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that Alice and Betty are identical twins, labeled at birth by bracelets reading A and B. Because the girls are identical, nobody would be able to tell if they had been switched at birth; the labels A and B are arbitrary, and can be interchanged. Such a permanent interchanging of their identities is like a global gauge symmetry. There is also a corresponding local gauge symmetry, which describes the fact that from one moment to the next, Alice and Betty could swap roles while nobody was looking, and nobody would be able to tell. If we observe that Mom's favorite vase is broken, we can only infer that the blame belongs to one twin or the other, but we cannot tell whether the blame is 100% Alice's and 0% Betty's, or vice versa. If Alice and Betty are in fact quantum-mechanical particles rather than people, then they also have wave properties, including the property of
2323:). A wave with a shorter wavelength oscillates more rapidly, and therefore changes more rapidly between nearby points. Now suppose that we arbitrarily fix a gauge at one point in space, by saying that the energy at that location is 20% A's and 80% B's. We then measure the two waves at some other, nearby point, in order to determine their wavelengths. But there are two entirely different reasons that the waves could have changed. They could have changed because they were oscillating with a certain wavelength, or they could have changed because the gauge function changed from a 20â80 mixture to, say, 21â79. If we ignore the second possibility, the resulting theory does not work; strange discrepancies in momentum will show up, violating the principle of conservation of momentum. Something in the theory must be changed.
2292:, which allows waves to be added, subtracted, and mixed arbitrarily. It follows that we are not even restricted to complete swaps of identity. For example, if we observe that a certain amount of energy exists in a certain location in space, there is no experiment that can tell us whether that energy is 100% A's and 0% B's, 0% A's and 100% B's, or 20% A's and 80% B's, or some other mixture. The fact that the symmetry is local means that we cannot even count on these proportions to remain fixed as the particles propagate through space. The details of how this is represented mathematically depend on technical issues relating to the
2343:, which turns out to patch up the discrepancies that otherwise would have broken conservation of momentum. In the context of electromagnetism, the particles A and B would be charged particles such as electrons, and the quantum mechanical wave represented by θ would be the electromagnetic field. (Here we ignore the technical issues raised by the fact that electrons actually have spin 1/2, not spin zero. This oversimplification is the reason that the gauge field θ comes out to be a scalar, whereas the electromagnetic field is actually represented by a vector consisting of
1183:. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. For example, say you cannot measure the diameter of a lead ball, but you can determine how many lead balls, which are equal in every way, are required to make a pound. Using the number of balls, the density of lead, and the formula for calculating the volume of a sphere from its diameter, one could indirectly determine the diameter of a single lead ball.
38:
2197:
calculation falls outside the range of 0â¤Î¸<360°, we force it to "wrap around" into the allowed range, which covers a circle. Another way of putting this is that a phase angle of, say, 5° is considered to be completely equivalent to an angle of 365°. Experiments have verified this testable statement about the interference patterns formed by electron waves. Except for the "wrap-around" property, the algebraic properties of this mathematical structure are exactly the same as those of the ordinary real numbers.
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2184:, gauge symmetry applies to both electromagnetic waves and electron waves. These two gauge symmetries are in fact intimately related. If a gauge transformation θ is applied to the electron waves, for example, then one must also apply a corresponding transformation to the potentials that describe the electromagnetic waves. Gauge symmetry is required in order to make quantum electrodynamics a
1256:, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (
2351:.) The result is that we have an explanation for the presence of electromagnetic interactions: if we try to construct a gauge-symmetric theory of identical, non-interacting particles, the result is not self-consistent, and can only be repaired by adding electric and magnetic fields that cause the particles to interact.
2362:, where the term "boson" refers to a particle with integer spin. In the simplest versions of the theory gauge bosons are massless, but it is also possible to construct versions in which they have mass. This is the case for the gauge bosons that carry the weak interaction: the force responsible for nuclear decay.
1808:
and the otherâ100 years ago (or any other time in the past or in the future), the two experiments would again produce completely identical results. The invariance of the properties of a hydrogen atom with respect to the time and place where these properties were investigated is called translation invariance.
2287:
Surprisingly, gauge symmetry can give a deeper explanation for the existence of interactions, such as the electric and nuclear interactions. This arises from a type of gauge symmetry relating to the fact that all particles of a given type are experimentally indistinguishable from one another. Imagine
2171:
amount θ, where θ could depend on both the position in space and on time. This would have no effect on the result of the experiment, since the final observation of the location of the electron occurs at a single place and time, so that the phase shift in each electron's "clock" would be the same, and
2121:
to exist within the solenoid. But the solenoid has been positioned so that the electron cannot possibly pass through its interior. If one believed that the fields were the fundamental quantities, then one would expect that the results of the experiment would be unchanged. In reality, the results are
1807:
Suppose, for example, that one observer examines the properties of a hydrogen atom on Earth, the otherâon the Moon (or any other place in the universe), the observer will find that their hydrogen atoms exhibit completely identical properties. Again, if one observer had examined a hydrogen atom today
2162:
in phase between the two parts of the electron wave. Suppose we imagine the two parts of the electron wave as tiny clocks, each with a single hand that sweeps around in a circle, keeping track of its own phase. Although this cartoon ignores some technical details, it retains the physical phenomena
2078:
But now suppose that the electrons in the experiment are subject to electric or magnetic fields. For example, if an electric field were imposed on one side of the axis but not on the other, the results of the experiment would be affected. The part of the electron wave passing through that side
1762:
The
Cartesian coordinate grid on this square has been distorted by a coordinate transformation, so that there is a nonlinear relationship between the old (x,y) coordinates and the new ones. Einstein's equations of general relativity are still valid in the new coordinate system. Such changes of
2091:
the electrical potential. The results of the experiment will be different, because phase relationships between the two parts of the electron wave have changed, and therefore the locations of constructive and destructive interference will be shifted to one side or the other. It is the electric
2196:
The description of the electrons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the
2075:. If there are no electric or magnetic fields present in this experiment, then the electron's energy is constant, and, for example, there will be a high probability of detecting the electron along the central axis of the experiment, where by symmetry the two parts of the wave are in phase.
