13373: 13350: 13347: 13342: 13338: 13334: 13331: 13328: 13323: 13305: 13304: 13293: 13286: 13282: 13276: 13272: 13268: 13265: 13262: 13259: 13255: 13251: 13248: 13245: 13242: 13239: 13204: 13203: 13192: 13189: 13186: 13183: 13178: 13174: 13169: 13163: 13159: 13153: 13150: 13146: 13142: 13139: 13134: 13130: 13126: 13123: 13120: 13115: 13110: 13106: 13101: 13097: 13094: 13091: 13088: 13085: 13080: 13076: 13072: 13069: 13066: 13061: 13056: 13052: 13047: 13044: 13041: 13038: 13033: 13029: 13025: 13022: 13019: 13014: 13011: 13007: 13003: 13000: 12997: 12994: 12989: 12985: 12942: 12941: 12930: 12925: 12920: 12916: 12910: 12905: 12901: 12895: 12892: 12889: 12885: 12881: 12878: 12873: 12868: 12864: 12858: 12854: 12850: 12845: 12840: 12836: 12830: 12826: 12822: 12817: 12812: 12809: 12805: 12776:are the eight 12762: 12761: 12750: 12745: 12741: 12735: 12730: 12726: 12722: 12719: 12716: 12711: 12707: 12703: 12698: 12694: 12672: 12666: 12665: 12654: 12649: 12645: 12639: 12632: 12629: 12622: 12619: 12614: 12611: 12608: 12605: 12601: 12595: 12590: 12587: 12583: 12577: 12574: 12569: 12564: 12560: 12554: 12551: 12547: 12541: 12537: 12533: 12528: 12524: 12518: 12511: 12508: 12501: 12498: 12493: 12405: 12378: 12375: 12372: 12369: 12366: 12362: 12328: 12325: 12322: 12319: 12315: 12312: 12308: 12305: 12302: 12299: 12296: 12293: 12290: 12286: 12263: 12260: 12257: 12253: 12250: 12246: 12242: 12219: 12216: 12213: 12210: 12207: 12203: 12187:gauge symmetry 12168: 12167: 12156: 12151: 12148: 12145: 12142: 12139: 12135: 12129: 12125: 12119: 12116: 12113: 12110: 12107: 12104: 12100: 12094: 12091: 12087: 12083: 12080: 12077: 12074: 12071: 12066: 12062: 12058: 12055: 12052: 12049: 12044: 12040: 12035: 12032: 12029: 12026: 12023: 12018: 12015: 12012: 12009: 12006: 12002: 11998: 11995: 11992: 11989: 11986: 11962:Main article: 11959: 11958:Gauge symmetry 11956: 11906: 11902: 11896: 11892: 11888: 11883: 11879: 11873: 11869: 11865: 11860: 11857: 11853: 11830: 11826: 11820: 11816: 11812: 11806: 11803: 11789:Dirac matrices 11779: 11778: 11767: 11762: 11758: 11754: 11749: 11745: 11738: 11735: 11729: 11726: 11721: 11718: 11714: 11708: 11705: 11701: 11695: 11692: 11687: 11684: 11680: 11676: 11673: 11668: 11664: 11658: 11654: 11650: 11646: 11639: 11636: 11630: 11625: 11552: 11551:Other theories 11549: 11467: 11457: 11441: 11435: 11434: 11423: 11417: 11412: 11408: 11404: 11398: 11395: 11391: 11385: 11381: 11375: 11372: 11367: 11364: 11361: 11357: 11348: 11344: 11340: 11334: 11330: 11324: 11318: 11315: 11310: 11307: 11299: 11296: 11291: 11288: 11285: 11282: 11279: 11228: 11227: 11216: 11210: 11205: 11201: 11197: 11191: 11188: 11180: 11176: 11172: 11166: 11162: 11156: 11150: 11147: 11142: 11139: 11131: 11128: 11123: 11120: 11117: 11114: 11111: 11070:Kenneth Wilson 11058:Main article: 11055: 11052: 11044:Standard Model 11029: 11020: 11011: 10986: 10985: 10972: 10968: 10964: 10959: 10955: 10951: 10948: 10943: 10939: 10934: 10929: 10924: 10920: 10916: 10913: 10908: 10904: 10900: 10895: 10891: 10886: 10883: 10880: 10877: 10874: 10869: 10865: 10847: 10838: 10832: 10831: 10820: 10815: 10810: 10806: 10799: 10796: 10790: 10786: 10780: 10775: 10770: 10766: 10760: 10756: 10750: 10747: 10742: 10739: 10734: 10730: 10724: 10720: 10716: 10713: 10708: 10704: 10698: 10694: 10690: 10685: 10681: 10675: 10672: 10667: 10662: 10657: 10653: 10646: 10643: 10637: 10633: 10627: 10622: 10617: 10613: 10607: 10602: 10598: 10592: 10589: 10584: 10581: 10576: 10572: 10566: 10562: 10558: 10555: 10550: 10546: 10540: 10536: 10532: 10527: 10524: 10519: 10514: 10489: 10477: 10476: 10465: 10460: 10456: 10450: 10446: 10442: 10438: 10434: 10431: 10403:regularization 10348:Main article: 10345: 10342: 10295: 10294: 10283: 10280: 10276: 10272: 10269: 10264: 10260: 10256: 10253: 10250: 10247: 10242: 10238: 10234: 10231: 10228: 10225: 10221: 10217: 10214: 10164: 10153: 10152: 10149: 10137: 10132: 10128: 10124: 10119: 10115: 10111: 10106: 10102: 10079: 10075: 10052: 10048: 10036: 10022: 10018: 10012: 10008: 10004: 10001: 9998: 9995: 9973: 9969: 9943: 9932: 9931: 9925: 9913: 9899: 9887: 9867: 9849: 9848: 9836: 9835: 9824: 9821: 9818: 9815: 9812: 9809: 9804: 9800: 9796: 9793: 9790: 9787: 9784: 9779: 9775: 9771: 9768: 9765: 9762: 9759: 9754: 9750: 9745: 9740: 9736: 9732: 9726: 9723: 9718: 9715: 9712: 9706: 9703: 9689: 9688: 9677: 9674: 9671: 9667: 9663: 9660: 9657: 9654: 9651: 9648: 9645: 9642: 9639: 9636: 9633: 9630: 9627: 9624: 9621: 9618: 9615: 9612: 9609: 9606: 9603: 9600: 9597: 9594: 9591: 9588: 9585: 9581: 9577: 9574: 9570: 9565: 9561: 9557: 9551: 9548: 9543: 9540: 9537: 9499:Main article: 9496: 9493: 9488: 9487: 9472: 9469: 9466: 9462: 9458: 9455: 9450: 9446: 9442: 9439: 9436: 9431: 9427: 9423: 9420: 9417: 9414: 9410: 9406: 9403: 9400: 9397: 9393: 9389: 9386: 9381: 9377: 9373: 9370: 9367: 9362: 9358: 9354: 9351: 9348: 9345: 9341: 9337: 9334: 9331: 9328: 9326: 9324: 9321: 9318: 9314: 9310: 9307: 9302: 9298: 9294: 9291: 9288: 9283: 9279: 9275: 9272: 9269: 9266: 9262: 9258: 9255: 9252: 9249: 9245: 9241: 9238: 9233: 9229: 9225: 9222: 9219: 9214: 9210: 9206: 9203: 9200: 9197: 9193: 9189: 9186: 9183: 9180: 9178: 9176: 9173: 9170: 9166: 9162: 9159: 9154: 9150: 9146: 9143: 9140: 9135: 9131: 9127: 9124: 9121: 9118: 9114: 9110: 9107: 9104: 9101: 9097: 9093: 9090: 9085: 9081: 9077: 9074: 9071: 9066: 9062: 9058: 9055: 9052: 9049: 9045: 9041: 9038: 9035: 9032: 9030: 9028: 9025: 9021: 9017: 9014: 9009: 9005: 9001: 8998: 8995: 8990: 8986: 8982: 8979: 8976: 8971: 8967: 8963: 8960: 8957: 8952: 8948: 8944: 8941: 8938: 8935: 8931: 8927: 8924: 8921: 8920: 8900:Wick's theorem 8877: 8864: 8863: 8852: 8845: 8839: 8833: 8828: 8824: 8818: 8813: 8810: 8806: 8802: 8798: 8794: 8791: 8787: 8782: 8777: 8771: 8765: 8759: 8754: 8750: 8744: 8739: 8736: 8732: 8728: 8724: 8720: 8717: 8714: 8711: 8708: 8705: 8702: 8699: 8696: 8693: 8689: 8684: 8679: 8671: 8668: 8665: 8662: 8659: 8656: 8653: 8650: 8647: 8643: 8639: 8636: 8633: 8629: 8625: 8622: 8619: 8616: 8613: 8610: 8607: 8604: 8601: 8598: 8595: 8591: 8587: 8584: 8554: 8551: 8547: 8543: 8539: 8535: 8532: 8512: 8506: 8503: 8499: 8495: 8490: 8486: 8480: 8476: 8472: 8467: 8463: 8459: 8454: 8450: 8444: 8440: 8436: 8431: 8427: 8423: 8420: 8416: 8412: 8409: 8404: 8400: 8394: 8390: 8386: 8383: 8378: 8374: 8368: 8364: 8360: 8354: 8351: 8346: 8342: 8338: 8335: 8332: 8325: 8321: 8317: 8314: 8311: 8308: 8300: 8295: 8292: 8289: 8285: 8279: 8276: 8272: 8268: 8263: 8259: 8253: 8249: 8245: 8240: 8236: 8232: 8227: 8223: 8217: 8213: 8209: 8204: 8200: 8196: 8193: 8190: 8185: 8181: 8177: 8174: 8171: 8166: 8162: 8158: 8155: 8151: 8147: 8144: 8139: 8135: 8129: 8125: 8121: 8118: 8113: 8109: 8103: 8099: 8095: 8089: 8086: 8081: 8077: 8073: 8070: 8067: 8060: 8056: 8052: 8049: 8046: 8043: 8035: 8030: 8027: 8024: 8020: 8013: 8010: 8007: 8003: 7999: 7996: 7993: 7990: 7987: 7984: 7981: 7978: 7975: 7972: 7969: 7965: 7961: 7958: 7938: 7916: 7912: 7908: 7905: 7900: 7896: 7889: 7886: 7882: 7876: 7871: 7867: 7863: 7860: 7857: 7854: 7851: 7846: 7842: 7819: 7815: 7792: 7776: 7775: 7764: 7758: 7754: 7750: 7745: 7740: 7736: 7733: 7730: 7725: 7721: 7716: 7713: 7708: 7703: 7700: 7696: 7692: 7689: 7685: 7681: 7678: 7674: 7670: 7666: 7662: 7658: 7653: 7649: 7645: 7640: 7635: 7631: 7628: 7625: 7620: 7616: 7611: 7608: 7603: 7598: 7595: 7591: 7587: 7584: 7580: 7576: 7573: 7570: 7567: 7564: 7559: 7555: 7551: 7548: 7545: 7540: 7536: 7531: 7527: 7523: 7519: 7515: 7507: 7504: 7501: 7498: 7495: 7492: 7489: 7486: 7483: 7479: 7475: 7472: 7469: 7465: 7461: 7458: 7455: 7452: 7449: 7446: 7443: 7440: 7437: 7434: 7431: 7427: 7423: 7420: 7389: 7386: 7383: 7378: 7374: 7353: 7350: 7347: 7342: 7338: 7326: 7325: 7314: 7309: 7304: 7300: 7296: 7291: 7287: 7283: 7278: 7274: 7270: 7267: 7263: 7256: 7253: 7250: 7245: 7241: 7237: 7232: 7228: 7222: 7218: 7213: 7203: 7199: 7195: 7192: 7189: 7184: 7179: 7175: 7168: 7163: 7160: 7157: 7153: 7149: 7146: 7143: 7140: 7137: 7134: 7129: 7125: 7121: 7118: 7115: 7111: 7107: 7104: 7101: 7098: 7095: 7092: 7089: 7086: 7083: 7080: 7077: 7073: 7069: 7066: 7012: 7009: 7005: 6984: 6981: 6977: 6954: 6950: 6927: 6923: 6898: 6874: 6854: 6834: 6831: 6827: 6823: 6820: 6817: 6814: 6811: 6808: 6805: 6802: 6799: 6796: 6793: 6789: 6785: 6782: 6772: 6769: 6765: 6761: 6758: 6755: 6752: 6749: 6746: 6743: 6740: 6737: 6734: 6731: 6727: 6723: 6720: 6705:Main article: 6702: 6699: 6694: 6693: 6682: 6677: 6673: 6669: 6666: 6663: 6660: 6657: 6653: 6648: 6644: 6640: 6637: 6634: 6631: 6628: 6592: 6591: 6580: 6576: 6572: 6568: 6565: 6560: 6555: 6551: 6547: 6543: 6539: 6536: 6532: 6529: 6526: 6523: 6518: 6513: 6510: 6507: 6502: 6498: 6493: 6487: 6484: 6481: 6478: 6474: 6469: 6463: 6459: 6455: 6434: 6433: 6422: 6419: 6414: 6410: 6405: 6399: 6395: 6391: 6388: 6385: 6382: 6378: 6373: 6367: 6363: 6359: 6356: 6351: 6347: 6342: 6336: 6332: 6328: 6325: 6322: 6319: 6315: 6310: 6304: 6300: 6296: 6293: 6290: 6285: 6282: 6279: 6275: 6270: 6264: 6260: 6256: 6253: 6250: 6247: 6243: 6238: 6232: 6228: 6224: 6218: 6215: 6212: 6208: 6204: 