8398:
33:
118:
8653:
4176:
2189:. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory. The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from
8665:
4412:
4199:
6928:. However, because of the subtleties imposed by the gauge constraints (see section on Mathematical formalism, above) there are many technical problems to be solved which do not arise in other field theories. At the same time, the richer structure of gauge theories allows simplification of some computations: for example
5155:
2418:
5949:
4688:
1918:
In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the
2122:
These assumptions have enough validity across a wide range of energy scales and experimental conditions to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life: light, heat, and electricity, eclipses, spaceflight, etc. They fail only at the
1955:
of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber
1963:
of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is
4547:
at all spacetime points. Instead of manually specifying the values of this field, it can be given as the solution to a field equation. Further requiring that the
Lagrangian that generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is
2073:
We cannot express the mathematical descriptions of the "setup information" and the "possible measurement outcomes", or the "boundary conditions" of the experiment, without reference to a particular coordinate system, including a choice of gauge. One assumes an adequate experiment isolated from
4119:
1854:
When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one
2963:
1345:, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).
2547:
5632:
5295:
4949:
2013:
limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.
1923:
in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a
4407:{\displaystyle \ {\mathcal {L}}_{\mathrm {int} }=i{\frac {g}{2}}\Phi ^{\mathsf {T}}A_{\mu }^{\mathsf {T}}\partial ^{\mu }\Phi +i{\frac {g}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi -{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi }
3589:
The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the
Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of
2165:. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a
9467:
3348:
1846:" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model.
1932:" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or
5051:
2000:
with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its
6868:
2325:
7215:
5798:
3898:
5727:
4554:
3056:
6168:
6574:
2169:
prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group).
3990:
3455:
6377:
2849:
2838:
3584:
5446:
2461:
5490:
2610:
The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between originally non-interacting fields.
5183:
4795:
6761:
2097:
given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these
6499:
7279:
3770:
3698:
1951:. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the
6454:
3972:
6629:
6022:
satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a
2466:
2330:
3182:
1939:
In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional Lie group. The simplest such group is
6300:
9412:
5340:
4467:
2101:
fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.
7542:
The discovery of the symmetry under gauge transformations (1 a,b,c) of the quantum mechanical system of a charged particle interacting with electromagnetic fields is due to Fock (1926b)
1839:" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group.
2310:
3089:
2050:
This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as
1867:
direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees states that the fluid velocity in the neighborhood of (
1842:
This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "
1651:. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as
9457:
4726:
6903:
1893:
is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the
9462:
8703:
6956:
method was also developed to handle this problem. Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on
6651:
6270:
6221:
5150:{\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal {L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal {L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}
4758:
2671:
2453:
2260:
5782:
3613:
3236:
6195:
4985:
3268:
1959:
A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a
6248:
4425:
scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator
4146:
3633:
3256:
3113:
2021:
of a gauge theory, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. In addition to its
1744:
The importance of gauge theories in physics is exemplified in the tremendous success of the mathematical formalism in providing a unified framework to describe the
6795:
4787:
4545:
3133:
2413:{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}
5944:{\displaystyle {\mathcal {L}}_{\text{QED}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }}
2123:
smallest and largest scales due to omissions in the theories themselves, and when the mathematical techniques themselves break down, most notably in the case of
5032:
4166:
2632:
2556:
is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation.
6412:
4683:{\displaystyle {\mathcal {L}}_{\text{gf}}=-{\frac {1}{2}}\operatorname {tr} \left(F^{\mu \nu }F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}
1431:
of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation,
9081:
2046:
reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.
5640:
2978:
9450:
9086:
6122:
4514:
The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives
2086:
The two gauge theories mentioned above, continuum electrodynamics and general relativity, are continuum field theories. The techniques of calculation in a
7112:
2074:"external" influence that is itself a gauge-dependent statement. Mishandling gauge dependence calculations in boundary conditions is a frequent source of
8283:
7001:
However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes suited to these problems (such as
6026:(called by physicists gauge transformations of the second kind, an affine representation)âand other more general representations, such as the B field in
3796:
2173:
More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of
2040:
listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
1721:
4-dimensional space. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994,
6507:
4114:{\displaystyle \ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{\mathsf {T}}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }
2958:{\displaystyle \ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }
1976:
is qualitatively different from that on expressions that do not. (This is analogous to a non-inertial change of reference frame, which can produce a
8696:
3384:
7464:
6319:
8810:
2678:
1551:
1554:") suggested the possibility of invariance, when he stated that any vector field whose curl vanishesâand can therefore normally be written as a
9445:
8427:
8061:
1204:
8251:
2542:{\displaystyle {\begin{aligned}\mathbf {A} &\mapsto \mathbf {A} +\nabla f\\V&\mapsto V-{\frac {\partial f}{\partial t}}\end{aligned}}}
8535:
3522:
5385:
2109:
establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information
9263:
7106:
on the null-field configuration, i.e., a gauge transform of zero. So it is a particular "gauge orbit" in the field configuration's space.
2157:
of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant
8689:
287:
5627:{\displaystyle {\mathcal {L}}_{\text{int}}={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}
9308:
1816:
1253:
5290:{\displaystyle {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}
4944:{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum _{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}
8669:
7392:
5038:
classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated
327:
3779:
is called the coupling constant; a quantity defining the strength of an interaction. After a simple calculation we can see that the
1647:
theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called
1593:
generalized the gauge invariance of electromagnetism, constructing a theory based on the action of the (non-abelian) SU(2) symmetry
8815:
8412:
9224:
8276:
6700:
2174:
2066:
limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
2118:
these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information
1574:
under a general coordinate transformation. The importance of these symmetry invariances remained unnoticed until Weyl's work.
8825:
8530:
8148:
7644:
7432:
6459:
6010:), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows that
3713:
9391:
6420:
3909:
1062:
8845:
8840:
8613:
7417:
6582:
2094:
given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described
1819:; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in
1667:
unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.
8269:
7220:
7021:
Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an
3640:
1197:
7751:(1994a), "Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory",
6991:
8437:
8112:
8077:
8050:
7979:
7437:
6275:
1286:. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called
76:
54:
5306:
4436:
47:
9258:
8942:
8618:
8042:
1326:
under some symmetry transformation groups. When they are invariant under a transformation identically performed at
832:
3144:
1577:
Inspired by Pauli's descriptions of connection between charge conservation and field theory driven by invariance,
9005:
8765:
8022:. Johns Hopkins University Press. Esp. chpt. 8. A serious attempt by a physicist to explain gauge theory and the
1424:
130:
2043:
computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
8932:
8578:
8162:
7372:
6086:. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the
1447:
1365:
17:
6305:
Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie-algebra-valued
9521:
8937:
8830:
8469:
8417:
8244:
7412:
6908:
The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that a
1944:
1377:
1190:
395:
244:
204:
2281:
9491:
8835:
8628:
3064:
2069:
computing the probability distribution of the possible outcomes that the experiment is designed to measure.
1738:
932:
862:
234:
93:
8234:
8877:
8239:
7971:
7362:
6069:
1632:
390:
332:
7469:
Studies in
History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
4991:
of the Lie algebra of the generators of the gauge group. This formulation of the
Lagrangian is called a
2230:. Knowledge of one makes it possible to find the other, except that potentials differing by a constant,
9495:
8959:
7442:
7322:
6916:; when one does so, one finds that the metric connection satisfies the YangâMills equations of motion.
5974:
3704:
1439:
239:
100:
4696:
1885:
direction. The coordinate transformation has affected both the coordinate system used to identify the
1042:
99:
This article discusses the physics of gauge theories. For the mathematical field of gauge theory, see
9355:
9288:
9283:
9246:
8777:
8750:
8720:
8583:
7337:
6967:
for various processes allowed by the theory. Technically, they reduce to the computations of certain
5458:
4996:
3465:
2579:
2313:
2162:
1769:
1741:). These contributions to mathematics from gauge theory have led to a renewed interest in this area.
1547:
1319:
1230:
380:
6884:
2112:
establishing a probability distribution of measurement outcomes for each possible physical situation
1831:
change of reference frame) they represent the same physical situation. These transformations form a
9212:
9207:
9116:
9076:
8792:
8588:
7332:
7307:
6083:
2198:
1714:
1590:
1567:
1459:
1416:
1381:
867:
497:
342:
307:
41:
6998:. At present some of these methods lead to the most precise experimental tests of gauge theories.
6634:
6253:
6204:
5165:
As a simple application of the formalism developed in the previous sections, consider the case of
4731:
4433:) needs to propagate in space. That is dealt with in the next section by adding yet another term,
2649:
2426:
2233:
1538:. Weyl's 1929 paper introduced the modern concept of gauge invariance subsequently popularized by
1435:, replaces the principle of general covariance with a true gauge principle with new gauge fields.
857:
9516:
9401:
8657:
8502:
8370:
8360:
8069:
7367:
7342:
6945:
5963:
5789:
5751:
4418:
2178:
2150:
2022:
1824:
1683:
1621:
1455:
1353:
1142:
612:
420:
415:
277:
9037:
3343:{\displaystyle \ (\partial _{\mu }\Phi )\mapsto (\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi }
9406:
9396:
9370:
9192:
9187:
9182:
9177:
9135:
9098:
9044:
8770:
8760:
8738:
8375:
8348:
7708:
7377:
7352:
7044:
6957:
6949:
5988:
5959:
5785:
5004:
5000:
4474:
4470:
2598:
2571:
2146:
1929:
1652:
1291:
1226:
1012:
992:
492:
425:
337:
302:
58:
2145:
Other than these classical continuum field theories, the most widely known gauge theories are
1855:
description of a pattern of fluid flow states that the fluid velocity in the neighborhood of (
9323:
9303:
9241:
9170:
8949:
8927:
8902:
8802:
8623:
8497:
8464:
7427:
6968:
6964:
6863:{\displaystyle \chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)}
6180:
6035:
6031:
6011:
5980:
5481:
4957:
4509:
3375:
implies that invariance under this group of transformations leads to the conservation of the
1523:
1312:
902:
762:
458:
322:
184:
3593:
3216:
2312:. Generalizing from static electricity to electromagnetism, we have a second potential, the
1498:(or "gauge") might also be a local symmetry of general relativity. After the development of
1256:
in the
Lagrangian of a physical system. The transformations between possible gauges, called
9481:
9375:
9360:
9275:
9202:
9165:
9160:
9150:
8910:
8864:
8573:
8512:
8507:
8481:
8328:
8184:
7932:
7892:
7872:
7833:
7813:
7790:
7770:
7730:
7674:
7609:
7476:
7382:
7103:
7002:
6925:
6878:
6874:
6306:
6226:
6079:
6015:
4131:
3618:
3372:
3241:
3195:
3098:
2575:
2182:
2140:
2029:
in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:
1997:
1993:
1745:
1636:
1432:
1349:
1147:
927:
463:
317:
109:
4763:
4521:
3118:
1988:
The "gauge covariant" version of a gauge theory accounts for this effect by introducing a
1427:
of general relativity in which the coordinate system can be chosen freely under arbitrary
229:
8:
9350:
9219:
9130:
9017:
8988:
8712:
8540:
7402:
6987:
6924:
Gauge theories may be quantized by specialization of methods which are applicable to any
6782:
6672:
6174:
6007:
4988:
2190:
2087:
1836:
1832:
1594:
1543:
1535:
1495:
1467:
1348:
Gauge theories are important as the successful field theories explaining the dynamics of
1323:
1242:
1157:
487:
8188:
7936:
7876:
7817:
7774:
7678:
7613:
7480:
7058:
in either chiral or vector field theories with fermions. This has close connection with
5788:
exactly as in electrodynamics, one obtains the
Lagrangian used as the starting point in
5014:
In this
Lagrangian term there is no field whose transformation counterweighs the one of
872:
602:
9487:
9140:
9126:
9108:
9091:
9071:
9054:
8964:
8787:
8568:
8338:
8213:
8200:
8174:
8130:
7948:
7922:
7896:
7862:
7794:
7760:
7515:
7317:
7082:
7048:
5017:
4151:
3092:
2617:
2262:, correspond to the same electric field. This is because the electric field relates to
2223:
2051:
2025:
with other objects via the covariant derivative, the gauge field typically contributes
2018:
1820:
1777:
1648:
1483:
1443:
1404:
1167:
1092:
1022:
907:
892:
822:
787:
777:
737:
652:
542:
502:
292:
254:
209:
197:
174:
169:
7853:(1994b), "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD",
7488:
6385:
5722:{\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}
3051:{\displaystyle \ \Phi ^{\mathsf {T}}=(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})}
9313:
9049:
9027:
8954:
8915:
8892:
8681:
8454:
8256:
8204:
8108:
8073:
8046:
8018:
7975:
7884:
7825:
7782:
7640:
7575:
7533:
7503:
7397:
7023:
6953:
6913:
6771:
6163:{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} +\mathbf {A} \wedge \mathbf {A} }
6113:
6094:
5465:(this is the origin of the usage of the term in gauge theories), and the gauge field
5008:
3977:
There are therefore as many gauge fields as there are generators of the Lie algebra.
