3352:, since motions of electric charge or magnetic field appear everywhere instantaneously as changes to the potentials. This is justified by noting that the scalar and vector potentials themselves do not affect the motions of charges, only the combinations of their derivatives that form the electromagnetic field strength. Although one can compute the field strengths explicitly in the Coulomb gauge and demonstrate that changes in them propagate at the speed of light, it is much simpler to observe that the field strengths are unchanged under gauge transformations and to demonstrate causality in the manifestly Lorentz covariant Lorenz gauge described below.
4407:
no effect on any physical observable. A quantity or expression that does not depend on the gauge function is said to be gauge invariant: All physical observables are required to be gauge invariant. A gauge transformation from the
Coulomb gauge to another gauge is made by taking the gauge function to be the sum of a specific function which will give the desired gauge transformation and the arbitrary function. If the arbitrary function is then set to zero, the gauge is said to be fixed. Calculations may be carried out in a fixed gauge but must be done in a way that is gauge invariant.
865:
7373:
47:
2572:
4248:
3786:
2411:
4389:
5239:
waves in the four-potential. The transverse polarizations correspond to classical radiation, i.e., transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one
4406:
is an arbitrary scalar field called the gauge function. The fields that are the derivatives of the gauge function are known as pure gauge fields and the arbitrariness associated with the gauge function is known as gauge freedom. In a calculation that is carried out correctly the pure gauge terms have
3904:
at infinity is a consequence of the unphysical assumption that the magnetic field is uniform throughout the whole of space. Although this vector potential is unrealistic in general it can provide a good approximation to the potential in a finite volume of space in which the magnetic field is uniform.
2845:
As an illustration of gauge fixing, one may look at a cylindrical rod and attempt to tell whether it is twisted. If the rod is perfectly cylindrical, then the circular symmetry of the cross section makes it impossible to tell whether or not it is twisted. However, if there were a straight line drawn
2739:
Although classical electromagnetism is now often spoken of as a gauge theory, it was not originally conceived in these terms. The motion of a classical point charge is affected only by the electric and magnetic field strengths at that point, and the potentials can be treated as a mere mathematical
5768:). For this reason, and because their appearance in spin sums can be seen as a mere mathematical device in QED (much like the electromagnetic four-potential in classical electrodynamics), they are often spoken of as "unphysical". But unlike the constraint-based gauge fixing procedures above, the
3088:
2030:
Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical
2225:
2740:
device for simplifying some proofs and calculations. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the
4090:
3613:
3202:
4836:
4943:
3574:
3801:
The
Coulomb gauge admits a natural Hamiltonian formulation of the evolution equations of the electromagnetic field interacting with a conserved current, which is an advantage for the quantization of the theory. The Coulomb gauge is, however, not Lorentz covariant. If a
3806:
to a new inertial frame is carried out, a further gauge transformation has to be made to retain the
Coulomb gauge condition. Because of this, the Coulomb gauge is not used in covariant perturbation theory, which has become standard for the treatment of relativistic
4252:
6584:
2567:{\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial {\mathbf {A} }}{\partial t}}-\nabla {\frac {\partial {\psi }}{\partial t}}=-\nabla \left(\varphi +{\frac {\partial {\psi }}{\partial t}}\right)-{\frac {\partial {\mathbf {A} }}{\partial t}}.}
6683:
2400:
4507:
4580:
5446:
3894:
2136:
5596:
2969:
2026:
along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
2630:
5032:
2035:
detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to
4735:
5657:
in which infinitesimal perturbations along "physical" directions in configuration space are entirely uncoupled from those along "unphysical" directions, allowing the latter to be absorbed into the physically meaningless
4840:
3815:(QED). Lorentz covariant gauges such as the Lorenz gauge are usually used in these theories. Amplitudes of physical processes in QED in the noncovariant Coulomb gauge coincide with those in the covariant Lorenz gauge.
5111:
3092:
4023:
5704:. An expansion of this factor as a sum over photon polarizations involves terms containing all four possible polarizations. Transversely polarized radiation can be expressed mathematically as a sum over either a
2793:
3374:
2102:
degree of freedom in an electromagnetic field configuration has a separately measurable effect on the motions of test charges in the vicinity. These "field strength" variables can be expressed in terms of the
2271:
6016:
5666:. When Ο is finite, each physical configuration (orbit of the group of gauge transformations) is represented not by a single solution of a constraint equation but by a Gaussian distribution centered on the
2945:
6340:
3260:
5229:
5791:
of the embedding of gauge freedom axes within the space of detailed configurations. This leads to the explicit appearance of forward and backward polarized gauge bosons in
Feynman diagrams, along with
5787:. The couplings between physical and unphysical perturbation axes do not entirely disappear under the corresponding change of variables; to obtain correct results, one must account for the non-trivial
4243:{\displaystyle \varphi (\mathbf {r} ,t)=\int {\frac {\nabla '\cdot {\mathbf {E} }(\mathbf {r} ',t)}{4\pi R}}\operatorname {d} \!^{3}\mathbf {r} '-{\frac {\partial {\psi (\mathbf {r} ,t)}}{\partial t}}}
6486:
5727:. Spin sums can be very helpful both in simplifying expressions and in obtaining a physical understanding of the experimental effects associated with different terms in a theoretical calculation.
3781:{\displaystyle \int \mathbf {A} ^{2}(\mathbf {r} ,t)d^{3}\mathbf {r} =\iint {\frac {\mathbf {B} (\mathbf {r} ,t)\cdot \mathbf {B} (\mathbf {r} ',t)}{4\pi R}}d^{3}\mathbf {r} \,d^{3}\mathbf {r} '.}
5733:
used arguments along approximately these lines largely to justify calculation procedures that produced consistent, finite, high precision results for important observable parameters such as the
6479:
6588:
4082:
3603:, no gauge arbitrariness remains. Because of this, the Coulomb gauge is said to be a complete gauge, in contrast to gauges where some gauge arbitrariness remains, like the Lorenz gauge below.
2052:
and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of
2331:
4436:
4954:
in some sense: there remains a subspace of gauge transformations which can also preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the
2326:
5288:
4625:
4516:
3296:
5369:
3954:
6819:
3829:
6281:
5953:
6749:
4384:{\displaystyle \mathbf {A} (\mathbf {r} ,t)=\nabla \times \int {\frac {\mathbf {B} (\mathbf {r} ',t)}{4\pi R}}\operatorname {d} \!^{3}\mathbf {r} '+\nabla \psi (\mathbf {r} ,t)}
5524:
2862:. In summary, to tell whether the rod is twisted, the gauge must be known. Physical quantities, such as the energy of the torsion, do not depend on the gauge, i.e., they are
5601:
5353:
5169:
4691:
6401:
2584:
4960:
5824:
2124:
3324:
5468: â 0 but postpones taking that limit until after the theory has been quantized. It improves the rigor of certain existence and equivalence proofs. Most
4087:
As a consequence of the considerations above, the electromagnetic potentials may be expressed in their most general forms in terms of the electromagnetic fields as
2830:
2022:
of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a certain transformation, equivalent to a
5046:
604:
5737:
of the electron. Although his arguments sometimes lacked mathematical rigor even by physicists' standards and glossed over details such as the derivation of
3959:
5799:. The relationship between these entities, and the reasons why they do not appear as particles in the quantum mechanical sense, becomes more evident in the
5296:
can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances.
5712:
basis. Similarly, one can combine the longitudinal and time-like gauge polarizations to obtain "forward" and "backward" polarizations; these are a form of
5646:
2755:
5761:
5675:
4424:
577:
5609:
2237:
5958:
2893:
2744:, which has no classical counterpart. Nevertheless, gauge freedom is still true in these theories. For example, the AharonovâBohm effect depends on a
7148:
5036:
These remaining gauge degrees of freedom propagate at the speed of light. To obtain a fully fixed gauge, one must add boundary conditions along the
1114:
589:
6286:
4947:
It can be seen from these equations that, in the absence of current and charge, the solutions are potentials which propagate at the speed of light.
2220:{\displaystyle {\mathbf {E} }=-\nabla \varphi -{\frac {\partial {\mathbf {A} }}{\partial t}}\,,\quad {\mathbf {B} }=\nabla \times {\mathbf {A} }.}
5604:, can be eliminated by "completing the square" to obtain the previous form. From a mathematical perspective the auxiliary field is a variety of
5181:
3083:{\displaystyle \varphi (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\mathbf {\rho } (\mathbf {r} ',t)}{R}}d^{3}\mathbf {r} '}
1981:
2846:
along the length of the rod, then one could easily say whether or not there is a twist by looking at the state of the line. Drawing a line is
3338:
4831:{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}-\nabla ^{2}{\varphi }={\frac {\rho }{\varepsilon _{0}}}}
1034:
6448:
5724:
2011:
840:
609:
6024:
gauge theory in D dimensions, the maximal abelian subgroup is a U(1)ĂU(1) subgroup. If this is chosen to be the one generated by the
1074:
4938:{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}-\nabla ^{2}{\mathbf {A} }=\mu _{0}\mathbf {J} }
2879:
3610:
over all space is minimal for this gauge: All other gauges give a larger integral. The minimum value given by the
Coulomb gauge is
619:
5250:
4587:
3197:{\displaystyle \mathbf {A} (\mathbf {r} ,t)=\nabla \times \int {\frac {\mathbf {B} (\mathbf {r} ',t)}{4\pi R}}d^{3}\mathbf {r} '}
6345:
This applies regularly in higher algebras (of groups in the algebras), for example the
Clifford Algebra and as it is regularly.