6201: 6198: 6193: 6189: 6185: 6182: 6177: 6173: 6169: 6166: 6163: 6160: 6155: 6151: 6146: 6140: 6137: 6134: 6131: 6127: 6122: 6116: 6112: 6108: 6046: 6041: 6037: 6032: 6004: 5999: 5995: 5990: 5962:Main article: 5959: 5958:Path integrals 5956: 5925: 5921: 5917: 5913: 5909: 5906: 5901: 5897: 5890: 5887: 5884: 5880: 5876: 5873: 5868: 5864: 5846: 5845: 5834: 5829: 5825: 5818: 5815: 5811: 5806: 5801: 5797: 5791: 5787: 5781: 5778: 5773: 5769: 5765: 5760: 5756: 5751: 5747: 5744: 5739: 5735: 5731: 5726: 5723: 5718: 5713: 5658: 5657: 5646: 5643: 5640: 5636: 5630: 5625: 5620: 5613: 5608: 5599: 5596: 5589: 5582: 5575: 5570: 5561: 5558: 5549: 5544: 5539: 5532: 5527: 5518: 5515: 5508: 5481: 5475: 5467: 5464: 5440: 5437: 5433: 5421: 5420: 5409: 5405: 5395: 5392: 5389: 5386: 5383: 5379: 5372: 5364: 5361: 5345:is defined by 5334: 5331: 5327: 5305: 5304: 5293: 5290: 5287: 5283: 5277: 5271: 5263: 5260: 5253: 5248: 5242: 5234: 5231: 5223: 5219: 5215: 5208: 5200: 5197: 5190: 5184: 5176: 5173: 5165: 5160: 5157: 5153: 5149: 5145: 5141: 5138: 5133: 5129: 5125: 5122: 5119: 5116: 5112: 5106: 5100: 5092: 5089: 5082: 5076: 5068: 5065: 5057: 5042: 5041: 5030: 5026: 5019: 5015: 5011: 5007: 5004: 5001: 4995: 4990: 4986: 4982: 4976: 4970: 4962: 4959: 4952: 4946: 4942: 4938: 4934: 4931: 4928: 4922: 4917: 4913: 4910: 4906: 4899: 4891: 4888: 4880: 4870: 4865: 4861: 4857: 4847: 4843: 4839: 4836: 4833: 4828: 4823: 4819: 4812: 4809: 4806: 4803: 4800: 4796: 4792: 4786: 4783: 4748: 4743: 4720: 4714: 4706: 4703: 4676: 4668: 4665: 4638: 4635: 4615: 4614: 4603: 4600: 4597: 4593: 4587: 4582: 4577: 4570: 4567: 4560: 4555: 4552: 4549: 4545: 4519: 4512: 4509: 4485: 4482: 4478: 4457: 4453: 4449: 4446: 4443: 4423: 4420: 4416: 4410: 4403: 4400: 4376: 4373: 4351: 4344: 4341: 4317: 4312: 4305: 4302: 4295: 4292: 4289: 4286: 4281: 4274: 4271: 4264: 4258: 4255: 4249: 4229: 4226: 4223: 4218: 4215: 4202: 4201: 4190: 4187: 4184: 4181: 4177: 4170: 4167: 4141: 4138: 4134: 4119: 4118: 4107: 4104: 4101: 4096: 4093: 4088: 4082: 4079: 4071: 4064: 4061: 4054: 4051: 4048: 4042: 4039: 4018: 4017: 4006: 4003: 3999: 3993: 3986: 3983: 3976: 3970: 3967: 3960: 3922: 3915: 3912: 3882: 3879: 3850: 3849: 3838: 3833: 3830: 3827: 3823: 3817: 3810: 3807: 3797: 3794: 3790: 3785: 3780: 3777: 3774: 3771: 3767: 3760: 3757: 3748: 3745: 3741: 3736: 3733: 3730: 3727: 3721: 3718: 3692: 3689: 3686: 3680: 3677: 3622: 3621: 3610: 3605: 3602: 3599: 3595: 3589: 3585: 3578: 3575: 3571: 3566: 3561: 3558: 3555: 3552: 3548: 3544: 3538: 3535: 3531: 3526: 3523: 3520: 3517: 3514: 3489:Main article: 3486: 3483: 3475: 3458: 3457: 3446: 3439: 3435: 3431: 3426: 3421: 3415: 3410: 3404: 3398: 3393: 3374: 3359:complex number 3349: 3348: 3337: 3333: 3326: 3322: 3318: 3314: 3311: 3308: 3302: 3297: 3293: 3289: 3283: 3277: 3272: 3268: 3262: 3258: 3254: 3250: 3247: 3244: 3238: 3233: 3229: 3226: 3222: 3215: 3210: 3205: 3195: 3190: 3186: 3182: 3172: 3168: 3164: 3161: 3158: 3153: 3148: 3144: 3137: 3134: 3131: 3128: 3125: 3121: 3117: 3114: 3102:) as follows: 3098:(obtained via 3081: 3080: 3069: 3066: 3063: 3059: 3053: 3049: 3045: 3040: 3036: 3032: 3024: 3020: 3016: 3010: 3006: 2999: 2982:we obtain the 2980: 2979: 2968: 2965: 2962: 2956: 2953: 2946: 2941: 2935: 2931: 2925: 2920: 2916: 2912: 2908: 2904: 2901: 2898: 2895: 2888: 2883: 2877: 2868: 2864: 2860: 2856: 2849: 2844: 2841: 2838: 2834: 2830: 2826: 2820: 2817: 2814: 2810: 2806: 2803: 2800: 2797: 2790: 2785: 2779: 2772: 2769: 2765: 2723: 2720: 2695: 2682: 2681: 2670: 2666: 2660: 2656: 2650: 2646: 2640: 2637: 2632: 2627: 2623: 2619: 2616: 2613: 2608: 2605: 2600: 2595: 2588: 2585: 2576: 2573: 2567: 2562: 2557: 2553: 2549: 2546: 2541: 2535: 2530: 2526: 2522: 2519: 2516: 2493: 2484:of the field, 2440:path integrals 2397:magnetic field 2379:electric field 2338: 2335: 2326:speed of light 2308: 2305: 2267:superconductor 2263:Yoichiro Nambu 2247: 2244: 2162: 2159: 2139:Standard Model 2094:Luciano Maiani 1992:local symmetry 1984:Yang Chen-Ning 1956:Standard Model 1944: 1943:Standard model 1941: 1895: 1892: 1840: 1837: 1746: 1739: 1678: 1675: 1618:excited states 1614:quantum fields 1598:fine structure 1583:Dirac equation 1508:Pascual Jordan 1500: 1497: 1463:Wolfgang Pauli 1375:atomic spectra 1368:speed of light 1352:magnetic field 1348:electric field 1338:The theory of 1264: 1261: 1257:Standard Model 1221:Main article: 1218: 1215: 1183:excited states 1174:. The current 1172:quasiparticles 1126: 1125: 1123: 1122: 1115: 1108: 1100: 1097: 1096: 1089: 1088: 1083: 1078: 1073: 1068: 1063: 1058: 1053: 1048: 1043: 1038: 1033: 1028: 1023: 1018: 1013: 1008: 1003: 998: 993: 988: 983: 978: 973: 968: 963: 958: 953: 948: 943: 938: 933: 928: 923: 918: 913: 908: 903: 898: 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: 663: 658: 653: 648: 643: 638: 633: 628: 623: 618: 613: 608: 603: 598: 593: 588: 583: 578: 573: 568: 563: 558: 553: 548: 543: 538: 533: 528: 523: 518: 513: 508: 503: 498: 493: 488: 483: 478: 473: 468: 463: 458: 453: 448: 443: 438: 433: 428: 423: 418: 413: 408: 403: 397: 396: 393: 392: 389: 388: 385: 384: 379: 374: 369: 364: 358: 355: 354: 351: 350: 347: 346: 341: 336: 331: 325: 322:Standard Model 320: 319: 316: 315: 312: 311: 306: 301: 296: 291: 289:Dirac equation 285: 282: 281: 278: 277: 274: 273: 271:Wick's theorem 268: 263: 258: 256:Regularization 253: 248: 243: 238: 233: 228: 223: 218: 213: 208: 203: 198: 193: 188: 182: 179: 178: 175: 174: 171: 170: 165: 163:Noether charge 160: 155: 150: 148:Gauge symmetry 145: 140: 135: 130: 125: 120: 114: 109: 108: 105: 104: 101: 100: 95: 90: 85: 80: 75: 70: 65: 60: 54: 51: 50: 47: 46: 40: 39: 34: 26: 25: 15: 9: 6: 4: 3: 2: 22090: 22079: 22076: 22074: 22071: 22069: 22066: 22065: 22063: 22048: 22047: 22038: 22036: 22035: 22026: 22025: 22022: 22016: 22013: 22011: 22008: 22006: 22003: 22001: 21998: 21996: 21993: 21991: 21988: 21987: 21985: 21981: 21971: 21968: 21966: 21963: 21961: 21958: 21956: 21953: 21951: 21948: 21946: 21943: 21941: 21940:String theory 21938: 21937: 21935: 21933: 21929: 21923: 21920: 21918: 21915: 21913: 21910: 21908: 21905: 21904: 21902: 21900: 21899:Supersymmetry 21896: 21890: 21887: 21885: 21882: 21880: 21877: 21875: 21872: 21871: 21869: 21865: 21859: 21856: 21854: 21851: 21847: 21844: 21843: 21842: 21839: 21837: 21834: 21832: 21829: 21828: 21826: 21822: 21819: 21817: 21811: 21805: 21802: 21800: 21797: 21795: 21792: 21790: 21787: 21786: 21784: 21780: 21763: 21760: 21758: 21755: 21753: 21750: 21748: 21745: 21743: 21740: 21739: 21738: 21735: 21731: 21728: 21726: 21723: 21721: 21718: 21717: 21716: 21713: 21711: 21708: 21706: 21703: 21699: 21696: 21694: 21691: 21689: 21686: 21685: 21684: 21681: 21680: 21678: 21674: 21670: 21663: 21658: 21656: 21651: 21649: 21644: 21643: 21640: 21628: 21625: 21623: 21620: 21618: 21615: 21613: 21610: 21608: 21605: 21604: 21602: 21598: 21592: 21589: 21587: 21586:Ocean physics 21584: 21582: 21579: 21577: 21574: 21572: 21569: 21567: 21564: 21562: 21559: 21557: 21554: 21552: 21549: 21547: 21544: 21543: 21541: 21539: 21535: 21529: 21526: 21522: 21521:Modern optics 21519: 21517: 21514: 21512: 21509: 21508: 21507: 21504: 21502: 21499: 21497: 21494: 21492: 21489: 21485: 21482: 21480: 21477: 21476: 21475: 21472: 21471: 21469: 21467: 21463: 21455: 21452: 21450: 21447: 21446: 21445: 21442: 21438: 21435: 21433: 21430: 21429: 21428: 21425: 21423: 21420: 21418: 21415: 21411: 21408: 21406: 21403: 21401: 21398: 21396: 21393: 21392: 21391: 21388: 21387: 21385: 21383: 21379: 21371: 21370:Computational 21368: 21367: 21366: 21363: 21361: 21358: 21357: 21355: 21351: 21343: 21340: 21339: 21338: 21335: 21333: 21330: 21329: 21327: 21323: 21319: 21311: 21306: 21304: 21299: 21297: 21292: 21291: 21288: 21275: 21267: 21264: 21263: 21259: 21251: 21248: 21246: 21243: 21241: 21238: 21237: 21236: 21233: 21232: 21230: 21226: 21216: 21213: 21211: 21208: 21206: 21203: 21201: 21198: 21196: 21193: 21191: 21190:Dual graviton 21188: 21186: 21183: 21181: 21178: 21177: 21175: 21171: 21165: 21162: 21161: 21159: 21157: 21153: 21147: 21144: 21142: 21139: 21138: 21136: 21134: 21130: 21124: 21121: 21119: 21116: 21114: 21111: 21109: 21106: 21105: 21103: 21101: 21100:String theory 21097: 21094: 21090: 21084: 21081: 21079: 21076: 21074: 21071: 21069: 21066: 21064: 21061: 21059: 21056: 21054: 21051: 21049: 21046: 21044: 21041: 21040: 21038: 21036: 21032: 21026: 21023: 21021: 21018: 21016: 21013: 21011: 21008: 21007: 21005: 21003: 20997: 20991: 20988: 20986: 20983: 20981: 20978: 20976: 20973: 20971: 20968: 20966: 20963: 20962: 20960: 20958: 20954: 20948: 20945: 20943: 20940: 20938: 20935: 20933: 20930: 20928: 20925: 20923: 20920: 20918: 20915: 20913: 20910: 20908: 20905: 20903: 20900: 20898: 20895: 20893: 20890: 20888: 20885: 20883: 20880: 20879: 20877: 20873: 20869: 20862: 20857: 20855: 20850: 20848: 20843: 20842: 20839: 20827: 20819: 20818: 20815: 20809: 20806: 20804: 20801: 20799: 20796: 20792: 20789: 20788: 20787: 20784: 20783: 20781: 20777: 20771: 20768: 20766: 20763: 20759: 20756: 20755: 20754: 20751: 20749: 20746: 20744: 