2969:
2186:
2075:
1765:
1628:
1499:
1306:
1278:. For each group generator there necessarily arises a corresponding field (usually a
1122:
1052:
937:
782:
717:
687:
657:
517:
512:
357:
282:
272:
164:
7900:
7798:
7721:
7703:
7210:{\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)}
6948:(QED). The first methods developed for this involved gauge fixing and then applying
3487:
matrices (which were earlier constant) should be allowed to become functions of the
1304:. If the symmetry group is non-commutative, then the gauge theory is referred to as
842:
9231:
9022:
9010:
8993:
8971:
8920:
8872:
8733:
8633:
8380:
8365:
8306:
8192:
7952:
7940:
7880:
7821:
7778:
7716:
7682:
7617:
7567:
7555:
7525:
7484:
7407:
6983:
6042:
5992:
5742:
5477:
4494:
3903:
The gauge field is an element of the Lie algebra, and can therefore be expanded as
3893:{\displaystyle \ A'_{\mu }=GA_{\mu }G^{-1}-{\frac {i}{g}}(\partial _{\mu }G)G^{-1}}
3357:
symmetry of this particular
Lagrangian, and the symmetry group is often called the
3206:
2643:
1928:
of the fiber bundle) and express the values of the objects of the theory (usually "
1749:
1702:
1571:
1487:
1330:
1162:
1077:
1057:
1027:
977:
802:
797:
792:
677:
662:
592:
297:
224:
214:
149:
7597:
89:
Physical theory with fields invariant under the action of local "gauge" Lie groups
9337:
9293:
9120:
9064:
9059:
8998:
8820:
8745:
8555:
8422:
7888:
7829:
7786:
7726:
7422:
7302:
6976:
6933:
6929:
6087:
6075:
6023:
6003:
5738:
5462:
5166:
4493:. The interpretation of the interaction Lagrangian in quantum field theory is of
4179:
3363:
2635:
2271:
2194:
2158:
2128:
1977:
1960:
1815:, the mathematical description of any physical situation usually contains excess
1718:
1687:
1679:
1617:
1531:
1463:
1412:
1338:
1275:
1107:
1102:
1072:
1067:
1037:
997:
967:
947:
852:
747:
742:
707:
692:
667:
647:
627:
557:
547:
468:
430:
347:
312:
144:
122:
117:
6569:{\displaystyle DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+\mathbf {A} X}
3138:
It is now transparent that the
Lagrangian is invariant under the transformation
8981:
8593:
8292:
8023:
7846:
7744:
7593:
7387:
7055:
7010:
6986:
of the theory is small enough, then all required quantities may be computed in
6106:
5955:
5734:
5350:
5174:
2270:
would cancel out when subtracting to find the change in potential. In terms of
2215:
2154:
2002:
1969:
1948:
1761:
1726:
1675:
1671:
1664:
1582:
1578:
1559:
1539:
1511:
1428:
1408:
1373:
1342:
1262:
1234:
1132:
1082:
1002:
987:
972:
952:
922:
917:
912:
897:
882:
847:
837:
767:
702:
672:
607:
572:
562:
537:
408:
375:
249:
7944:
7621:
7529:
5034:. Invariance of this term under gauge transformations is a particular case of
1897:
of some quantity along some path in space and time as it passes through point
887:
9510:
9298:
9253:
9236:
8882:
8755:
8563:
8522:
8087:
7850:
7748:
7579:
7537:
7312:
7078:
7032:
6909:
6665:
6657:
6310:
6198:
3450:{\displaystyle \ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{\mathsf {T}}T^{a}\Phi }
2105:
Determination of the likelihood of possible measurement outcomes proceed by:
1773:
1730:
1722:
1698:
1563:
1527:
1515:
1503:
1361:
1357:
1137:
1127:
1117:
1112:
1032:
1017:
817:
812:
642:
637:
597:
567:
532:
522:
453:
448:
7571:
6372:{\displaystyle \delta _{\varepsilon }\mathbf {A} =-\mathrm {d} \varepsilon }
2062:
Gauge theories used to model the results of physical experiments engage in:
1764:, accurately describes experimental predictions regarding three of the four
9318:
8976:
8887:
8782:
8545:
8476:
8385:
8343:
8165:(2002). "From Lorenz to Coulomb and other explicit gauge transformations".
7687:
7661:
7347:
7327:
6972:
6661:
6061:
6049:
5039:
2833:{\displaystyle {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left}
2166:
2037:
without the gauge field (in which the derivatives appear in a "bare" form);
1920:
1757:
1656:
1558:
of a functionâcould be added to the vector potential without affecting the
1507:
1479:
1446:. However, the modern importance of gauge symmetries appeared first in the
1279:
1172:
1097:
982:
957:
942:
877:
757:
752:
632:
622:
617:
582:
577:
552:
507:
385:
352:
159:
9429:
9345:
8459:
8353:
8321:
8316:
8144:
7357:
7040:
6786:
6105:. This is evidently not an intrinsic but a frame-dependent quantity. The
5730:
4490:
3368:
3262:
in the
Lagrangian (orthogonal transformations preserve the dot product).
3259:
2602:
2594:
2590:
2115:
1644:
1400:
1300:
1271:
1152:
1087:
1047:
1007:
962:
827:
732:
727:
712:
697:
682:
92:
For a more accessible and less technical introduction to this topic, see
8218:
8179:
6116:
that is an intrinsic quantity, is constructed from a connection form by
1407:. Its case is somewhat unusual in that the gauge field is a tensor, the
9365:
8135:
7927:
7867:
7765:
7666:
7520:
6990:. Quantization schemes intended to simplify such computations (such as
6767:
6019:
3579:{\displaystyle \ \partial _{\mu }(G\Phi )\neq G(\partial _{\mu }\Phi )}
3491:
2124:
1973:
1753:
1640:
1389:
807:
722:
587:
527:
219:
154:
8196:
7913:
Jackson, JD; Okun, LB (2001). "Historical roots of gauge invariance".
7013:, and are therefore less well-developed currently than other schemes.
5441:{\displaystyle D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }}
5353:
of the field, with the particular rotation determined by the constant
2266:
in the potential from one point in space to another, and the constant
1631:
became the prototype theory to resolve some of the great confusion in
9413:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
8728:
8095:
7063:
6684:
6057:
6027:
5042:, but even after restriction, gauge transformations may be possible.
3488:
2006:
1706:
1334:
1238:
7465:"Which Symmetry? Noether, Weyl, and Conservation of Electric Charge"
3213:). This is seen to preserve the Lagrangian, since the derivative of
2161:
that characterizes "allowable" physical situations according to the
1913:
1482:
in 1918. Weyl, in an attempt to generalize the geometrical ideas of
1252:
refers to any specific mathematical formalism to regulate redundant
8638:
7059:
6680:
6676:
6668:
5170:
2275:
1843:
1828:
1734:
1691:
1555:
1451:
1420:
1233:, and hence the dynamics of the system itself, do not change under
772:
6018:) of the gauge transformations can be classified as sections of a
4175:
1478:
The concept and the name of gauge theory derives from the work of
8311:
8261:
6102:
5173:
field. The bare-bones action that generates the electron field's
4478:
2078:, and approaches to anomaly avoidance classifies gauge theories.
1812:
1606:
1598:
1586:
1295:
1218:
4503:
9468:
European Community on Computational Methods in Applied Sciences
9032:
8850:
7556:"Gauge theory: Historical origins and some modern developments"
7090:
7071:
2034:
2026:
1791:
for more about the history of gauge and quantum field theories.
1660:
1614:
1602:
1393:
1385:
1369:
1337:
in which the physical processes occur, they are said to have a
7704:"Self-dual connections and the topology of smooth 4-manifolds"
7102:
A pure gauge is the set of field configurations obtained by a
7081:, which must cancel in any consistent physical theory. In the
1655:, is a gauge theory with the action of the SU(3) group on the
1514:
quantity and turned the scale transformation into a change of
1438:
Historically, these ideas were first stated in the context of
7662:"Conservation of Isotopic Spin and Isotopic Gauge Invariance"
7086:
6053:
4497:
3472:) group. There is one conserved current for every generator.
2570:
The remainder of this section requires some familiarity with
1695:
1674:
began studying the mathematics of solutions to the classical
1419:, also postulate the existence of a gauge boson known as the
1376:
is a non-abelian gauge theory with the symmetry group U(1) Ă
9463:
International Council for Industrial and Applied Mathematics
7554:
OâRaifeartaigh, Lochlainn; Straumann, Norbert (2000-01-01).
6223:
is an element of the vector space spanned by the generators
6078:(gauge connection) define this principal bundle, yielding a
6034:(realizations), but these are extremely complicated. Still,
2646:
that is the sum of the (usual) action for each scalar field
1901:
is the same as the effect on values that are truly local to
8212:
Svetlichny, George (1999). "Preparation for Gauge Theory".
7067:
6756:{\displaystyle {\frac {1}{4g^{2}}}\int \operatorname {Tr} }
5346:
1940:
1610:
1519:
1458:, elaborated on below. Today, gauge theories are useful in
7217:, the pure gauge is just the set of field configurations
6660:
gauge transformations in general. An example is when the
6272:
do not commute with one another. Hence the wedge product
6963:
The main point to quantization is to be able to compute
6789:, which is defined over any closed path, Îł, as follows:
5979:
Gauge theories are usually discussed in the language of
4186:
The difference between this Lagrangian and the original
7659:
7553:
6494:{\displaystyle \delta _{\varepsilon }DX=\varepsilon DX}
5741:
is therefore seen to naturally introduce the so-called
2559:
8711:
7639:(Impression: 3 ed.). Oxford: Oxford Univ. Press.
5362:"Localising" this symmetry implies the replacement of
3765:{\displaystyle \ D_{\mu }=\partial _{\mu }-igA_{\mu }}
2423:
The general gauge transformations now become not just
1996:) and formulating all rates of change in terms of the
1768:
of nature, and is a gauge theory with the gauge group
9082:
Numerical methods for ordinary differential equations
8129:
Becchi, C. (1997). "Introduction to Gauge Theories".
7598:"Relativistic Field Theories of Elementary Particles"
7223:
7115:
7009:. Precise computations in such schemes often require
6887:
6798:
6703:
6637:
6585:
6510:
6462:
6449:{\displaystyle \delta _{\varepsilon }X=\varepsilon X}
6423:
6388:
6322:
6278:
6256:
6229:
6207:
6183:
6125:
5801:
5754:
5643:
5493:
5388:
5309:
5186:
5054:
5020:
4960:
4798:
4766:
4734:
4699:
4557:
4524:
4439:
4202:
4154:
4134:
3993:
3967:{\displaystyle \ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}}
3912:
3799:
3716:
3643:
3621:
3596:
3525:
3504:
matrices do not "pass through" the derivatives, when
3387:
3271:
3244:
3219:
3147:
3121:
3101:
3067:
2981:
2852:
2843:
The Lagrangian (density) can be compactly written as
2681:
2652:
2620:
2464:
2429:
2328:
2284:
2236:
1423:. Gauge symmetries can be viewed as analogues of the
9458:
Société de Mathématiques Appliquées et Industrielles
9451:
Japan Society for Industrial and Applied Mathematics
9087:
Numerical methods for partial differential equations
8801:
5999:
is just a transformation between two such sections.
5745:
of the electromagnetic field to the electron field.
5045:
The complete Lagrangian for the gauge theory is now
1270:
of the theory. Associated with any Lie group is the
1237:
according to certain smooth families of operations (
9274:
6624:{\displaystyle \delta _{\varepsilon }\mathbf {F} =}
6002:Although gauge theory is dominated by the study of
5457:(not to be confused with the mathematical constant
4500:interacting by the exchange of these gauge bosons.
1643:, and its unification with electromagnetism in the
1318:Many powerful theories in physics are described by
7273:
7209:
7039:. In QED this gives rise to the phenomenon of the
6897:
6862:
6755:
6656:Not all gauge transformations can be generated by
6645:
6623:
6568:
6493:
6448:
6406:
6371:
6294:
6264:
6242:
6215:
6189:
6162:
6038:transform nonlinearly, so there are applications.
5943:
5776:
5721:
5626:
5440:
5334:
5289:
5149:
5026:
4995:. Other gauge invariant actions also exist (e.g.,
4979:
4943:
4781:
4752:
4720:
4682:
4539:
4461:
4406:
4160:
4140:
4113:
3966:
3892:
3764:
3692:
3627:
3607:
3578:
3449:
3342:
3250:
3230:
3176:
3127:
3107:
3083:
3050:
2957:
2832:
2665:
2626:
2541:
2447:
2412:
2304:
2254:
7274:{\displaystyle A'_{\mu }(x)=\partial _{\mu }f(x)}
6919:
4518:, one needs to know the value of the gauge field
3693:{\displaystyle \ (D_{\mu }\Phi )'=GD_{\mu }\Phi }
1914:Use of fiber bundles to describe local symmetries
9508:
3475:Now, demanding that this Lagrangian should have
1399:Gauge theories are also important in explaining
6295:{\displaystyle \mathbf {A} \wedge \mathbf {A} }
1552:A Dynamical Theory of the Electromagnetic Field
9446:Society for Industrial and Applied Mathematics
8428:Mathematical formulation of the Standard Model
7845:
7743:
7085:this cancellation requires an equal number of
5379:. An appropriate covariant derivative is then
5335:{\displaystyle \psi \mapsto e^{i\theta }\psi }
5160:
4462:{\displaystyle {\mathcal {L}}_{\mathrm {gf} }}
4182:of scalar bosons interacting via a gauge boson
1889:of the measurement and the basis in which its
1801:
1780:, are, in one way or another, gauge theories.
8697:
8277:
8036:
7547:
4504:The YangâMills Lagrangian for the gauge field
2218:, one can either discuss the electric field,
2209:
1943:, which appears in the modern formulation of
1849:
1729:invented gauge-theoretic techniques based on
1198:
9264:Supersymmetric theory of stochastic dynamics
7066:. In QCD this anomaly causes the decay of a
5461:in the symmetry description) with the usual
3177:{\displaystyle \ \Phi \mapsto \Phi '=G\Phi }
1694:is very different from their classification
1384:and has a total of twelve gauge bosons: the
8066:Gauge Theory of Elementary Particle Physics
7912:
7906:
7501:
4148:, while the compensating transformation in
1972:; its effect on expressions that involve a
1635:. This idea later found application in the
8704:
8690:
8284:
8270:
8211:
7637:The quantum story: a history in 40 moments
7502:Jackson, J. D.; Okun, L. B. (2001-09-14).
6006:(primarily because it's mainly studied by
1205:
1191:
116:
8217:
8178:
8149:"Gauge theory â Past, Present and Future"
8134:
7965:
7959:
7926:
7866:
7764:
7720:
7701:
7686:
7519:
7393:Standard Model (mathematical formulation)
5968:
5845:
5271:
5226:
2695:
2204:
2134:
1823:, if two configurations are related by a
1585:binding based on conservation of nuclear
77:Learn how and when to remove this message
8086:
8037:Greiner, Walter; MĂŒller, Berndt (2000).
7628:
5748:Adding a Lagrangian for the gauge field
4174:
2009:) is zero everywhere; a gauge theory is
1968:a constant function is referred to as a
1788:for early history of gauge and Pickering
1733:that enabled the calculation of certain
1510:replaced the simple scale factor with a
40:This article includes a list of general
8161:
8059:
7634:
7462:
6072:of the group of gauge transformations.
5300:The global symmetry for this system is
4170:gauge transformation of the second type
2057:
1609:. This is similar to the action of the
14:
9509:
8252:YangâMills equations on DispersiveWiki
8128:
7504:"Historical roots of gauge invariance"
7456:
6905:represents the path-ordered operator.