1517:
is also a solution to
Maxwell's equations and no experiment can distinguish between these two solutions. In other words, the laws of physics governing electricity and magnetism (that is, Maxwell equations) are invariant under gauge transformation. Maxwell's equations have a gauge symmetry.
1612:
measures the extra energy stored in the electric field because of the existence of a charge at a certain point. Outside of the interval during which the particle exists, conservation of energy would be satisfied, because the net energy released by creation and destruction of the particle,
1819:+100 years. Both observers discover the same laws of physics. Because light from hydrogen atoms in distant galaxies may reach the earth after having traveled across space for billions of years, in effect one can do such observations covering periods of time almost all the way back to the
2247:
As a way of visualizing the choice of a gauge, consider whether it is possible to tell if a cylinder has been twisted. If the cylinder has no bumps, marks, or scratches on it, we cannot tell. We could, however, draw an arbitrary curve along the cylinder, defined by some function
1460:
must have two probes, and can only report the voltage difference between them. Thus one could choose to define all voltage differences relative to some other standard, rather than the Earth, resulting in the addition of a constant offset. If the potential
2050:
is performed with electrons, then a wave-like interference pattern is observed. The electron has the highest probability of being detected at locations where the parts of the wave passing through the two slits are in phase with one another, resulting in
1579:
at a certain point in space, 1, moving it to some other point 2, and then destroying it. We might imagine that this process was consistent with conservation of energy. We could posit a rule stating that creating the charge required an input of energy
2326:
Again there are technical issues relating to spin, but in several important cases, including electrically charged particles and particles interacting via nuclear forces, the solution to the problem is to impute physical reality to the gauge function
2163:
that are important here. If both clocks are sped up by the same amount, the phase relationship between them is unchanged, and the results of experiments are the same. Not only that, but it is not even necessary to change the speed of each clock by a
2007:
Until the advent of quantum mechanics, the only well known example of gauge symmetry was in electromagnetism, and the general significance of the concept was not fully understood. For example, it was not clear whether it was the fields
2358:) describes a wave, the laws of quantum mechanics require that it also have particle properties. In the case of electromagnetism, the particle corresponding to electromagnetic waves is the photon. In general, such particles are called
2099:
Schematic of double-slit experiment in which
AharonovâBohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field
2113:
It is even possible to have cases in which an experiment's results differ when the potentials are changed, even if no charged particle is ever exposed to a different field. One such example is the
2092:
potential that occurs here, not the electric field, and this is a manifestation of the fact that it is the potentials and not the fields that are of fundamental significance in quantum mechanics.
2232:
such that the sum of a phase and its inverse is 0. Other examples of abelian groups are the integers under addition, 0, and negation, and the nonzero fractions under product, 1, and reciprocal.
1198:. For example, if you could measure the color of lead balls and discover that when you change the color, you still fit the same number of balls in a pound, the property of "color" would show
1951:
2267:
proposed to generalize these ideas to noncommutative groups. A noncommutative gauge group can describe a field that, unlike the electromagnetic field, interacts with itself. For example,
2256:
measures distance along the axis of the cylinder. Once this arbitrary choice (the choice of gauge) has been made, it becomes possible to detect it if someone later twists the cylinder.
1330:
of railroads) might also be a local symmetry of electromagnetism. Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. After the development of
1564:, along with the precise statement of the nature of the gauge transformation. The relevant point here is that the fields remain the same under the gauge transformation, and therefore
2319:. In terms of empirical measurements, the wavelength can only be determined by observing a change in the wave between one point in space and another nearby point (mathematically, by
1673:, which implies that no experiment should be able to measure the absolute potential, without reference to some external standard such as an electrical ground. But the proposed rules
1776:
are arbitrary coordinate transformations. Technically, the transformations must be invertible, and both the transformation and its inverse must be smooth, in the sense of being
1641:. But although this scenario salvages conservation of energy, it violates gauge symmetry. Gauge symmetry requires that the laws of physics be invariant under the transformation
1892:
1671:
1515:
1448:) that is defined at every point in space, and in practical work it is conventional to take the Earth as a physical reference that defines the zero level of the potential, or
2303:
According to the principles of quantum mechanics, particles do not actually have trajectories through space. Motion can only be described in terms of waves, and the momentum
2296:
of the particles, but for our present purposes we consider a spinless particle, for which it turns out that the mixing can be specified by some arbitrary choice of gauge θ(
1894:, rotations, etc., but become completely arbitrary, so that, for example, one can define an entirely new time-like coordinate according to some arbitrary rule such as
1994:
Maxwell's equations can also be expressed in a generally covariant form, which is as invariant under general coordinate transformation as
Einstein's field equation.
2236:
1978:
2456:
JĂźrgen Renn and John
Stachel (2007), "Hilbert's Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity", in Renn, JĂźrgen (ed.),
1811:
Recalling our two observers from different ages: the time in their experiments is shifted by 100 years. If the time when the older observer did the experiment was
1314:
had derived
Einstein's equations of general relativity by postulating a symmetry under any change of coordinates, just as Einstein was completing his work. Later
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In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i.e.,
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the two effects would cancel out. This is another example of a gauge transformation: it is local, and it does not change the results of experiments.
2300:), where an angle θ = 0° represents 100% A and 0% B, θ = 90° means 0% A and 100% B, and intermediate angles represent mixtures.
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2024:
that were the fundamental quantities; if the former, then the gauge transformations could be considered as nothing more than a mathematical trick.
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1991:, and equations with this property are referred to as written in the covariant form. General covariance is a special case of gauge invariance.