20741: 20739: 20736: 20735: 20733: 20729: 20723: 20720: 20718: 20715: 20713: 20710: 20708: 20705: 20703: 20700: 20698: 20695: 20693: 20690: 20688: 20685: 20683: 20680: 20678: 20675: 20673: 20670: 20668: 20665: 20663: 20662:Quantum logic 20660: 20658: 20655: 20653: 20650: 20648: 20645: 20643: 20640: 20638: 20635: 20633: 20630: 20628: 20625: 20621: 20618: 20617: 20616: 20613: 20611: 20608: 20606: 20603: 20601: 20598: 20594: 20591: 20590: 20589: 20586: 20584: 20581: 20579: 20576: 20574: 20571: 20570: 20568: 20566: 20562: 20556: 20553: 20551: 20548: 20546: 20543: 20541: 20538: 20536: 20533: 20531: 20528: 20526: 20523: 20521: 20518: 20516: 20515:Quantum chaos 20513: 20511: 20508: 20506: 20503: 20502: 20500: 20498: 20494: 20488: 20485: 20483: 20482:Stern–Gerlach 20480: 20478: 20475: 20473: 20470: 20468: 20465: 20463: 20460: 20458: 20455: 20453: 20450: 20448: 20445: 20443: 20440: 20438: 20435: 20434: 20432: 20428: 20422: 20419: 20417: 20416:Transactional 20414: 20412: 20409: 20407: 20406:Quantum logic 20404: 20402: 20399: 20397: 20394: 20388: 20385: 20384: 20383: 20380: 20379: 20378: 20375: 20373: 20370: 20368: 20365: 20363: 20360: 20358: 20355: 20353: 20350: 20349: 20347: 20345: 20341: 20335: 20332: 20330: 20327: 20325: 20322: 20320: 20317: 20315: 20312: 20311: 20309: 20305: 20299: 20296: 20294: 20291: 20289: 20286: 20284: 20281: 20279: 20276: 20274: 20271: 20269: 20266: 20265: 20263: 20259: 20251: 20248: 20246: 20243: 20242: 20241: 20240:Wave function 20238: 20236: 20233: 20231: 20228: 20226: 20223: 20221: 20218: 20216: 20215:Superposition 20213: 20211: 20210:Quantum state 20208: 20206: 20203: 20201: 20198: 20196: 20193: 20191: 20188: 20186: 20183: 20181: 20178: 20174: 20171: 20169: 20166: 20164: 20163:Excited state 20161: 20159: 20156: 20155: 20154: 20151: 20149: 20146: 20144: 20141: 20139: 20136: 20134: 20131: 20130: 20128: 20124: 20118: 20115: 20113: 20110: 20108: 20105: 20101: 20098: 20097: 20096: 20093: 20091: 20088: 20087: 20085: 20081: 20077: 20070: 20065: 20063: 20058: 20056: 20051: 20050: 20047: 20037: 20029: 20024: 20018: 20015: 20013: 20012:Quantum logic 20010: 20006: 20003: 20002: 20001: 19998: 19996: 19993: 19991: 19988: 19986: 19983: 19979: 19976: 19975: 19974: 19971: 19967: 19964: 19963: 19962: 19959: 19957: 19954: 19952: 19949: 19947: 19946:Quantum chaos 19944: 19942: 19939: 19937: 19934: 19932: 19929: 19927: 19924: 19922: 19921:Cosmic string 19919: 19917: 19914: 19913: 19911: 19907: 19897: 19894: 19892: 19889: 19887: 19884: 19882: 19879: 19877: 19874: 19872: 19869: 19867: 19864: 19863: 19861: 19857: 19851: 19848: 19846: 19843: 19842: 19840: 19836: 19830: 19827: 19825: 19822: 19820: 19817: 19815: 19812: 19810: 19807: 19805: 19802: 19800: 19797: 19795: 19794:Pure 4D N = 1 19792: 19791: 19789: 19785: 19779: 19776: 19774: 19771: 19769: 19766: 19765: 19763: 19759: 19753: 19750: 19748: 19745: 19743: 19740: 19738: 19735: 19733: 19730: 19729: 19727: 19723: 19717: 19714: 19712: 19709: 19707: 19704: 19702: 19699: 19697: 19694: 19693: 19691: 19687: 19681: 19678: 19676: 19675:Thirring–Wess 19673: 19671: 19668: 19666: 19663: 19661: 19658: 19656: 19653: 19651: 19650:Bullough–Dodd 19648: 19646: 19645:2D Yang–Mills 19643: 19642: 19640: 19636: 19630: 19627: 19625: 19622: 19620: 19617: 19615: 19612: 19610: 19607: 19605: 19602: 19600: 19597: 19595: 19592: 19590: 19587: 19585: 19582: 19580: 19577: 19575: 19572: 19570: 19567: 19565: 19562: 19561: 19559: 19555: 19552: 19548: 19542: 19539: 19537: 19534: 19532: 19529: 19527: 19524: 19522: 19521:String theory 19519: 19517: 19514: 19512: 19509: 19507: 19504: 19502: 19499: 19497: 19494: 19492: 19491:Axiomatic QFT 19489: 19487: 19486:Algebraic QFT 19484: 19483: 19481: 19477: 19473: 19466: 19461: 19459: 19454: 19452: 19447: 19446: 19443: 19436: 19433: 19430: 19426: 19423: 19422: 19417: 19414: 19410: 19409: 19405: 19401: 19397: 19396: 19391: 19387: 19386: 19382: 19377: 19373: 19372: 19362: 19356: 19352: 19347: 19346: 19340: 19336: 19332: 19326: 19322: 19318: 19314: 19313:Logunov, A.A. 19309: 19305: 19299: 19295: 19291: 19287: 19283: 19282: 19278: 19277: 19272: 19266: 19262: 19257: 19256: 19250: 19246: 19242: 19236: 19232: 19228: 19223: 19213: 19209: 19205: 19201: 19195: 19191: 19187: 19186: 19180: 19176: 19170: 19166: 19161: 19157: 19151: 19147: 19143: 19142: 19137: 19133: 19129: 19123: 19118: 19117: 19111: 19107: 19103: 19097: 19093: 19089: 19085: 19081: 19077: 19073: 19069: 19059:on 2018-03-22 19058: 19054: 19048: 19044: 19043: 19037: 19033: 19027: 19023: 19019: 19018: 19013: 19009: 19005: 18999: 18995: 18991: 18987: 18983: 18979: 18973: 18969: 18965: 18961: 18959: 18958:9780199699339 18955: 18951: 18949: 18944: 18940: 18934: 18927: 18926: 18920: 18916: 18910: 18906: 18902: 18901: 18896: 18892: 18888: 18882: 18878: 18877:Perseus Group 18874: 18870: 18866: 18862: 18856: 18852: 18847: 18846: 18840: 18836: 18832: 18828: 18822: 18818: 18814: 18810: 18806: 18802: 18798: 18792: 18788: 18787: 18778: 18774: 18770: 18766: 18762: 18756: 18752: 18748: 18744: 18740: 18736: 18732: 18726: 18722: 18718: 18713: 18712: 18708: 18707: 18702: 18696: 18692: 18688: 18684: 18680: 18676: 18670: 18666: 18661: 18657: 18651: 18647: 18642: 18641: 18635: 18634:Feynman, R.P. 18631: 18627: 18625:9780691033273 18621: 18617: 18612: 18611: 18605: 18601: 18597: 18591: 18587: 18583: 18579: 18575: 18574: 18570: 18569: 18559: 18555: 18551: 18547: 18543: 18539: 18535: 18531: 18530: 18525: 18520: 18516: 18512: 18508: 18504: 18500: 18496: 18493:(2): 83–112. 18492: 18488: 18487: 18482: 18477: 18472: 18471: 18464: 18463: 18459: 18458: 18444:on 2015-03-30 18440: 18436: 18429: 18425: 18421: 18420:Jaffe, Arthur 18415: 18406: 18401: 18397: 18390: 18382: 18378: 18374: 18368: 18364: 18357: 18349: 18345: 18341: 18339:0-691-08144-1 18335: 18331: 18324: 18315: 18310: 18303: 18301: 18299: 18290: 18286: 18282: 18276: 18272: 18268: 18264: 18260: 18255: 18250: 18246: 18242: 18236: 18234: 18225: 18221: 18217: 18213: 18209: 18205: 18200: 18195: 18192:(9): 092301. 18191: 18187: 18186:J. Math. Phys 18180: 18173: 18169: 18165: 18159: 18153: 18149: 18145: 18139: 18131: 18127: 18120: 18116: 18110: 18102: 18098: 18094: 18090: 18087:(1): 91–112. 18086: 18082: 18081: 18076: 18070: 18062: 18056: 18052: 18051: 18043: 18035: 18031: 18027: 18023: 18019: 18015: 18010: 18005: 18001: 17997: 17996: 17988: 17980: 17976: 17972: 17968: 17964: 17960: 17956: 17952: 17948: 17944: 17943: 17938: 17934: 17928: 17920: 17916: 17912: 17908: 17903: 17898: 17894: 17890: 17883: 17875: 17873:9780511564192 17869: 17865: 17861: 17856: 17851: 17847: 17846: 17841: 17835: 17826: 17821: 17814: 17806: 17800: 17796: 17791: 17790: 17781: 17773: 17767: 17763: 17762: 17754: 17746: 17742: 17738: 17734: 17730: 17726: 17721: 17716: 17712: 17708: 17701: 17693: 17689: 17685: 17681: 17677: 17673: 17666: 17658: 17652: 17648: 17647:String Theory 17644: 17638: 17629: 17624: 17617: 17609: 17603: 17598: 17597: 17591: 17585: 17577: 17573: 17568: 17563: 17559: 17555: 17551: 17547: 17540: 17533: 17527: 17518: 17513: 17507: 17499: 17491: 17487: 17483: 17479: 17475: 17471: 17466: 17461: 17457: 17453: 17446: 17437: 17432: 17425: 17417: 17411: 17407: 17402: 17401: 17395: 17391: 17384: 17376: 17372: 17368: 17364: 17360: 17356: 17352: 17349:(in German). 17348: 17344: 17338: 17330: 17324: 17319: 17318: 17312: 17306: 17304: 17302: 17300: 17298: 17296: 17294: 17292: 17290: 17288: 17286: 17284: 17282: 17280: 17278: 17276: 17274: 17272: 17270: 17268: 17257: 17250: 17245: 17243: 17234: 17230: 17226: 17222: 17218: 17214: 17209: 17204: 17200: 17196: 17192: 17186: 17184: 17168: 17164: 17157: 17151: 17142: 17137: 17133: 17127: 17119: 17113: 17109: 17108:String Theory 17105: 17099: 17097: 17095: 17085: 17080: 17076: 17070: 17056: 17052: 17048: 17044: 17038: 17030: 17026: 17022: 17018: 17014: 17010: 17006: 17002: 16998: 16994: 16993: 16988: 16982: 16980: 16978: 16968: 16963: 16959: 16955: 16951: 16947: 16946: 16941: 16937: 16933: 16927: 16919: 16915: 16911: 16905: 16901: 16897: 16893: 16889: 16884: 16879: 16875: 16871: 16865: 16863: 16861: 16859: 16850: 16844: 16840: 16833: 16825: 16818: 16810: 16803: 16795: 16788: 16780: 16774: 16770: 16763: 16755: 16749: 16745: 16738: 16730: 16728:0-7382-0053-0 16724: 16720: 16713: 16711: 16709: 16700: 16696: 16692: 16688: 16684: 16680: 16673: 16665: 16661: 16656: 16651: 16647: 16643: 16638: 16633: 16629: 16625: 16621: 16614: 16606: 16602: 16597: 16592: 16588: 16584: 16579: 16574: 16570: 16566: 16562: 16558: 16554: 16547: 16539: 16535: 16530: 16525: 16521: 16517: 16512: 16507: 16503: 16499: 16495: 16491: 16487: 16480: 16472: 16465: 16457: 16453: 16449: 16445: 16441: 16437: 16433: 16429: 16425: 16418: 16410: 16404: 16400: 16396: 16390: 16388: 16386: 16384: 16382: 16380: 16378: 16376: 16374: 16372: 16363: 16357: 16353: 16352: 16347: 16341: 16333: 16329: 16325: 16321: 16318:(11): 69–85. 