4385:
4324:
4267:
4250:
4126:gauge transformation of the first type
4102:
4054:
3428:
2991:
2946:
2898:
2305:{\displaystyle \mathbf {E} =-\nabla V}
2177:by introducing additional fields (the
1260:, form a Lie groupâreferred to as the
8685:
8265:
8143:
7592:
7007:non-perturbative quantization schemes
6944:The first gauge theory quantized was
6064:is a Lie group, then the sections of
5484:results in an interaction Lagrangian
3084:{\displaystyle \partial _{\mu }\Phi }
2081:
1678:equations. In 1983, Atiyah's student
1570:by postulating the invariance of the
1360:gauge theory with the symmetry group
8664:
8102:
6785:under gauge transformations) is the
6683:to the Lie group is nontrivial. See
6501:where D is the covariant derivative
1682:built on this work to show that the
26:
6939:
1542:in his 1941 review. In retrospect,
1522:gauge symmetry. This explained the
24:
8713:Industrial and applied mathematics
8291:
8105:Modern Elementary Particle Physics
7998:, Addison-Wesley, 1967, sect. 1â4.
7250:
7186:
6890:
6822:
6770:and the integral is defined as in
6548:
6536:
6533:
6530:
6362:
6135:
5805:
5497:
5403:
5274:
5238:
5189:
5136:
5119:
5102:
5085:
5068:
5057:
4849:
4821:
4561:
4453:
4450:
4443:
4401:
4376:
4340:
4315:
4306:
4283:
4274:
4245:
4222:
4219:
4216:
4209:
4135:
4108:
4097:
4070:
4045:
4013:
4010:
4007:
4000:
3862:
3734:
3703:This new "derivative" is called a
3687:
3660:
3622:
3615:again transforms identically with
3598:
3570:
3561:
3545:
3530:
3444:
3423:
3413:
3337:
3328:
3310:
3301:
3288:
3279:
3258:and both quantities appear inside
3245:
3221:
3171:
3158:
3151:
3102:
3078:
3069:
2986:
2952:
2941:
2914:
2905:
2889:
2880:
2858:
2768:
2748:
2698:
2684:
2526:
2518:
2489:
2395:
2370:
2360:
2348:
2296:
1806:
1494:or invariance under the change of
46:it lacks sufficient corresponding
25:
9533:
8943:Stochastic differential equations
8227:
8039:Gauge Theory of Weak Interactions
7438:1964 PRL symmetry breaking papers
7433:YangâMills existence and mass gap
7109:Thus, in the abelian case, where
7027:. Among the most well known are:
6996:perturbative quantization schemes
5839:
5671:
5516:
5423:
5220:
2642:. This system is described by an
1908:
1705:used Donaldson's work to exhibit
9259:Supersymmetric quantum mechanics
8663:
8652:
8651:
8396:
7660:Yang C. N., Mills R. L. (1954).
6639:
6614:
6597:
6559:
6351:
6334:
6288:
6280:
6258:
6209:
6156:
6148:
6140:
6127:
4721:{\displaystyle F_{\mu \nu }^{a}}
2482:
2470:
2402:
2384:
2364:
2334:
2286:
2181:) and counterterms motivated by
1881:) is 1 m/s in the negative
31:
9141:Stochastic variational calculus
8933:Ordinary differential equations
8026:with little formal mathematics.
8005:
7988:
7722:10.1090/S0273-0979-1983-15090-5
3483:)-invariance requires that the
2005:(in mathematical language, its
1983:
1546:'s formulation, in 1864â65, of
1425:principle of general covariance
1298:of the gauge fields are called
1241:). Formally, the Lagrangian is
8938:Partial differential equations
8811:Arbitrary-precision arithmetic
8579:Causal dynamical triangulation
8257:Gauge theories on Scholarpedia
7839:
7737:
7695:
7653:
7586:
7495:
7373:Electromagnetic four-potential
7268:
7262:
7243:
7237:
7204:
7198:
7179:
7173:
7157:
7151:
7135:
7132:
7126:
6920:Quantization of gauge theories
6898:{\displaystyle {\mathcal {P}}}
6810:
6804:
6750:
6735:
6618:
6604:
6401:
6389:
6355:
6341:
5987:is just a choice of a (local)
5825:
5771:
5765:
5716:
5710:
5694:
5688:
5682:
5660:
5654:
5621:
5615:
5602:
5596:
5580:
5574:
5561:
5555:
5539:
5533:
5527:
5313:
5206:
4776:
4770:
4534:
4528:
4380:
4363:
4319:
4302:
4128:to mean the transformation of
4049:
4032:
3874:
3858:
3664:
3647:
3573:
3557:
3548:
3539:
3367:, especially in the theory of
3314:
3297:
3294:
3291:
3275:
3154:
3045:
3000:
2893:
2876:
2506:
2478:
2433:
2240:
1992:(in mathematical language, an
1796:
1448:relativistic quantum mechanics
1366:electromagnetic four-potential
1311:, the usual example being the
1245:under these transformations.
13:
1:
8826:Interactive geometry software
8418:Spontaneous symmetry breaking
7489:10.1016/S1355-2198(01)00033-8
7449:
7413:Symmetry in quantum mechanics
7097:
4728:are obtained from potentials
4190:Lagrangian is seen to be the
2589:Definitions in this section:
1945:quantum electrodynamics (QED)
1863:=0) is 1 m/s in the positive
1364:and has one gauge field, the
245:Spontaneous symmetry breaking
205:Symmetry in quantum mechanics
7885:10.1016/0550-3213(94)90214-3
7826:10.1016/0550-3213(94)00449-8
7783:10.1016/0550-3213(94)90124-4
7702:Donaldson, Simon K. (1983).
7016:
6679:class of mappings from that
6646:{\displaystyle \mathbf {F} }
6265:{\displaystyle \mathbf {A} }
6216:{\displaystyle \mathbf {A} }
4753:{\displaystyle A_{\mu }^{a}}
4469:, to the Lagrangian. In the
3790:) must transform as follows
3705:(gauge) covariant derivative
2666:{\displaystyle \varphi _{i}}
2448:{\displaystyle V\mapsto V+C}
2274:, the electric field is the
2255:{\displaystyle V\mapsto V+C}
1760:. This theory, known as the
94:Introduction to gauge theory
7:
8878:Computational number theory
8841:Numerical-analysis software
8240:Encyclopedia of Mathematics
7972:University of Chicago Press
7463:Brading, Katherine (1941).
7363:Gluon field strength tensor
7295:
6250:, and so the components of
6070:principal homogeneous space
5777:{\displaystyle A_{\mu }(x)}
5161:An example: Electrodynamics
3361:; the mathematical term is
1802:Global and local symmetries
1633:elementary particle physics
1372:being the gauge boson. The
10:
9538:
7996:Advanced Quantum Mechanics
7443:Gauge theory (mathematics)
7323:Gauge covariant derivative
6417:One nice thing is that if
5975:Gauge theory (mathematics)
5972:
5953:
4760:, being the components of
4507:
3238:transforms identically to
2210:Classical electromagnetism
2185:, in an approach known as
2138:
1850:Example of global symmetry
1581:sought a field theory for
1473:
1440:classical electromagnetism
240:Explicit symmetry breaking
101:Gauge theory (mathematics)
98:
91:
9476:
9438:
9422:
9384:
9336:
9284:Algebra of physical space
9149:
9107:
8901:
8863:
8751:Automated theorem proving
8719:
8647:
8606:
8584:Canonical quantum gravity
8554:
8521:
8490:
8447:
8436:
8405:
8394:
8299:
7945:10.1103/RevModPhys.73.663
7915:Reviews of Modern Physics
7622:10.1103/revmodphys.13.203
7560:Reviews of Modern Physics
7530:10.1103/RevModPhys.73.663
7508:Reviews of Modern Physics
7338:Gauge group (mathematics)
7037:running coupling constant
6032:nonlinear representations
6030:. There are more general
5451:Identifying the "charge"
4997:nonlinear electrodynamics
2163:principle of least action
1739:SeibergâWitten invariants
1715:differentiable structures
1290:). When such a theory is
396:BargmannâWigner equations
9077:Numerical linear algebra
8589:Superfluid vacuum theory
7333:Gauge gravitation theory
7035:, which gives rise to a
6653:transforms covariantly.
6084:associated vector bundle
5729:is the electric current
5349:, just rotations of the
5345:The gauge group here is
4473:version of the obtained
4188:globally gauge-invariant
2199:low-dimensional topology
2090:implicitly assume that:
1568:Einstein field equations
1417:gauge gravitation theory
8816:Finite element analysis
8766:Constraint satisfaction
8371:Quantum electrodynamics
8361:Electroweak interaction
8070:Oxford University Press
7635:Baggott, J. E. (2013).
7572:10.1103/RevModPhys.72.1
7368:Quantum electrodynamics
6946:quantum electrodynamics
6766:where * stands for the
6190:{\displaystyle \wedge }
6173:where d stands for the
6112:, a Lie algebra-valued
6093:, a Lie algebra-valued
5964:Quantum electrodynamics
5790:quantum electrodynamics
4980:{\displaystyle f^{abc}}
3982:locally gauge invariant
3980:Finally, we now have a
3353:This characterizes the
2222:, or its corresponding
2151:quantum electrodynamics
1825:Galilean transformation
1772:. Modern theories like
1622:quantum electrodynamics
1562:. Similarly unnoticed,
1470:among other subfields.
1456:quantum electrodynamics
1354:Quantum electrodynamics
421:Electroweak interaction
416:Quantum electrodynamics
391:WheelerâDeWitt equation
278:Background field method
61:more precise citations.
9371:Mathematical economics
9045:Multivariable calculus
8928:Differential equations
8771:Constraint programming
8761:Computational geometry
8349:Quantum chromodynamics
8235:"Gauge transformation"
7966:Pickering, A. (1984).
7709:Bull. Amer. Math. Soc.
7688:10.1103/PhysRev.96.191
7378:Electromagnetic tensor
7353:Quantum chromodynamics
7275:
7211:
7062:through the notion of
7045:quantum chromodynamics
6992:canonical quantization
6950:canonical quantization
6899:
6864:
6757:
6647:
6625:
6570:
6495:
6450:
6408:
6373:
6313:gauge transformation,
6296:
6266:
6244:
6217:
6191:
6164:
6097:, which is called the
6036:nonlinear sigma models
6012:affine representations
6008:high-energy physicists
5969:Mathematical formalism
5945:
5778:
5723:
5628:
5442:
5336:
5291:
5151:
5028:
4981:
4945:
4783:
4754:
4722:
4684:
4541:
4475:classical field theory
4463:
4408:
4192:interaction Lagrangian
4183:
4162:
4142:
4115:
3968:
3894:
3766:
3694:
3629:
3609:
3608:{\displaystyle \Phi '}
3580:
3451:
3344:
3252:
3232:
3231:{\displaystyle \Phi '}
3178:
3129:
3109:
3085:
3052:
2959:
2834:
2731:
2667:
2628:
2599:interaction Lagrangian
2543:
2449:
2414:
2306:
2256:
2205:Classical gauge theory
2147:quantum field theories
2135:Quantum field theories
2033:starting from a naĂŻve
1746:quantum field theories
1653:quantum chromodynamics
426:Quantum chromodynamics
303:Effective field theory
9324:Supersymmetry algebra
9309:Representation theory
9304:Renormalization group
8950:Differential geometry
8831:Optimization software
8803:Mathematical software
8465:Cosmological constant
8016:Schumm, Bruce (2004)
7276:
7212:
6969:correlation functions
6900:
6865:
6772:differential geometry
6758:
6648:
6626:
6571:
6496:
6451:
6409:
6374:
6297:
6267:
6245:
6243:{\displaystyle T^{a}}
6218:
6192:
6165:
5981:differential geometry
5946:
5786:field strength tensor
5779:
5724:
5629:
5482:electromagnetic field
5443:
5337:
5292:
5152:
5029:
4982:
4946:
4784:
4755:
4723:
4685:
4542:
4464:
4417:This term introduces
4409:
4178:
4163:
4143:
4141:{\displaystyle \Phi }
4116:
3969:
3895:
3767:
3695:
3630:
3628:{\displaystyle \Phi }
3610:
3581:
3452:
3345:
3253:
3251:{\displaystyle \Phi }
3233:
3179:
3130:
3110:
3108:{\displaystyle \Phi }
3086:
3053:
2960:
2835:
2711:
2668:
2634:non-interacting real
2629:
2560:An example: Scalar O(
2544:
2450:
2415:
2307:
2257:
1524:electromagnetic field
1258:gauge transformations
1235:local transformations
381:KleinâGordon equation
323:LSZ reduction formula
9522:Mathematical physics
9376:Mathematical finance
9361:Social choice theory
9276:Algebraic structures
9225:in quantum mechanics
9161:Analytical mechanics
9127:Stochastic processes
9099:Variational calculus
8911:Approximation theory
8836:Statistical software
8574:Loop quantum gravity
8513:Theory of everything
8508:Grand Unified Theory
8482:Neutrino oscillation
8329:Quantum field theory
8092:Gauge Field Theories
7812:(2): 485â486, 1994,
7383:Quantum field theory
7308:AharonovâBohm effect
7221:
7113:
7104:gauge transformation
7047:(QCD) this leads to
7003:lattice gauge theory
6926:quantum field theory
6885:
6796:
6777:A quantity which is
6701:
6635:
6583:
6508:
6460:
6421:
6414:is the Lie bracket.
6386:
6320:
6276:
6254:
6227:
6205:
6181:
6123:
6080:covariant derivative
5997:gauge transformation
5983:. Mathematically, a
5799:
5752:
5641:
5491:
5386:
5307:
5184:
5052:
5018:
4958:
4796:
4782:{\displaystyle A(x)}
4764:
4732:
4697:
4555:
4540:{\displaystyle A(x)}
4522:
4437:
4200:
4152:
4132:
4124:Pauli uses the term
3991:
3910:
3797:
3714:
3641:
3619:
3594:
3523:
3385:
3269:
3242:
3217:
3145:
3128:{\displaystyle \mu }
3119:
3099:
3065:
2979:
2850:
2679:
2650:
2638:, with equal masses
2618:
2576:quantum field theory
2462:
2427:
2326:
2282:
2234:
2183:anomaly cancellation
2179:FaddeevâPopov ghosts
2141:Quantum field theory
2058:Physical experiments
1998:covariant derivative
1994:Ehresmann connection
1934:gauge transformation
1785:See Jackson and Okun
1770:SU(3) Ă SU(2) Ă U(1)
1637:quantum field theory
1589:. In 1954, Yang and
1433:gauge theory gravity
1350:elementary particles
464:Theory of everything
318:Lattice field theory
288:Correlation function
110:Quantum field theory
9351:Operations research
9220:Perturbation theory
9018:Multilinear algebra
8989:Functional analysis
8846:Numerical libraries
8778:Computational logic
8541:Split supersymmetry
8503:KaluzaâKlein theory
8376:Fermi's interaction
8189:2002AmJPh..70..917J
8103:Kane, G.L. (1987).
7968:Constructing Quarks
7937:2001RvMP...73..663J
7877:1994NuPhB.431..484S
7818:1994NuPhB.430..485.