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2331:). We say that if the function θ oscillates, it represents a new type of quantum-mechanical wave, and this new wave has its own momentum
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The importance of gauge theories for physics stems from their tremendous success in providing a unified framework to describe the
1282:. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the
2153:
2725:
1528:, which can also undergo gauge transformations. These transformations may be local. That is, rather than adding a constant onto
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allow an experimenter to determine the absolute potential, simply by comparing the energy input required to create the charge
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Some global symmetries under changes of coordinate predate both general relativity and the concept of a gauge. For example,
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1575:. Suppose that there existed some process by which one could briefly violate conservation of charge by creating a charge
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2967:
2781:. Princeton University Press. A nontechnical description of quantum field theory (not specifically about gauge theory).
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respectively. The conclusion is that if gauge symmetry holds, and energy is conserved, then charge must be conserved.
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Another example of a symmetry: the invariance of
Einstein's field equation under arbitrary coordinate transformations
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concludes that energy is equivalent to mass. Hence a gravitational field induces a further gravitational field. The
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2769:. Johns Hopkins University Press. A serious attempt by a physicist to explain gauge theory and the Standard Model.
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Invariance of the form of an equation under an arbitrary coordinate transformation is customarily referred to as
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concept that different places in space, such as the earth versus the heavens, obeyed different physical rules.
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Gauge invariance: the results of the experiments are independent of the choice of the gauge for the potentials
2126:
in the region that the electrons do pass through. Now that it has been established that it is the potentials
1346:, and applying it successfully to electromagnetism. Gauge symmetry was generalized mathematically in 1954 by
1194:; the lack of change in the measurable quantities, despite the field being transformed, is a property called
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theory, i.e., one in which the calculated predictions of all physically measurable quantities are finite.
1310:"). The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed,
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310:
252:
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1278:(usually employed for scattering theory) describes forces in terms of force-mediating particles called
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In quantum mechanics, a particle such as an electron is also described as a wave. For example, if the
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1163:), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a
300:
2529:; Straumann, Norbert (2000-01-01). "Gauge theory: Historical origins and some modern developments".
19:
This article is a non-technical introduction to the subject. For the main encyclopedia article, see
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2446:, Addison Wesley Longman, 1970, II-15-7,8,12, but this is partly a matter of personal preference.
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1210:. Generally, any theory that has the property of gauge invariance is considered a gauge theory.
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In field theories, different configurations of the unobservable fields can result in identical
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1286:, a quantum field theory that accurately predicts all of the fundamental interactions except
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These books are intended for general readers and employ the barest minimum of mathematics.
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As discussed above, the gauge transformations for classical (i.e., non-quantum mechanical)
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Generalizing from static electricity to electromagnetism, we have a second potential, the
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Note that in these experiments, the only quantity that affects the result is the
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Historically, the first example of gauge symmetry to be discovered was classical
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modified their gauge choice by replacing the scale factor with a change of wave
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1190:. A transformation from one such field configuration to another is called a
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different, because turning on the solenoid changed the vector potential
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is turned on in the cylindrical solenoid, marked in blue on the diagram.
1233:) are not. Under a gauge transformation in which a constant is added to
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2417:, p. 451. For an alternative formulation in terms of symmetries of the
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coordinate system are the gauge transformations of general relativity.
1627:, would be equal to the work done in moving the particle from 1 to 2,
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An example of a symmetry in a physical theory: translation invariance
1457:
1268:
1823:, and they show that the laws of physics have always been the same.
3188:
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in potential are physically measurable, which is the reason that a
692:
2578:. Gerstein - University of Toronto. London, Methuen & co. ltd.
1709:
at a particular point in space in the case where the potential is
2167:
amount. We could change the angle of the hand on each clock by a
1862:
are not only "relative" in the global sense of translations like
1789:
1287:
1146:
1267:, in which case the transformations vary from point to point in
2748:: "Gauge Theories of the Force between Elementary Particles,"
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1536:
is also changed in certain corresponding ways, then the same
2455:
2307:
of an individual particle is related to its wavelength Îť by
2621:
2063:
is related to the kinetic energy of an individual electron
2479:(1919), "Eine neue Erweiterung der RelativitĂatstheorie,"
2142:, we can see that the gauge transformations, which change
2079:
oscillates at a different rate, since its energy has had â
1424:. A static electric field can be described in terms of an
2572:
Weyl, Hermann; Brose, Henry Herman
Leopold Adolf (1922).
1485:
then, after this gauge transformation, the new potential
2442:
are more fundamental, see
Feynman, Leighton, and Sands,
2235:
1167:). Modern theories describe physical forces in terms of
2525:
2216:, so that θ + Ď = Ď + θ.
1298:
The earliest field theory having a gauge symmetry was
2200:
In mathematical terminology, electron phases form an
1959:
1900:
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1735:
1715:
1647:
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1467:
1434:
2228:, namely "0". Also, for every phase there exists an
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In other words, if in the theory we change the time
1213:
For example, in electromagnetism the electric field
1370:, and its unification with electromagnetism in the
2497:
2271:states that gravitational fields have energy, and
1972:
1945:
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1747:
1721:
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1509:
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1440:
3418:
3518:
2756:"Press Release: The 1999 Nobel Prize in Physics"
2036:Double-slit diffraction and interference pattern
2693:. Addison Wesley, vol. II, chpt. 15, section 5.
1308:A Dynamical Theory of the Electromagnetic Field
2108:
1946:{\displaystyle t\rightarrow t+t^{3}/t_{0}^{2}}
1252:in the 1920s, and with successive advances in
1179:, and fields that describe forces between the
3404:
2800:
1701:for the energies of creation and destruction
1165:number of lead balls in a pound of ammunition
1159:, a thickness, an in-between distance (as in
1118:
2537:(1). American Physical Society (APS): 1â23.