16317: 16313: 16312: 16311:Physics Today 16307: 16301: 16299: 16297: 16295: 16293: 16291: 16289: 16287: 16285: 16283: 16281: 16279: 16277: 16268: 16264: 16258: 16250: 16244: 16240: 16236: 16232: 16226: 16218: 16214: 16210: 16206: 16202: 16196: 16194: 16192: 16190: 16188: 16186: 16184: 16182: 16180: 16178: 16176: 16174: 16172: 16170: 16168: 16166: 16157: 16153: 16149: 16145: 16141: 16137: 16132: 16127: 16123: 16119: 16118: 16110: 16108: 16106: 16097: 16091: 16087: 16086: 16081: 16075: 16073: 16071: 16069: 16067: 16065: 16063: 16061: 16059: 16057: 16055: 16053: 16051: 16049: 16047: 16045: 16043: 16041: 16039: 16037: 16035: 16033: 16031: 16029: 16027: 16025: 16023: 16021: 16019: 16017: 16015: 16013: 16011: 16009: 16007: 16005: 16003: 16001: 15999: 15997: 15995: 15993: 15991: 15989: 15987: 15985: 15983: 15981: 15979: 15977: 15975: 15973: 15968: 15957: 15954: 15952: 15949: 15947: 15944: 15942: 15939: 15937: 15934: 15931: 15929: 15926: 15924: 15921: 15919: 15916: 15914: 15911: 15909: 15906: 15904: 15901: 15899: 15896: 15894: 15891: 15889: 15886: 15884: 15881: 15879: 15876: 15874: 15871: 15869: 15865: 15862: 15860: 15857: 15855: 15852: 15850: 15847: 15845: 15842: 15840: 15837: 15835: 15832: 15830: 15827: 15825: 15822: 15820: 15817: 15815: 15812: 15810: 15807: 15805: 15802: 15800: 15797: 15795: 15792: 15790: 15787: 15785: 15782: 15780: 15777: 15776: 15771: 15765: 15760: 15757: 15751: 15746: 15738: 15736: 15732: 15728: 15721: 15703: 15687: 15683: 15680: 15677: 15671: 15669: 15665: 15662:, one of the 15661: 15657: 15655: 15654:Wick rotation 15651: 15647: 15643: 15639: 15635: 15631: 15627: 15625: 15621: 15617: 15613: 15608: 15606: 15602: 15598: 15594: 15590: 15586: 15581: 15578: 15574: 15570: 15566: 15562: 15559:'s monograph 15558: 15554: 15549: 15547: 15543: 15539: 15535: 15525: 15523: 15519: 15515: 15511: 15507: 15503: 15499: 15494: 15490: 15486: 15482: 15477: 15473: 15463: 15461: 15457: 15453: 15449: 15445: 15441: 15440: 15435: 15431: 15427: 15422: 15412: 15402: 15400: 15394: 15385: 15378: 15358: 15354: 15348: 15344: 15338: 15334: 15328: 15325: 15320: 15317: 15312: 15304: 15299: 15289: 15286: 15282: 15276: 15273: 15267: 15256: 15246: 15232: 15231: 15230: 15227: 15218: 15198: 15195: 15190: 15182: 15177: 15167: 15164: 15160: 15156: 15153: 15148: 15140: 15135: 15126: 15121: 15117: 15111: 15107: 15101: 15098: 15094: 15090: 15085: 15081: 15075: 15071: 15063: 15062: 15061: 15059: 15056:describing a 15055: 15051: 15046: 15039: 15030: 15021: 15001: 14998: 14993: 14985: 14980: 14970: 14967: 14963: 14959: 14956: 14951: 14943: 14938: 14929: 14924: 14920: 14914: 14910: 14904: 14901: 14897: 14893: 14888: 14884: 14878: 14874: 14866: 14865: 14864: 14863: 14859: 14854: 14847: 14842: 14840: 14836: 14832: 14831:string theory 14828: 14824: 14820: 14815: 14813: 14809: 14804: 14793: 14789: 14787: 14783: 14779: 14775: 14771: 14767: 14762: 14760: 14755: 14753: 14749: 14745: 14739: 14734: 14732: 14726: 14719: 14713: 14709: 14705: 14697: 14693: 14692:Weyl fermions 14689: 14684: 14675: 14668: 14663: 14659: 14655: 14650: 14648: 14647:supersymmetry 14644: 14640: 14634: 14633:Supersymmetry 14627:Supersymmetry 14624: 14622: 14618: 14614: 14610: 14606: 14601: 14597: 14594: 14587: 14580: 14573: 14569: 14561: 14555: 14549: 14545: 14538: 14532: 14528: 14524: 14519: 14513: 14505: 14501: 14481: 14476: 14471: 14465: 14461: 14455: 14451: 14446: 14439: 14436: 14431: 14426: 14422: 14416: 14412: 14406: 14402: 14396: 14393: 14388: 14385: 14380: 14376: 14370: 14366: 14362: 14357: 14352: 14347: 14343: 14339: 14334: 14329: 14324: 14320: 14315: 14309: 14305: 14301: 14297: 14291: 14288: 14283: 14279: 14275: 14270: 14261: 14256: 14252: 14247: 14238: 14232: 14229: 14224: 14220: 14214: 14210: 14204: 14195: 14190: 14184: 14180: 14174: 14165: 14159: 14156: 14151: 14137: 14136: 14135: 14118: 14114: 14107: 14101: 14098: 14092: 14088: 14083: 14077: 14069: 14066: 14063: 14059: 14055: 14052: 14049: 14043: 14035: 14031: 14026: 14022: 14016: 14008: 14004: 13996: 13995: 13994: 13991: 13987:The original 13971: 13967: 13960: 13956: 13951: 13948: 13945: 13942: 13939: 13936: 13932: 13928: 13923: 13918: 13914: 13906: 13905: 13904: 13901: 13881: 13876: 13871: 13867: 13861: 13856: 13851: 13847: 13841: 13836: 13832: 13824: 13823: 13822: 13816: 13796: 13790: 13779: 13776: 13772: 13767: 13763: 13757: 13754: 13750: 13741: 13737: 13729: 13728: 13727: 13724: 13722: 13714: 13708: 13688: 13683: 13678: 13672: 13668: 13662: 13658: 13653: 13646: 13643: 13638: 13633: 13629: 13623: 13619: 13613: 13609: 13603: 13600: 13595: 13591: 13585: 13581: 13575: 13566: 13561: 13555: 13551: 13545: 13536: 13530: 13527: 13522: 13508: 13507: 13506: 13503: 13498: 13493: 13491: 13486: 13476: 13474: 13469: 13466: 13465:Faddeev–Popov 13461: 13457: 13452: 13448: 13443: 13437: 13435: 13427: 13423: 13419: 13415: 13413: 13412:Lorentz group 13409: 13405: 13400: 13381: 13371: 13364: 13345: 13340: 13332: 13329: 13311: 13291: 13284: 13280: 13274: 13266: 13260: 13257: 13253: 13249: 13243: 13237: 13230: 13229: 13228: 13225: 13214: 13210: 13190: 13184: 13176: 13172: 13167: 13161: 13151: 13148: 13144: 13140: 13137: 13132: 13128: 13121: 13113: 13108: 13104: 13099: 13092: 13086: 13078: 13074: 13067: 13059: 13054: 13050: 13045: 13039: 13031: 13027: 13020: 13012: 13009: 13005: 12995: 12987: 12983: 12975: 12974: 12973: 12970: 12966: 12962: 12953: 12948: 12928: 12923: 12918: 12914: 12908: 12903: 12899: 12893: 12890: 12887: 12883: 12879: 12876: 12871: 12866: 12862: 12856: 12848: 12843: 12838: 12834: 12828: 12820: 12815: 12810: 12807: 12803: 12795: 12794: 12793: 12787: 12779: 12774: 12768: 12748: 12743: 12739: 12733: 12728: 12724: 12720: 12717: 12714: 12709: 12701: 12696: 12692: 12684: 12683: 12682: 12680: 12677:is the gauge 12675: 12652: 12647: 12643: 12637: 12627: 12620: 12617: 12612: 12609: 12606: 12603: 12599: 12593: 12588: 12585: 12581: 12575: 12572: 12567: 12562: 12558: 12552: 12549: 12539: 12535: 12526: 12522: 12516: 12506: 12499: 12496: 12482: 12481: 12480: 12474: 12471:representing 12468: 12464: 12459: 12456:representing 12453: 12449: 12443: 12438: 12434: 12430: 12426: 12425:Abelian group 12418: 12416: 12408: 12395: 12373: 12367: 12364: 12360: 12349: 12345: 12320: 12313: 12310: 12306: 12300: 12294: 12288: 12284: 12258: 12251: 12248: 12244: 12240: 12214: 12208: 12205: 12201: 12192: 12188: 12184: 12178: 12174: 12154: 12146: 12140: 12137: 12133: 12127: 12114: 12108: 12105: 12102: 12098: 12092: 12089: 12085: 12081: 12078: 12072: 12064: 12060: 12050: 12042: 12038: 12033: 12027: 12021: 12013: 12007: 12004: 12000: 11990: 11984: 11977: 11976: 11975: 11972: 11965: 11955: 11952: 11943: 11939: 11936: 11932: 11927: 11922: 11904: 11900: 11894: 11886: 11881: 11877: 11871: 11863: 11858: 11855: 11851: 11828: 11824: 11818: 11814: 11810: 11801: 11790: 11785: 11765: 11760: 11756: 11752: 11747: 11743: 11733: 11727: 11724: 11719: 11716: 11712: 11706: 11703: 11699: 11693: 11690: 11685: 11682: 11678: 11674: 11671: 11666: 11656: 11652: 11648: 11644: 11634: 11628: 11614: 11613: 11612: 11610: 11606: 11601: 11596: 11591: 11586: 11581: 11579: 11575: 11571: 11567: 11563: 11560: 11548: 11538: 11525: 11516: 11514: 11510: 11504: 11503:string theory 11500: 11496: 11492: 11488: 11484: 11482: 11477: 11470: 11460: 11452: 11449: 11444: 11421: 11415: 11410: 11406: 11402: 11396: 11393: 11389: 11383: 11379: 11373: 11370: 11365: 11362: 11359: 11355: 11346: 11342: 11338: 11332: 11328: 11322: 11313: 11308: 11305: 11294: 11289: 11283: 11277: 11270: 11269: 11268: 11266: 11261: 11256: 11251: 11245: 11243: 11238: 11214: 11208: 11203: 11199: 11195: 11189: 11186: 11178: 11174: 11170: 11164: 11160: 11154: 11145: 11140: 11137: 11126: 11121: 11115: 11109: 11102: 11101: 11100: 11098: 11093: 11089: 11084: 11082: 11078: 11076: 11071: 11067: 11061: 11051: 11049: 11045: 11039: 11032: 11023: 11014: 10998: 10992: 10970: 10966: 10962: 10957: 10953: 10949: 10946: 10941: 10937: 10932: 10927: 10922: 10918: 10914: 10911: 10906: 10902: 10898: 10893: 10889: 10884: 10881: 10878: 10875: 10872: 10867: 10863: 10855: 10854: 10853: 10850: 10841: 10818: 10813: 10808: 10804: 10797: 10794: 10788: 10784: 10778: 10773: 10768: 10764: 10758: 10754: 10748: 10745: 10740: 10732: 10728: 10722: 10706: 10702: 10696: 10683: 10679: 10673: 10670: 10665: 10660: 10655: 10651: 10644: 10641: 10635: 10631: 10625: 10620: 10615: 10611: 10605: 10600: 10596: 10590: 10587: 10582: 10574: 10570: 10564: 10548: 10544: 10538: 10525: 10522: 10517: 10503: 10502: 10501: 10498: 10492: 10483: 10463: 10458: 10454: 10448: 10444: 10440: 10436: 10432: 10429: 10422: 10421: 10420: 10417: 10410: 10404: 10390: 10384: 10378: 10372: 10366: 10360: 10358: 10351: 10341: 10339: 10334: 10328: 10322: 10320: 10316: 10311: 10308: 10301: 10262: 10258: 10251: 10248: 10240: 10236: 10229: 10223: 10205: 10204: 10203: 10200: 10194: 10188: 10182: 10176: 10162: 10150: 10130: 10126: 10122: 10117: 10113: 10104: 10100: 10077: 10073: 10050: 10046: 10037: 10020: 10016: 10010: 10006: 10002: 9999: 9996: 9993: 9971: 9967: 9958: 9957: 9956: 9941: 9929: 9926: 9911: 9903: 9900: 9885: 9865: 9857: 9854: 9853: 9852: 9845: 9841: 9840: 9839: 9822: 9816: 9813: 9810: 9802: 9798: 9791: 9788: 9785: 9777: 9773: 9766: 9763: 9760: 9752: 9748: 