7775:1994NuPhB.426...19S
7679:1954PhRv...96..191Y
7614:1941RvMP...13..203P
7481:2002SHPMP..33....3B
7403:Symmetry in physics
7343:KaluzaâKlein theory
7236:
7150:
6988:perturbation theory
6309:, Δ. Under such an
6175:exterior derivative
5960:Maxwell's equations
4989:structure constants
4940:
4925:
4872:
4844:
4816:
4749:
4717:
4679:
4481:of the gauge field
4272:
3953:
3815:
3707:and takes the form
3405:
2824:
2191:solid-state physics
2175:perturbation theory
2017:When analyzing the
1534:quantum mechanical
1490:, conjectured that
1468:high energy physics
443:Incomplete theories
9488:Mathematics portal
9385:Other applications
9109:Probability theory
9092:Validated numerics
9072:Numerical analysis
8965:Geometric analysis
8955:Differential forms
8788:Information theory
8569:Superstring theory
8339:Strong interaction
7318:Electroweak theory
7271:
7224:
7207:
7138:
7083:electroweak theory
7049:asymptotic freedom
6975:. This involves a
6965:quantum amplitudes
6932:connect different
6895:
6860:
6753:
6643:
6621:
6566:
6491:
6446:
6404:
6369:
6302:does not vanish.)
6292:
6262:
6240:
6213:
6187:
6160:
5941:
5774:
5719:
5624:
5438:
5332:
5287:
5147:
5024:
5005:ChernâSimons model
5001:BornâInfeld action
4977:
4941:
4926:
4911:
4894:
4858:
4830:
4799:
4779:
4750:
4735:
4718:
4700:
4680:
4662:
4537:
4459:
4404:
4256:
4184:
4158:
4138:
4111:
3964:
3939:
3938:
3890:
3803:
3762:
3690:
3625:
3605:
3576:
3500:In this case, the
3447:
3391:
3340:
3248:
3228:
3174:
3125:
3105:
3093:partial derivative
3081:
3048:
2955:
2830:
2810:
2663:
2624:
2614:Consider a set of
2539:
2537:
2445:
2410:
2408:
2302:
2278:of the potential,
2252:
2224:electric potential
2082:Continuum theories
2052:general relativity
1919:theory involves a
1821:Newtonian dynamics
1817:degrees of freedom
1778:general relativity
1766:fundamental forces
1713:, that is, exotic
1686:classification of
1649:asymptotic freedom
1484:general relativity
1454: –
1444:general relativity
1405:general relativity
1254:degrees of freedom
328:Partition function
255:Topological charge
175:General relativity
170:Special relativity
9504:
9503:
9338:Decision sciences
9332:
9331:
9314:Spacetime algebra
9006:Harmonic analysis
8972:Dynamical systems
8916:Clifford analysis
8893:Discrete geometry
8859:
8858:
8679:
8678:
8602:
8601:
8477:Strong CP problem
8455:Hierarchy problem
8197:10.1119/1.1491265
8107:. Perseus Books.
7855:Nuclear Physics B
7806:Nuclear Physics B
7753:Nuclear Physics B
7646:978-0-19-956684-6
7428:YangâMills theory
7398:Symmetry breaking
6914:metric connection
6724:
6692:YangâMills action
6546:
6541:
6519:
5913:
5828:
5813:
5685:
5674:
5530:
5519:
5505:
5426:
5209:
5144:
5127:
5110:
5093:
5076:
5027:{\displaystyle A}
4993:YangâMills action
4879:
4644:
4586:
4569:
4510:YangâMills theory
4361:
4300:
4242:
4205:
4161:{\displaystyle A}
4084:
4030:
3996:
3929:
3915:
3856:
3802:
3719:
3646:
3528:
3390:
3373:Noether's theorem
3274:
3198:belonging to the
3150:
2984:
2968:by introducing a
2928:
2874:
2855:
2798:
2745:
2627:{\displaystyle n}
2578:, and the use of
2533:
2377:
2187:BRST quantization
1956:over that point.
1629:Yang-Mills theory
1500:quantum mechanics
1415:, beginning with
1403:in the theory of
1313:YangâMills theory
1215:
1214:
308:Expectation value
283:BRST quantization
230:Poincaré symmetry
185:YangâMills theory
165:Quantum mechanics
87:
86:
79:
16:(Redirected from
9529:
9289:Feynman integral
9272:
9271:
9232:Potential theory
9121:random variables
9011:Fourier analysis
8994:Operator algebra
8921:Clifford algebra
8873:Computer algebra
8799:
8798:
8706:
8699:
8692:
8683:
8682:
8667:
8666:
8655:
8654:
8445:
8444:
8400:
8399:
8381:Weak hypercharge
8366:Weak interaction
8307:Particle physics
8286:
8279:
8272:
8263:
8262:
8248:
8223:
8221:
8208:
8182:
8158:
8156:
8155:
8140:
8138:
8118:
8099:
8094:(3rd ed.).
8083:
8056:
8019:Deep Down Things
7999:
7992:
7986:
7985:
7963:
7957:
7956:
7930:
7910:
7904:
7903:
7870:
7843:
7837:
7836:
7801:
7768:
7741:
7735:
7734:
7724:
7699:
7693:
7692:
7690:
7657:
7651:
7650:
7632:
7626:
7625:
7590:
7584:
7583:
7551:
7545:
7544:
7523:
7499:
7493:
7492:
7460:
7408:Charge (physics)
7291:
7280:
7278:
7277:
7272:
7258:
7257:
7232:
7216:
7214:
7213:
7208:
7194:
7193:
7172:
7171:
7146:
7125:
7124:
7005:) may be called
6994:) may be called
6984:running coupling
6940:Methods and aims
6904:
6902:
6901:
6896:
6894:
6893:
6869:
6867:
6866:
6861:
6859:
6855:
6854:
6850:
6849:
6845:
6844:
6826:
6825:
6814:
6813:
6762:
6760:
6759:
6754:
6725:
6723:
6722:
6721:
6705:
6694:is now given by
6687:for an example.
6652:
6650:
6649:
6644:
6642:
6630:
6628:
6627:
6622:
6617:
6600:
6595:
6594:
6575:
6573:
6572:
6567:
6562:
6551:
6544:
6543:
6542:
6540:
6539:
6527:
6522:
6517:
6500:
6498:
6497:
6492:
6472:
6471:
6455:
6453:
6452:
6447:
6433:
6432:
6413:
6411:
6410:
6407:{\displaystyle }
6405:
6378:
6376:
6375:
6370:
6365:
6354:
6337:
6332:
6331:
6301:
6299:
6298:
6293:
6291:
6283:
6271:
6269:
6268:
6263:
6261:
6249:
6247:
6246:
6241:
6239:
6238:
6222:
6220:
6219:
6214:
6212:
6196:
6194:
6193:
6188:
6169:
6167:
6166:
6161:
6159:
6151:
6143:
6138:
6130:
6043:principal bundle
5993:principal bundle
5950:
5948:
5947:
5942:
5940:
5939:
5927:
5926:
5914:
5912:
5911:
5910:
5894:
5886:
5882:
5881:
5880:
5865:
5864:
5855:
5854:
5830:
5829:
5821:
5815:
5814:
5811:
5809:
5808:
5784:in terms of the
5783:
5781:
5780:
5775:
5764:
5763:
5743:minimal coupling
5728:
5726:
5725:
5720:
5706:
5705:
5687:
5686:
5678:
5675:
5667:
5653:
5652:
5633:
5631:
5630:
5625:
5614:
5613:
5595:
5594:
5573:
5572:
5551:
5550:
5532:
5531:
5523:
5520:
5512:
5507:
5506:
5503:
5501:
5500:
5478:vector potential
5475:
5456:
5447:
5445:
5444:
5439:
5437:
5436:
5427:
5419:
5411:
5410:
5398:
5397:
5378:
5367:
5358:
5341:
5339:
5338:
5333:
5328:
5327:
5296:
5294:
5293:
5288:
5283:
5282:
5277:
5267:
5263:
5262:
5261:
5246:
5245:
5236:
5235:
5211:
5210:
5202:
5193:
5192:
5169:, with only the
5156:
5154:
5153:
5148:
5146:
5145:
5142:
5140:
5139:
5129:
5128:
5125:
5123:
5122:
5112:
5111:
5108:
5106:
5105:
5095:
5094:
5091:
5089:
5088:
5078:
5077:
5074:
5072:
5071:
5061:
5060:
5033:
5031:
5030:
5025:
4986:
4984:
4983:
4978:
4976:
4975:
4950:
4948:
4947:
4942:
4939:
4934:
4924:
4919:
4910:
4909:
4893:
4871:
4866:
4857:
4856:
4843:
4838:
4829:
4828:
4815:
4810:
4788:
4786:
4785:
4780:
4759:
4757:
4756:
4751:
4748:
4743:
4727:
4725:
4724:
4719:
4716:
4711:
4689:
4687:
4686:
4681:
4678:
4673:
4661:
4660:
4645:
4637:
4629:
4625:
4624:
4623:
4611:
4610:
4587:
4579:
4571:
4570:
4567:
4565:
4564:
4546:
4544:
4543:
4538:
4468:
4466:
4465:
4460:
4458:
4457:
4456:
4447:
4446:
4413:
4411:
4410:
4405:
4400:
4399:
4390:
4389:
4388:
4375:
4374:
4362:
4357:
4356:
4347:
4339:
4338:
4329:
4328:
4327:
4314:
4313:
4301:
4293:
4282:
4281:
4271:
4270:
4264:
4255:
4254:
4253:
4243:
4235:
4227:
4226:
4225:
4213:
4212:
4203:
4167:
4165:
4164:
4159:
4147:
4145:
4144:
4139:
4120:
4118:
4117:
4112:
4107:
4106:
4105:
4095:
4094:
4085:
4077:
4069:
4068:
4059:
4058:
4057:
4044:
4043:
4031:
4023:
4018:
4017:
4016:
4004:
4003:
3994:
3973:
3971:
3970:
3965:
3963:
3962:
3952:
3947:
3937:
3925:
3924:
3913:
3899:
3897:
3896:
3891:
3889:
3888:
3870:
3869:
3857:
3849:
3844:
3843:
3831:
3830:
3811:
3800:
3771:
3769:
3768:
3763:
3761:
3760:
3742:
3741:
3729:
3728:
3717:
3699:
3697:
3696:
3691:
3686:
3685:
3670:
3659:
3658:
3644:
3634:
3632:
3631:
3626:
3614:
3612:
3611:
3606:
3604:
3585:
3583:
3582:
3577:
3569:
3568:
3538:
3537:
3526:
3456:
3454:
3453:
3448:
3443:
3442:
3433:
3432:
3431:
3421:
3420:
3404:
3399:
3388:
3371:. Incidentally,
3349:
3347:
3346:
3341:
3336:
3335:
3320:
3309:
3308:
3287:
3286:
3272:
3257:
3255:
3254:
3249:
3237:
3235:
3234:
3229:
3227:
3207:orthogonal group
3183:
3181:
3180:
3175:
3164:
3148:
3134:
3132:
3131:
3126:
3115:along dimension
3114:
3112:
3111:
3106:
3090:
3088:
3087:
3082:
3077:
3076:
3057:
3055:
3054:
3049:
3044:
3043:
3025:
3024:
3012:
3011:
2996:
2995:
2994:
2982:
2964:
2962:
2961:
2956:
2951:
2950:
2949:
2939:
2938:
2929:
2921:
2913:
2912:
2903:
2902:
2901:
2888:
2887:
2875:
2867:
2862:
2861:
2853:
2839:
2837:
2836:
2831:
2829:
2825:
2823:
2818:
2809:
2808:
2799:
2791:
2786:
2785:
2776:
2775:
2766:
2765:
2756:
2755:
2746:
2738:
2730:
2725:
2707:
2706:
2701:
2688:
2687:
2672:
2670:
2669:
2664:
2662:
2661:
2633:
2631:
2630:
2625:
2548:
2546:
2545:
2540:
2538:
2534:
2532:
2524:
2516:
2485:
2473:
2454:
2452:
2451:
2446:
2419:
2417:
2416:
2411:
2409:
2405:
2387:
2378:
2376:
2368:
2367:
2358:
2337:
2314:vector potential
2311:
2309:
2308:
2303:
2289:
2261:
2259:
2258:
2253:
2088:continuum theory
1880:
1873:
1750:electromagnetism
1737:invariants (the
1703:Michael Freedman
1566:had derived the
1488:electromagnetism
1460:condensed matter
1288:gauge invariance
1276:group generators
1207:
1200:
1193:
298:Effective action
225:Lorentz symmetry
150:Electromagnetism
120:
106:
105:
82:
75:
71:
68:
62:
57:this article by
48:inline citations
35:
34:
27:
21:
9537:
9536:
9532:
9531:
9530:
9528:
9527:
9526:
9507:
9506:
9505:
9500:
9472:
9434:
9418:
9380:
9328:
9294:Poisson algebra
9270:
9152:
9145:
9103:
8999:Operator theory
8897:
8855:
8821:Tensor software
8797:
8746:Automata theory
8715:
8710:
8680:
8675:
8643:
8598:
8556:Quantum gravity
8550:
8517:
8486:
8439:
8432:
8423:Higgs mechanism
8401:
8397:
8392:
8295:
8290:
8233:
8230:
8219:math-ph/9902027
8180:physics/0204034
8153:
8151:
8115:
8080:
8053:
8011:General readers
8008:
8003:
8002:
7994:J. J. Sakurai,
7993:
7989:
7982:
7964:
7960:
7911:
7907:
7844:
7840:
7803:
7742:
7738:
7700:
7696:
7658:
7654:
7647:
7633:
7629:
7594:Pauli, Wolfgang
7591:
7587:
7552:
7548:
7500:
7496:
7461:
7457:
7452:
7447:
7423:Ward identities
7303:Gauge principle
7298:
7282:
7253:
7249:
7228:
7222:
7219:
7218:
7189:
7185:
7167:
7163:
7142:
7120:
7116:
7114:
7111:
7110:
7100:
7019:
6979:of the theory.