2421:, see p. 489. Also see J. D. Jackson (1975)
2191:
2778:QED: The Strange Theory of Light and Matter
2708:QED: The Strange Theory of Light and Matter
2495:
2489:
2027:
1326:or "gauge" (a term inspired by the various
3411:
3397:
2814:
2807:
2793:
2279:also have this self-interacting property.
2002:
1125:
1111:
36:
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2465:, vol. 4, Springer, pp. 857â973
2372:
2134:that are fundamental, and not the fields
1293:
2234:
2094:
2031:
1757:
1260:) arise from the constraints imposed by
1997:
1815:, the time of the modern experiment is
1404:
1206:, gauge invariance is sometimes called
3519:
2885:Two-dimensional conformal field theory
3392:
2788:
2500:An Elementary Primer for Gauge Theory
1767:
1571:Gauge symmetry is closely related to
1221:are observable, while the potentials
2687:Feynman, Leighton, and Sands (1970)
2212:(1). "Abelian" means that addition
2067:via the quantum-mechanical relation
1318:, inspired by success in Einstein's
2762:. Nobel Media AB 2013. 20 Aug 2013.
2726:Introduction to High-Energy Physics
2672:Misner, Thorne, and Wheeler (1973)
2637:. University of Chicago Press: 260.
2616:Introduction to High-Energy Physics
2397:Introduction to High-Energy Physics
1409:
13:
2736:
2087:is the charge of the electron and
1393:. This gauge theory, known as the
14:
3553:
3380:Template:Quantum mechanics topics
2524:For a review and references, see
2459:The Genesis of General Relativity
1608:, which would seem natural since
1362:, later found application in the
1237:, no observable change occurs in
3375:
3374:
1980:has dimensions of time, and yet
1887:{\displaystyle t\rightarrow t+C}
1666:{\displaystyle V\rightarrow V+C}
1510:{\displaystyle V\rightarrow V+C}
2714:
2696:
2690:The Feynman Lectures on Physics
2681:
2666:
2640:
2608:
2595:
2582:
2282:
1302:'s formulation, in 1864â65, of
2605:, 2nd ed. Wiley and Sons: 176.
2565:
2518:
2470:
2449:
2428:
2425:, 2nd ed. Wiley and Sons: 176.
2402:
2386:
1904:
1872:
1780:an arbitrary number of times.
1651:
1495:
1354:in an attempt to describe the
1:
3419:Introductory science articles
2752:, 242(6):104â138 (June 1980).
2711:. Princeton University Press.
2365:
165:Spontaneous symmetry breaking
125:Symmetry in quantum mechanics
7:
3344:Quantum information science
2204:under addition, called the
2109:Explanation with potentials
1594:and destroying it released
1379:quantum-mechanical behavior
10:
3558:
2175:
2039:
1800:, an advancement from the
1413:
160:Explicit symmetry breaking
18:
3498:
3450:
3424:
3369:
3252:
3202:
3181:
3130:
3104:
3068:
3032:
2981:
2900:
2893:
2822:
2614:Donald H. Perkins (1982)
2603:Classical Electrodynamics
2590:Electricity and Magnetism
2531:Reviews of Modern Physics
2527:OâRaifeartaigh, Lochlainn
2423:Classical Electrodynamics
2192:Types of gauge symmetries
2053:constructive interference
1984:will have the same form.
1796:introduced the notion of
1523:magnetic vector potential
316:BargmannâWigner equations
2354:Although the function θ(
2028:AharonovâBohm experiment
1560:is given in the article
1258:fundamental interactions
3040:2D free massless scalar
2933:Quantum electrodynamics
2860:QFT in curved spacetime
2551:10.1103/revmodphys.72.1
2182:quantum electrodynamics
2003:Quantum electrodynamics
1217:and the magnetic field
341:Electroweak interaction
336:Quantum electrodynamics
311:WheelerâDeWitt equation
198:Background field method
3532:Quantum chromodynamics
3478:mathematical formalism
3361:Quantum thermodynamics
3285:On shell and off shell
3280:Loop quantum cosmology
3122:N = 4 super YangâMills
3081:N = 1 super YangâMills
2948:Scalar electrodynamics
2938:Quantum chromodynamics
2840:Conformal field theory
2816:Quantum field theories
2729:. Addison-Wesley: 332.
2588:Edward Purcell (1963)
2244:
2105:
2048:double-slit experiment
2037:
1974:
1947:
1888:
1798:translation invariance
1764:
1749:
1723:
1667:
1511:
1475:
1442:
1294:History and importance
346:Quantum chromodynamics
223:Effective field theory
3537:Differential topology
3334:Quantum hydrodynamics
3329:Quantum hadrodynamics
2953:Scalar chromodynamics
2765:Schumm, Bruce (2004)
2678:. W. H. Freeman: 967.
2618:. Addison-Wesley: 92.
2434:For an argument that
2399:. Addison-Wesley: 22.
2380:"Definition of Gauge"
2238:
2098:
2035:
1975:
1973:{\displaystyle t_{0}}
1948:
1889:
1761:
1750:
1724:
1668:
1568:are still satisfied.
1512:
1476:
1443:
1356:strong nuclear forces
1188:observable quantities
1173:electromagnetic field
301:KleinâGordon equation
243:LSZ reduction formula
3305:Quantum fluctuations
3275:Loop quantum gravity
2845:Lattice field theory
2663:. W. H. Freeman: 68.
2601:J.D. Jackson (1975)
2504:. World Scientific.
2496:K. Moriyasu (1983).