9743: 9738: 9734: 9730: 9724: 9721: 9716: 9713: 9710: 9704: 9701: 9694: 9693: 9692: 9675: 9669: 9655: 9649: 9643: 9637: 9631: 9625: 9619: 9613: 9607: 9601: 9595: 9589: 9583: 9575: 9568: 9563: 9559: 9555: 9549: 9546: 9541: 9538: 9535: 9525: 9524: 9523: 9520: 9514: 9509: 9502: 9492: 9470: 9464: 9448: 9444: 9437: 9429: 9425: 9418: 9412: 9404: 9395: 9379: 9375: 9368: 9360: 9356: 9349: 9343: 9335: 9329: 9327: 9316: 9300: 9296: 9289: 9281: 9277: 9270: 9264: 9256: 9247: 9231: 9227: 9220: 9212: 9208: 9201: 9195: 9187: 9181: 9179: 9168: 9152: 9148: 9141: 9133: 9129: 9122: 9116: 9108: 9099: 9083: 9079: 9072: 9064: 9060: 9053: 9047: 9039: 9033: 9031: 9023: 9007: 9003: 8996: 8988: 8984: 8977: 8969: 8965: 8958: 8950: 8946: 8939: 8933: 8925: 8911: 8910: 8909: 8906: 8901: 8897: 8894: 8850: 8843: 8831: 8826: 8822: 8816: 8811: 8808: 8804: 8800: 8796: 8792: 8789: 8785: 8775: 8769: 8757: 8752: 8748: 8742: 8737: 8734: 8730: 8726: 8722: 8718: 8715: 8709: 8703: 8697: 8691: 8687: 8677: 8666: 8663: 8660: 8657: 8645: 8637: 8617: 8611: 8605: 8599: 8593: 8575: 8574: 8573: 8570: 8568: 8549: 8541: 8533: 8510: 8501: 8488: 8478: 8474: 8465: 8461: 8457: 8452: 8442: 8438: 8429: 8425: 8418: 8410: 8402: 8398: 8392: 8388: 8384: 8381: 8376: 8372: 8366: 8362: 8358: 8352: 8349: 8344: 8336: 8333: 8323: 8315: 8312: 8309: 8293: 8290: 8287: 8283: 8274: 8261: 8251: 8247: 8238: 8234: 8230: 8225: 8215: 8211: 8202: 8198: 8191: 8183: 8179: 8172: 8164: 8160: 8153: 8145: 8137: 8133: 8127: 8123: 8119: 8116: 8111: 8107: 8101: 8097: 8093: 8087: 8084: 8079: 8071: 8068: 8058: 8050: 8047: 8044: 8028: 8025: 8022: 8018: 8011: 7991: 7985: 7979: 7973: 7967: 7936: 7914: 7906: 7898: 7894: 7887: 7884: 7880: 7874: 7869: 7865: 7861: 7858: 7852: 7844: 7840: 7817: 7813: 7804: 7800: 7795: 7787: 7786:infinitesimal 7782: 7762: 7756: 7752: 7748: 7743: 7738: 7731: 7723: 7719: 7714: 7711: 7706: 7701: 7698: 7694: 7690: 7687: 7683: 7679: 7676: 7672: 7668: 7664: 7660: 7656: 7651: 7647: 7643: 7638: 7633: 7626: 7618: 7614: 7609: 7606: 7601: 7596: 7593: 7589: 7585: 7582: 7578: 7574: 7571: 7565: 7557: 7553: 7546: 7538: 7534: 7529: 7525: 7521: 7517: 7513: 7502: 7499: 7496: 7493: 7481: 7473: 7453: 7447: 7441: 7435: 7429: 7411: 7410: 7409: 7406: 7404: 7384: 7376: 7372: 7348: 7340: 7336: 7312: 7302: 7298: 7294: 7289: 7285: 7276: 7272: 7268: 7265: 7261: 7254: 7251: 7248: 7243: 7239: 7235: 7230: 7226: 7220: 7216: 7211: 7201: 7193: 7190: 7182: 7177: 7173: 7166: 7161: 7155: 7147: 7141: 7138: 7135: 7127: 7123: 7119: 7113: 7099: 7093: 7087: 7081: 7075: 7067: 7057: 7056: 7055: 7053: 7048: 7046: 7042: 7038: 7033: 7027: 7007: 6952: 6948: 6925: 6921: 6912: 6911:time ordering 6896: 6888: 6885:are position 6872: 6852: 6815: 6809: 6803: 6797: 6791: 6767: 6753: 6747: 6741: 6735: 6729: 6721: 6708: 6698: 6680: 6675: 6671: 6667: 6661: 6655: 6651: 6646: 6642: 6638: 6632: 6626: 6619: 6618: 6617: 6615: 6610: 6604: 6598: 6578: 6574: 6570: 6566: 6563: 6558: 6553: 6549: 6545: 6541: 6537: 6534: 6527: 6521: 6511: 6508: 6500: 6496: 6485: 6482: 6479: 6476: 6472: 6461: 6457: 6446: 6445: 6444: 6440: 6420: 6412: 6408: 6397: 6393: 6389: 6386: 6383: 6380: 6376: 6365: 6361: 6349: 6345: 6334: 6330: 6326: 6323: 6320: 6317: 6313: 6302: 6298: 6291: 6283: 6280: 6277: 6273: 6262: 6258: 6254: 6251: 6248: 6245: 6241: 6230: 6226: 6216: 6213: 6210: 6206: 6202: 6199: 6196: 6191: 6187: 6183: 6180: 6175: 6171: 6167: 6164: 6161: 6153: 6149: 6138: 6135: 6132: 6129: 6125: 6114: 6110: 6099: 6098: 6097: 6095: 6092: 6088: 6083: 6077: 6071: 6065: 6061: 6039: 6035: 6019: 5997: 5993: 5979: 5975: 5971: 5965: 5955: 5952: 5947: 5942: 5923: 5919: 5911: 5904: 5899: 5888: 5885: 5878: 5871: 5866: 5852: 5832: 5827: 5823: 5816: 5813: 5809: 5804: 5799: 5795: 5789: 5785: 5779: 5776: 5771: 5767: 5763: 5758: 5749: 5742: 5737: 5724: 5721: 5716: 5702: 5701: 5700: 5698: 5693: 5691: 5687: 5683: 5679: 5678:vector fields 5675: 5670: 5668: 5664: 5644: 5638: 5628: 5623: 5618: 5611: 5594: 5587: 5580: 5573: 5556: 5547: 5542: 5537: 5530: 5513: 5506: 5497: 5496: 5495: 5479: 5462: 5435: 5407: 5399:for all 5393: 5390: 5387: 5381: 5359: 5348: 5347: 5346: 5329: 5316: 5311: 5291: 5288: 5285: 5281: 5275: 5258: 5251: 5246: 5229: 5221: 5217: 5213: 5195: 5188: 5171: 5163: 5158: 5147: 5136: 5131: 5123: 5120: 5114: 5110: 5104: 5087: 5080: 5063: 5055: 5047: 5046: 5045: 5028: 5024: 5013: 5005: 5002: 4999: 4988: 4984: 4980: 4974: 4957: 4950: 4940: 4932: 4929: 4926: 4915: 4911: 4908: 4904: 4886: 4878: 4863: 4859: 4855: 4845: 4837: 4834: 4826: 4821: 4817: 4810: 4807: 4801: 4798: 4781: 4771: 4770: 4769: 4766: 4741: 4718: 4701: 4663: 4633: 4621: 4601: 4595: 4585: 4580: 4575: 4565: 4558: 4553: 4547: 4535: 4534: 4533: 4517: 4507: 4480: 4455: 4451: 4447: 4441: 4418: 4408: 4398: 4374: 4371: 4349: 4339: 4315: 4310: 4300: 4293: 4287: 4279: 4269: 4262: 4253: 4227: 4224: 4216: 4213: 4188: 4185: 4179: 4165: 4155: 4154: 4153: 4136: 4124: 4105: 4102: 4094: 4091: 4086: 4077: 4069: 4059: 4052: 4046: 4037: 4027: 4026: 4025: 4023: 4004: 4001: 3997: 3991: 3981: 3974: 3965: 3958: 3950: 3949: 3948: 3946: 3942: 3920: 3910: 3899: 3877: 3867: 3862: 3856: 3836: 3831: 3828: 3825: 3821: 3815: 3805: 3795: 3792: 3788: 3783: 3778: 3775: 3772: 3769: 3765: 3755: 3746: 3743: 3739: 3734: 3728: 3716: 3706: 3705: 3704: 3687: 3675: 3665: 3659: 3655: 3649: 3646: 3640: 3634: 3628: 3608: 3603: 3600: 3597: 3593: 3587: 3583: 3576: 3573: 3569: 3564: 3559: 3556: 3553: 3550: 3546: 3542: 3536: 3533: 3529: 3524: 3518: 3512: 3505: 3504: 3503: 3500: 3498: 3492: 3482: 3478: 3474: 3469: 3464: 3444: 3437: 3433: 3429: 3424: 3402: 3391: 3383: 3382: 3381: 3377: 3373: 3368: 3360: 3355: 3335: 3331: 3320: 3312: 3309: 3306: 3295: 3291: 3287: 3281: 3270: 3266: 3256: 3248: 3245: 3242: 3231: 3227: 3224: 3220: 3208: 3203: 3188: 3184: 3180: 3170: 3162: 3159: 3151: 3146: 3142: 3135: 3132: 3126: 3123: 3112: 3105: 3104: 3103: 3101: 3097: 3093: 3092:wave equation 3088: 3086: 3067: 3064: 3061: 3057: 3051: 3047: 3043: 3038: 3030: 3022: 3018: 3008: 2997: 2989: 2988: 2987: 2985: 2966: 2963: 2960: 2954: 2933: 2929: 2918: 2914: 2906: 2902: 2875: 2866: 2862: 2847: 2842: 2839: 2836: 2832: 2828: 2824: 2815: 2808: 2804: 2777: 2770: 2754: 2753: 2752: 2750: 2745: 2721: 2718: 2668: 2664: 2658: 2654: 2648: 2644: 2638: 2635: 2630: 2625: 2617: 2606: 2603: 2598: 2593: 2586: 2583: 2574: 2571: 2565: 2560: 2555: 2551: 2547: 2544: 2533: 2528: 2524: 2520: 2517: 2514: 2507: 2506: 2505: 2491: 2483: 2478: 2472: 2465: 2461: 2457: 2452: 2448: 2443: 2441: 2437: 2433: 2431: 2427: 2422: 2420: 2416: 2410: 2406: 2402: 2398: 2392: 2388: 2384: 2380: 2374: 2370: 2366: 2362: 2358: 2354: 2350: 2344: 2334: 2331: 2327: 2322: 2318: 2314: 2313:natural units 2304: 2302: 2298: 2294: 2290: 2289:magnetic flux 2285: 2283: 2279: 2278:quasiparticle 2274: 2272: 2268: 2264: 2259: 2257: 2253: 2243: 2241: 2237: 2233: 2229: 2225: 2224:string theory 2221: 2216: 2214: 2210: 2206: 2202: 2198: 2194: 2193:Supersymmetry 2190: 2188: 2184: 2180: 2176: 2172: 2168: 2158: 2156: 2152: 2148: 2144: 2140: 2135: 2133: 2129: 2125: 2124:Frank Wilczek 2121: 2117: 2113: 2109: 2105: 2101: 2097: 2095: 2091: 2087: 2083: 2079: 2075: 2071: 2067: 2062: 2060: 2056: 2054: 2049: 2045: 2041: 2037: 2033: 2029: 2025: 2023: 2019: 2015: 2011: 2009: 2005: 2001: 1997: 1993: 1989: 1985: 1977: 1973: 1969: 1965: 1961: 1957: 1953: 1949: 1940: 1938: 1934: 1928: 1924: 1920: 1916: 1914: 1910: 1909:source theory 1906: 1902: 1894:Source theory 1891: 1889: 1885: 1880: 1877: 1871: 1867: 1863: 1857: 1855: 1851: 1845: 1836: 1833: 1831: 1827: 1823: 1818: 1816: 1812: 1810: 1805: 1799: 1794: 1792: 1788: 1784: 1783:Freeman Dyson 1780: 1776: 1771: 1768: 1764: 1760: 1756: 1752: 1745: 1738: 1734: 1730: 1725: 1723: 1719: 1715: 1711: 1707: 1703: 1698: 1695: 1690: 1687: 1683: 1674: 1672: 1668: 1664: 1660: 1656: 1652: 1646: 1644: 1640: 1639:Atomic nuclei 1636: 1632: 1627: 1623: 1622:Eugene Wigner 1619: 1615: 1610: 1607: 1603: 1602:hydrogen atom 1599: 1595: 1593: 1588: 1584: 1580: 1579:wave equation 1575: 1573: 1569: 1565: 1560: 1557:. It is this 1556: 1552: 1548: 1544: 1540: 1536: 1532: 1528: 1524: 1519: 1517: 1513: 1509: 1504: 1496: 1494: 1490: 1486: 1482: 1477: 1475: 1471: 1466: 1464: 1460: 1456: 1452: 1448: 1444: 1440: 1436: 1432: 1428: 1423: 1419: 1415: 1410: 1408: 1404: 1400: 1396: 1392: 1388: 1384: 1380: 1376: 1371: 1369: 1365: 1361: 1357: 1353: 1349: 1345: 1341: 1336: 1333: 1329: 1324: 1322: 1318: 1314: 1310: 1306: 1305: 1300: 1295: 1293: 1289: 1285: 1277: 1273: 1269: 1260: 1258: 1254: 1250: 1246: 1242: 1238: 1234: 1230: 1224: 1214: 1212: 1209:according to 1208: 1204: 1200: 1196: 1192: 1188: 1185:(also called 1184: 1179: 1177: 1173: 1169: 1165: 1161: 1158:to construct 1157: 1153: 1149: 1145: 1141: 1137: 1133: 1121: 1116: 1114: 1109: 1107: 1102: 1101: 1099: 1098: 1092: 1087: 1084: 1082: 1079: 1077: 1074: 1072: 1069: 1067: 1064: 1062: 1061:Zamolodchikov 1059: 1057: 1056:Zamolodchikov 1054: 1052: 1049: 1047: 1044: 1042: 1039: 1037: 1034: 1032: 1029: 1027: 1024: 1022: 1019: 1017: 1014: 1012: 1009: 1007: 1004: 1002: 999: 997: 994: 992: 989: 987: 984: 