5835:
gauge theory in D dimensions, the maximal abelian subgroup is a U(1) subgroup. If this is chosen to be the one generated by the
444:
7141:
7098:
3222:
459:
454:
81:
6774:
3569:{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\,\nabla \times \int \leftd^{3}\mathbf {r} '.}
469:
1839:
6044:
5848:
4725:; it is often misspelled "Lorentz gauge". (Neither was the first to use it in calculations; it was introduced in 1888 by
6711:
3578:
Further gauge transformations that retain the
Coulomb gauge condition might be made with gauge functions that satisfy
1974:
5178:
Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation
4028:
3896:
plus the gradient of any scalar field (the gauge function), which can be confirmed by calculating the div and curl of
7376:
7117:
71:
6483:
This is another gauge in which the potentials can be expressed in a simple way in terms of the instantaneous fields
339:
7408:
7248:
7134:
3588:, but as the only solution to this equation that vanishes at infinity (where all fields are required to vanish) is
2949:
It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is
1604:
2296:
7171:
5788:
3265:
877:
254:
6579:{\displaystyle \mathbf {A} (\mathbf {r} ,t)=-\mathbf {r} \times \int _{0}^{1}\mathbf {B} (u\mathbf {r} ,t)u\,du}
6380:
2018:
variables. By definition, a gauge theory represents each physically distinct configuration of the system as an
7062:
4694:
2077:
833:
599:
76:
6940:
Gubarev, F. V.; Stodolsky, L.; Zakharov, V. I. (2001). "On the
Significance of the Vector Potential Squared".
3919:
7403:
2962:
1967:
1152:
991:
951:
614:
319:
17:
1709:
1634:
981:
479:
219:
86:
209:
7298:
7181:
6887:
6839:
6678:{\displaystyle \varphi (\mathbf {r} ,t)=-\mathbf {r} \cdot \int _{0}^{1}\mathbf {E} (u\mathbf {r} ,t)du.}
5796:
5765:
5738:
1147:
1084:
772:
647:
544:
519:
439:
7002:
Adkins, Gregory S. (1987-09-15). "Feynman rules of
Coulomb-gauge QED and the electron magnetic moment".
2395:{\displaystyle {\mathbf {B} }=\nabla \times ({\mathbf {A} }+\nabla \psi )=\nabla \times {\mathbf {A} }.}
4502:{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0}
986:
272:
6353:
Various other gauges, which can be beneficial in specific situations have appeared in the literature.
1819:
7398:
7319:
7258:
7186:
5734:
5689:
The photon propagator, which is the multiplicative factor corresponding to an internal photon in the
5232:
4726:
3795:
2127:
1137:
1079:
826:
787:
314:
304:
244:
239:
179:
5334:
5116:
4630:
4575:{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c}}{\frac {\partial \varphi }{\partial t}}=0.}
3355:
Another expression for the vector potential, in terms of the time-retarded electric current density
5776:
2887:
2798:
2741:
1639:
1264:
1157:
1094:
1054:
324:
1629:
757:
259:
7413:
7212:
7157:
6942:
5792:
5623:
5441:{\displaystyle \delta {\mathcal {L}}=-{\frac {\left(\partial _{\mu }A^{\mu }\right)^{2}}{2\xi }}}
5293:
5235:, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like"
3812:
2961:
The potentials can be expressed in terms of instantaneous values of the fields and densities (in
2814:
2023:
1919:
1379:
1187:
1182:
1024:
637:
164:
154:
149:
4084:
are unitary vectors of the
Cartesian coordinate system (z-axis aligned with the magnetic field).
762:
732:
5784:
5754:
5713:
4418:
3808:
3803:
2950:
2095:
2045:
1789:
1769:
1259:
1192:
1089:
1049:
584:
354:
129:
7334:
2109:
3889:{\displaystyle {\mathbf {A} }(\mathbf {r} ,t)=-{\frac {1}{2}}\mathbf {r} \times \mathbf {B} }
3309:
2802:
1674:
1529:
1225:
1162:
1069:
931:
682:
369:
359:
309:
299:
3790:
In regions far from electric charge the scalar potential becomes zero. This is known as the
7011:
6960:
6906:
6848:
6756:
5709:
5683:
5659:
5469:
5236:
2858:; it need not be straight. Almost any line is a valid gauge fixing, i.e., there is a large
2069:
2053:
2049:
2037:
1924:
1699:
1230:
1064:
856:
807:
707:
672:
424:
289:
189:
174:
109:
7344:
6885:
Jackson, J. D. (2002). "From Lorenz to Coulomb and other explicit gauge transformations".
976:
46:
8:
7288:
5723:
factor in terms of circularly polarized (spin ±1) and light-cone coordinates is called a
5705:
5663:
5654:
5324:
are a generalization of the Lorenz gauge applicable to theories expressed in terms of an
5245:
4732:
The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:
1934:
1254:
767:
747:
742:
549:
534:
419:
389:
284:
214:
7015:
6964:
6910:
6852:
5591:{\displaystyle \delta {\mathcal {L}}=B\,\partial _{\mu }A^{\mu }+{\frac {\xi }{2}}B^{2}}
1644:
1369:
7354:
7329:
7253:
6984:
6950:
6922:
6896:
6864:
6414:
6025:
5750:
5631:
5329:
4714:
2806:
2736:
of this theory. Gauge fixing can be done in many ways, some of which we exhibit below.
2104:
1944:
1869:
1799:
1679:
1664:
1594:
1554:
1544:
1504:
1419:
1309:
1269:
1039:
1001:
956:
944:
921:
916:
642:
382:
184:
144:
464:
7314:
7283:
7113:
7094:
7058:
7035:
7027:
6976:
6926:
6868:
6860:
5231:
In this form it is clear that the components of the potential separately satisfy the
2883:
2019:
1899:
1829:
1714:
1704:
1549:
1484:
1454:
1424:
1284:
1279:
1109:
1029:
1019:
911:
702:
6988:
6442:
1614:
7349:
7232:
7019:
6968:
6914:
6856:
6701:
5746:
5701:
5492:
5325:
2810:
2625:{\displaystyle \varphi \rightarrow \varphi -{\frac {\partial {\psi }}{\partial t}}}
2065:
1939:
1854:
1834:
1804:
1754:
1574:
1564:
1559:
1444:
1429:
1359:
1044:
971:
961:
896:
802:
717:
677:
667:
554:
509:
492:
409:
344:
114:
38:
5027:{\displaystyle {\frac {\partial ^{2}\psi }{\partial t^{2}}}=c^{2}\nabla ^{2}\psi }
7293:
7278:
7176:
7086:
6410:
5730:
5690:
5605:
5514:
5241:
4722:
4717:. Note, however, that this gauge was originally named after the Danish physicist
3348:
The instantaneous nature of these potentials appears, at first sight, to violate
2073:
2041:
2015:
1884:
1879:
1849:
1844:
1814:
1774:
1744:
1724:
1624:
1514:
1509:
1474:
1459:
1434:
1414:
1394:
1324:
1314:
1235:
1197:
1099:
1059:
891:
869:
864:
737:
662:
657:
524:
399:
364:
224:
124:
6972:
777:
27:
Procedure of coping with redundant degrees of freedom in physical field theories
7324:
5800:
5679:
5627:
4510:
2890:
and is defined by the gauge condition (more precisely, gauge fixing condition)
2854:
of the cross section at each point of the rod. The line is the equivalent of a
2088:
2081:
1909:
1859:
1779:
1764:
1749:
1729:
1694:
1689:
1684:
1669:
1654:
1619:
1609:
1534:
1469:
1439:
1374:
1339:
1329:
1304:
1175:
1132:
996:
697:
692:
514:
404:
329:
279:
229:
202:
159:
134:
104:
97:
1659:
7392:
7031:
7023:
6697:
5742:
5671:
4955:
4718:
2745:
1914:
1904:
1894:
1889:
1809:
1794:
1589:
1584:
1409:
1404:
1364:
1334:
1299:
1289:
1220:
1215:
812:
797:
782:
722:
434:
349:
334:
249:
234:
139:
3822:
the vector potential in the Coulomb gauge can be expressed in the so-called
6980:
6406:
5836:
5812:
5745:
soon demonstrated that his method was substantially equivalent to those of
5172:
2818:
2080:, which is presented here in space/time asymmetric Heaviside notation. The
1999:
1949:
1874:
1759:
1734:
1719:
1649:
1569:
1524:
1519:
1399:
1389:
1384:
1349:
1344:
1319:
1274:
1142:
1104:
926:
906:
792:
687:
652:
594:
529:
449:
414:
294:
169:
7126:
7039:
2850:. Drawing the line spoils the gauge symmetry, i.e., the circular symmetry
7273:
7263:
7227:
7202:
5356:
5106:{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=\mu _{0}j^{\nu }}
1929:
1864:
1824:
1784:
1739:
1599:
1499:
1494:
1479:
1464:
1449:
712:
564:
394:
56:
6901:
7339:
7268:
7082:
6955:
6768:
5464:
is classically equivalent to Lorenz gauge: it is obtained in the limit
5037:
4707:
4018:{\displaystyle \mathbf {A} =B(\mathbf {r} \cdot {\hat {x}}){\hat {y}},}
3606:
The Coulomb gauge is a minimal gauge in the sense that the integral of
1579:
1489:
1354:
1294:
966:
901:
429:
6918:
6041:, then the maximal abelian gauge is that which maximizes the function
5845:, then the maximal abelian gauge is that which maximizes the function
3349:
752:
727:
539:
61:
2788:{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +\nabla \psi \,.}
2098:
contain only "physical" degrees of freedom, in the sense that every
7222:
7207:
5667:
1539:
504:
499:
119:
5674:
of the gauge-fixed theory, this appears as a contribution to the
2266:{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +\nabla \psi }
1995:
474:
6837:
Stewart, A. M. (2003). "Vector potential of the Coulomb gauge".
6417:, and requires longitudinal photons and a constraint on states.