2444:The Feynman Lectures
2220:means that addition
2115:AharonovâBohm effect
2083:added to it, where â
2042:AharonovâBohm effect
1998:In quantum mechanics
1982:Einstein's equations
1957:
1898:
1866:
1733:
1713:
1645:
1489:
1465:
1432:
1405:In classical physics
1364:quantum field theory
1358:. This idea, dubbed
1276:quantum field theory
1254:quantum field theory
1192:gauge transformation
1181:elementary particles
384:Theory of everything
238:Lattice field theory
208:Correlation function
30:Quantum field theory
16:Introductory article
3339:Quantum information
2943:Quartic interaction
2750:Scientific American
2543:2000RvMP...72....1O
2414:The Road to Reality
1942:
1846:, coordinates like
1748:{\displaystyle V+C}
1573:charge conservation
1566:Maxwell's equations
1552:and the potentials
1483:Maxwell's equations
1248:With the advent of
1177:gravitational field
363:Incomplete theories
3473:general relativity
3225:NambuâJona-Lasinio
3153:Higher dimensional
3060:WessâZuminoâWitten
2850:Noncommutative QFT
2635:General Relativity
2592:. McGraw-Hill: 38.
2419:Lagrangian density
2273:special relativity
2269:general relativity
2245:
2239:Gauge fixing of a
2106:
2059:, of the electron
2038:
2016:or the potentials
1989:general covariance
1970:
1943:
1928:
1884:
1844:general relativity
1774:general relativity
1768:General relativity
1765:
1745:
1719:
1663:
1507:
1471:
1438:
1426:electric potential
1399:fundamental forces
1320:general relativity
248:Partition function
175:Topological charge
95:General relativity
90:Special relativity
3514:
3513:
3506:systolic geometry
3490:quantum mechanics
3386:
3385:
3248:
3247:
2721:Donald H. Perkins
2575:Space-time-matter
2511:978-9971-950-83-5
2393:Donald H. Perkins
2055:. The frequency,
1722:{\displaystyle V}
1481:is a solution to
1474:{\displaystyle V}
1441:{\displaystyle V}
1360:YangâMills theory
1332:quantum mechanics
1250:quantum mechanics
1135:
1134:
228:Expectation value
203:BRST quantization
150:PoincarĂŠ symmetry
105:YangâMills theory
85:Quantum mechanics
3549:
3458:electromagnetism
3413:
3406:
3399:
3390:
3389:
3378:
3377:
3295:Quantum dynamics
2968:YangâMillsâHiggs
2923:Non-linear sigma
2913:EulerâHeisenberg
2898:
2897:
2809:
2802:
2795:
2786:
2785:
2773:Feynman, Richard
2767:Deep Down Things
2746:'t Hooft, Gerard
2730:
2718:
2712:
2700:
2694:
2685:
2679:
2670:
2664:
2644:
2638:
2628:
2619:
2612:
2606:
2599:
2593:
2586:
2580:
2579:
2569:
2563:
2562:
2522:
2516:
2515:
2503:
2493:
2487:
2474:
2468:
2466:
2464:
2453:
2447:
2432:
2426:
2406:
2400:
2390:
2384:
2383:
2376:
2226:identity element
1979:
1977:
1976:
1971:
1969:
1968:
1952:
1950:
1949:
1944:
1941:
1936:
1927:
1922:
1921:
1893:
1891:
1890:
1885:
1754:
1752:
1751:
1746:
1728:
1726:
1725:
1720:
1672:
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1669:
1664:
1516:
1514:
1513:
1508:
1480:
1478:
1477:
1472:
1447:
1445:
1444:
1439:
1422:electromagnetism
1410:Electromagnetism
1383:electromagnetism
1265:gauge symmetries
1231:vector potential
1225:("voltage") and
1200:gauge invariance
1196:gauge invariance
1127:
1120:
1113:
218:Effective action
145:Lorentz symmetry
70:Electromagnetism
40:
26:
25:
3557:
3556:
3552:
3551:
3550:
3548:
3547:
3546:
3517:
3516:
3515:
3510:
3494:
3446:
3420:
3417:
3387:
3382:
3365:
3317:Quantum gravity
3244:
3203:Particle theory
3198:
3177:
3126:
3100:
3064:
3028:
2982:Low dimensional
2977:
2918:GinzburgâLandau
2889:
2880:Topological QFT
2818:
2813:
2739:
2737:Further reading
2734:
2733:
2719:
2715:
2703:Richard Feynman
2701:
2697:
2686:
2682:
2671:
2667:
2655:John A. Wheeler
2645:
2641:
2629:
2622:
2613:
2609:
2600:
2596:
2587:
2583:
2570:
2566:
2523:
2519:
2512:
2494:
2490:
2481:Ann. der Physik
2475:
2471:
2462:
2454:
2450:
2433:
2429:
2407:
2403:
2391:
2387:
2378:
2377:
2373:
2368:
2321:differentiation
2285:
2194:
2178:
2156:
2111:
2044:
2030:
2005:
2000:
1964:
1960:
1958:
1955:
1954:
1937:
1932:
1923:
1917:
1913:
1899:
1896:
1895:
1867:
1864:
1863:
1840:
1786:
1770:
1734:
1731:
1730:
1714:
1711:
1710:
1700:
1693:
1686:
1679:
1646:
1643:
1642:
1640:
1633:
1626:
1619:
1607:
1600:
1593:
1586:
1540:(electric) and
1490:
1487:
1486:
1466:
1463:
1462:
1433:
1430:
1429:
1418:
1412:
1407:
1304:electrodynamics
1296:
1161:railroad tracks
1131:
1102:
1101:
1100:
1098:
402:
394:
393:
389:Quantum gravity
364:
356:
355:
351:Higgs mechanism
331:
321:
320:
306:Proca equations
291:
283:
282:
268:Renormalization
233:Feynman diagram
188:
180:
179:
120:
110:
109:
60:
45:
43:Feynman diagram
24:
17:
12:
11:
5:
3555:
3545:
3544:
3539:
3534:
3529:
3527:Gauge theories
3512:
3511:
3509:
3508:
3502:
3500:
3496:
3495:
3493:
3492:
3487:
3482:
3481:
3480:
3470:
3465:
3460:
3454:
3452:
3448:
3447:
3445:
3444:
3439:
3434:
3428:
3426:
3422:
3421:
3416:
3415:
3408:
3401:
3393:
3384:
3383:
3370:
3367:
3366:
3364:
3363:
3358:
3353:
3352:
3351:
3341:
3336:
3331:
3326:
3325:
3324:
3314:
3313:
3312:
3302:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3260:Casimir effect
3256:
3254:
3250:
3249:
3246:
3245:
3243:
3242:
3237:
3235:Standard Model
3232:
3227:
3222:
3217:
3212:
3206:
3204:
3200:
3199:
3197:
3196:
3191:
3185:
3183:
3179:
3178:
3176:
3175:
3170:
3165:
3160:
3155:
3150:
3145:
3140:
3134:
3132:
3128:
3127:
3125:
3124:
3119:
3114:
3108:
3106:
3105:Superconformal
3102:
3101:
3099:
3098:
3093:
3088:
3086:SeibergâWitten
3083:
3078:
3072:
3070:
3069:Supersymmetric
3066:
3065:
3063:
3062:
3057:
3052:
3047:
3042:
3036:
3034:
3030:
3029:
3027:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2985:
2983:
2979:
2978:
2976:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2904:
2902:
2895:
2891:
2890:
2888:
2887:
2882:
2877:
2872:
2867:
2862:
2857:
2852:
2847:
2842:
2837:
2832:
2826:
2824:
2820:
2819:
2812:
2811:
2804:
2797:
2789:
2783:
2782:
2770:
2763:
2760:Nobelprize.org
2753:
2738:
2735:
2732:
2731:
2713:
2695:
2680:
2665:
2647:Charles Misner
2639:
2631:Robert M. Wald
2620:
2607:
2594:
2581:
2564:
2517:
2510:
2488:
2469:
2448:
2427:
2401:
2385:
2370:
2369:
2367:
2364:
2284:
2281:
2277:nuclear forces
2261:Chen Ning Yang
2193:
2190:
2186:renormalizable
2177:
2174:
2155:
2152:
2110:
2107:
2040:Main article:
2029:
2026:
2004:
2001:
1999:
1996:
1967:
1963:
1940:
1935:
1931:
1926:
1920:
1916:
1912:
1909:
1906:
1903:
1883:
1880:
1877:
1874:
1871:
1842:In Einstein's
1839:
1836:
1785:
1782:
1778:differentiable
1769:
1766:
1744:
1741:
1738:
1718:
1698:
1691:
1684:
1677:
1662:
1659:
1656:
1653:
1650:
1638:
1631:
1624:
1617:
1605:
1598:
1591:
1584:
1506:
1503:
1500:
1497:
1494:
1470:
1437:
1414:Main article:
1411:
1408:
1406:
1403:
1395:Standard Model
1348:Chen Ning Yang
1295:
1292:
1284:Standard Model
1269:space and time
1208:gauge symmetry
1133:
1132:
1130:
1129:
1122:
1115:
1107:
1104:
1103:
1096:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
410:
404:
403:
400:
399:
396:
395:
392:
391:
386:
381:
376:
371:
365:
362:
361:
358:
357:
354:
353:
348:
343:
338:
332:
329:Standard Model
327:
326:
323:
322:
319:
318:
313:
308:
303:
298:
296:Dirac equation
292:
289:
288:
285:
284:
281:
280:
278:Wick's theorem
275:
270:
265:
263:Regularization
260:
255:
250:
245:
240:
235:
230:
225:
220:
215:
210:
205:
200:
195:
189:
186:
185:
182:
181:
178:
177:
172:
170:Noether charge
167:
162:
157:
155:Gauge symmetry
152:
147:
142:
137:
132:
127:
121:
116:
115:
112:
111:
108:
107:
102:
97:
92:
87:
82:
77:
72:
67:
61:
58:
57:
54:
53:
47:
46:
41:
33:
32:
15:
9:
6:
4:
3:
2:
3554:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3524:
3522:
3507:
3504:
3503:
3501:
3497:
3491:
3488:
3486:
3483:
3479:
3476:
3475:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3455:
3453:
3449:
3443:
3440:
3438:
3435:
3433:
3430:
3429:
3427:
3423:
3414:
3409:
3407:
3402:
3400:
3395:
3394:
3391:
3381:
3373:
3368:
3362:
3359:
3357:
3356:Quantum logic
3354:
3350:
3347:
3346:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3323:
3320:
3319:
3318:
3315:
3311:
3308:
3307:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3290:Quantum chaos
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3265:Cosmic string
3263:
3261:
3258:
3257:
3255:
3251:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3207:
3205:
3201:
3195:
3192:
3190:
3187:
3186:
3184:
3180:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3138:Pure 4D N = 1
3136:
3135:
3133:
3129:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3107:
3103:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3071:
3067:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3037:
3035:
3031:
3025:
3022:
3020:
3019:ThirringâWess
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2994:BulloughâDodd
2992:
2990:
2989:2D YangâMills
2987:
2986:
2984:
2980:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2905:
2903:
2899:
2896:
2892:
2886:
2883:
2881:
2878:
2876:
2873:
2871:
2868:
2866:
2865:String theory
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2835:Axiomatic QFT
2833:
2831:
2830:Algebraic QFT
2828:
2827:
2825:
2821:
2817:
2810:
2805:
2803:
2798:
2796:
2791:
2790:
2787:
2780:
2779:
2774:
2771:
2768:
2764:
2761:
2757:
2754:
2751:
2747:
2744:
2743:
2742:
2728:
2727:
2722:
2717:
2710:
2709:
2704:
2699:
2692:
2691:
2684:
2677:
2676:
2669:
2662:
2661:
2656:
2652:
2648:
2643:
2636:
2632:
2627:
2625:
2617:
2611:
2604:
2598:
2591:
2585:
2577:
2576:
2568:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
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2410:
2409:Roger Penrose
2405:
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2346:
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2338:
2335: =
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2324:
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2318:
2314:
2311: =
2310:
2306:
2301:
2299:
2295:
2291:
2290:superposition
2280:
2278:
2274:
2270:
2266:
2262:
2257:
2255:
2251:
2242:
2237:
2233:
2231:
2227:
2223:
2219:
2215:
2211:
2207:
2203:
2202:Abelian group
2198:
2189:
2187:
2183:
2173:
2170:
2166:
2161:
2151:
2149:
2145:
2141:
2137:
2133:
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2125:
2120:
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2103:
2097:
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2076:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2043:
2034:
2025:
2023:
2019:
2015:
2011:
1995:
1992:
1990:
1985:
1983:
1965:
1961:
1938:
1933:
1929:
1924:
1918:
1914:
1910:
1907:
1901:
1881:
1878:
1875:
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1321:
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1274:
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1189:
1184:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1153:
1148:
1144:
1141:is a type of
1140:
1128:
1123:
1121:
1116:
1114:
1109:
1108:
1106:
1105:
1099:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1068:Zamolodchikov
1066:
1064:
1063:Zamolodchikov
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
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1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
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979:
976:
974:
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949:
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409:
406:
405:
398:
397:
390:
387:
385:
382:
380:
377:
375:
374:Supersymmetry
372:
370:
369:String theory
367:
366:
360:
359:
352:
349:
347:
344:
342:
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325:
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317:
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138:
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128:
126:
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122:
119:
114:
113:
106:
103:
101:
98:
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83:
81:
78:
76:
73:
71:
68:
66:
63:
62:
56:
55:
52:
49:
48:
44:
39:
35:
34:
31:
28:
27:
22:
3468:gauge theory
3467:
3371:
3300:Quantum foam
3240:Stueckelberg
3194:ChernâSimons
3131:Supergravity
2870:Supergravity
2855:Gauge theory
2776:
2766:
2759:
2749:
2740:
2724:
2716:
2706:
2698:
2688:
2683:
2673:
2668:
2658:
2642:
2634:
2615:
2610:
2602:
2597:
2589:
2584:
2574:
2567:
2534:
2530:
2520:
2499:
2491:
2483:
2480:
2477:Hermann Weyl
2472:
2458:
2451:
2443:
2439:
2435:
2430:
2422:
2412:
2404:
2396:
2388:
2374:
2360:gauge bosons
2355:
2353:
2348:
2344:
2340:
2336:
2332:
2328:
2325:
2316:
2312:
2308:
2304:
2302:
2297:
2286:
2283:Gauge bosons
2265:Robert Mills
2258:
2253:
2249:
2246:
2240:
2209:
2206:circle group
2199:
2195:
2179:
2168:
2164:
2159:
2157:
2147:
2143:
2139:
2135:
2131:
2127:
2123:
2118:
2112:
2101:
2088:
2084:
2080:
2077:
2072:
2068:
2064:
2060:
2056:
2045:
2021:
2017:
2013:
2009:
2006:
1993:
1986:
1859:
1855:
1851:
1847:
1841:
1831:
1827:
1825:
1816:
1812:
1810:
1806:
1802:Aristotelian
1787:
1771:
1706:
1702:
1695:
1688:
1681:
1674:
1635:
1628:
1621:
1614:
1609:
1602:
1595:
1588:
1581:
1576:
1570:
1562:Gauge fixing
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1520:
1453:
1419:
1416:Gauge fixing
1391:strong force
1376:
1352:Robert Mills
1328:track gauges
1316:Hermann Weyl
1297:
1280:gauge bosons
1273:Perturbative
1262:
1247:
1242:
1238:
1234:
1226:
1222:
1218:
1214:
1212:
1207:
1199:
1195:
1191:
1185:
1171:, e.g., the
1151:
1150:
1139:gauge theory
1138:
1136:
1097:
943:Stueckelberg
683:Jona-Lasinio
273:Vacuum state
258:Quantization
100:Gauge theory
80:Strong force
65:Field theory
21:Gauge theory
3499:Mathematics
3182:Topological
3096:WessâZumino
3009:Sine-Gordon
2999:GrossâNeveu
2908:BornâInfeld
2875:Thermal QFT
2675:Gravitation
2660:Gravitation
2224:and has an
1454:differences
1452:. But only
1401:of nature.
1372:electroweak
1157:measurement
1149:. The word
1083:Zinn-Justin
933:Sommerfield
858:Pomeranchuk
828:Osterwalder
823:Oppenheimer
753:ĹopuszaĹski
578:Fredenhagen
379:Technicolor
3521:Categories
2963:YangâMills
2651:Kip Thorne
2486:, 101â133.
2366:References
2222:associates
2160:difference
1428:(voltage,
1387:weak force
1368:weak force
1078:Zimmermann
973:Vainshtein
718:Kontsevich
663:Iliopoulos
638:Heisenberg
463:Bogoliubov
401:Scientists
253:Propagator
140:T-symmetry
135:P-symmetry
130:C-symmetry
118:Symmetries
75:Weak force
59:Background
3432:evolution
3372:See also:
3091:Super QCD
3045:Liouville
3033:Conformal
3004:Schwinger
2559:0034-6861
2259:In 1954,
2252:), where
1905:→
1873:→
1652:→
1496:→
1458:voltmeter
1013:Wetterich
998:Weisskopf
948:Sudarshan
898:Schwinger
813:Nishijima
778:Maldacena
743:Leutwyler
708:Kinoshita
608:Goldstone
598:Gell-Mann
513:Doplicher
290:Equations
3542:Symmetry
3485:M-theory
3437:genetics
3168:Type IIB
3163:Type IIA
3148:4D N = 8
3143:4D N = 1
3112:6D (2,0)
3076:4D N = 1
3055:Polyakov
3014:Thirring
2823:Theories
2243:cylinder
2214:commutes
2065:particle
1953:, where
1821:Big Bang
1389:and the
1374:theory.