6977:renormalization
6942:
6934:renormalization
6930:Ward identities
6922:
6889:
6888:
6886:
6883:
6882:
6873:where Ï is the
6840:
6836:
6835:
6831:
6827:
6821:
6820:
6819:
6815:
6803:
6799:
6797:
6794:
6793:
6779:gauge-invariant
6717:
6713:
6709:
6704:
6702:
6699:
6698:
6638:
6636:
6633:
6632:
6613:
6596:
6590:
6586:
6584:
6581:
6580:
6558:
6547:
6529:
6528:
6523:
6521:
6520:
6509:
6506:
6505:
6467:
6463:
6461:
6458:
6457:
6428:
6424:
6422:
6419:
6418:
6387:
6384:
6383:
6361:
6350:
6333:
6327:
6323:
6321:
6318:
6317:
6287:
6279:
6277:
6274:
6273:
6257:
6255:
6252:
6251:
6234:
6230:
6228:
6225:
6224:
6208:
6206:
6203:
6202:
6197:stands for the
6182:
6179:
6178:
6155:
6147:
6139:
6134:
6126:
6124:
6121:
6120:
6099:gauge potential
6088:connection form
6062:structure group
6024:connection form
5977:
5971:
5966:
5932:
5928:
5919:
5915:
5906:
5902:
5898:
5893:
5876:
5872:
5860:
5856:
5850:
5846:
5835:
5831:
5820:
5819:
5810:
5804:
5803:
5802:
5800:
5797:
5796:
5759:
5755:
5753:
5750:
5749:
5739:gauge principle
5701:
5697:
5677:
5676:
5666:
5648:
5644:
5642:
5639:
5638:
5609:
5605:
5590:
5586:
5568:
5564:
5546:
5542:
5522:
5521:
5511:
5502:
5496:
5495:
5494:
5492:
5489:
5488:
5466:
5463:electric charge
5452:
5432:
5428:
5418:
5406:
5402:
5393:
5389:
5387:
5384:
5383:
5369:
5363:
5354:
5320:
5316:
5308:
5305:
5304:
5278:
5273:
5272:
5257:
5253:
5241:
5237:
5231:
5227:
5216:
5212:
5201:
5200:
5188:
5187:
5185:
5182:
5181:
5167:electrodynamics
5163:
5141:
5135:
5134:
5133:
5124:
5118:
5117:
5116:
5107:
5101:
5100:
5099:
5090:
5084:
5083:
5082:
5073:
5067:
5066:
5065:
5056:
5055:
5053:
5050:
5049:
5019:
5016:
5015:
4965:
4961:
4959:
4956:
4955:
4935:
4930:
4920:
4915:
4899:
4895:
4883:
4867:
4862:
4852:
4848:
4839:
4834:
4824:
4820:
4811:
4803:
4797:
4794:
4793:
4765:
4762:
4761:
4744:
4739:
4733:
4730:
4729:
4712:
4704:
4698:
4695:
4694:
4674:
4666:
4650:
4646:
4636:
4616:
4612:
4603:
4599:
4598:
4594:
4578:
4566:
4560:
4559:
4558:
4556:
4553:
4552:
4523:
4520:
4519:
4512:
4506:
4449:
4448:
4442:
4441:
4440:
4438:
4435:
4434:
4395:
4391:
4384:
4383:
4379:
4370:
4366:
4352:
4348:
4346:
4334:
4330:
4323:
4322:
4318:
4309:
4305:
4292:
4277:
4273:
4266:
4265:
4260:
4249:
4248:
4244:
4234:
4215:
4214:
4208:
4207:
4206:
4201:
4198:
4197:
4180:Feynman diagram
4153:
4150:
4149:
4133:
4130:
4129:
4101:
4100:
4096:
4090:
4086:
4076:
4064:
4060:
4053:
4052:
4048:
4039:
4035:
4022:
4006:
4005:
3999:
3998:
3997:
3992:
3989:
3988:
3958:
3954:
3948:
3943:
3933:
3920:
3916:
3911:
3908:
3907:
3881:
3877:
3865:
3861:
3848:
3836:
3832:
3826:
3822:
3807:
3798:
3795:
3794:
3756:
3752:
3737:
3733:
3724:
3720:
3715:
3712:
3711:
3681:
3677:
3663:
3654:
3650:
3642:
3639:
3638:
3620:
3617:
3616:
3597:
3595:
3592:
3591:
3564:
3560:
3533:
3529:
3524:
3521:
3520:
3438:
3434:
3427:
3426:
3422:
3416:
3412:
3400:
3395:
3386:
3383:
3382:
3364:structure group
3331:
3327:
3313:
3304:
3300:
3282:
3278:
3270:
3267:
3266:
3243:
3240:
3239:
3220:
3218:
3215:
3214:
3157:
3146:
3143:
3142:
3120:
3117:
3116:
3100:
3097:
3096:
3072:
3068:
3066:
3063:
3062:
3039:
3035:
3020:
3016:
3007:
3003:
2990:
2989:
2985:
2980:
2977:
2976:
2945:
2944:
2940:
2934:
2930:
2920:
2908:
2904:
2897:
2896:
2892:
2883:
2879:
2866:
2857:
2856:
2851:
2848:
2847:
2819:
2814:
2804:
2800:
2790:
2781:
2777:
2771:
2767:
2761:
2757:
2751:
2747:
2737:
2736:
2732:
2726:
2715:
2702:
2697:
2696:
2683:
2682:
2680:
2677:
2676:
2657:
2653:
2651:
2648:
2647:
2619:
2616:
2615:
2566:
2536:
2535:
2525:
2517:
2515:
2502:
2496:
2495:
2481:
2474:
2469:
2465:
2463:
2460:
2459:
2428:
2425:
2424:
2407:
2406:
2401:
2388:
2383:
2380:
2379:
2369:
2363:
2359:
2357:
2338:
2333:
2329:
2327:
2324:
2323:
2285:
2283:
2280:
2279:
2272:vector calculus
2235:
2232:
2231:
2212:
2207:
2195:crystallography
2159:action integral
2143:
2137:
2084:
2060:
1986:
1978:Coriolis effect
1961:global symmetry
1949:complex numbers
1947:via its use of
1916:
1911:
1875:
1868:
1852:
1809:
1807:Global symmetry
1804:
1799:
1680:Simon Donaldson
1548:electrodynamics
1476:
1429:diffeomorphisms
1413:quantum gravity
1339:global symmetry
1211:
1182:
1181:
1180:
1178:
482:
474:
473:
469:Quantum gravity
444:
436:
435:
431:Higgs mechanism
411:
401:
400:
386:Proca equations
371:
363:
362:
348:Renormalization
313:Feynman diagram
268:
260:
259:
200:
190:
189:
140:
125:
123:Feynman diagram
104:
97:
90:
83:
72:
66:
63:
53:Please help to
52:
36:
32:
23:
22:
15:
12:
11:
5:
9535:
9525:
9524:
9519:
9517:Gauge theories
9502:
9501:
9499:
9498:
9485:
9477:
9474:
9473:
9471:
9470:
9465:
9460:
9455:
9454:
9453:
9442:
9440:
9436:
9435:
9433:
9432:
9426:
9424:
9420:
9419:
9417:
9416:
9409:
9404:
9399:
9394:
9388:
9386:
9382:
9381:
9379:
9378:
9373:
9368:
9363:
9358:
9353:
9348:
9342:
9340:
9334:
9333:
9330:
9329:
9327:
9326:
9321:
9316:
9311:
9306:
9301:
9296:
9291:
9286:
9280:
9278:
9269:
9268:
9267:
9266:
9261:
9251:
9250:
9249:
9244:
9234:
9229:
9228:
9227:
9217:
9216:
9215:
9210:
9205:
9200:
9195:
9190:
9185:
9175:
9174:
9173:
9168:
9157:
9155:
9147:
9146:
9144:
9143:
9138:
9133:
9124:
9113:
9111:
9105:
9104:
9102:
9101:
9096:
9095:
9094:
9089:
9084:
9079:
9069:
9068:
9067:
9062:
9057:
9052:
9042:
9041:
9040:
9035:
9030:
9025:
9015:
9014:
9013:
9003:
9002:
9001:
8996:
8986:
8985:
8984:
8982:Control theory
8979:
8969:
8968:
8967:
8962:
8957:
8947:
8946:
8945:
8940:
8935:
8925:
8924:
8923:
8913:
8907:
8905:
8899:
8898:
8896:
8895:
8890:
8885:
8880:
8875:
8869:
8867:
8861:
8860:
8857:
8856:
8854:
8853:
8848:
8843:
8838:
8833:
8828:
8823:
8818:
8813:
8807:
8805:
8796:
8795:
8790:
8785:
8780:
8775:
8774:
8773:
8763:
8758:
8753:
8748:
8743:
8742:
8741:
8736:
8725:
8723:
8717:
8716:
8709:
8708:
8701:
8694:
8686:
8677:
8676:
8674:
8673:
8661:
8648:
8645:
8644:
8642:
8641:
8636:
8631:
8626:
8621:
8616:
8610:
8608:
8604:
8603:
8600:
8599:
8597:
8596:
8594:Twistor theory
8591:
8586:
8581:
8576:
8571:
8566:
8560:
8558:
8552:
8551:
8549:
8548:
8543:
8538:
8533:
8527:
8525:
8519:
8518:
8516:
8515:
8510:
8505:
8500:
8494:
8492:
8488:
8487:
8485:
8484:
8479:
8474:
8473:
8472:
8462:
8457:
8451:
8449:
8442:
8440:Standard Model
8434:
8433:
8431:
8430:
8425:
8420:
8415:
8409:
8407:
8403:
8402:
8395:
8393:
8391:
8390:
8389:
8388:
8383:
8378:
8373:
8368:
8358:
8357:
8356:
8351:
8346:
8336:
8331:
8326:
8325:
8324:
8319:
8314:
8303:
8301:
8297:
8296:
8293:Standard Model
8289:
8288:
8281:
8274:
8266:
8260:
8259:
8254:
8249:
8229:
8228:External links
8226:
8225:
8224:
8209:
8173:(9): 917â928.
8159:
8141:
8136:hep-ph/9705211
8125:
8124:
8120:
8119:
8113:
8100:
8084:
8078:
8060:Cheng, T.-P.;
8057:
8051:
8033:
8032:
8028:
8027:
8024:Standard Model
8013:
8012:
8007:
8004:
8001:
8000:
7987:
7980:
7958:
7928:hep-ph/0012061
7905:
7868:hep-th/9408099
7861:(3): 484â550,
7838:
7766:hep-th/9407087
7736:
7694:
7673:(1): 191â195.
7652:
7645:
7627:
7602:Rev. Mod. Phys
7585:
7546:
7521:hep-ph/0012061
7514:(3): 663â680.
7494:
7454:
7453:
7451:
7448:
7446:
7445:
7440:
7435:
7430:
7425:
7420:
7415:
7410:
7405:
7400:
7395:
7390:
7388:Standard Model
7385:
7380:
7375:
7370:
7365:
7360:
7355:
7350:
7345:
7340:
7335:
7330:
7325:
7320:
7315:
7310:
7305:
7299:
7297:
7294:
7270:
7267:
7264:
7261:
7256:
7252:
7248:
7245:
7242:
7239:
7235:
7231:
7227:
7206:
7203:
7200:
7197:
7192:
7188:
7184:
7181:
7178:
7175:
7170:
7166:
7162:
7159:
7156:
7153:
7149:
7145:
7141:
7137:
7134:
7131:
7128:
7123:
7119:
7099:
7096:
7095:
7094:
7075:
7056:chiral anomaly
7052:
7018:
7015:
7011:supercomputing
6941:
6938:
6921:
6918:
6892:
6879:representation
6871:
6870:
6858:
6853:
6848:
6843:
6839:
6834:
6830:
6824:
6818:
6812:
6809:
6806:
6802:
6764:
6763:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6728:
6720:
6716:
6712:
6708:
6675:such that the
6641:
6631:, which means
6620:
6616:
6612:
6609:
6606:
6603:
6599:
6593:
6589:
6577:
6576:
6565:
6561:
6557:
6554:
6550:
6538:
6535:
6532:
6526:
6516:
6513:
6490:
6487:
6484:
6481:
6478:
6475:
6470:
6466:
6445:
6442:
6439:
6436:
6431:
6427:
6403:
6400:
6397:
6394:
6391:
6380:
6379:
6368:
6364:
6360:
6357:
6353:
6349:
6346:
6343:
6340:
6336:
6330:
6326:
6290:
6286:
6282:
6260:
6237:
6233:
6211:
6186:
6171:
6170:
6158:
6154:
6150:
6146:
6142:
6137:
6133:
6129:
6107:curvature form
6041:If there is a
6014:(i.e., affine
5970:
5967:
5956:Dirac equation
5952:
5951:
5938:
5935:
5931:
5925:
5922:
5918:
5909:
5905:
5901:
5897:
5892:
5889:
5885:
5879:
5875:
5871:
5868:
5863:
5859:
5853:
5849:
5844:
5841:
5838:
5834:
5827:
5824:
5818:
5807:
5773:
5770:
5767:
5762:
5758:
5718:
5715:
5712:
5709:
5704:
5700:
5696:
5693:
5690:
5684:
5681:
5673:
5670:
5665:
5662:
5659:
5656:
5651:
5647:
5635:
5634:
5623:
5620:
5617:
5612:
5608:
5604:
5601:
5598:
5593:
5589:
5585:
5582:
5579:
5576:
5571:
5567:
5563:
5560:
5557:
5554:
5549:
5545:
5541:
5538:
5535:
5529:
5526:
5518:
5515:
5510:
5499:
5476:with the four-
5449:
5448:
5435:
5431:
5425:
5422:
5417:
5414:
5409:
5405:
5401:
5396:
5392:
5343:
5342:
5331:
5326:
5323:
5319:
5315:
5312:
5298:
5297:
5286:
5281:
5276:
5270:
5266:
5260:
5256:
5252:
5249:
5244:
5240:
5234:
5230:
5225:
5222:
5219:
5215:
5208:
5205:
5199:
5196:
5191:
5175:Dirac equation
5162:
5159:
5158:
5157:
5138:
5132:
5121:
5115:
5104:
5098:
5087:
5081:
5070:
5064:
5059:
5023:
4974:
4971:
4968:
4964:
4952:
4951:
4938:
4933:
4929:
4923:
4918:
4914:
4908:
4905:
4902:
4898:
4892:
4889:
4886:
4882:
4878:
4875:
4870:
4865:
4861:
4855:
4851:
4847:
4842:
4837:
4833:
4827:
4823:
4819:
4814:
4809:
4806:
4802:
4778:
4775:
4772:
4769:
4747:
4742:
4738:
4715:
4710:
4707:
4703:
4691:
4690:
4677:
4672:
4669:
4665:
4659:
4656:
4653:
4649:
4643:
4640:
4635:
4632:
4628:
4622:
4619:
4615:
4609:
4606:
4602:
4597:
4593:
4590:
4585:
4582:
4577:
4574:
4563:
4536:
4533:
4530:
4527:
4508:Main article:
4505:
4502:
4455:
4452:
4445:
4415:
4414:
4403:
4398:
4394:
4387:
4382:
4378:
4373:
4369:
4365:
4360:
4355:
4351:
4345:
4342:
4337:
4333:
4326:
4321:
4317:
4312:
4308:
4304:
4299:
4296:
4291:
4288:
4285:
4280:
4276:
4269:
4263:
4259:
4252:
4247:
4241:
4238:
4233:
4230:
4224:
4221:
4218:
4211:
4157:
4137:
4122:
4121:
4110:
4104:
4099:
4093:
4089:
4083:
4080:
4075:
4072:
4067:
4063:
4056:
4051:
4047:
4042:
4038:
4034:
4029:
4026:
4021:
4015:
4012:
4009:
4002:
3975:
3974:
3961:
3957:
3951:
3946:
3942:
3936:
3932:
3928:
3923:
3919:
3901:
3900:
3887:
3884:
3880:
3876:
3873:
3868:
3864:
3860:
3855:
3852:
3847:
3842:
3839:
3835:
3829:
3825:
3821:
3818:
3814:
3810:
3806:
3773:
3772:
3759:
3755:
3751:
3748:
3745:
3740:
3736:
3732:
3727:
3723:
3701:
3700:
3689:
3684:
3680:
3676:
3673:
3669:
3666:
3662:
3657:
3653:
3649:
3624:
3603:
3600:
3587:
3586:
3575:
3572:
3567:
3563:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3536:
3532:
3458:
3457:
3446:
3441:
3437:
3430:
3425:
3419:
3415:
3411:
3408:
3403:
3398:
3394:
3351:
3350:
3339:
3334:
3330:
3326:
3323:
3319:
3316:
3312:
3307:
3303:
3299:
3296:
3293:
3290:
3285:
3281:
3277:
3247:
3226:
3223:
3185:
3184:
3173:
3170:
3167:
3163:
3160:
3156:
3153:
3124:
3104:
3080:
3075:
3071:
3059:
3058:
3047:
3042:
3038:
3034:
3031:
3028:
3023:
3019:
3015:
3010:
3006:
3002:
2999:
2993:
2988:
2966:
2965:
2954:
2948:
2943:
2937:
2933:
2927:
2924:
2919:
2916:
2911:
2907:
2900:
2895:
2891:
2886:
2882:
2878:
2873:
2870:
2865:
2860:
2841:
2840:
2828:
2822:
2817:
2813:
2807:
2803:
2797:
2794:
2789:
2784:
2780:
2774:
2770:
2764:
2760:
2754:
2750:
2744:
2741:
2735:
2729:
2724:
2721:
2718:
2714:
2710:
2705:
2700:
2694:
2691:
2686:
2660:
2656:
2623:
2608:
2607:
2585:
2584:
2565:
2564:) gauge theory
2558:
2550:
2549:
2531:
2528:
2523:
2520:
2514:
2511:
2508:
2505:
2503:
2501:
2498:
2497:
2494:
2491:
2488:
2484:
2480:
2477:
2475:
2472:
2468:
2467:
2444:
2441:
2438:
2435:
2432:
2421:
2420:
2404:
2400:
2397:
2394:
2391:
2389:
2386:
2382:
2381:
2375:
2372:
2366:
2362:
2356:
2353:
2350:
2347:
2344:
2341:
2339:
2336:
2332:
2331:
2301:
2298:
2295:
2292:
2288:
2251:
2248:
2245:
2242:
2239:
2216:electrostatics
2211:
2208:
2206:
2203:
2155:Standard Model
2139:Main article:
2136:
2133:
2120:
2119:
2113:
2110:
2103:
2102:
2099:
2095:
2083:
2080:
2071:
2070:
2067:
2059:
2056:
2048:
2047:
2044:
2041:
2038:
2003:field strength
1985:
1982:
1970:local symmetry
1915:
1912:
1910:
1909:Local symmetry
1907:
1895:rate of change
1851:
1848:
1808:
1805:
1803:
1800:
1798:
1795:
1794:
1793:
1762:Standard Model
1727:Nathan Seiberg
1684:differentiable
1672:Michael Atiyah
1670:In the 1970s,
1665:Standard Model
1579:Chen Ning Yang
1560:magnetic field
1526:effect on the
1475:
1472:
1411:. Theories of
1409:Lanczos tensor
1374:Standard Model
1343:Local symmetry
1263:symmetry group
1213:
1212:
1210:
1209:
1202:
1195:
1187:
1184:
1183:
1176:
1175:
1170:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1100:
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:
484:
483:
480:
479:
476:
475:
472:
471:
466:
461:
456:
451:
445:
442:
441:
438:
437:
434:
433:
428:
423:
418:
412:
409:Standard Model
407:
406:
403:
402:
399:
398:
393:
388:
383:
378:
376:Dirac equation
372:
369:
368:
365:
364:
361:
360:
358:Wick's theorem
355:
350:
345:
343:Regularization
340:
335:
330:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
275:
269:
266:
265:
262:
261:
258:
257:
252:
250:Noether charge
247:
242:
237:
235:Gauge symmetry
232:
227:
222:
217:
212:
207:
201:
196:
195:
192:
191:
188:
187:
182:
177:
172:
167:
162:
157:
152:
147:
141:
138:
137:
134:
133:
127:
126:
121:
113:
112:
88:
85:
84:
67:September 2016
39:
37:
30:
18:Gauge symmetry
9:
6:
4:
3:
2:
9534:
9523:
9520:
9518:
9515:
9514:
9512:
9497:
9493:
9489:
9486:
9484:
9483:
9479:
9478:
9475:
9469:
9466:
9464:
9461:
9459:
9456:
9452:
9449:
9448:
9447:
9444:
9443:
9441:
9439:Organizations
9437:
9431:
9428:
9427:
9425:
9421:
9414:
9410:
9408:
9405:
9403:
9400:
9398:
9395:
9393:
9390:
9389:
9387:
9383:
9377:
9374:
9372:
9369:
9367:
9364:
9362:
9359:
9357:
9354:
9352:
9349:
9347:
9344:
9343:
9341:
9339:
9335:
9325:
9322:
9320:
9317:
9315:
9312:
9310:
9307:
9305:
9302:
9300:
9299:Quantum group
9297:
9295:
9292:
9290:
9287:
9285:
9282:
9281:
9279:
9277:
9273:
9265:
9262:
9260:
9257:
9256:
9255:
9254:Supersymmetry
9252:
9248:
9245:
9243:
9240:
9239:
9238:
9237:String theory
9235:
9233:
9230:
9226:
9223:
9222:
9221:
9218:
9214:
9211:
9209:
9206:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9184:
9181:
9180:
9179:
9176:
9172:
9169:
9167:
9164:
9163:
9162:
9159:
9158:
9156:
9154:
9148:
9142:
9139:
9137:
9136:Path integral
9134:
9132:
9128:
9125:
9122:
9118:
9117:Distributions
9115:
9114:
9112:
9110:
9106:
9100:
9097:
9093:
9090:
9088:
9085:
9083:
9080:
9078:
9075:
9074:
9073:
9070:
9066:
9063:
9061:
9058:
9056:
9053:
9051:
9048:
9047:
9046:
9043:
9039:
9036:
9034:
9031:
9029:
9026:
9024:
9021:
9020:
9019:
9016:
9012:
9009:
9008:
9007:
9004:
9000:
8997:
8995:
8992:
8991:
8990:
8987:
8983:
8980:
8978:
8975:
8974:
8973:
8970:
8966:
8963:
8961:
8958:
8956:
8953:
8952:
8951:
8948:
8944:
8941:
8939:
8936:
8934:
8931:
8930:
8929:
8926:
8922:
8919:
8918:
8917:
8914:
8912:
8909:
8908:
8906:
8904:
8900:
8894:
8891:
8889:
8886:
8884:
8883:Combinatorics
8881:
8879:
8876:
8874:
8871:
8870:
8868:
8866:
8862:
8852:
8849:
8847:
8844:
8842:
8839:
8837:
8834:
8832:
8829:
8827:
8824:
8822:
8819:
8817:
8814:
8812:
8809:
8808:
8806:
8804:
8800:
8794:
8791:
8789:
8786:
8784:
8781:
8779:
8776:
8772:
8769:
8768:
8767:
8764:
8762:
8759:
8757:
8756:Coding theory
8754:
8752:
8749:
8747:
8744:
8740:
8737:
8735:
8732:
8731:
8730:
8727:
8726:
8724:
8722:
8721:Computational
8718:
8714:
8707:
8702:
8700:
8695:
8693:
8688:
8687:
8684:
8672:
8671:
8662:
8660:
8659:
8650:
8649:
8646:
8640:
8637:
8635:
8632:
8630:
8627:
8625:
8622:
8620:
8617:
8615:
8612:
8611:
8609:
8605:
8595:
8592:
8590:
8587:
8585:
8582:
8580:
8577:
8575:
8572:
8570:
8567:
8565:
8564:String theory
8562:
8561:
8559:
8557:
8553:
8547:
8544:
8542:
8539:
8537:
8534:
8532:
8529:
8528:
8526:
8524:
8523:Supersymmetry
8520:
8514:
8511:
8509:
8506:
8504:
8501:
8499:
8496:
8495:
8493:
8489:
8483:
8480:
8478:
8475:
8471:
8468:
8467:
8466:
8463:
8461:
8458:
8456:
8453:
8452:
8450:
8446:
8443:
8441:
8435:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8410:
8408:
8404:
8387:
8384:
8382:
8379:
8377:
8374:
8372:
8369:
8367:
8364:
8363:
8362:
8359:
8355:
8352:
8350:
8347:
8345:
8342:
8341:
8340:
8337:
8335:
8332:
8330:
8327:
8323:
8320:
8318:
8315:
8313:
8310:
8309:
8308:
8305:
8304:
8302:
8298:
8294:
8287:
8282:
8280:
8275:
8273:
8268:
8267:
8264:
8258:
8255:
8253:
8250:
8246:
8242:
8241:
8236:
8232:
8231:
8220:
8215:
8210:
8206:
8202:
8198:
8194:
8190:
8186:
8181:
8176:
8172:
8168:
8164:
8163:Jackson, J.D.
8160:
8150:
8146:
8142:
8137:
8132:
8127:
8126:
8122:
8121:
8116:
8114:0-201-11749-5
8110:
8106:
8101:
8097:
8093:
8089:
8085:
8081:
8079:0-19-851961-3
8075:
8071:
8067:
8063:
8058:
8054:
8052:3-540-67672-4
8048:
8044:
8040:
8035:
8034:
8030:
8029:
8025:
8021:
8020:
8015:
8014:
8010:
8009:
7997:
7991:
7983:
7981:0-226-66799-5
7977:
7973:
7969:
7962:
7954:
7950:
7946:
7942:
7938:
7934:
7929:
7924:
7920:
7916:
7909:
7902:
7898:
7894:
7890:
7886:
7882:
7878:
7874:
7869:
7864:
7860:
7856:
7852:
7848:
7842:
7835:
7831:
7827:
7823:
7819:
7815:
7811:
7807:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7767:
7762:
7758:
7754:
7750:
7746:
7740:
7732:
7728:
7723:
7718:
7714:
7711:
7710:
7705:
7698:
7689:
7684:
7680:
7676:
7672:
7669:
7668:
7663:
7656:
7648:
7642:
7638:
7631:
7623:
7619:
7615:
7611:
7608:(3): 203â32.