6011:{\displaystyle {\mathbf {A} }_{\mu }=A_{\mu }^{a}\sigma _{a}\,.}
2940:{\displaystyle \nabla \cdot {\mathbf {A} }(\mathbf {r} ,t)=0\,.}
7217:
5760:
Forward and backward polarized radiation can be omitted in the
4713:
It is unique among the constraint gauges in retaining manifest
3306:′ is a point in the charge or current distribution), the
559:
66:
6335:{\displaystyle {\mathbf {A} }_{\mu }=A_{\mu }^{a}\lambda _{a}}
6021:
5832:
5780:
3905:
Another common choice for homogeneous constant fields is the
2698:
A particular choice of the scalar and vector potentials is a
2010:) denotes a mathematical procedure for coping with redundant
5795:, which are even more "unphysical" in that they violate the
5612:
of the theory, and especially when generalizing beyond QED.
5224:{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=0.}
2851:
2752:
around a closed loop, and this integral is not changed by
2730:
5716:
in which the metric is off-diagonal. An expansion of the
2044:, especially when the computation is continued to higher
6939:
5622:
gauges was a significant technical advance in extending
2829:
2714:. The existence of arbitrary numbers of gauge functions
5244:. Classically, these identities are equivalent to the
4430:
3255:{\displaystyle \mathbf {R} =\mathbf {r} -\mathbf {r} '}
4654:
2809:, is a rather more complicated topic; for details see
6777:
6714:
6591:
6489:
6451:
6383:
6289:
6047:
5961:
5851:
5527:
5372:
5337:
5253:
5184:
5119:
5049:
5043:
Maxwell's equations in the Lorenz gauge simplify to
4963:
4843:
4738:
4633:
4590:
4519:
4439:
4255:
4093:
4031:
3962:
3922:
3832:
3616:
3377:
3312:
3268:
3225:
3095:
2972:
2896:
2758:
2587:
2414:
2334:
2299:
2240:
2139:
2112:
5741:
of the quantum theory, his calculations worked, and
5366:
term to the "physical" (gauge invariant) Lagrangian
7055:
Quantum field theory of point particles and strings
5823:gauge which fixes the gauge freedom outside of the
5641:prescription breaks the symmetry under local gauge
4425:
Covariant formulation of classical electromagnetism
6813:
6743:
6677:
6578:
6473:
6395:
6334:
6275:
6010:
5947:
5693:expansion of a QED calculation, contains a factor
5608:, and its use has advantages when identifying the
5590:
5440:
5355:. Instead of fixing the gauge by constraining the
5347:
5282:
5223:
5163:
5105:
5026:
4937:
4830:
4685:
4619:
4574:
4501:
4383:
4242:
4076:
4017:
3948:
3888:
3780:
3568:
3318:
3290:
3254:
3196:
3082:
2939:
2787:
2624:
2566:
2394:
2320:
2265:
2219:
2118:
7010:(6). American Physical Society (APS): 1929â1932.
6767:The nonlinear Dirac gauge condition (named after
6474:{\displaystyle \mathbf {r} \cdot \mathbf {A} =0.}
4338:
4181:
2056:from the late nineteenth century to the present.
7390:
5240:must also employ auxiliary constraints known as
4077:{\displaystyle {\hat {x}},{\hat {y}},{\hat {z}}}
2660:fields are unchanged if one takes any function
7081:
5362:, via an auxiliary equation, one adds a gauge
5292:Many of the differences between classical and
2956:The Coulomb gauge has a number of properties:
7142:
7093:. Amsterdam: Elsevier Butterworth Heinemann.
6880:
6878:
1975:
834:
6348:
5670:of the gauge breaking term. In terms of the
2321:{\displaystyle \nabla \times \nabla \psi =0}
2293:remains unchanged, since (with the identity
7156:
5283:{\displaystyle \partial _{\mu }j^{\mu }=0.}
4620:{\displaystyle \partial _{\mu }A^{\mu }=0.}
3291:{\displaystyle R=\left|\mathbf {R} \right|}
7149:
7135:
6875:
6830:
3818:For a uniform and constant magnetic field
1982:
1968:
863:
841:
827:
45:
6954:
6900:
6569:
6269:
6004:
5941:
5544:
5155:
5149:
5142:
5138:
4677:
4671:
4652:
3916:Landau gauge of the next section), where
3754:
3531:
3469:
3428:
2933:
2781:
2186:
2040:is fraught with complications related to
7052:
5806:
5600:The auxiliary field, sometimes called a
3949:{\displaystyle \mathbf {B} =B{\hat {z}}}
2828:
7107:
6884:
6836:
6687:
590:Electromagnetism and special relativity
14:
7391:
7001:
6814:{\displaystyle A_{\mu }A^{\mu }=k^{2}}
7130:
5649:of any two physically distinct gauge
2710:used to change the gauge is called a
2404:However, this transformation changes
2064:The archetypical gauge theory is the
610:Maxwell equations in curved spacetime
7057:. Addison-Wesley. pp. 210â213.
5753:, with whom Feynman shared the 1965
5630:. In addition to retaining manifest
5483:; a few are more tractable in other
5454:determines the choice of gauge. The
3302:is any position vector in space and
2953:but the Coulomb interaction is not.
2837:cylinder. (Note: the line is on the
2578:
2231:
6420:
6409:. It eliminates the negative-norm
6276:{\displaystyle \int d^{D}x\left\,,}
5948:{\displaystyle \int d^{D}x\left\,,}
24:
7075:
6744:{\displaystyle x^{\mu }A_{\mu }=0}
5546:
5533:
5398:
5378:
5340:
5255:
5196:
5186:
5061:
5051:
5012:
4982:
4968:
4901:
4881:
4865:
4794:
4774:
4760:
4592:
4557:
4549:
4520:
4484:
4476:
4440:
4358:
4334:
4281:
4231:
4204:
4177:
4124:
3798:was first quantized in this gauge.
3429:
3313:
3121:
2897:
2824:
2775:
2652:also remains the same. Hence, the
2613:
2603:
2552:
2540:
2520:
2510:
2493:
2478:
2468:
2462:
2450:
2438:
2426:
2376:
2364:
2345:
2306:
2300:
2257:
2201:
2177:
2165:
2153:
25:
7425:
7377:Template:Quantum mechanics topics
7112:(3rd ed.). New York: Wiley.
5472:computations are simplest in the
7372:
7371:
7249:Anomalous magnetic dipole moment
6653:
6642:
6619:
6599:
6553:
6542:
6519:
6499:
6491:
6461:
6453:
6293:
5965:
5151:
4931:
4912:
4875:
4673:
4528:
4448:
4368:
4347:
4303:
4294:
4265:
4257:
4215:
4190:
4146:
4136:
4101:
3978:
3964:
3924:
3882:
3874:
3844:
3835:
3767:
3750:
3708:
3699:
3682:
3674:
3660:
3636:
3622:
3555:
3513:
3484:
3475:
3387:
3379:
3280:
3244:
3235:
3227:
3219:is the electric charge density,
3186:
3143:
3134:
3105:
3097:
3072:
3037:
2980:
2914:
2905:
2869:
2841:of the cylinder, not inside it.)
2768:
2760:
2545:
2443:
2416:
2384:
2356:
2337:
2250:
2242:
2209:
2193:
2170:
2142:
2059:
2048:. Historically, the search for
5764:of a quantum field theory (see
4412:
2190:
7091:The classical theory of fields
7046:
6995:
6933:
6762:
6663:
6646:
6609:
6595:
6563:
6546:
6509:
6495:
5645:while preserving the ratio of
5521:with no independent dynamics:
5348:{\displaystyle {\mathcal {L}}}
5164:{\displaystyle j^{\nu }=\left}
4695:electromagnetic four-potential
4686:{\displaystyle A^{\mu }=\left}
4429:The Lorenz gauge is given, in
4378:
4364:
4317:
4298:
4275:
4261:
4225:
4211:
4160:
4141:
4111:
4097:
4068:
4053:
4038:
4006:
3997:
3991:
3974:
3940:
3854:
3840:
3722:
3703:
3692:
3678:
3646:
3632:
3504:
3479:
3397:
3383:
3157:
3138:
3115:
3101:
3051:
3032:
2990:
2976:
2924:
2910:
2764:
2675:and simultaneously transforms
2591:
2370:
2351:
2246:
2078:electromagnetic four-potential
1163:Renormalization group equation
13:
1:
6823:
6377:gauge obtained by the choice
6356:
5506:An equivalent formulation of
3909:(not to be confused with the
2963:International System of Units
992:Spontaneous symmetry breaking
952:Symmetry in quantum mechanics
615:Relativistic electromagnetism
6704:; sometimes also called the
5450:The choice of the parameter
5299:
5040:of the experimental region.
7:
7172:EulerâHeisenberg Lagrangian
6973:10.1103/PhysRevLett.86.2220
6888:American Journal of Physics
6840:European Journal of Physics
6706:relativistic Poincaré gauge
6692:The gauge condition of the
6425:The gauge condition of the
3370:, has been obtained to be:
2691:
2685:
2638:
2279:
10:
7430:
6861:10.1088/0143-0807/24/5/308
6396:{\displaystyle \varphi =0}
5775:gauge generalizes well to
4584:This may be rewritten as:
4422:
4416:
987:Explicit symmetry breaking
340:LiĂ©nardâWiechert potential
7363:
7307:
7241:
7195:
7187:Path integral formulation
7164:
7110:Classical Electrodynamics
6349:Less commonly used gauges
5779:gauge groups such as the
5739:WardâTakahashi identities
5735:anomalous magnetic moment
5615:Historically, the use of
4727:George Francis FitzGerald
3796:Electromagnetic radiation
2683:via the transformations (
2128:magnetic vector potential
2105:electric scalar potential
2072:formulation of continuum
1153:BargmannâWigner equations
1080:Path Integral Formulation
605:Mathematical descriptions
315:Electromagnetic radiation
305:Electromagnetic induction
245:Magnetic vector potential
240:Magnetic scalar potential
7355:Photon-photon scattering
7053:Hatfield, Brian (1992).