1334:, Weyl,
1204:symmetry
1155:means a
1028:Wightman
993:Weinberg
983:Virasoro
963:Tomonaga
958:Thirring
953:Symanzik
913:Semenoff
888:Schrader
853:Polyakov
773:Majorana
713:Klebanov
668:Ivanenko
658:'t Hooft
628:Guralnik
573:FrĂśhlich
568:Fritzsch
563:Frampton
478:Buchholz
423:Bargmann
413:Anderson
213:Crossing
3463:entropy
3451:Physics
3442:viruses
3425:Biology
3270:History
3253:Related
3050:Minimal
2901:Regular
2775:(2006)
2723:(1982)
2705:(1985)
2657:(1973)
2633:(1984)
2539:Bibcode
2411:(2004)
2395:(1982)
2241:twisted
2230:inverse
2176:Summary
2169:varying
1790:Galileo
1366:of the
1312:Hilbert
1300:Maxwell
1288:gravity
1147:physics
1038:Wilczek
1003:Wentzel
978:Veltman
923:Shirkov
918:Shifman
908:Seiberg
893:Schwarz
873:Rubakov
798:Naimark
748:Lipatov
738:Lehmann
703:Kendall
593:Gelfand
588:Glashow
548:Feynman
528:Faddeev
523:Englert
493:Coleman
483:Cachazo
468:Brodsky
453:Bjorken
443:Berezin
433:Belavin
193:Anomaly
51:History
3210:Chiral
3158:Type I
2973:Yukawa
2894:Models
2653:, and
2557:
2508:
1858:, and
1794:Newton
1450:ground
1385:, the
1340:London
1175:, the
1169:fields
1143:theory
1093:Zumino
1058:Yukawa
1048:Witten
1043:Wilson
1033:Wigner
968:Tyutin
928:Skyrme
878:Ruelle
848:Plefka
843:Peskin
833:Parisi
793:Møller
783:Migdal
768:Maiani
763:LĂźders
728:Landau
723:Kuraev
698:KällÊn
688:Jordan
673:Jackiw
613:Gribov
503:DeWitt
498:Dashen
488:Callan
458:Bleuer
428:Becchi
418:Anselm
3349:links
3322:links
3310:links
3230:NMSSM
3215:Fermi
2958:Soler
2928:Proca
2463:(PDF)
2294:spins
2218:Group
2165:fixed
1703:would
1344:phase
1324:scale
1263:local
1229:(the
1152:gauge
1088:Zuber
938:Stora
903:Segal
883:Salam
868:Proca
863:Popov
838:Pauli
818:Oehme
808:Neveu
803:Nambu
788:Mills
678:Jaffe
653:Hagen
648:Higgs
623:Gupta
618:Gross
603:Glimm
583:Furry
553:Fierz
543:Fermi
538:Fayet
533:Fadin
518:Dyson
508:Dirac
473:Brout
448:Bethe
408:Adler
187:Tools
3220:MSSM
3117:ABJM
3024:Toda
2555:ISSN
2506:ISBN
2438:and
2347:and
2263:and
2146:and
2138:and
2130:and
2061:wave
2020:and
2012:and
1792:and
1729:and
1687:and
1556:and
1548:and
1350:and
1338:and
1336:Fock
1053:Yang
1023:Wick
1018:Weyl
1008:Wess
988:Ward
693:Jost
643:Hepp
633:Haag
558:Fock
438:Bell
3173:11D
2547:doi
2208:or
1830:to
1381:of
1241:or
1145:in
1073:Zee
758:Low
733:Lee
3523::
3189:BF
2758:.
2649:,
2623:^
2553:.
2545:.
2535:72
2533:.
2484:59
2327:θ(
2248:θ(
2081:eV
2073:hf
2071:=
1854:,
1850:,
1696:qV
1682:qV
1636:qV
1629:qV
1622:qV
1615:qV
1610:qV
1603:qV
1589:qV
1306:("
1290:.
1271:.
1245:.
1137:A
3412:e
3405:t
3398:v
2808:e
2801:t
2794:v
2561:.
2549::
2541::
2514:.
2467:.
2440:A
2436:V
2382:.
2356:x
2349:A
2345:V
2341:Îť
2339:/
2337:h
2333:p
2329:x
2317:Îť
2315:/
2313:h
2309:p
2305:p
2298:x
2254:x
2250:x
2210:U
2148:A
2144:V
2140:B
2136:E
2132:A
2128:V
2124:A
2119:B
2102:B
2089:V
2085:e
2069:E
2057:f
2022:A
2018:V
2014:B
2010:E
1966:0
1962:t
1939:2
1934:0
1930:t
1925:/
1919:3
1915:t
1911:+
1908:t
1902:t
1882:C
1879:+
1876:t
1870:t
1860:t
1856:z
1852:y
1848:x
1832:t
1828:t
1817:t
1813:t
1743:C
1740:+
1737:V
1717:V
1707:q
1699:2
1694:=
1692:2
1689:E
1685:1
1680:=
1678:1
1675:E
1661:C
1658:+
1655:V
1649:V
1639:1
1634:-
1632:2
1625:1
1620:-
1618:2
1606:2
1601:=
1599:2
1596:E
1592:1
1587:=
1585:1
1582:E
1577:q
1558:A
1554:V
1550:B
1546:E
1542:B
1538:E
1534:A
1530:V
1526:A
1505:C
1502:+
1499:V
1493:V
1469:V
1436:V
1243:B
1239:E
1235:V
1227:A
1223:V
1219:B
1215:E
1126:e
1119:t
1112:v
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