982: 979: 977: 974: 972: 969: 967: 964: 962: 959: 957: 954: 952: 949: 947: 944: 942: 939: 937: 934: 932: 929: 927: 924: 922: 919: 917: 914: 912: 909: 907: 904: 902: 899: 897: 894: 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: 662: 659: 657: 654: 652: 649: 647: 644: 642: 639: 637: 634: 632: 629: 627: 624: 622: 619: 617: 614: 612: 609: 607: 604: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 574: 572: 569: 567: 564: 562: 559: 557: 554: 552: 549: 547: 544: 542: 539: 537: 534: 532: 529: 527: 524: 522: 519: 517: 514: 512: 509: 507: 504: 502: 499: 497: 494: 492: 489: 487: 484: 482: 479: 477: 474: 472: 469: 467: 464: 462: 459: 457: 454: 452: 449: 447: 444: 442: 439: 437: 434: 432: 429: 427: 424: 422: 419: 417: 414: 412: 409: 407: 404: 402: 399: 398: 391: 390: 383: 380: 378: 375: 373: 370: 368: 367:Supersymmetry 365: 363: 362:String theory 360: 359: 353: 352: 345: 342: 340: 337: 335: 332: 330: 327: 326: 323: 318: 317: 310: 307: 305: 302: 300: 297: 295: 292: 290: 287: 286: 280: 279: 272: 269: 267: 264: 262: 259: 257: 254: 252: 249: 247: 244: 242: 239: 237: 234: 232: 229: 227: 224: 222: 219: 217: 214: 212: 209: 207: 204: 202: 199: 197: 194: 192: 189: 187: 184: 183: 177: 176: 169: 166: 164: 161: 159: 156: 154: 151: 149: 146: 144: 141: 139: 136: 134: 131: 129: 126: 124: 121: 119: 116: 115: 112: 107: 106: 99: 96: 94: 91: 89: 86: 84: 81: 79: 76: 74: 71: 69: 66: 64: 61: 59: 56: 55: 49: 48: 45: 42: 41: 37: 32: 28: 27: 24: 21: 20: 22044: 22032: 21922:Supergravity 21782:Constituents 21762:Weak isospin 21720:Color charge 21710:Gauge theory 21704: 21546:Astrophysics 21360:Experimental 21265: 21228:Applications 21118:Supergravity 21025:Unruh effect 20932:Quantum foam 20927:Planck units 20922:IR/UV mixing 20897:Causal patch 20752: 20545:Quantum mind 20457:Franck–Hertz 20319:Klein–Gordon 20268:Formulations 20261:Formulations 20190:Interference 20180:Entanglement 20158:Ground state 20153:Energy level 20126:Fundamentals 20090:Introduction 20027: 19956:Quantum foam 19896:Stueckelberg 19850:Chern–Simons 19787:Supergravity 19526:Supergravity 19511:Gauge theory 19471: 19419: 19406: 19393: 19344: 19339:Weinberg, S. 19316: 19289: 19254: 19249:Zee, Anthony 19226: 19215:. Retrieved 19184: 19164: 19140: 19120:. Springer. 19115: 19075: 19061:. Retrieved 19057:the original 19041: 19016: 18989: 18967: 18952:OUP Oxford. 18947: 18924: 18899: 18895:Kleinert, H. 18872: 18844: 18839:Zuber, J.-B. 18835:Itzykson, C. 18812: 18785: 18772: 18746: 18716: 18686: 18664: 18639: 18609: 18581: 18533: 18527: 18490: 18484: 18469: 18460:Bibliography 18446:. Retrieved 18439:the original 18414: 18389: 18362: 18356: 18329: 18323: 18244: 18189: 18185: 18179: 18163: 18158: 18143: 18138: 18129: 18125: 18115:Haag, Rudolf 18109: 18084: 18078: 18075:Thirring, W. 18069: 18053:. Springer. 18049: 18042: 17999: 17993: 17987: 17946: 17940: 17927: 17892: 17888: 17882: 17844: 17834: 17813: 17788: 17780: 17764:. Springer. 17760: 17753: 17710: 17706: 17700: 17675: 17671: 17665: 17646: 17637: 17616: 17595: 17584: 17549: 17545: 17539: 17531: 17526: 17505: 17498: 17455: 17451: 17445: 17424: 17399: 17383: 17350: 17346: 17337: 17316: 17256: 17198: 17194: 17171:. Retrieved 17169:. 2015-02-02 17162: 17150: 17126: 17107: 17069: 17058:. Retrieved 17050: 17037: 16996: 16990: 16949: 16943: 16936:Mills, R. L. 16926: 16873: 16838: 16832: 16823: 16817: 16808: 16802: 16793: 16787: 16768: 16762: 16743: 16737: 16718: 16682: 16678: 16672: 16627: 16623: 16613: 16560: 16556: 16546: 16493: 16489: 16479: 16470: 16464: 16431: 16427: 16417: 16398: 16350: 16340: 16315: 16309: 16266: 16257: 16234: 16225: 16211:(4): 17–35. 16208: 16204: 16121: 16115: 16084: 15864:Quantization 15685: 15673: 15658: 15628: 15609: 15582: 15560: 15553:perturbative 15552: 15550: 15531: 15469: 15437: 15429: 15417: 15414: 15397:denotes the 15392: 15380: 15376: 15373: 15222: 15216: 15213: 15041: 15034: 15025: 15019: 15016: 14861: 14849: 14843: 14816: 14811: 14802: 14799: 14790: 14774:Planck scale 14763: 14759:supergravity 14756: 14752:superpartner 14743: 14735: 14724: 14717: 14711: 14707: 14700: 14688:supercharges 14679: 14670: 14661: 14658:translations 14651: 14636: 14602: 14598: 14592: 14585: 14578: 14571: 14567: 14559: 14547: 14543: 14536: 14529: 14522: 14511: 14503: 14499: 14496: 14133: 13989: 13986: 13899: 13896: 13814: 13811: 13720: 13712: 13706: 13703: 13501: 13494: 13488: 13468:gauge fixing 13462: 13455: 13451:polarization 13441: 13438: 13421: 13416: 13401: 13306: 13223: 13212: 13208: 13205: 12968: 12964: 12960: 12946: 12943: 12772: 12766: 12763: 12670: 12667: 12466: 12462: 12451: 12447: 12419: 12403: 12347: 12343: 12176: 12172: 12169: 11970: 11967: 11964:Gauge theory 11947: 11934: 11925: 11783: 11780: 11599: 11589: 11582: 11570:vector field 11558: 11554: 11523: 11517: 11508: 11498: 11494: 11485: 11475: 11465: 11455: 11439: 11436: 11264: 11249: 11246: 11236: 11229: 11096: 11091: 11085: 11074: 11063: 11040: 11027: 11018: 11009: 10990: 10987: 10845: 10836: 10833: 10496: 10487: 10481: 10478: 10415: 10411: 10388: 10382: 10376: 10370: 10364: 10361: 10353: 10332: 10326: 10323: 10312: 10306: 10299: 10296: 10198: 10192: 10186: 10180: 10177: 10154: 9933: 9927: 9901: 9855: 9850: 9837: 9690: 9518: 9512: 9504: 9489: 8904: 8898: 8892: 8865: 8571: 8566: 7803:power series 7790: 7780: 7777: 7407: 7402: 7327: 7049: 7043:, two-point 7039:, two-point 7031: 7025: 6887:four-vectors 6710: 6695: 6608: 6602: 6596: 6593: 6438: 6435: 6090: 6081: 6075: 6069: 6063: 6059: 6017: 5967: 5950: 5940: 5850: 5847: 5694: 5681: 5674:Dirac fields 5671: 5659: 5422: 5309: 5306: 5043: 4764: 4619: 4616: 4203: 4123:vacuum state 4120: 4019: 3860: 3854: 3851: 3657: 3653: 3650: 3644: 3638: 3632: 3626: 3623: 3501: 3494: 3476: 3472: 3462: 3459: 3375: 3371: 3353: 3350: 3096:normal modes 3089: 3082: 2981: 2743: 2683: 2476: 2470: 2463: 2459: 2455: 2447:scalar field 2444: 2434: 2423: 2408: 2404: 2400: 2390: 2386: 2382: 2372: 2368: 2364: 2347:A classical 2346: 2329: 2320: 2310: 2286: 2275: 2260: 2249: 2239: 2236:John Schwarz 2217: 2213:Bruno Zumino 2205:Iron Curtain 2197:Yuri Golfand 2191: 2164: 2136: 2098: 2072:between all 2063: 2052: 2032:Robert Brout 2026: 2012: 2008:gauge bosons 1988:Robert Mills 1981: 1968:gauge bosons 1930: 1926: 1921: 1917: 1912: 1900: 1897: 1881: 1869: 1858: 1850:Fermi theory 1846: 1842: 1834: 1819: 1813:from 2) and 1808: 1801: 1796: 1772: 1743: 1736: 1726: 1713: 1699: 1691: 1680: 1647: 1642: 1626:Enrico Fermi 1611: 1591: 1576: 1566:of photons, 1547:ground state 1522: 1520: 1505: 1502: 1478: 1467: 1411: 1372: 1337: 1325: 1302: 1296: 1281: 1276:iron filings 1253:gauge theory 1226: 1180: 1139: 1135: 1129: 1090: 936:Stueckelberg 676:Jona-Lasinio 266:Vacuum state 251:Quantization 93:Gauge theory 73:Strong force 58:Field theory 22: 21983:Experiments 21874:Technicolor 21836:Dark matter 21730:Quark model 21698:Higgs boson 21693:Gauge boson 21449:Statistical 21365:Theoretical 21342:Engineering 21185:Causal sets 21035:Black holes 20803:EPR paradox 20583:Quantum bus 20452:Double-slit 20430:Experiments 20396:Many-worlds 20334:Schrödinger 20298:Phase space 20288:Schrödinger 20278:Interaction 20235:Uncertainty 20205:Nonlocality 20200:Measurement 20195:Decoherence 20185:Hamiltonian 19838:Topological 19752:Wess–Zumino 19665:Sine-Gordon 19655:Gross–Neveu 19564:Born–Infeld 19531:Thermal QFT 19208:Tong, David 19110:Greiner, W. 19012:Ryder, L.H. 18851:McGraw-Hill 18805:Greiner, W. 18743:Shirkov, D. 18721:McGraw-Hill 18683:Gribbin, J. 18405:1109.0955v2 18314:1203.3991v2 18174:| chapter=8 17825:0810.0344v5 17552:(1): 3–22. 17251:, Chapter 1 16932:Yang, C. N. 16685:(4): 1219. 16395:Shifman, M. 15682:gauge group 15597:CPT theorem 15502:domain wall 15060:) is used: 15032:satisfying 14856:is used to 14786:dark matter 14778:Higgs field 14617:Higgs boson 14554:Lie algebra 13821:satisfying 13499:containing 13497:sigma model 13447:four-vector 12792:matrices), 12415:gauge boson 11574:Dirac field 7799:exponential 7788:number and 6087:Hamiltonian 4022:Hamiltonian 2451:real number 2293:resistivity 2273:in matter. 2232:Joël Scherk 2209:Julius Wess 2183:Poul Olesen 2151:Higgs boson 2120:David Gross 2078:Higgs boson 2028:Peter Higgs 2018:Abdus Salam 1976:Higgs boson 1970:that carry 1937:S. Weinberg 1729:Willis Lamb 1710:observables 1689:developed. 1686:self-energy 1663:cosmic rays 1391:oscillators 1076:Zinn-Justin 926:Sommerfield 851:Pomeranchuk 821:Osterwalder 816:Oppenheimer 746:Łopuszański 571:Fredenhagen 372:Technicolor 22062:Categories 21990:Gran Sasso 21814:Beyond the 21789:CKM matrix 21676:Background 21566:Geophysics 21556:Biophysics 21400:Analytical 21353:Approaches 21250:Multiverse 21092:Approaches 20970:CGHS model 20957:Toy models 20892:CA-duality 20731:Extensions 20565:Technology 20411:Relational 20362:Copenhagen 20273:Heisenberg 20220:Tunnelling 20083:Background 19619:Yang–Mills 19217:2016-02-09 19136:Peskin, M. 19063:2020-05-13 18964:Loudon, R. 18869:Kane, G.L. 18809:Müller, B. 18448:2018-07-18 18199:1505.04809 18172:0821847058 18009:1612.09298 17902:1705.05734 17855:2312.12596 17720:1701.05259 17678:(2): 177. 