7024:10.1103/physrevd.36.1929
5825:maximal abelian subgroup
5678:for internal lines from
2888:condensed matter physics
2801:gauge theories, such as
2706:) and a scalar function
2119:{\displaystyle \varphi }
1158:Schwinger-Dyson equation
7409:Quantum electrodynamics
7299:WardâTakahashi identity
7182:GuptaâBleuler formalism
7158:Quantum electrodynamics
7108:Jackson, J. D. (1999).
5797:spinâstatistics theorem
5766:WardâTakahashi identity
5624:quantum electrodynamics
5602:NakanishiâLautrup field
5294:quantum electrodynamics
3813:quantum electrodynamics
3319:{\displaystyle \nabla }
2133:through the relations:
1188:Electroweak interaction
1183:Quantum electrodynamics
1148:WheelerâDeWitt equation
1025:Background field method
155:Electrostatic induction
150:Electrostatic discharge
6815:
6745:
6679:
6580:
6475:
6397:
6336:
6277:
6012:
5949:
5755:Nobel Prize in Physics
5714:light-cone coordinates
5592:
5474:Feynmanâ't Hooft gauge
5442:
5349:
5284:
5225:
5165:
5107:
5028:
4939:
4832:
4687:
4621:
4576:
4503:
4419:Lorenz gauge condition
4385:
4244:
4078:
4019:
3950:
3890:
3809:quantum field theories
3804:Lorentz transformation
3782:
3570:
3320:
3292:
3256:
3198:
3084:
2941:
2842:
2789:
2626:
2568:
2396:
2322:
2267:
2229:If the transformation
2221:
2120:
1193:Quantum chromodynamics
1050:Effective field theory
585:Electromagnetic tensor
7320:BreitâWheeler process
7259:KleinâNishina formula
6816:
6746:
6680:
6581:
6476:
6398:
6337:
6278:
6013:
5950:
5817:maximal abelian gauge
5807:Maximal abelian gauge
5700:corresponding to the
5593:
5443:
5350:
5285:
5233:KleinâGordon equation
5226:
5166:
5108:
5029:
4940:
4833:
4688:
4622:
4577:
4504:
4386:
4245:
4079:
4020:
3951:
3891:
3783:
3571:
3321:
3293:
3257:
3199:
3085:
2942:
2832:
2790:
2627:
2569:
2397:
2323:
2268:
2222:
2121:
1138:KleinâGordon equation
1070:LSZ reduction formula
578:Covariant formulation
370:Synchrotron radiation
310:Electromagnetic pulse
300:Electromagnetic field
7404:Quantum field theory
6775:
6757:position four-vector
6712:
6694:FockâSchwinger gauge
6688:FockâSchwinger gauge
6589:
6487:
6449:
6381:
6287:
6045:
5959:
5849:
5813:abelian gauge theory
5793:FaddeevâPopov ghosts
5710:circularly polarized
5626:computations beyond
5525:
5490:gauges, such as the
5470:quantum field theory
5370:
5335:
5251:
5182:
5117:
5047:
4961:
4950:The Lorenz gauge is
4841:
4736:
4631:
4588:
4517:
4437:
4253:
4091:
4029:
3960:
3920:
3900:. The divergence of
3830:
3614:
3375:
3310:
3266:
3223:
3093:
2970:
2894:
2756:
2742:AharonovâBohm effect
2585:
2412:
2332:
2297:
2238:
2137:
2110:
2054:mathematical physics
2050:logically consistent
2038:quantum field theory
1231:Theory of everything
1065:Lattice field theory
1035:Correlation function
857:Quantum field theory
620:Stressâenergy tensor
545:Reluctance (complex)
290:Displacement current
7335:DelbrĂŒck scattering
7289:Vacuum polarization
7213:FaddeevâPopov ghost
7016:1987PhRvD..36.1929A
6965:2001PhRvL..86.2220G
6911:2002AmJPh..70..917J
6853:2003EJPh...24..519S
6640:
6540:
6429:(also known as the
6365:(also known as the
6321:
6253:
6220:
6187:
6154:
6121:
6088:
5993:
5925:
5892:
5664:functional integral
5655:change of variables
5647:functional measures
5246:continuity equation
3465:
2878:(also known as the
2815:FaddeevâPopov ghost
2729:corresponds to the
2096:Maxwell's equations
2031:configuration by a
1210:Incomplete theories
535:Magnetomotive force
420:Electromotive force
390:Alternating current
325:Jefimenko equations
285:Cyclotron radiation
7330:Compton scattering
6811:
6741:
6675:
6626:
6576:
6526:
6471:
6415:Lorentz invariance
6405:It is named after
6393:
6332:
6307:
6273:
6239:
6206:
6173:
6140:
6107:
6074:
6026:Gell-Mann matrices
6008:
5979:
5945:
5911:
5878:
5751:Sin-Itiro Tomonaga
5653:. This permits a
5632:Lorentz invariance
5588:
5438:
5345:
5330:Lagrangian density
5280:
5221:
5161:
5103:
5024:
4935:
4828:
4715:Lorentz invariance
4683:
4663:
4617:
4572:
4499:
4381:
4240:
4074:
4015:
3946:
3886:
3778:
3566:
3443:
3316:
3288:
3252:
3194:
3080:
2937:
2843:
2833:Gauge fixing of a
2807:general relativity
2785:
2622:
2576:If another change
2564:
2392:
2318:
2263:
2217:
2116:
2012:degrees of freedom
1075:Partition function
1002:Topological charge
922:General relativity
917:Special relativity
383:Electrical network
220:Gauss magnetic law
185:Static electricity
145:Electric potential
7386:
7385:
7345:MĂžller scattering
7315:Bhabha scattering
7284:Uehling potential
7233:Virtual particles
7100:978-0-7506-2768-9
7004:Physical Review D
6949:(11): 2220â2222.
6919:10.1119/1.1491265
6413:, lacks manifest
5803:of quantization.
5762:asymptotic states
5676:photon propagator
5610:asymptotic states
5576:
5517:, a scalar field
5436:
4996:
4895:
4859:
4826:
4788:
4754:
4662:
4564:
4544:
4491:
4471:
4332:
4238:
4175:
4071:
4056:
4041:
4009:
3994:
3943:
3871:
3737:
3529:
3426:
3172:
3058:
3019:
2884:quantum chemistry
2803:YangâMills theory
2702:(more precisely,
2646:
2645:
2620:
2559:
2527:
2485:
2457:
2287:
2286:
2184:
2020:equivalence class
1992:
1991:
1055:Expectation value
1030:BRST quantization
977:Poincaré symmetry
932:YangâMills theory
912:Quantum mechanics
851:
850:
550:Reluctance (real)
520:Gyratorâcapacitor
465:Resonant cavities
355:Maxwell equations
16:(Redirected from
7421:
7399:Electromagnetism
7375:
7374:
7350:Schwinger effect
7151:
7144:
7137:
7128:
7127:
7123:
7104:
7087:Lifshitz, Evgeny
7069:
7068:
7050:
7044:
7043:
6999:
6993:
6992:
6958:
6943:Phys. Rev. Lett.