7607:
7603:
7599:
7595:
7589:
7581:
7577:
7573:
7569:
7565:
7561:
7557:
7550:
7543:
7539:
7535:
7531:
7527:
7522:
7517:
7513:
7509:
7505:
7498:
7490:
7486:
7482:
7478:
7474:
7470:
7466:
7459:
7455:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7419:
7418:Fock symmetry
7416:
7414:
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7364:
7361:
7359:
7356:
7354:
7351:
7349:
7346:
7344:
7341:
7339:
7336:
7334:
7331:
7329:
7326:
7324:
7321:
7319:
7316:
7314:
7313:Coulomb gauge
7311:
7309:
7306:
7304:
7301:
7300:
7293:
7289:
7285:
7265:
7259:
7254:
7246:
7240:
7233:
7229:
7225:
7201:
7195:
7190:
7182:
7176:
7168:
7164:
7160:
7154:
7147:
7143:
7139:
7129:
7121:
7117:
7107:
7105:
7092:
7088:
7084:
7080:
7079:gauge anomaly
7076:
7073:
7069:
7065:
7061:
7057:
7053:
7050:
7046:
7042:
7038:
7034:
7033:scale anomaly
7030:
7029:
7028:
7026:
7025:
7014:
7012:
7008:
7004:
6999:
6997:
6993:
6989:
6985:
6980:
6978:
6974:
6970:
6966:
6961:
6959:
6955:
6954:GuptaâBleuler
6951:
6947:
6937:
6935:
6931:
6927:
6917:
6915:
6911:
6910:vector bundle
6906:
6880:
6877:of a complex
6876:
6856:
6851:
6846:
6841:
6837:
6832:
6828:
6816:
6807:
6800:
6792:
6791:
6790:
6788:
6784:
6780:
6775:
6773:
6769:
6747:
6744:
6741:
6738:
6732:
6729:
6726:
6718:
6714:
6710:
6706:
6697:
6696:
6695:
6693:
6688:
6686:
6682:
6678:
6674:
6670:
6667:
6663:
6662:base manifold
6659:
6658:infinitesimal
6654:
6610:
6607:
6601:
6591:
6587:
6563:
6555:
6552:
6524:
6514:
6511:
6504:
6503:
6502:
6488:
6485:
6482:
6479:
6476:
6473:
6468:
6464:
6443:
6440:
6437:
6434:
6429:
6425:
6415:
6398:
6395:
6392:
6366:
6358:
6347:
6344:
6338:
6328:
6324:
6316:
6315:
6314:
6312:
6311:infinitesimal
6308:
6303:
6284:
6235:
6231:
6200:
6199:wedge product
6184:
6176:
6152:
6144:
6131:
6119:
6118:
6117:
6115:
6111:
6108:
6104:
6100:
6096:
6092:
6089:
6085:
6081:
6077:
6073:
6071:
6067:
6063:
6059:
6055:
6051:
6047:
6044:
6039:
6037:
6033:
6029:
6025:
6021:
6017:
6013:
6009:
6005:
6000:
5998:
5994:
5990:
5986:
5982:
5976:
5965:
5961:
5957:
5936:
5933:
5929:
5923:
5920:
5916:
5907:
5903:
5899:
5895:
5890:
5887:
5883:
5877:
5873:
5869:
5866:
5861:
5857:
5851:
5847:
5842:
5836:
5832:
5822:
5816:
5795:
5794:
5793:
5791:
5787:
5768:
5760:
5756:
5746:
5744:
5740:
5736:
5732:
5713:
5707:
5702:
5698:
5691:
5679:
5668:
5663:
5657:
5649:
5645:
5618:
5610:
5606:
5599:
5591:
5587:
5583:
5577:
5569:
5565:
5558:
5552:
5547:
5543:
5536:
5524:
5513:
5508:
5487:
5486:
5485:
5483:
5479:
5473:
5469:
5464:
5460:
5455:
5433:
5429:
5420:
5415:
5412:
5407:
5399:
5394:
5390:
5382:
5381:
5380:
5376:
5372:
5366:
5360:
5357:
5352:
5348:
5329:
5324:
5321:
5317:
5310:
5303:
5302:
5301:
5284:
5279:
5268:
5264:
5258:
5254:
5250:
5247:
5242:
5232:
5228:
5223:
5217:
5213:
5203:
5197:
5194:
5180:
5179:
5178:
5176:
5172:
5168:
5130:
5113:
5096:
5079:
5062:
5048:
5047:
5046:
5043:
5041:
5037:
5021:
5012:
5010:
5006:
5002:
4998:
4994:
4990:
4972:
4969:
4966:
4962:
4936:
4931:
4927:
4921:
4916:
4912:
4906:
4903:
4900:
4896:
4890:
4887:
4884:
4880:
4876:
4873:
4868:
4863:
4859:
4853:
4845:
4840:
4835:
4831:
4825:
4817:
4812:
4807:
4804:
4800:
4792:
4791:
4790:
4773:
4767:
4745:
4740:
4736:
4713:
4708:
4705:
4701:
4675:
4670:
4667:
4663:
4657:
4654:
4651:
4647:
4641:
4638:
4633:
4630:
4626:
4620:
4617:
4613:
4607:
4604:
4600:
4595:
4591:
4588:
4583:
4580:
4575:
4572:
4551:
4550:
4549:
4531:
4525:
4517:
4511:
4501:
4499:
4496:
4492:
4489:) are called
4488:
4484:
4480:
4476:
4472:
4432:
4428:
4424:
4420:
4396:
4392:
4371:
4367:
4358:
4353:
4349:
4343:
4335:
4331:
4310:
4297:
4294:
4289:
4286:
4278:
4261:
4257:
4239:
4236:
4231:
4228:
4196:
4195:
4194:
4193:
4189:
4181:
4177:
4173:
4171:
4155:
4127:
4091:
4087:
4081:
4078:
4073:
4065:
4061:
4040:
4036:
4027:
4024:
4019:
3987:
3986:
3985:
3983:
3978:
3959:
3955:
3949:
3944:
3940:
3934:
3930:
3926:
3921:
3917:
3906:
3905:
3904:
3885:
3882:
3878:
3871:
3866:
3853:
3850:
3845:
3840:
3837:
3833:
3827:
3823:
3819:
3816:
3812:
3808:
3804:
3793:
3792:
3791:
3789:
3785:
3782:
3778:
3757:
3753:
3749:
3746:
3743:
3738:
3730:
3725:
3721:
3710:
3709:
3708:
3706:
3682:
3678:
3674:
3671:
3667:
3655:
3651:
3637:
3636:
3635:
3601:
3565:
3554:
3551:
3542:
3534:
3519:
3518:
3517:
3515:
3511:
3507:
3503:
3498:
3496:
3493:
3490:
3486:
3482:
3478:
3473:
3471:
3467:
3464:matrices are
3463:
3439:
3435:
3417:
3409:
3406:
3401:
3396:
3392:
3381:
3380:
3379:
3378:
3374:
3370:
3366:
3365:
3360:
3356:
3332:
3324:
3321:
3317:
3305:
3283:
3265:
3264:
3263:
3261:
3224:
3212:
3208:
3205:
3201:
3197:
3194:
3190:
3168:
3165:
3161:
3141:
3140:
3139:
3136:
3122:
3094:
3073:
3040:
3036:
3032:
3029:
3026:
3021:
3017:
3013:
3008:
3004:
2997:
2975:
2974:
2973:
2971:
2935:
2931:
2925:
2922:
2917:
2909:
2884:
2871:
2868:
2863:
2846:
2845:
2844:
2826:
2820:
2815:
2811:
2805:
2801:
2795:
2792:
2787:
2782:
2778:
2772:
2762:
2758:
2752:
2742:
2739:
2733:
2727:
2722:
2719:
2716:
2712:
2708:
2703:
2692:
2689:
2675:
2674:
2673:
2658:
2654:
2645:
2641:
2637:
2636:scalar fields
2621:
2612:
2606:
2604:
2600:
2596:
2592:
2587:
2586:
2583:
2581:
2577:
2573:
2568:
2567:
2563:
2557:
2555:
2529:
2521:
2512:
2509:
2504:
2499:
2492:
2486:
2476:
2458:
2457:
2456:
2442:
2439:
2436:
2430:
2398:
2392:
2390:
2373:
2354:
2351:
2345:
2342:
2340:
2322:
2321:
2320:
2318:
2315:
2299:
2293:
2290:
2277:
2273:
2269:
2265:
2249:
2246:
2243:
2237:
2229:
2225:
2221:
2217:
2202:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2171:
2168:
2164:
2160:
2156:
2152:
2148:
2142:
2132:
2130:
2126:
2117:
2114:
2111:
2108:
2107:
2106:
2100:
2096:
2093:
2092:
2091:
2089:
2079:
2077:
2068:
2065:
2064:
2063:
2055:
2053:
2045:
2042:
2039:
2036:
2032:
2031:
2030:
2028:
2024:
2020:
2015:
2012:
2008:
2004:
1999:
1995:
1991:
1981:
1979:
1975:
1971:
1967:
1962:
1957:
1954:
1950:
1946:
1942:
1937:
1935:
1931:
1927:
1926:local section
1922:
1906:
1904:
1900:
1896:
1892:
1888:
1884:
1878:
1871:
1866:
1862:
1858:
1847:
1845:
1840:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1792:
1789:
1786:
1783:
1782:
1781:
1779:
1776:, as well as
1775:
1774:string theory
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1742:
1740:
1736:
1732:
1731:supersymmetry
1728:
1724:
1723:Edward Witten
1720:
1716:
1712:
1710:
1704:
1700:
1699:homeomorphism
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1668:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1625:
1623:
1619:
1616:
1613:group on the
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1584:
1583:atomic nuclei
1580:
1575:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1528:wave function
1525:
1521:
1518:, which is a
1517:
1513:
1509:
1505:
1504:Vladimir Fock
1501:
1497:
1493:
1492:Eichinvarianz
1489:
1485:
1481:
1471:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1442:and later in
1441:
1436:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1397:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1346:
1344:
1340:
1336:
1332:
1329:
1325:
1321:
1316:
1314:
1310:
1308:
1303:
1302:
1297:
1293:
1289:
1285:
1282:) called the
1281:
1277:
1273:
1269:
1265:
1264:
1259:
1255:
1251:
1246:
1244:
1240:
1236:
1232:
1229:in which the
1228:
1225:is a type of
1224:
1220:
1208:
1203:
1201:
1196:
1194:
1189:
1188:
1186:
1185:
1179:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1148:Zamolodchikov
1146:
1144:
1143:Zamolodchikov
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
519:
516:
514:
511:
509:
506:
504:
501:
499:
496:
494:
491:
489:
486:
485:
478:
477:
470:
467:
465:
462:
460:
457:
455:
454:Supersymmetry
452:
450:
449:String theory
447:
446:
440:
439:
432:
429:
427:
424:
422:
419:
417:
414:
413:
410:
405:
404:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
373:
367:
366:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
270:
264:
263:
256:
253:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
202:
199:
194:
193:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
142:
136:
135:
132:
129:
128:
124:
119:
115:
114:
111:
108:
107:
102:
95:
81:
78:
70:
60:
56:
50:
49:
43:
38:
29:
28:
19:
9494: /
9490: /
9480:
9356:Optimization
9319:Superalgebra
9197:
9178:Field theory
9151:Mathematical
9129: /
8977:Chaos theory
8960:Gauge theory
8888:Graph theory
8783:Cryptography
8668:
8656:
8546:Supergravity
8406:Constituents
8386:Weak isospin
8344:Color charge
8334:Gauge theory
8333:
8238:
8170:
8166:
8152:. Retrieved
8104:
8091:
8088:Frampton, P.
8065:
8038:
8017:
8006:Bibliography
7995:
7990:
7967:
7961:
7918:
7914:
7908:
7858:
7854:
7841:
7809:
7805:
7759:(1): 19â52,
7756:
7752:
7739:
7715:(1): 81â83.
7712:
7707:
7697:
7670:
7665:
7655:
7636:
7630:
7605:
7601:
7588:
7563:
7559:
7549:
7541:
7511:
7507:
7497:
7472:
7468:
7458:
7348:Lorenz gauge
7328:Gauge fixing
7287:
7283:
7108:
7101:
7036:
7022:
7020:
7006:
7000:
6995:
6981:
6973:vacuum state
6962:
6958:quantization
6943:
6923:
6907:
6872:
6778:
6776:
6765:
6691:
6689:
6655:
6578:
6416:
6381:
6304:
6172:
6109:
6098:
6090:
6074:
6065:
6045:
6040:
6001:
5996:
5984:
5978:
5747:
5636:
5471:
5467:
5453:
5450:
5374:
5370:
5364:
5361:
5355:
5344:
5299:
5164:
5044:
5040:gauge fixing
5035:
5013:
4992:
4953:
4692:
4515:
4513:
4491:gauge bosons
4486:
4482:
4430:
4426:
4422:
4421:between the
4419:interactions
4416:
4191:
4187:
4185:
4169:
4168:is called a
4125:
4123:
3981:
3979:
3976:
3902:
3787:
3783:
3780:
3776:
3774:
3702:
3588:
3513:
3509:
3505:
3501:
3499:
3494:
3484:
3480:
3476:
3474:
3469:
3461:
3459:
3376:
3369:G-structures
3362:
3358:
3354:
3352:
3260:dot products
3210:
3203:
3199:
3192:
3188:
3186:
3137:
3060:
2967:
2842:
2639:
2613:
2609:
2588:
2569:
2561:
2553:
2551:
2422:
2316:
2267:
2263:
2227:
2219:
2213:
2172:
2167:gauge fixing
2149:, including
2144:
2121:
2104:
2085:
2072:
2061:
2049:
2016:
2010:
1989:
1987:
1984:Gauge fields
1965:
1958:
1952:
1938:
1933:
1925:
1921:fiber bundle
1917:
1902:
1898:
1894:
1890:
1886:
1882:
1876:
1869:
1864:
1860:
1856:
1853:
1841:
1810:
1790:
1787:
1784:
1758:strong force
1743:
1708:
1669:
1626:
1591:Robert Mills
1576:
1508:Fritz London
1491:
1480:Hermann Weyl
1477:
1437:
1398:
1347:
1327:
1317:
1309:gauge theory
1305:
1301:gauge bosons
1299:
1287:
1283:
1280:vector field
1267:
1261:
1257:
1249:
1247:
1227:field theory
1223:gauge theory
1222:
1216:
1177:
1023:Stueckelberg
763:Jona-Lasinio
353:Vacuum state
338:Quantization
180:Gauge theory
179:
160:Strong force
145:Field theory
73:
64:
45:
9496:topics list
9430:Mathematics
9346:Game theory
9247:Topological
9213:Topological
9208:Statistical
9171:Hamiltonian
8607:Experiments
8498:Technicolor
8460:Dark matter
8354:Quark model
8322:Higgs boson
8317:Gauge boson
8167:Am. J. Phys
7847:Seiberg, N.
7804:"Erratum",
7745:Seiberg, N.
7566:(1): 1â23.
7475:(1): 3â22.
7358:Gluon field
7041:Landau pole
6936:constants.
6787:Wilson loop
6076:Connections
6004:connections
5735:Dirac field
5731:four vector
5351:phase angle
3984:Lagrangian
3781:gauge field
3492:coordinates
3359:gauge group
2603:gauge boson
2595:gauge field
2591:gauge group
2580:Lagrangians
2131:phenomena.
2023:interaction
1990:gauge field
1953:gauge group
1797:Description
1735:topological
1659:triplet of
1645:electroweak
1601:doublet of
1486:to include
1401:gravitation
1390:weak bosons
1368:, with the
1320:Lagrangians
1307:non-abelian
1284:gauge field
1272:Lie algebra
1268:gauge group
1163:Zinn-Justin
1013:Sommerfield
938:Pomeranchuk
908:Osterwalder
903:Oppenheimer
833:ĆopuszaĆski
658:Fredenhagen
459:Technicolor
59:introducing
9511:Categories
9402:Psychology
9366:Statistics
9166:Lagrangian
8793:Statistics
8729:Algorithms
8614:Gran Sasso
8438:Beyond the
8413:CKM matrix
8300:Background
8154:2009-04-23
7921:(3): 663.
7851:Witten, E.
7749:Witten, E.
7667:Phys. Rev.
7450:References
7098:Pure gauge
7064:instantons
6768:Hodge dual
6082:â in each
6050:base space
6020:jet bundle
5973:See also:
5954:See also:
5009:theta term
4693:where the
3468:of the SO(
3466:generators
3460:where the
2972:of fields
2127:and other
2125:turbulence
2116:convolving
1974:derivative
1837:symmetries
1754:weak force
1676:YangâMills
1641:weak force
1392:and eight
1239:Lie groups
1231:Lagrangian
1158:Zimmermann
1053:Vainshtein
798:Kontsevich
743:Iliopoulos
718:Heisenberg
543:Bogoliubov
481:Scientists
333:Propagator
220:T-symmetry
215:P-symmetry
210:C-symmetry
198:Symmetries
155:Weak force
139:Background
42:references
9407:Sociology
9397:Chemistry
9193:Effective
9188:Conformal
9183:Classical
9055:Geometric
9028:Geometric
8245:EMS Press
8205:119652556
8145:Gross, D.
8096:Wiley-VCH
8062:Li, L.-F.