17465:1501.06664 17208:1604.05669 17173:2018-07-18 17060:2018-08-14 16883:1503.05007 16269:. Dawsons. 16080:Peskin, M. 15963:References 15718:and has a 15648:theory by 15624:cobordisms 15573:Polchinski 15058:black hole 14710:= 1, ..., 14502:= 1, ..., 13445:, being a 12778:generators 12341:. For any 11572:, and the 11267:function: 11099:function: 10315:decay rate 9522:theory is 7037:propagator 5663:Fock space 2482:Lagrangian 2341:See also: 2307:Principles 2187:instantons 2175:flux tubes 2048:Tom Kibble 2044:Carl Hagen 1755:Hans Bethe 1751:Lamb shift 1651:antimatter 1631:beta decay 1564:scattering 1459:Paul Dirac 1439:Max Planck 1418:Bohr model 1414:Niels Bohr 1389:, as tiny 1383:Max Planck 1203:Lagrangian 1071:Zimmermann 966:Vainshtein 711:Kontsevich 656:Iliopoulos 631:Heisenberg 456:Bogoliubov 394:Scientists 246:Propagator 133:T-symmetry 128:P-symmetry 123:C-symmetry 111:Symmetries 68:Weak force 52:Background 21516:Molecular 21417:Acoustics 21410:Continuum 21405:Celestial 21395:Newtonian 21382:Classical 21325:Divisions 21266:See also: 21205:Spin foam 20985:RST model 20437:Bell test 20307:Equations 20133:Born rule 20028:See also: 19747:Super QCD 19701:Liouville 19689:Conformal 19660:Schwinger 19400:EMS Press 18986:Mandl, F. 18646:MIT Press 18636:(2001) . 18580:(1994) . 18558:119389461 18515:189829853 18381:852790676 18348:905864308 18034:119578849 17919:119166976 17713:: 01002. 17562:CiteSeerX 17490:115167208 17458:(5): 31. 17375:186238995 17249:Tong 2015 17233:118439678 17141:1201.0981 17084:1412.4094 16918:119198452 16646:0027-8424 16587:0027-8424 16520:0027-8424 16131:1204.4616 15551:However, 15510:instanton 15506:flux tube 15345:ϕ 15321:− 15318:ϕ 15313:ν 15309:∇ 15305:ϕ 15300:μ 15296:∇ 15290:ν 15287:μ 15196:ϕ 15191:ν 15187:∂ 15183:ϕ 15178:μ 15174:∂ 15168:ν 15165:μ 15154:ϕ 15149:μ 15145:∂ 15141:ϕ 15136:μ 15132:∂ 15122:ν 15112:μ 15102:ν 15099:μ 15086:μ 15076:μ 14999:ϕ 14994:ν 14990:∂ 14986:ϕ 14981:μ 14977:∂ 14971:ν 14968:μ 14964:η 14957:ϕ 14952:μ 14948:∂ 14944:ϕ 14939:μ 14935:∂ 14925:ν 14915:μ 14905:ν 14902:μ 14898:η 14889:μ 14879:μ 14686:, called 14556:), while 14462:π 14452:π 14437:λ 14432:− 14423:σ 14413:π 14403:π 14394:λ 14389:− 14386:σ 14377:π 14367:π 14363:μ 14358:λ 14353:− 14344:σ 14340:μ 14335:λ 14330:− 14321:σ 14306:μ 14284:− 14276:σ 14271:μ 14267:∂ 14253:σ 14248:μ 14244:∂ 14211:π 14205:μ 14201:∂ 14181:π 14175:μ 14171:∂ 14102:σ 14093:λ 14089:μ 14067:− 14060:π 14053:⋯ 14032:π 14005:ϕ 13961:λ 13957:μ 13943:⋯ 13915:ϕ 13877:λ 13868:μ 13848:ϕ 13833:ϕ 13780:∈ 13764:ϕ 13747:→ 13738:ϕ 13669:ϕ 13659:ϕ 13644:λ 13639:− 13630:ϕ 13620:ϕ 13610:μ 13582:ϕ 13576:μ 13572:∂ 13552:ϕ 13546:μ 13542:∂ 13382:ϕ 13372:∫ 13346:ϕ 13341:μ 13337:∂ 13330:ϕ 13261:α 13177:† 13162:μ 13158:∂ 13149:− 13109:μ 13084:→ 13055:μ 13028:ψ 13002:→ 12984:ψ 12919:ν 12904:μ 12867:μ 12857:ν 12853:∂ 12849:− 12839:ν 12829:μ 12825:∂ 12811:ν 12808:μ 12729:μ 12715:− 12710:μ 12706:∂ 12697:μ 12644:ψ 12631:¯ 12628:ψ 12618:− 12613:ν 12610:μ 12589:ν 12586:μ 12568:− 12559:ψ 12540:μ 12527:μ 12523:γ 12510:¯ 12507:ψ 12469:= 1,...,8 12368:α 12311:α 12295:α 12249:α 12209:α 12141:α 12128:μ 12124:∂ 12109:α 12103:− 12090:− 12065:μ 12057:→ 12043:μ 12022:ψ 12008:α 11997:→ 11985:ψ 11951:off-shell 11905:μ 11895:ν 11891:∂ 11887:− 11882:ν 11872:μ 11868:∂ 11859:ν 11856:μ 11825:γ 11819:† 11815:ψ 11805:¯ 11802:ψ 11761:μ 11753:ψ 11748:μ 11744:γ 11737:¯ 11734:ψ 11725:− 11720:ν 11717:μ 11707:ν 11704:μ 11686:− 11683:ψ 11672:− 11667:μ 11663:∂ 11657:μ 11653:γ 11638:¯ 11635:ψ 11360:− 11343:π 11317:Λ 11298:Λ 11290:≡ 11278:β 11175:π 11149:Λ 11130:Λ 11122:≡ 11110:β 10967:λ 10963:− 10950:λ 10942:λ 10938:δ 10915:− 10890:δ 10879:− 10864:δ 10805:ϕ 10789:λ 10785:δ 10779:− 10765:ϕ 10755:δ 10741:− 10729:ϕ 10723:μ 10719:∂ 10703:ϕ 10697:μ 10693:∂ 10680:δ 10652:ϕ 10632:λ 10626:− 10612:ϕ 10583:− 10571:ϕ 10565:μ 10561:∂ 10545:ϕ 10539:μ 10535:∂ 10455:ϕ 10430:ϕ 10282:⟩ 10279:Ω 10252:ϕ 10249:⋯ 10230:ϕ 10216:Ω 10213:⟨ 10123:− 10003:∫ 10000:λ 9994:− 9942:ϕ 9814:− 9789:− 9764:− 9731:∫ 9717:λ 9711:− 9705:⋅ 9673:⟩ 9650:ϕ 9638:ϕ 9626:ϕ 9614:ϕ 9602:ϕ 9590:ϕ 9573:⟨ 9556:∫ 9542:λ 9536:− 9468:⟩ 9438:ϕ 9419:ϕ 9402:⟨ 9399:⟩ 9369:ϕ 9350:ϕ 9333:⟨ 9320:⟩ 9290:ϕ 9271:ϕ 9254:⟨ 9251:⟩ 9221:ϕ 9202:ϕ 9185:⟨ 9172:⟩ 9142:ϕ 9123:ϕ 9106:⟨ 9103:⟩ 9073:ϕ 9054:ϕ 9037:⟨ 9027:⟩ 8997:ϕ 8978:ϕ 8959:ϕ 8940:ϕ 8923:⟨ 8809:− 8805:∫ 8793:⁡ 8786:ϕ 8776:∫ 8735:− 8731:∫ 8719:⁡ 8704:ϕ 8692:ϕ 8688:ϕ 8678:∫ 8667:ϵ 8661:− 8652:∞ 8649:→ 8635:⟩ 8632:Ω 8612:ϕ 8600:ϕ 8586:Ω 8583:⟨ 8553:⟩ 8542:⋯ 8531:⟨ 8505:⟩ 8462:ϕ 8458:⋯ 8426:ϕ 8408:⟨ 8385:∫ 8382:⋯ 8359:∫ 8316:λ 8310:− 8299:∞ 8284:∑ 8278:⟩ 8235:ϕ 8231:⋯ 8199:ϕ 8180:ϕ 8161:ϕ 8143:⟨ 8120:∫ 8117:⋯ 8094:∫ 8051:λ 8045:− 8034:∞ 8019:∑ 8009:⟩ 8006:Ω 7986:ϕ 7974:ϕ 7960:Ω 7957:⟨ 7937:λ 7895:ϕ 7881:λ 7862:∫ 7814:ϕ 7699:− 7695:∫ 7688:− 7680:⁡ 7594:− 7590:∫ 7583:− 7575:⁡ 7554:ϕ 7535:ϕ 7503:ϵ 7497:− 7488:∞ 7485:→ 7471:⟩ 7468:Ω 7448:ϕ 7436:ϕ 7422:Ω 7419:⟨ 7303:μ 7295:− 7290:μ 7277:μ 7266:− 7255:ϵ 7236:− 7231:μ 7221:μ 7194:π 7167:∫ 7159:→ 7156:ϵ 7139:− 7120:≡ 7117:⟩ 7094:ϕ 7082:ϕ 7065:⟨ 7011:⟩ 6983:⟩ 6980:Ω 6833:⟩ 6830:Ω 6810:ϕ 6798:ϕ 6784:Ω 6781:⟨ 6771:⟩ 6748:ϕ 6736:ϕ 6719:⟨ 6672:ϕ 6656:ϕ 6643:ϕ 6627:ϕ 6550:∫ 6538:⁡ 6522:ϕ 6512:∫ 6506:⟩ 6497:ϕ 6477:− 6458:ϕ 6454:⟨ 6418:⟩ 6409:ϕ 6381:− 6362:ϕ 6358:⟨ 6355:⟩ 6346:ϕ 6318:− 6299:ϕ 6295:⟨ 6292:⋯ 6289:⟩ 6281:− 6274:ϕ 6246:− 6227:ϕ 6223:⟨ 6214:− 6207:ϕ 6200:∫ 6197:⋯ 6188:ϕ 6181:∫ 6172:ϕ 6165:∫ 6159:⟩ 6150:ϕ 6130:− 6111:ϕ 6107:⟨ 6045:⟩ 6036:ϕ 6003:⟩ 5994:ϕ 5916:∂ 5908:∂ 5896:∂ 5883:∂ 5875:∂ 5863:∂ 5824:ϕ 5810:λ 5805:− 5796:ϕ 5772:− 5764:ϕ 5759:μ 5755:∂ 5743:ϕ 5738:μ 5734:∂ 5642:⟩ 5619:† 5598:^ 5581:† 5560:^ 5538:† 5517:^ 5480:† 5466:^ 5439:⟩ 5385:⟩ 5363:^ 5333:⟩ 5276:† 5262:^ 5247:† 5233:^ 5199:^ 5175:^ 5148:− 5137:δ 5124:π 5105:† 5091:^ 5067:^ 5014:⋅ 5003:− 4989:ω 4975:† 4961:^ 4941:⋅ 4916:ω 4909:− 4890:^ 4864:ω 4838:π 4811:∫ 4785:^ 4782:ϕ 4742:ω 4719:† 4705:^ 4667:^ 4637:^ 4634:ϕ 4599:⟩ 4576:† 4569:^ 4554:∝ 4551:⟩ 4518:† 4511:^ 4484:⟩ 4448:ω 4445:ℏ 4422:⟩ 4409:† 4402:^ 4375:ω 4372:ℏ 4350:† 4343:^ 4311:† 4304:^ 4294:ω 4291:ℏ 4280:† 4273:^ 4257:^ 4225:ω 4222:ℏ 4183:⟩ 4169:^ 4140:⟩ 4103:ω 4100:ℏ 4081:^ 4070:† 4063:^ 4053:ω 4050:ℏ 4041:^ 3992:† 3985:^ 3969:^ 3921:† 3914:^ 3881:^ 3829:ω 3816:† 3809:^ 3796:ω 3776:ω 3770:− 3759:^ 3747:ω 3720:^ 3679:^ 3601:ω 3588:∗ 3577:ω 3557:ω 3551:− 3537:ω 3392:ω 3321:⋅ 3310:− 3296:ω 3282:∗ 3257:⋅ 3232:ω 3225:− 3189:ω 3163:π 3136:∫ 3113:ϕ 3062:ϕ 3035:∇ 3031:− 3015:∂ 3005:∂ 2955:ϕ 2952:∂ 2940:∂ 2934:− 2911:∂ 2903:ϕ 2900:∂ 2894:∂ 2882:∂ 2859:∂ 2855:∂ 2833:∑ 2813:∂ 2805:ϕ 2802:∂ 2796:∂ 2784:∂ 2768:∂ 2764:∂ 2722:˙ 2719:ϕ 2655:ϕ 2631:− 2618:ϕ 2615:∇ 2599:− 2587:˙ 2584:ϕ 2548:∫ 2521:∫ 2215:in 1973. 1982:In 1954, 1727:In 1947, 1655:positrons 1633:known as 1422:electrons 1412:In 1913, 1233:electrons 1006:Wetterich 991:Weisskopf 941:Sudarshan 891:Schwinger 806:Nishijima 771:Maldacena 736:Leutwyler 701:Kinoshita 601:Goldstone 591:Gell-Mann 506:Doplicher 283:Equations 22034:Category 22015:Tevatron 21867:Theories 21824:Evidence 21688:Fermions 21113:M-theory 21073:ER = EPR 20912:Graviton 20826:Category 20620:Timeline 20372:Ensemble 20352:Bayesian 20245:Collapse 20117:Glossary 20100:Timeline 19824:Type IIB 19819:Type IIA 19804:4D N = 8 19799:4D N = 1 19768:6D (2,0) 19732:4D N = 1 19711:Polyakov 19670:Thirring 19479:Theories 19341:(1995). 19288:(1994). 19251:(2010). 19210:(2015). 19074:(1996). 19014:(1985). 18966:(1983). 18871:(1987). 18841:(1980). 18817:Springer 18811:(2000). 18771:(2000). 18745:(1982). 18685:(1998). 18606:(1994). 18578:Pais, A. 18224:54813572 18117:(1955). 17979:14951363 17935:(1989). 17895:: 9–47. 17842:(1998). 17745:55199323 17645:(2005). 17592:(1995). 17396:(2007). 17343:Fock, V. 17313:(2010). 17106:(2005). 17029:17799637 16679:Phys Rev 16664:15930139 16605:16578384 16538:16578383 16456:17744604 16397:(2012). 16348:(1999). 16265:(1893). 16217:20024506 16205:Daedalus 16156:18254182 15742:See also 15723:Δ > 0 15720:mass gap 15565:Kadanoff 15520:and the 15439:topology 15430:geometry 14812:a priori 14782:Higgsino 14740:(MSSM), 14665:and the 14643:fermions 14518:subgroup 12950:are the 12314:′ 12252:′ 11595:electron 11451:flavours 11077:function 10319:S-matrix 8569:theory. 7757:⟩ 7657:⟨ 7652:⟩ 7514:⟨ 6096:), then 6015:at time 3939:denotes 3896:and the 3365:denotes 2468:, where 2377:and the 2353:function 2324:and the 1884:symmetry 1761:, Lamb, 1533:and the 1451:Max Born 1021:Wightman 986:Weinberg 976:Virasoro 956:Tomonaga 951:Thirring 946:Symanzik 906:Semenoff 881:Schrader 846:Polyakov 766:Majorana 706:Klebanov 661:Ivanenko 651:'t Hooft 621:Guralnik 566:Fröhlich 561:Fritzsch 556:Frampton 471:Buchholz 416:Bargmann 406:Anderson 206:Crossing 22046:Commons 22010:Super-K 21846:problem 21600:Related 21484:General 21479:Special 21337:Applied 20779:Related 20758:History 20497:Science 20329:Rydberg 20095:History 19926:History 19909:Related 19706:Minimal 19557:Regular 19421:Fields. 