6937:
6931:
6930:
6904:
6882:
6873:
6872:
6834:
6820:
6818:
6817:
6812:
6810:
6809:
6797:
6796:
6787:
6786:
6750:
6748:
6747:
6742:
6734:
6733:
6724:
6723:
6702:Julian Schwinger
6684:
6682:
6681:
6676:
6656:
6645:
6639:
6634:
6622:
6602:
6585:
6583:
6582:
6577:
6556:
6545:
6539:
6534:
6522:
6502:
6494:
6480:
6478:
6477:
6472:
6464:
6456:
6427:multipolar gauge
6421:Multipolar gauge
6402:
6400:
6399:
6394:
6341:
6339:
6338:
6333:
6331:
6330:
6320:
6315:
6303:
6302:
6297:
6296:
6282:
6280:
6279:
6274:
6268:
6264:
6263:
6262:
6257:
6252:
6247:
6230:
6229:
6224:
6219:
6214:
6197:
6196:
6191:
6186:
6181:
6164:
6163:
6158:
6153:
6148:
6131:
6130:
6125:
6120:
6115:
6098:
6097:
6092:
6087:
6082:
6060:
6059:
6017:
6015:
6014:
6009:
6003:
6002:
5992:
5987:
5975:
5974:
5969:
5968:
5954:
5952:
5951:
5946:
5940:
5936:
5935:
5934:
5929:
5924:
5919:
5902:
5901:
5896:
5891:
5886:
5864:
5863:
5747:Julian Schwinger
5702:Minkowski metric
5597:
5595:
5594:
5589:
5587:
5586:
5577:
5569:
5564:
5563:
5554:
5553:
5537:
5536:
5502:
5482:
5447:
5445:
5444:
5439:
5437:
5435:
5427:
5426:
5421:
5417:
5416:
5415:
5406:
5405:
5390:
5382:
5381:
5354:
5352:
5351:
5346:
5344:
5343:
5326:action principle
5289:
5287:
5286:
5281:
5273:
5272:
5263:
5262:
5230:
5228:
5227:
5222:
5214:
5213:
5204:
5203:
5194:
5193:
5170:
5168:
5167:
5162:
5160:
5156:
5154:
5129:
5128:
5112:
5110:
5109:
5104:
5102:
5101:
5092:
5091:
5079:
5078:
5069:
5068:
5059:
5058:
5033:
5031:
5030:
5025:
5020:
5019:
5010:
5009:
4997:
4995:
4994:
4993:
4980:
4976:
4975:
4965:
4944:
4942:
4941:
4936:
4934:
4929:
4928:
4916:
4915:
4909:
4908:
4896:
4894:
4893:
4892:
4879:
4878:
4873:
4872:
4862:
4860:
4858:
4857:
4845:
4837:
4835:
4834:
4829:
4827:
4825:
4824:
4812:
4807:
4802:
4801:
4789:
4787:
4786:
4785:
4772:
4768:
4767:
4757:
4755:
4753:
4752:
4740:
4705:
4692:
4690:
4689:
4684:
4682:
4678:
4676:
4664:
4655:
4643:
4642:
4626:
4624:
4623:
4618:
4610:
4609:
4600:
4599:
4581:
4579:
4578:
4573:
4565:
4563:
4555:
4547:
4545:
4537:
4532:
4531:
4508:
4506:
4505:
4500:
4492:
4490:
4482:
4474:
4472:
4470:
4469:
4457:
4452:
4451:
4405:
4390:
4388:
4387:
4382:
4371:
4354:
4350:
4344:
4343:
4333:
4331:
4320:
4310:
4306:
4297:
4291:
4268:
4260:
4249:
4247:
4246:
4241:
4239:
4237:
4229:
4228:
4218:
4202:
4197:
4193:
4187:
4186:
4176:
4174:
4163:
4153:
4149:
4140:
4139:
4130:
4121:
4104:
4083:
4081:
4080:
4075:
4073:
4072:
4064:
4058:
4057:
4049:
4043:
4042:
4034:
4024:
4022:
4021:
4016:
4011:
4010:
4002:
3996:
3995:
3987:
3981:
3967:
3955:
3953:
3952:
3947:
3945:
3944:
3936:
3927:
3895:
3893:
3892:
3887:
3885:
3877:
3872:
3864:
3847:
3839:
3838:
3787:
3785:
3784:
3779:
3774:
3770:
3764:
3763:
3753:
3748:
3747:
3738:
3736:
3725:
3715:
3711:
3702:
3685:
3677:
3671:
3663:
3658:
3657:
3639:
3631:
3630:
3625:
3602:
3587:
3575:
3573:
3572:
3567:
3562:
3558:
3552:
3551:
3542:
3538:
3530:
3528:
3527:
3518:
3517:
3516:
3507:
3491:
3487:
3478:
3471:
3464:
3460:
3451:
3427:
3425:
3424:
3423:
3404:
3390:
3382:
3369:
3325:
3323:
3322:
3317:
3297:
3295:
3294:
3289:
3287:
3283:
3261:
3259:
3258:
3253:
3251:
3247:
3238:
3230:
3218:
3203:
3201:
3200:
3195:
3193:
3189:
3183:
3182:
3173:
3171:
3160:
3150:
3146:
3137:
3131:
3108:
3100:
3089:
3087:
3086:
3081:
3079:
3075:
3069:
3068:
3059:
3054:
3044:
3040:
3031:
3025:
3020:
3018:
3017:
3016:
2997:
2983:
2946:
2944:
2943:
2938:
2917:
2909:
2908:
2880:transverse gauge
2811:Gribov ambiguity
2797:Gauge fixing in
2794:
2792:
2791:
2786:
2771:
2763:
2728:
2674:
2640:
2631:
2629:
2628:
2623:
2621:
2619:
2611:
2610:
2601:
2579:
2573:
2571:
2570:
2565:
2560:
2558:
2550:
2549:
2548:
2538:
2533:
2529:
2528:
2526:
2518:
2517:
2508:
2486:
2484:
2476:
2475:
2466:
2458:
2456:
2448:
2447:
2446:
2436:
2419:
2401:
2399:
2398:
2393:
2388:
2387:
2360:
2359:
2341:
2340:
2327:
2325:
2324:
2319:
2281:
2272:
2270:
2269:
2264:
2253:
2245:
2232:
2226:
2224:
2223:
2218:
2213:
2212:
2197:
2196:
2185:
2183:
2175:
2174:
2173:
2163:
2146:
2145:
2125:
2123:
2122:
2117:
2008:choosing a gauge
1984:
1977:
1970:
1045:Effective action
972:Lorentz symmetry
897:Electromagnetism
867:
853:
852:
843:
836:
829:
510:Electric machine
493:Magnetic circuit
455:Parallel circuit
445:Network analysis
410:Electric current
345:London equations
190:Triboelectricity
180:Potential energy
49:
39:Electromagnetism
30:
29:
21:
7429:
7428:
7424:
7423:
7422:
7420:
7419:
7418:
7389:
7388:
7387:
7382:
7381:
7359:
7303:
7294:Vertex function
7279:Schwinger limit
7254:Furry's theorem
7237:
7191:
7177:Feynman diagram
7160:
7155:
7120:
7101:
7078:
7076:Further reading
7073:
7072:
7065:
7051:
7047:
7000:
6996:
6938:
6934:
6902:physics/0204034
6883:
6876:
6835:
6831:
6826:
6805:
6801:
6792:
6788:
6782:
6778:
6776:
6773:
6772:
6765:
6729:
6725:
6719:
6715:
6713:
6710:
6709:
6690:
6652:
6641:
6635:
6630:
6618:
6598:
6590:
6587:
6586:
6552:
6541:
6535:
6530:
6518:
6498:
6490:
6488:
6485:
6484:
6460:
6452:
6450:
6447:
6446:
6423:
6382:
6379:
6378:
6359:
6351:
6326:
6322:
6316:
6311:
6298:
6292:
6291:
6290:
6288:
6285:
6284:
6258:
6248:
6243:
6235:
6234:
6225:
6215:
6210:
6202:
6201:
6192:
6182:
6177:
6169:
6168:
6159:
6149:
6144:
6136:
6135:
6126:
6116:
6111:
6103:
6102:
6093:
6083:
6078:
6070:
6069:
6068:
6064:
6055:
6051:
6046:
6043:
6042:
6040:
6033:
5998:
5994:
5988:
5983:
5970:
5964:
5963:
5962:
5960:
5957:
5956:
5930:
5920:
5915:
5907:
5906:
5897:
5887:
5882:
5874:
5873:
5872:
5868:
5859:
5855:
5850:
5847:
5846:
5844:
5827:. Examples are
5809:
5773:
5731:Richard Feynman
5722:
5699:
5691:Feynman diagram
5680:virtual photons
5643:transformations
5639:
5621:
5606:Goldstone boson
5582:
5578:
5568:
5559:
5555:
5549:
5545:
5532:
5531:
5526:
5523:
5522:
5515:auxiliary field
5512:
5497:
5489:
5477:
5461:
5428:
5422:
5411:
5407:
5401:
5397:
5396:
5392:
5391:
5389:
5377:
5376:
5371:
5368:
5367:
5339:
5338:
5336:
5333:
5332:
5321:
5310:
5307:
5268:
5264:
5258:
5254:
5252:
5249:
5248:
5242:Ward identities
5209:
5205:
5199:
5195:
5189:
5185:
5183:
5180:
5179:
5150:
5137:
5133:
5124:
5120:
5118:
5115:
5114:
5097:
5093:
5087:
5083:
5074:
5070:
5064:
5060:
5054:
5050:
5048:
5045:
5044:
5015:
5011:
5005:
5001:
4989:
4985:
4981:
4971:
4967:
4966:
4964:
4962:
4959:
4958:
4930:
4924:
4920:
4911:
4910:
4904:
4900:
4888:
4884:
4880:
4874:
4868:
4864:
4863:
4861:
4853:
4849:
4844:
4842:
4839:
4838:
4820:
4816:
4811:
4803:
4797:
4793:
4781:
4777:
4773:
4763:
4759:
4758:
4756:
4748:
4744:
4739:
4737:
4734:
4733:
4723:Hendrik Lorentz
4704:
4698:
4672:
4653:
4651:
4647:
4638:
4634:
4632:
4629:
4628:
4605:
4601:
4595:
4591:
4589:
4586:
4585:
4556:
4548:
4546:
4536:
4527:
4526:
4518:
4515:
4514:
4483:
4475:
4473:
4465:
4461:
4456:
4447:
4446:
4438:
4435:
4434:
4427:
4421:
4415:
4410:
4392:
4367:
4346:
4345:
4339:
4337:
4321:
4302:
4301:
4293:
4292:
4290:
4264:
4256:
4254:
4251:
4250:
4230:
4214:
4207:
4203:
4201:
4189:
4188:
4182:
4180:
4164:
4145:
4144:
4135:
4134:
4123:
4122:
4120:
4100:
4092:
4089:
4088:
4063:
4062:
4048:
4047:
4033:
4032:
4030:
4027:
4026:
4001:
4000:
3986:
3985:
3977:
3963:
3961:
3958:
3957:
3935:
3934:
3923:
3921:
3918:
3917:
3915:
3881:
3873:
3863:
3843:
3834:
3833:
3831:
3828:
3827:
3824:symmetric gauge
3792:radiation gauge
3766:
3765:
3759:
3755:
3749:
3743:
3739:
3726:
3707:
3706:
3698:
3681:
3673:
3672:
3670:
3659:
3653:
3649:
3635:
3626:
3621:
3620:
3615:
3612:
3611:
3589:
3579:
3554:
3553:
3547:
3543:
3523:
3519:
3512:
3511:
3483:
3482:
3474:
3473:
3472:
3470:
3456:
3452:
3447:
3442:
3438:
3419:
3415:
3408:
3403:
3386:
3378:
3376:
3373:
3372:
3356:
3311:
3308:
3307:
3279:
3275:
3267:
3264:
3263:
3243:
3242:
3234:
3226:
3224:
3221:
3220:
3205:
3185:
3184:
3178:
3174:
3161:
3142:
3141:
3133:
3132:
3130:
3104:
3096:
3094:
3091:
3090:
3071:
3070:
3064:
3060:
3036:
3035:
3027:
3026:
3024:
3012:
3008:
3001:
2996:
2979:
2971:
2968:
2967:
2913:
2904:
2903:
2895:
2892:
2891:
2872:
2864:gauge invariant
2827:
2825:An illustration
2767:
2759:
2757:
2754:
2753:
2715:
2704:gauge potential
2661:
2612:
2606:
2602:
2600:
2586:
2583:
2582:
2551:
2544:
2543:
2539:
2537:
2519:
2513:
2509:
2507:
2500:
2496:
2477:
2471:
2467:
2465:
2449:
2442:
2441:
2437:
2435:
2415:
2413:
2410:
2409:
2383:
2382:
2355:
2354:
2336:
2335:
2333:
2330:
2329:
2298:
2295:
2294:
2249:
2241:
2239:
2236:
2235:
2208:
2207:
2192:
2191:
2176:
2169:
2168:
2164:
2162:
2141:
2140:
2138:
2135:
2134:
2111:
2108:
2107:
2076:in terms of an
2074:electrodynamics
2062:
2042:renormalization
1988:
1959:
1958:
1957:
1955:
1249:
1241:
1240:
1236:Quantum gravity
1211:
1203:
1202:
1198:Higgs mechanism
1178:
1168:
1167:
1143:Proca equations
1128:
1120:
1119:
1115:Wightman Axioms
1100:Renormalization
1060:Feynman diagram
1015:
1007:
1006:
947:
937:
936:
887:
872:
870:Feynman diagram
847:
818:
817:
633:
625:
624:
580:
570:
569:
525:Induction motor
495:
485:
484:
400:Current density
385:
375:
374:
365:Poynting vector
275:
273:Electrodynamics
265:
264:
260:Right-hand rule
225:Magnetic dipole
215:BiotâSavart law
205:
195:
194:
130:Electric dipole
125:Electric charge
100:
28:
23:
22:
15:
12:
11:
5:
7427:
7417:
7416:
7414:Gauge theories
7411:
7406:
7401:
7384:
7383:
7380:
7379:
7365:
7364:
7361:
7360:
7358:
7357:
7352:
7347:
7342:
7337:
7332:
7327:
7325:Bremsstrahlung
7322:
7317:
7311:
7309:
7305:
7304:
7302:
7301:
7296:
7291:
7286:
7281:
7276:
7271:
7266:
7261:
7256:
7251:
7245:
7243:
7239:
7238:
7236:
7235:
7230:
7225:
7220:
7215:
7210:
7205:
7199:
7197:
7193:
7192:
7190:
7189:
7184:
7179:
7174:
7168:
7166:
7162:
7161:
7154:
7153:
7146:
7139:
7131:
7125:
7124:
7118:
7105:
7099:
7077:
7074:
7071:
7070:
7063:
7045:
6994:
6956:hep-ph/0010057
6932:
6895:(9): 917â928.