7580:0034-6861
7538:0034-6861
7255:μ
7251:∂
7230:μ
7191:μ
7187:∂
7169:μ
7144:μ
7136:→
7122:μ
7017:Anomalies
6982:When the
6875:character
6842:γ
6838:∫
6808:ρ
6801:χ
6783:invariant
6745:∧
6739:∗
6733:
6727:∫
6685:instanton
6608:ε
6592:ε
6588:δ
6483:ε
6469:ε
6465:δ
6441:ε
6430:ε
6426:δ
6399:⋅
6393:⋅
6367:ε
6359:−
6345:ε
6329:ε
6325:δ
6285:∧
6185:∧
6153:∧
6058:spacetime
6028:BF theory
5937:ν
5934:μ
5924:ν
5921:μ
5904:μ
5891:−
5888:ψ
5867:−
5862:μ
5852:μ
5848:γ
5840:ℏ
5826:¯
5823:ψ
5761:μ
5708:ψ
5703:μ
5699:γ
5683:¯
5680:ψ
5672:ℏ
5650:μ
5611:μ
5592:μ
5570:μ
5553:ψ
5548:μ
5544:γ
5528:¯
5525:ψ
5517:ℏ
5434:μ
5424:ℏ
5413:−
5408:μ
5404:∂
5395:μ
5330:ψ
5325:θ
5314:↦
5311:ψ
5269:ψ
5248:−
5243:μ
5239:∂
5233:μ
5229:γ
5221:ℏ
5207:¯
5204:ψ
5198:∫
5011:, etc.).
4932:ν
4917:μ
4881:∑
4864:μ
4854:ν
4850:∂
4846:−
4836:ν
4826:μ
4822:∂
4808:ν
4805:μ
4741:μ
4709:ν
4706:μ
4671:ν
4668:μ
4658:ν
4655:μ
4634:−
4621:ν
4618:μ
4608:ν
4605:μ
4592:
4576:−
4471:quantized
4402:Φ
4397:μ
4377:Φ
4372:μ
4344:−
4341:Φ
4336:μ
4316:Φ
4311:μ
4307:∂
4284:Φ
4279:μ
4275:∂
4262:μ
4246:Φ
4136:Φ
4109:Φ
4098:Φ
4074:−
4071:Φ
4066:μ
4046:Φ
4041:μ
3945:μ
3931:∑
3922:μ
3883:−
3867:μ
3863:∂
3846:−
3838:−
3828:μ
3809:μ
3758:μ
3744:−
3739:μ
3735:∂
3726:μ
3688:Φ
3683:μ
3661:Φ
3656:μ
3623:Φ
3599:Φ
3571:Φ
3566:μ
3562:∂
3552:≠
3546:Φ
3535:μ
3531:∂
3489:spacetime
3445:Φ
3424:Φ
3418:μ
3414:∂
3397:μ
3338:Φ
3333:μ
3329:∂
3311:Φ
3306:μ
3302:∂
3295:↦
3289:Φ
3284:μ
3280:∂
3246:Φ
3222:Φ
3187:whenever
3172:Φ
3159:Φ
3155:↦
3152:Φ
3123:μ
3103:Φ
3079:Φ
3074:μ
3070:∂
3061:The term
3037:φ
3030:…
3018:φ
3005:φ
2987:Φ
2953:Φ
2942:Φ
2918:−
2915:Φ
2910:μ
2906:∂
2890:Φ
2885:μ
2881:∂
2812:φ
2788:−
2779:φ
2773:μ
2769:∂
2759:φ
2753:μ
2749:∂
2713:∑
2693:∫
2655:φ
2572:classical
2527:∂
2519:∂
2513:−
2507:↦
2490:∇
2479:↦
2434:↦
2399:×
2396:∇
2371:∂
2361:∂
2355:−
2349:∇
2346:−
2297:∇
2294:−
2241:↦
2076:anomalies
2007:curvature
1719:Euclidean
1692:manifolds
1452:electrons
1335:spacetime
1324:invariant
1322:that are
1292:quantized
1248:The term
1243:invariant
1093:Wetterich
1078:Weisskopf
1028:Sudarshan
978:Schwinger
893:Nishijima
858:Maldacena
823:Leutwyler
788:Kinoshita
688:Goldstone
678:Gell-Mann
593:Doplicher
370:Equations
9482:Category
9131:analysis
9050:Exterior
9023:Exterior
8903:Analysis
8865:Discrete
8739:analysis
8658:Category
8639:Tevatron
8491:Theories
8448:Evidence
8312:Fermions
8147:(1992).
8123:Articles
8090:(2008).
8064:(1983).
8043:Springer
7901:17584951
7799:14361074
7596:(1941).
7296:See also
7281:for all
7234:′
7148:′
7060:topology
6681:manifold
6677:homotopy
6673:boundary
6671:without
6669:manifold
5991:of some
5171:electron
5036:a priori
4987:are the
4954:and the
3813:′
3668:′
3602:′
3377:currents
3318:′
3225:′
3193:constant
3162:′
2276:gradient
2153:and the
2019:dynamics
1887:location
1844:inertial
1829:inertial
1756:and the
1607:neutrons
1556:gradient
1536:particle
1502:, Weyl,
1421:graviton
1388:, three
1108:Wightman
1073:Weinberg
1063:Virasoro
1043:Tomonaga
1038:Thirring
1033:Symanzik
993:Semenoff
968:Schrader
933:Polyakov
853:Majorana
793:Klebanov
748:Ivanenko
738:'t Hooft
708:Guralnik
653:Fröhlich
648:Fritzsch
643:Frampton
558:Buchholz
503:Bargmann
493:Anderson
293:Crossing
9492:outline
9423:Related
9392:Biology
9242:Bosonic
9203:Quantum
9153:physics
9119: (
8851:Solvers
8670:Commons
8634:Super-K
8470:problem
8247:, 2001
8185:Bibcode
7953:8285663
7933:Bibcode
7893:1306869
7873:Bibcode
7834:1303306
7814:Bibcode
7791:1293681
7771:Bibcode
7731:0682827
7675:Bibcode
7610:Bibcode
7477:Bibcode
7091:leptons
7072:photons
7070:to two
7024:anomaly
6971:in the
6912:have a
6781:(i.e.,
6666:compact
6456:, then
6103:physics
6068:form a
6016:modules
5989:section
5733:in the
5480:of the
3091:is the
2319:, with
2264:changes
2129:chaotic
1813:physics
1707:exotic
1639:of the
1603:protons
1599:isospin
1597:on the
1587:isospin
1564:Hilbert
1544:Maxwell
1532:charged
1512:complex
1474:History
1464:nuclear
1358:abelian
1333:in the
1266:or the
1219:physics
1118:Wilczek
1083:Wentzel
1058:Veltman
1003:Shirkov
998:Shifman
988:Seiberg
973:Schwarz
953:Rubakov
878:Naimark
828:Lipatov
818:Lehmann
783:Kendall
673:Gelfand
668:Glashow
628:Feynman
608:Faddeev
603:Englert
573:Coleman
563:Cachazo
548:Brodsky
533:Bjorken
523:Berezin
513:Belavin
273:Anomaly
131:History
55:improve
9065:Vector
9060:Tensor
9038:Vector
9033:Tensor
8734:design
8203:
8111:
8076:
8049:
7978:
7951:
7899:
7891:
7832:
7797:
7789:
7729:
7643:
7578:
7536:
7087:quarks
6952:. The
6881:Ï and
6579:Also,
6545:
6518:
6382:where
6307:scalar
6114:2-form
6095:1-form
6048:whose
5962:, and
5737:. The
5637:where
5109:global
4498:bosons
4495:scalar
4479:quanta
4477:, the
4204:
3995:
3914:
3801:
3775:where
3718:
3645:
3527:
3389:
3355:global
3273:
3196:matrix
3149:
2983:
2970:vector
2854:
2644:action
2552:where
2098:bounds
2035:ansatz
2027:energy
1930:fields
1752:, the
1688:smooth
1663:. The
1661:quarks
1618:fields
1615:spinor
1572:action
1394:gluons
1386:photon
1370:photon
1356:is an
1296:quanta
1294:, the
1173:Zumino
1138:Yukawa
1128:Witten
1123:Wilson
1113:Wigner
1048:Tyutin
1008:Skyrme
958:Ruelle
928:Plefka
923:Peskin
913:Parisi
873:MĂžller
863:Migdal
848:Maiani
843:LĂŒders
808:Landau
803:Kuraev
778:KÀllén
768:Jordan
753:Jackiw
693:Gribov
583:DeWitt
578:Dashen
568:Callan
538:Bleuer
508:Becchi
498:Anselm
44:, but
9198:Gauge
8536:NMSSM
8214:arXiv
8201:S2CID
8175:arXiv
8131:arXiv
8031:Texts
7949:S2CID
7923:arXiv
7897:S2CID
7863:arXiv
7795:S2CID
7761:arXiv
7516:arXiv
7043:. In
6664:is a
6054:space
5985:gauge
4789:, by
3477:local
3191:is a
1891:value
1833:group
1696:up to
1657:color
1595:group
1540:Pauli
1530:of a
1516:phase
1496:scale
1382:SU(3)
1378:SU(2)
1331:point
1328:every
1250:gauge
1168:Zuber
1018:Stora
983:Segal
963:Salam
948:Proca
943:Popov
918:Pauli
898:Oehme
888:Neveu
883:Nambu
868:Mills
758:Jaffe
733:Hagen
728:Higgs
703:Gupta
698:Gross
683:Glimm
663:Furry
633:Fierz
623:Fermi
618:Fayet
613:Fadin
598:Dyson
588:Dirac
553:Brout
528:Bethe
488:Adler
267:Tools
8531:MSSM
8109:ISBN
8074:ISBN
8047:ISBN
7976:ISBN
7641:ISBN
7576:ISSN
7534:ISSN
7089:and
7077:The
7068:pion
7054:The
7031:The
6690:The
6177:and
6060:and
5995:. A
5347:U(1)
3202:-by-
2455:but
2193:and
1941:U(1)
1879:= â1
1859:=1,
1835:of "
1827:(an
1725:and
1627:The
1611:U(1)
1605:and
1520:U(1)
1506:and
1466:and
1362:U(1)
1221:, a
1133:Yang
1103:Wick
1098:Weyl
1088:Wess
1068:Ward
773:Jost
723:Hepp
713:Haag
638:Fock
518:Bell
8629:SNO
8624:LHC
8619:INO
8193:doi
7941:doi
7881:doi
7859:431
7822:doi
7810:430
7779:doi
7757:426
7717:doi
7683:doi
7618:doi
7568:doi
7526:doi
7485:doi
6201:. (
6101:in
6056:or
6052:is
5812:QED
5504:int
5368:by
5177:is
5126:int
5075:loc
3516:),
3095:of
2574:or
2214:In
2197:to
2011:not
1980:.)
1966:not
1936:).
1872:= 0
1811:In
1748:of
1717:on
1620:of
1450:of
1274:of
1217:In
1153:Zee
838:Low
813:Lee
9513::
8243:,
8237:,
8199:.
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8171:70
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8072:.
8068:.
8045:.
8041:.
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7970:.
7947:.
7939:.
7931:.
7919:73
7917:.
7895:,
7889:MR
7887:,
7879:,
7871:,
7857:,
7849:;
7830:MR
7828:,
7820:,
7808:,
7802:;
7793:,
7787:MR
7785:,
7777:,
7769:,
7755:,
7747:;
7727:MR
7725:.
7706:.
7681:.
7671:96
7664:.
7616:.
7606:13
7604:.
7600:.
7574:.
7564:72
7562:.
7558:.
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7506:.
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7473:33
7471:.
7467:.
7292:.
6960:.
6774:.
6730:Tr
5958:,
5792:.
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5092:gf
5007:,
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4589:tr
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4172:.
3508:=
3497:.
3479:O(
3209:O(
3135:.
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2201:.
2054:.
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1624:.
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1462:,
1396:.
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8216::
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7288:x
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7263:(
7260:f
7247:=
7244:)
7241:x
7238:(
7226:A
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7183:+
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7093:.
7074:.
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6891:P
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6852:}
6847:A
6833:e
6829:{
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6817:(
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6805:(
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6748:F
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6736:[
6719:2
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6707:1
6640:F
6619:]
6615:F
6611:,
6605:[
6602:=
6598:F
6564:X
6560:A
6556:+
6553:X
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6531:d
6525:=
6515:X
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6486:D
6480:=
6477:X
6474:D
6444:X
6438:=
6435:X
6402:]
6396:,
6390:[
6363:d
6356:]
6352:A
6348:,
6342:[
6339:=
6335:A
6289:A
6281:A
6259:A
6236:a
6232:T
6210:A
6157:A
6149:A
6145:+
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6136:d
6132:=
6128:F
6110:F
6091:A
6066:P
6046:P
5930:F
5917:F
5908:0
5900:4
5896:1
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5878:2
5874:c
5870:m
5858:D
5843:c
5837:i
5833:(
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5806:L
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5664:=
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5584:=
5581:)
5578:x
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5556:(
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5509:=
5498:L
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5468:A
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5421:e
5416:i
5400:=
5391:D
5377:)
5375:x
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5371:Ξ
5365:Ξ
5356:Ξ
5322:i
5318:e
5285:x
5280:4
5275:d
5265:)
5259:2
5255:c
5251:m
5224:c
5218:i
5214:(
5195:=
5190:S
5137:L
5131:+
5120:L
5114:+
5103:L
5097:=
5086:L
5080:+
5069:L
5063:=
5058:L
5022:A
4973:c
4970:b
4967:a
4963:f
4937:c
4928:A
4922:b
4913:A
4907:c
4904:b
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4891:c
4888:,
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4874:+
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4818:=
4813:a
4801:F
4777:)
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4771:(
4768:A
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4737:A
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4676:a
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4562:L
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4532:x
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4526:A
4516:D
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4451:g
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4364:(
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4320:)
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4298:2
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4251:T
4240:2
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4232:i
4229:=
4223:t
4220:n
4217:i
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4156:A
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4088:m
4082:2
4079:1
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4020:=
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4008:l
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3927:=
3918:A
3886:1
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3805:A
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3786:(
3784:A
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3731:=
3722:D
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3652:D
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3407:=
3402:a
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3322:=
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3298:(
3292:)
3276:(
3211:n
3204:n
3200:n
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3041:n
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3027:,
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2998:=
2992:T
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2877:(
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2743:2
2740:1
2734:[
2728:n
2723:1
2720:=
2717:i
2709:x
2704:4
2699:d
2690:=
2685:S
2659:i
2640:m
2622:n
2605:.
2582:.
2562:n
2554:f
2530:t
2522:f
2510:V
2500:V
2493:f
2487:+
2483:A
2471:A
2443:C
2440:+
2437:V
2431:V
2403:A
2393:=
2385:B
2374:t
2365:A
2352:V
2343:=
2335:E
2317:A
2300:V
2291:=
2287:E
2268:C
2250:C
2247:+
2244:V
2238:V
2228:V
2220:E
1903:P
1899:P
1883:y
1877:y
1870:x
1865:x
1861:y
1857:x
1711:s
1709:R
1206:e
1199:t
1192:v
103:.
96:.
80:)
74:(
69:)
65:(
51:.
20:)
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