19402:, 2001 19080:Bibcode 18538:Bibcode 18495:Bibcode 18289:5052535 18259:Bibcode 18204:Bibcode 18089:Bibcode 18014:Bibcode 17971:0990772 17951:Bibcode 17725:Bibcode 17680:Bibcode 17554:Bibcode 17470:Bibcode 17355:Bibcode 17311:Zee, A. 17213:Bibcode 17021:1749117 17001:Bibcode 16992:Science 16954:Bibcode 16888:Bibcode 16687:Bibcode 16655:1142349 16596:1063401 16565:Bibcode 16529:1063400 16498:Bibcode 16436:Bibcode 16428:Science 16320:Bibcode 16136:Bibcode 15676:compact 15048:. For 14772:or the 13363:measure 13310:anomaly 12784:in the 12454:= 1,2,3 11919:is the 11566:complex 7949:becomes 6909:is the 6085:be the 5686:strings 5313:is the 2430:photons 2295:in the 2282:phonons 2147:gravity 2145:except 2074:leptons 1964:leptons 1954:of the 1872:≈ 1/137 1852:of the 1811:-factor 1600:of the 1594:-factor 1407:photons 1217:History 1187:quantum 1166:and in 1031:Wilczek 996:Wentzel 971:Veltman 916:Shirkov 911:Shifman 901:Seiberg 886:Schwarz 866:Rubakov 791:Naimark 741:Lipatov 731:Lehmann 696:Kendall 586:Gelfand 581:Glashow 541:Feynman 521:Faddeev 516:Englert 486:Coleman 476:Cachazo 461:Brodsky 446:Bjorken 436:Berezin 426:Belavin 186:Anomaly 44:History 21511:Atomic 21466:Modern 21316:Major 21173:Others 20472:Popper 19866:Chiral 19814:Type I 19629:Yukawa 19550:Models 19357: 19327: 19300: 19267: 19237: 19196: 19171: 19152: 19124: 19098: 19049: 19028: 19000: 18974: 18956: 18935: 18911: 18883: 18857: 18823: 18793: 18757: 18727: 18697: 18671: 18652: 18622: 18592: 18556: 18513: 18379: 18369: 18346: 18336: 18287: 18277: 18222: 18170: 18150: 18057: 18032: 17977: 17969: 17917: 17870: 17801: 17768: 17743: 17653: 17604: 17564: 17488: 17412: 17373: 17325: 17231: 17114: 17027: 17019: 16916: 16906: 16845: 16775: 16750: 16725: 16662: 16652: 16644: 16603: 16593: 16585: 16536: 16526: 16518: 16454: 16405: 16358: 16245: 16215: 16154: 16092: 15679:simple 15603:, and 15585:axioms 15571:, and 15569:Wilson 15508:, and 15493:gluons 15483:. The 15470:Using 15460:anyons 15388:, and 15379:= det( 15374:where 15214:where 15017:where 14846:metric 14639:bosons 14574:− 2)/2 14550:− 1)/2 14497:where 13704:where 13406:, the 13206:where 12764:where 12668:where 12423:is an 12183:action 12170:where 11843:, and 11781:where 11605:photon 11562:theory 11437:where 11230:where 11025:, and 10834:where 10479:where 8866:where 7784:is an 7778:where 6594:where 5892: 5848:where 5307:where 3943:. 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9392:| 9388:} 9385:) 9380:4 9376:x 9372:( 9366:) 9361:1 9357:x 9353:( 9347:{ 9344:T 9340:| 9336:0 9330:+ 9317:0 9313:| 9309:} 9306:) 9301:4 9297:x 9293:( 9287:) 9282:2 9278:x 9274:( 9268:{ 9265:T 9261:| 9257:0 9248:0 9244:| 9240:} 9237:) 9232:3 9228:x 9224:( 9218:) 9213:1 9209:x 9205:( 9199:{ 9196:T 9192:| 9188:0 9182:+ 9169:0 9165:| 9161:} 9158:) 9153:4 9149:x 9145:( 9139:) 9134:3 9130:x 9126:( 9120:{ 9117:T 9113:| 9109:0 9100:0 9096:| 9092:} 9089:) 9084:2 9080:x 9076:( 9070:) 9065:1 9061:x 9057:( 9051:{ 9048:T 9044:| 9040:0 9034:= 9024:0 9020:| 9016:} 9013:) 9008:4 9004:x 9000:( 8994:) 8989:3 8985:x 8981:( 8975:) 8970:2 8966:x 8962:( 8956:) 8951:1 8947:x 8943:( 8937:{ 8934:T 8930:| 8926:0 8905:n 8893:λ 8876:L 8851:, 8844:] 8838:L 8832:z 8827:4 8823:d 8817:T 8812:T 8801:i 8797:[ 8781:D 8770:] 8764:L 8758:z 8753:4 8749:d 8743:T 8738:T 8727:i 8723:[ 8713:) 8710:y 8707:( 8701:) 8698:x 8695:( 8683:D 8670:) 8664:i 8658:1 8655:( 8646:T 8638:= 8628:| 8624:} 8621:) 8618:y 8615:( 8609:) 8606:x 8603:( 8597:{ 8594:T 8590:| 8550:0 8546:| 8538:| 8534:0 8511:. 8502:0 8498:| 8494:} 8489:4 8485:) 8479:n 8475:z 8471:( 8466:I 8453:4 8449:) 8443:1 8439:z 8435:( 8430:I 8422:{ 8419:T 8415:| 8411:0 8403:n 8399:z 8393:4 8389:d 8377:1 8373:z 8367:4 8363:d 8353:! 8350:n 8345:n 8341:) 8337:! 8334:4 8331:( 8324:n 8320:) 8313:i 8307:( 8294:0 8291:= 8288:n 8275:0 8271:| 8267:} 8262:4 8258:) 8252:n 8248:z 8244:( 8239:I 8226:4 8222:) 8216:1 8212:z 8208:( 8203:I 8195:) 8192:y 8189:( 8184:I 8176:) 8173:x 8170:( 8165:I 8157:{ 8154:T 8150:| 8146:0 8138:n 8134:z 8128:4 8124:d 8112:1 8108:z 8102:4 8098:d 8088:! 8085:n 8080:n 8076:) 8072:! 8069:4 8066:( 8059:n 8055:) 8048:i 8042:( 8029:0 8026:= 8023:n 8012:= 8002:| 7998:} 7995:) 7992:y 7989:( 7983:) 7980:x 7977:( 7971:{ 7968:T 7964:| 7915:4 7911:) 7907:x 7904:( 7899:I 7888:! 7885:4 7875:x 7870:3 7866:d 7859:= 7856:) 7853:t 7850:( 7845:I 7841:H 7818:4 7793:I 7791:ϕ 7781:ε 7763:, 7753:0 7749:| 7744:} 7739:] 7735:) 7732:t 7729:( 7724:I 7720:H 7715:t 7712:d 7707:T 7702:T 7691:i 7684:[ 7673:{ 7669:T 7665:| 7661:0 7648:0 7644:| 7639:} 7634:] 7630:) 7627:t 7624:( 7619:I 7615:H 7610:t 7607:d 7602:T 7597:T 7586:i 7579:[ 7569:) 7566:y 7563:( 7558:I 7550:) 7547:x 7544:( 7539:I 7530:{ 7526:T 7522:| 7518:0 7506:) 7500:i 7494:1 7491:( 7482:T 7474:= 7464:| 7460:} 7457:) 7454:y 7451:( 7445:) 7442:x 7439:( 7433:{ 7430:T 7426:| 7388:) 7385:t 7382:( 7377:I 7373:H 7352:) 7349:t 7346:( 7341:I 7337:L 7313:. 7308:) 7299:y 7286:x 7282:( 7273:p 7269:i 7262:e 7252:i 7249:+ 7244:2 7240:m 7227:p 7217:p 7212:i 7202:4 7198:) 7191:2 7188:( 7183:p 7178:4 7174:d 7162:0 7148:= 7145:) 7142:y 7136:x 7133:( 7128:F 7124:D 7114:0 7110:| 7106:} 7103:) 7100:y 7097:( 7091:) 7088:x 7085:( 7079:{ 7076:T 7072:| 7068:0 7032:x 7026:y 7008:0 7004:| 6976:| 6953:0 6949:y 6926:0 6922:x 6897:T 6873:y 6853:x 6826:| 6822:} 6819:) 6816:y 6813:( 6807:) 6804:x 6801:( 6795:{ 6792:T 6788:| 6768:0 6764:| 6760:} 6757:) 6754:y 6751:( 6745:) 6742:x 6739:( 6733:{ 6730:T 6726:| 6722:0 6681:. 6676:F 6668:= 6665:) 6662:T 6659:( 6652:, 6647:I 6639:= 6636:) 6633:0 6630:( 6609:H 6603:ϕ 6597:L 6579:, 6575:} 6571:L 6567:t 6564:d 6559:T 6554:0 6546:i 6542:{ 6531:) 6528:t 6525:( 6517:D 6509:= 6501:I 6492:| 6486:T 6483:H 6480:i 6473:e 6468:| 6462:F 6439:N 6421:. 6413:I 6404:| 6398:N 6394:/ 6390:T 6387:H 6384:i 6377:e 6372:| 6366:1 6350:1 6341:| 6335:N 6331:/ 6327:T 6324:H 6321:i 6314:e 6309:| 6303:2 6284:1 6278:N 6269:| 6263:N 6259:/ 6255:T 6252:H 6249:i 6242:e 6237:| 6231:F 6217:1 6211:N 6203:d 6192:2 6184:d 6176:1 6168:d 6162:= 6154:I 6145:| 6139:T 6136:H 6133:i 6126:e 6121:| 6115:F 6089:( 6082:H 6076:N 6070:T 6064:T 6060:t 6040:F 6031:| 6018:t 5998:I 5989:| 5951:λ 5941:μ 5924:1 5920:x 5912:/ 5905:= 5900:1 5889:, 5886:t 5879:/ 5872:= 5867:0 5851:μ 5833:, 5828:4 5817:! 5814:4 5800:2 5790:2 5786:m 5780:2 5777:1 5768:) 5750:( 5746:) 5730:( 5725:2 5722:1 5717:= 5712:L 5680:( 5645:. 5639:0 5635:| 5629:2 5624:) 5612:1 5607:p 5595:a 5588:( 5574:2 5569:p 5557:a 5548:3 5543:) 5531:3 5526:p 5514:a 5507:( 5474:p 5463:a 5436:0 5432:| 5408:. 5404:p 5394:, 5391:0 5388:= 5382:0 5378:| 5371:p 5360:a 5330:0 5326:| 5310:δ 5292:, 5289:0 5286:= 5282:] 5270:q 5259:a 5252:, 5241:p 5230:a 5222:[ 5218:= 5214:] 5207:q 5196:a 5189:, 5183:p 5172:a 5164:[ 5159:, 5156:) 5152:q 5144:p 5140:( 5132:3 5128:) 5121:2 5118:( 5115:= 5111:] 5099:q 5088:a 5081:, 5075:p 5064:a 5056:[ 5029:. 5025:) 5018:x 5010:p 5006:i 5000:t 4994:p 4985:i 4981:e 4969:p 4958:a 4951:+ 4945:x 4937:p 4933:i 4930:+ 4927:t 4921:p 4912:i 4905:e 4898:p 4887:a 4879:( 4869:p 4860:2 4856:1 4846:3 4842:) 4835:2 4832:( 4827:p 4822:3 4818:d 4808:= 4805:) 4802:t 4799:, 4795:x 4791:( 4765:p 4747:p 4713:p 4702:a 4675:p 4664:a 4620:ϕ 4602:. 4596:0 4592:| 4586:n 4581:) 4566:a 4559:( 4548:n 4544:| 4508:a 4481:0 4477:| 4456:2 4452:/ 4442:3 4419:0 4415:| 4399:a 4340:a 4316:, 4301:a 4288:= 4285:] 4270:a 4263:, 4254:H 4248:[ 4228:. 4217:2 4214:1 4189:0 4186:= 4180:0 4176:| 4166:a 4137:0 4133:| 4106:. 4095:2 4092:1 4087:+ 4078:a 4060:a 4047:= 4038:H 4002:= 3998:] 3982:a 3975:, 3966:a 3959:[ 3937:† 3911:a 3878:a 3861:a 3855:a 3837:. 3832:t 3826:i 3822:e 3806:a 3793:2 3789:1 3784:+ 3779:t 3773:i 3766:e 3756:a 3744:2 3740:1 3735:= 3732:) 3729:t 3726:( 3717:x 3691:) 3688:t 3685:( 3676:x 3660:) 3658:t 3656:( 3654:x 3645:x 3639:x 3633:ω 3627:a 3609:, 3604:t 3598:i 3594:e 3584:a 3574:2 3570:1 3565:+ 3560:t 3554:i 3547:e 3543:a 3534:2 3530:1 3525:= 3522:) 3519:t 3516:( 3513:x 3477:p 3473:ω 3463:p 3445:. 3438:2 3434:m 3430:+ 3425:2 3420:| 3414:p 3409:| 3403:= 3397:p 3376:p 3372:ω 3363:* 3354:a 3336:, 3332:) 3325:x 3317:p 3313:i 3307:t 3301:p 3292:i 3288:e 3276:p 3271:a 3267:+ 3261:x 3253:p 3249:i 3246:+ 3243:t 3237:p 3228:i 3221:e 3214:p 3209:a 3204:( 3194:p 3185:2 3181:1 3171:3 3167:) 3160:2 3157:( 3152:p 3147:3 3143:d 3133:= 3130:) 3127:t 3124:, 3120:x 3116:( 3065:= 3058:) 3052:2 3048:m 3044:+ 3039:2 3023:2 3019:t 3009:2 2998:( 2967:, 2964:0 2961:= 2945:L 2930:] 2924:) 2919:i 2915:x 2907:/ 2897:( 2887:L 2876:[ 2867:i 2863:x 2848:3 2843:1 2840:= 2837:i 2829:+ 2825:] 2819:) 2816:t 2809:/ 2799:( 2789:L 2778:[ 2771:t 2744:m 2739:∇ 2694:L 2669:, 2665:] 2659:2 2649:2 2645:m 2639:2 2636:1 2626:2 2622:) 2612:( 2607:2 2604:1 2594:2 2575:2 2572:1 2566:[ 2561:x 2556:3 2552:d 2545:= 2540:L 2534:x 2529:3 2525:d 2518:= 2515:L 2492:L 2477:t 2471:x 2466:) 2464:t 2460:x 2458:( 2456:ϕ 2411:) 2409:t 2405:x 2403:( 2401:B 2393:) 2391:t 2387:x 2385:( 2383:E 2375:) 2373:t 2369:x 2367:( 2365:g 2330:c 2321:ħ 2280:— 1870:α 1809:g 1744:P 1737:S 1712:( 1592:g 1138:( 1119:e 1112:t 1105:v

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Knowledge

Quantum field theory

Index