6874:
6847:(5): 519â524.
6828:
6827:
6825:
6822:
6808:
6804:
6800:
6795:
6791:
6785:
6781:
6764:
6761:
6740:
6737:
6732:
6728:
6722:
6718:
6689:
6686:
6674:
6671:
6668:
6665:
6662:
6659:
6655:
6651:
6648:
6644:
6638:
6633:
6629:
6625:
6621:
6617:
6614:
6611:
6608:
6605:
6601:
6597:
6594:
6575:
6572:
6568:
6565:
6562:
6559:
6555:
6551:
6548:
6544:
6538:
6533:
6529:
6525:
6521:
6517:
6514:
6511:
6508:
6505:
6501:
6497:
6493:
6470:
6467:
6463:
6459:
6455:
6443:Henri Poincaré
6439:Poincaré gauge
6422:
6419:
6392:
6389:
6386:
6371:temporal gauge
6358:
6355:
6350:
6347:
6343:
6342:
6329:
6325:
6319:
6314:
6310:
6306:
6301:
6295:
6272:
6267:
6261:
6256:
6251:
6246:
6242:
6238:
6233:
6228:
6223:
6218:
6213:
6209:
6205:
6200:
6195:
6190:
6185:
6180:
6176:
6172:
6167:
6162:
6157:
6152:
6147:
6143:
6139:
6134:
6129:
6124:
6119:
6114:
6110:
6106:
6101:
6096:
6091:
6086:
6081:
6077:
6073:
6067:
6063:
6058:
6054:
6050:
6038:
6031:
6018:
6007:
6001:
5997:
5991:
5986:
5982:
5978:
5973:
5967:
5944:
5939:
5933:
5928:
5923:
5918:
5914:
5910:
5905:
5900:
5895:
5890:
5885:
5881:
5877:
5871:
5867:
5862:
5858:
5854:
5842:
5808:
5805:
5801:BRST formalism
5771:
5720:
5697:
5682:of unphysical
5651:configurations
5637:
5628:one-loop order
5619:
5585:
5581:
5575:
5572:
5567:
5562:
5558:
5552:
5548:
5543:
5540:
5535:
5530:
5513:gauge uses an
5510:
5487:
5459:
5434:
5431:
5425:
5420:
5414:
5410:
5404:
5400:
5395:
5388:
5385:
5380:
5375:
5342:
5317:
5309:
5303:
5298:
5279:
5276:
5271:
5267:
5261:
5257:
5220:
5217:
5212:
5208:
5202:
5198:
5192:
5188:
5159:
5153:
5148:
5145:
5141:
5136:
5132:
5127:
5123:
5100:
5096:
5090:
5086:
5082:
5077:
5073:
5067:
5063:
5057:
5053:
5023:
5018:
5014:
5008:
5004:
5000:
4992:
4988:
4984:
4979:
4974:
4970:
4933:
4927:
4923:
4919:
4914:
4907:
4903:
4899:
4891:
4887:
4883:
4877:
4871:
4867:
4856:
4852:
4848:
4823:
4819:
4815:
4810:
4806:
4800:
4796:
4792:
4784:
4780:
4776:
4771:
4766:
4762:
4751:
4747:
4743:
4721:and not after
4700:
4681:
4675:
4670:
4667:
4661:
4658:
4650:
4646:
4641:
4637:
4616:
4613:
4608:
4604:
4598:
4594:
4571:
4568:
4562:
4559:
4554:
4551:
4543:
4540:
4535:
4530:
4525:
4522:
4511:Gaussian units
4498:
4495:
4489:
4486:
4481:
4478:
4468:
4464:
4460:
4455:
4450:
4445:
4442:
4417:Main article:
4414:
4411:
4409:
4408:
4380:
4377:
4374:
4370:
4366:
4363:
4360:
4357:
4353:
4349:
4342:
4336:
4330:
4327:
4324:
4319:
4316:
4313:
4309:
4305:
4300:
4296:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4267:
4263:
4259:
4236:
4233:
4227:
4224:
4221:
4217:
4213:
4210:
4206:
4200:
4196:
4192:
4185:
4179:
4173:
4170:
4167:
4162:
4159:
4156:
4152:
4148:
4143:
4138:
4133:
4129:
4126:
4119:
4116:
4113:
4110:
4107:
4103:
4099:
4096:
4085:
4070:
4067:
4061:
4055:
4052:
4046:
4040:
4037:
4014:
4008:
4005:
3999:
3993:
3990:
3984:
3980:
3976:
3973:
3970:
3966:
3942:
3939:
3933:
3930:
3926:
3913:
3884:
3880:
3876:
3870:
3867:
3862:
3859:
3856:
3853:
3850:
3846:
3842:
3837:
3816:
3799:
3788:
3777:
3773:
3769:
3762:
3758:
3752:
3746:
3742:
3735:
3732:
3729:
3724:
3721:
3718:
3714:
3710:
3705:
3701:
3697:
3694:
3691:
3688:
3684:
3680:
3676:
3669:
3666:
3662:
3656:
3652:
3648:
3645:
3642:
3638:
3634:
3629:
3624:
3619:
3604:
3576:
3565:
3561:
3557:
3550:
3546:
3541:
3537:
3534:
3526:
3522:
3515:
3510:
3506:
3503:
3500:
3497:
3494:
3490:
3486:
3481:
3477:
3468:
3463:
3459:
3455:
3450:
3446:
3441:
3437:
3434:
3431:
3422:
3418:
3414:
3411:
3407:
3402:
3399:
3396:
3393:
3389:
3385:
3381:
3339:volume element
3315:
3286:
3282:
3278:
3274:
3271:
3250:
3246:
3241:
3237:
3233:
3229:
3192:
3188:
3181:
3177:
3170:
3167:
3164:
3159:
3156:
3153:
3149:
3145:
3140:
3136:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3107:
3103:
3099:
3078:
3074:
3067:
3063:
3057:
3053:
3050:
3047:
3043:
3039:
3034:
3030:
3023:
3015:
3011:
3007:
3004:
3000:
2995:
2992:
2989:
2986:
2982:
2978:
2975:
2958:
2936:
2932:
2929:
2926:
2923:
2920:
2916:
2912:
2907:
2902:
2899:
2871:
2868:
2856:gauge function
2826:
2823:
2784:
2780:
2777:
2774:
2770:
2766:
2762:
2712:gauge function
2644:
2643:
2634:
2632:
2618:
2615:
2609:
2605:
2599:
2596:
2593:
2590:
2563:
2557:
2554:
2547:
2542:
2536:
2532:
2525:
2522:
2516:
2512:
2506:
2503:
2499:
2495:
2492:
2489:
2483:
2480:
2474:
2470:
2464:
2461:
2455:
2452:
2445:
2440:
2434:
2431:
2428:
2425:
2422:
2418:
2391:
2386:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2358:
2353:
2350:
2347:
2344:
2339:
2317:
2314:
2311:
2308:
2305:
2302:
2289:is made, then
2285:
2284:
2275:
2273:
2262:
2259:
2256:
2252:
2248:
2244:
2216:
2211:
2206:
2203:
2200:
2195:
2189:
2182:
2179:
2172:
2167:
2161:
2158:
2155:
2152:
2149:
2144:
2115:
2089:magnetic field
2082:electric field
2061:
2058:
2000:gauge theories
1990:
1989:
1987:
1986:
1979:
1972:
1964:
1961:
1960:
1953:
1952:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1712:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1452:
1447:
1442:
1437:
1432:
1427:
1422:
1417:
1412:
1407:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1362:
1357:
1352:
1347:
1342:
1337:
1332:
1327:
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1282:
1277:
1272:
1267:
1262:
1257:
1251:
1250:
1247:
1246:
1243:
1242:
1239:
1238:
1233:
1228:
1223:
1218:
1212:
1209:
1208:
1205:
1204:
1201:
1200:
1195:
1190:
1185:
1179:
1176:Standard Model
1174:
1173:
1170:
1169:
1166:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1133:Dirac equation
1129:
1126:
1125:
1122:
1121:
1118:
1117:
1112:
1110:Wick's theorem
1107:
1102:
1097:
1095:Regularization
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1016:
1013:
1012:
1009:
1008:
1005:
1004:
999:
997:Noether charge
994:
989:
984:
982:Gauge symmetry
979:
974:
969:
964:
959:
954:
948:
943:
942:
939:
938:
935:
934:
929:
924:
919:
914:
909:
904:
899:
894:
888:
885:
884:
881:
880:
874:
873:
868:
860:
859:
849:
848:
846:
845:
838:
831:
823:
820:
819:
816:
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:
634:
631:
630:
627:
626:
623:
622:
617:
612:
607:
602:
600:Four-potential
597:
592:
587:
581:
576:
575:
572:
571:
568:
567:
562:
557:
552:
547:
542:
537:
532:
527:
522:
517:
515:Electric motor
512:
507:
502:
496:
491:
490:
487:
486:
483:
482:
477:
472:
470:Series circuit
467:
462:
457:
452:
447:
442:
440:Kirchhoff laws
437:
432:
427:
422:
417:
412:
407:
405:Direct current
402:
397:
392:
386:
381:
380:
377:
376:
373:
372:
367:
362:
360:Maxwell tensor
357:
352:
347:
342:
337:
332:
330:Larmor formula
327:
322:
317:
312:
307:
302:
297:
292:
287:
282:
280:Bremsstrahlung
276:
271:
270:
267:
266:
263:
262:
257:
252:
247:
242:
237:
232:
230:Magnetic field
227:
222:
217:
212:
206:
203:Magnetostatics
201:
200:
197:
196:
193:
192:
187:
182:
177:
172:
167:
162:
157:
152:
147:
142:
137:
135:Electric field
132:
127:
122:
117:
112:
107:
105:Charge density
101:
98:Electrostatics
96:
95:
92:
91:
90:
89:
84:
79:
74:
69:
64:
59:
51:
50:
42:
41:
35:
34:
33:Articles about
26:
9:
6:
4:
3:
2:
7426:
7415:
7412:
7410:
7407:
7405:
7402:
7400:
7397:
7396:
7394:
7378:
7370:
7367:
7366:
7362:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7312:
7310:
7306:
7300:
7297:
7295:
7292:
7290:
7287:
7285:
7282:
7280:
7277:
7275:
7272:
7270:
7267:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7246:
7244:
7240:
7234:
7231:
7229:
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7200:
7198:
7194:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7169:
7167:
7163:
7159:
7152:
7147:
7145:
7140:
7138:
7133:
7132:
7129:
7121:
7119:0-471-30932-X
7115:
7111:
7106:
7102:
7096:
7092:
7088:
7084:
7080:
7079:
7066:
7060:
7056:
7049:
7041:
7037:
7033:
7029:
7025:
7021:
7017:
7013:
7009:
7005:
6998:
6990:
6986:
6982:
6978:
6974:
6970:
6966:
6962:
6957:
6952:
6948:
6945:
6944:
6936:
6928:
6924:
6920:
6916:
6912:
6908:
6903:
6898:
6894:
6890:
6889:
6881:
6879:
6870:
6866:
6862:
6858:
6854:
6850:
6846:
6842:
6841:
6833:
6829:
6821:
6806:
6802:
6798:
6793:
6789:
6783:
6779:
6770:
6760:
6758:
6754:
6738:
6735:
6730:
6726:
6720:
6716:
6707:
6703:
6699:
6698:Vladimir Fock
6696:(named after
6695:
6685:
6672:
6669:
6666:
6660:
6657:
6649:
6636:
6631:
6627:
6623:
6615:
6612:
6606:
6603:
6592:
6573:
6570:
6566:
6560:
6557:
6549:
6536:
6531:
6527:
6523:
6515:
6512:
6506:
6503:
6481:
6468:
6465:
6457:
6444:
6441:(named after
6440:
6436:
6432:
6428:
6418:
6416:
6412:
6408:
6403:
6390:
6387:
6384:
6376:
6372:
6368:
6364:
6354:
6346:
6327:
6323:
6317:
6312:
6308:
6304:
6299:
6270:
6265:
6259:
6254:
6249:
6244:
6240:
6236:
6231:
6226:
6221:
6216:
6211:
6207:
6203:
6198:
6193:
6188:
6183:
6178:
6174:
6170:
6165:
6160:
6155:
6150:
6145:
6141:
6137:
6132:
6127:
6122:
6117:
6112:
6108:
6104:
6099:
6094:
6089:
6084:
6079:
6075:
6071:
6065:
6061:
6056:
6052:
6048:
6037:
6030:
6027:
6023:
6019:
6005:
5999:
5995:
5989:
5984:
5980:
5976:
5971:
5942:
5937:
5931:
5926:
5921:
5916:
5912:
5908:
5903:
5898:
5893:
5888:
5883:
5879:
5875:
5869:
5865:
5860:
5856:
5852:
5841:
5838:
5834:
5830:
5829:
5828:
5826:
5822:
5818:
5814:
5804:
5802:
5798:
5794:
5790:
5786:
5782:
5778:
5774:
5767:
5763:
5758:
5756:
5752:
5748:
5744:
5743:Freeman Dyson
5740:
5736:
5732:
5728:
5726:
5719:
5715:
5711:
5707:
5703:
5696:
5692:
5687:
5685:
5681:
5677:
5673:
5672:Feynman rules
5669:
5665:
5661:
5660:normalization
5656:
5652:
5648:
5644:
5640:
5633:
5629:
5625:
5618:
5613:
5611:
5607:
5603:
5598:
5583:
5579:
5573:
5570:
5565:
5560:
5556:
5550:
5541:
5538:
5528:
5520:
5516:
5509:
5504:
5500:
5496:
5494:
5486:
5480:
5475:
5471:
5467:
5463:
5458:
5453:
5448:
5432:
5429:
5423:
5418:
5412:
5408:
5402:
5393:
5386:
5383:
5373:
5365:
5361:
5358:
5331:
5327:
5323:
5320:
5316:
5306:
5302:
5297:
5295:
5290:
5277:
5274:
5269:
5265:
5259:
5247:
5243:
5238:
5234:
5218:
5215:
5210:
5206:
5200:
5190:
5176:
5174:
5157:
5146:
5143:
5139:
5134:
5130:
5125:
5121:
5098:
5094:
5088:
5084:
5080:
5075:
5071:
5065:
5055:
5041:
5039:
5034:
5021:
5016:
5006:
5002:
4998:
4990:
4986:
4977:
4972:
4957:
4956:wave equation
4953:
4948:
4945:
4925:
4921:
4917:
4905:
4897:
4889:
4885:
4869:
4854:
4850:
4846:
4821:
4817:
4813:
4808:
4804:
4798:
4790:
4782:
4778:
4769:
4764:
4749:
4745:
4741:
4730:
4728:
4724:
4720:
4719:Ludvig Lorenz
4716:
4711:
4709:
4703:
4696:
4679:
4668:
4665:
4659:
4656:
4648:
4644:
4639:
4635:
4614:
4611:
4606:
4602:
4596:
4582:
4569:
4566:
4560:
4552:
4541:
4538:
4533:
4523:
4512:
4496:
4493:
4487:
4479:
4466:
4462:
4458:
4453:
4443:
4432:
4426:
4420:
4403:
4399:
4395:
4375:
4372:
4361:
4355:
4351:
4340:
4328:
4325:
4322:
4314:
4311:
4307:
4287:
4284:
4278:
4272:
4269:
4234:
4222:
4219:
4208:
4198:
4194:
4183:
4171:
4168:
4165:
4157:
4154:
4150:
4131:
4127:
4117:
4114:
4108:
4105:
4094:
4086:
4065:
4059:
4050:
4044:
4035:
4012:
4003:
3988:
3982:
3971:
3968:
3937:
3931:
3928:
3912:
3908:
3903:
3899:
3878:
3868:
3865:
3860:
3857:
3851:
3848:
3825:
3821:
3817:
3814:
3810:
3805:
3800:
3797:
3793:
3789:
3775:
3771:
3760:
3756:
3744:
3740:
3733:
3730:
3727:
3719:
3716:
3712:
3695:
3689:
3686:
3667:
3664:
3654:
3650:
3643:
3640:
3627:
3617:
3609:
3605:
3600:
3596:
3592:
3585:
3582:
3577:
3563:
3559:
3548:
3544:
3539:
3535:
3532:
3524:
3520:
3508:
3501:
3498:
3495:
3492:
3488:
3466:
3461:
3457:
3453:
3448:
3444:
3439:
3435:
3432:
3420:
3416:
3412:
3409:
3405:
3400:
3394:
3391:
3371:
3367:
3363:
3359:
3353:
3351:
3346:
3344:
3340:
3336:
3333:
3329:
3305:
3301:
3284:
3276:
3272:
3269:
3248:
3239:
3231:
3216:
3212:
3208:
3190:
3179:
3175:
3168:
3165:
3162:
3154:
3151:
3147:
3127:
3124:
3118:
3112:
3109:
3076:
3065:
3061:
3055:
3048:
3045:
3041:
3028:
3021:
3013:
3009:
3005:
3002:
2998:
2993:
2987:
2984:
2973:
2964:
2960:
2959:
2957:
2954:
2952:
2947:
2934:
2930:
2927:
2921:
2918:
2900:
2889:
2885:
2882:) is used in
2881:
2877:
2876:Coulomb gauge
2870:Coulomb gauge
2867:
2865:
2861:
2860:gauge freedom
2857:
2853:
2849:
2840:
2836:
2831:
2822:
2820:
2816:
2812:
2808:
2804:
2800:
2795:
2782:
2778:
2772:
2751:
2747:
2746:line integral
2743:
2737:
2735:
2734:gauge freedom
2732:
2726:
2722:
2718:
2713:
2709:
2705:
2701:
2696:
2694:
2693:
2688:
2687:
2682:
2678:
2672:
2668:
2664:
2659:
2655:
2651:
2648:is made then
2642:
2635:
2633:
2616:
2607:
2597:
2594:
2588:
2581:
2580:
2577:
2574:
2561:
2555:
2534:
2530:
2523:
2514:
2504:
2501:
2497:
2490:
2487:
2481:
2472:
2459:
2453:
2432:
2429:
2423:
2420:
2408:according to
2407:
2402:
2389:
2379:
2373:
2367:
2361:
2348:
2342:
2315:
2312:
2309:
2303:
2292:
2283:
2276:
2274:
2260:
2254:
2234:
2233:
2230:
2227:
2214:
2204:
2198:
2187:
2180:
2159:
2156:
2150:
2147:
2132:
2129:
2113:
2106:
2101:
2097:
2093:
2090:
2086:
2083:
2079:
2075:
2071:
2067:
2060:Gauge freedom
2057:
2055:
2051:
2047:
2043:
2039:
2034:
2028:
2025:
2021:
2017:
2013:
2009:
2006:(also called
2005:
2001:
1997:
1985:
1980:
1978:
1973:
1971:
1966:
1965:
1963:
1962:
1956:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1925:Zamolodchikov
1923:
1921:
1920:Zamolodchikov
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
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1741:
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1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
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1691:
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1681:
1678:
1676:
1673:
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1668:
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1663:
1661:
1658:
1656:
1653:
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1606:
1603:
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1496:
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1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
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1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1252:
1245:
1244:
1237:
1234:
1232:
1229:
1227:
1224:
1222:
1221:Supersymmetry
1219:
1217:
1216:String theory
1214:
1213:
1207:
1206:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1180:
1177:
1172:
1171:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1130:
1124:
1123:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1017:
1011:
1010:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
949:
946:
941:
940:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
889:
883:
882:
879:
876:
875:
871:
866:
862:
861:
858:
855:
854:
844:
839:
837:
832:
830:
825:
824:
822:
821:
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:
635:
629:
628:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
582:
579:
574:
573:
566:
563:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
531:
528:
526:
523:
521:
518:
516:
513:
511:
508:
506:
503:
501:
498:
497:
494:
489:
488:
481:
478:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
446:
443:
441:
438:
436:
435:Joule heating
433:
431:
428:
426:
423:
421:
418:
416:
413:
411:
408:
406:
403:
401:
398:
396:
393:
391:
388:
387:
384:
379:
378:
371:
368:
366:
363:
361:
358:
356:
353:
351:
350:Lorentz force
348:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
277:
274:
269:
268:
261:
258:
256:
253:
251:
250:Magnetization
248:
246:
243:
241:
238:
236:
235:Magnetic flux
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
207:
204:
199:
198:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
140:Electric flux
138:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
111:
108:
106:
103:
102:
99:
94:
93:
88:
85:
83:
80:
78:
77:Computational
75:
73:
70:
68:
65:
63:
60:
58:
55:
54:
53:
52:
48:
44:
43:
40:
37:
36:
32:
31:
19:
18:Feynman gauge
7368:
7109:
7090:
7054:
7048:
7007:
7003:
6997:
6946:
6941:
6935:
6892:
6886:
6844:
6838:
6832:
6766:
6752:
6705:
6693:
6691:
6482:
6438:
6434:
6430:
6426:
6424:
6407:Hermann Weyl
6404:
6374:
6370:
6366:
6362:
6360:
6352:
6344:
6035:
6028:
5839:
5837:Pauli matrix
5820:
5816:
5810:
5769:
5759:
5729:
5717:
5694:
5688:
5684:polarization
5650:
5642:
5635:
5616:
5614:
5599:
5518:
5507:
5505:
5498:
5491:
5484:
5478:
5473:
5465:
5462:Landau gauge
5456:
5455:
5451:
5449:
5363:
5359:
5318:
5314:
5313:
5311:
5304:
5300:
5291:
5177:
5173:four-current
5042:
5035:
4951:
4949:
4946:
4731:
4712:
4701:
4583:
4428:
4413:Lorenz gauge
4401:
4397:
4393:
3910:
3907:Landau gauge
3906:
3901:
3897:
3823:
3819:
3791:
3607:
3598:
3594:
3590:
3583:
3580:
3365:
3361:
3357:
3354:
3347:
3342:
3334:
3331:
3327:
3326:operates on
3303:
3299:
3214:
3210:
3206:
2966:
2955:
2948:
2875:
2873:
2863:
2859:
2855:
2848:gauge fixing
2847:
2844:
2838:
2834:
2819:frame bundle
2796:
2749:
2738:
2733:
2724:
2720:
2716:
2711:
2707:
2703:
2699:
2697:
2690:
2684:
2680:
2676:
2670:
2666:
2662:
2657:
2653:
2649:
2647:
2636:
2575:
2405:
2403:
2290:
2288:
2277:
2228:
2130:
2100:mathematical
2099:
2091:
2084:
2063:
2032:
2029:
2007:
2004:gauge fixing
2003:
1993:
1954:
1800:Stueckelberg
1530:Jona-Lasinio
1105:Vacuum state
1090:Quantization
927:Gauge theory
907:Strong force
892:Field theory
595:Four-current
530:Linear motor
415:Electrolysis
295:Eddy current
255:Permeability
175:Polarization
170:Permittivity
7274:Self-energy
7264:Landau pole
7228:Positronium
7203:Dual photon
7083:Landau, Lev
6763:Dirac gauge
6435:point gauge
6367:Hamiltonian
5811:In any non-
5777:non-abelian
5476:, in which
5357:gauge field
4433:units, by:
2799:non-abelian
1940:Zinn-Justin
1790:Sommerfield
1715:Pomeranchuk
1680:Osterwalder
1675:Oppenheimer
1605:ĆopuszaĆski
1425:Fredenhagen
1226:Technicolor
565:Transformer
395:Capacitance
320:Faraday law
115:Coulomb law
57:Electricity
7393:Categories
7340:Lamb shift
7269:QED vacuum
7064:0201360799
6824:References
6769:Paul Dirac
6431:line gauge
6375:incomplete
6363:Weyl gauge
6357:Weyl gauge
5821:incomplete
5038:light cone
4952:incomplete
4708:4-gradient
4423:See also:
2033:particular
1935:Zimmermann
1830:Vainshtein
1705:Polchinski
1565:Kontsevich
1510:Iliopoulos
1485:Heisenberg
1310:Bogoliubov
1248:Scientists
1085:Propagator
967:T-symmetry
962:P-symmetry
957:C-symmetry
945:Symmetries
902:Weak force
886:Background
632:Scientists
480:Waveguides
460:Resistance
430:Inductance
210:AmpĂšre law
7369:See also:
7308:Processes
7196:Particles
7165:Formalism
7032:0556-2821
6927:119652556
6869:250880504
6794:μ
6784:μ
6731:μ
6721:μ
6628:∫
6624:⋅
6616:−
6593:φ
6528:∫
6524:×
6516:−
6458:⋅
6385:φ
6324:λ
6313:μ
6300:μ
6245:μ
6212:μ
6179:μ
6146:μ
6113:μ
6080:μ
6049:∫
5996:σ
5985:μ
5972:μ
5917:μ
5884:μ
5853:∫
5571:ξ
5561:μ
5551:μ
5547:∂
5529:δ
5433:ξ
5413:μ
5403:μ
5399:∂
5387:−
5374:δ
5270:μ
5260:μ
5256:∂
5237:polarized
5211:ν
5201:μ
5197:∂
5191:μ
5187:∂
5144:ρ
5126:ν
5099:ν
5085:μ
5076:ν
5066:μ
5062:∂
5056:μ
5052:∂
5022:ψ
5013:∇
4983:∂
4978:ψ
4969:∂
4922:μ
4902:∇
4898:−
4882:∂
4866:∂
4818:ε
4814:ρ
4805:φ
4795:∇
4791:−
4775:∂
4770:φ
4761:∂
4666:φ
4640:μ
4607:μ
4597:μ
4593:∂
4558:∂
4553:φ
4550:∂
4524:⋅
4521:∇
4485:∂
4480:φ
4477:∂
4444:⋅
4441:∇
4362:ψ
4359:∇
4326:π
4288:∫
4285:×
4282:∇
4232:∂
4209:ψ
4205:∂
4199:−
4169:π
4132:⋅
4125:∇
4118:∫
4095:φ
4069:^
4054:^
4039:^
4007:^
3992:^
3983:⋅
3941:^
3879:×
3861:−
3731:π
3696:⋅
3668:∬
3618:∫
3536:τ
3509:×
3502:τ
3499:−
3467:τ
3445:∫
3436:∫
3433:×
3430:∇
3417:ε
3413:π
3350:causality
3314:∇
3240:−
3166:π
3128:∫
3125:×
3122:∇
3029:ρ
3022:∫
3010:ε
3006:π
2974:φ
2951:quantized
2901:⋅
2898:∇
2779:ψ
2776:∇
2765:→
2614:∂
2608:ψ
2604:∂
2598:−
2595:φ
2592:→
2589:φ
2553:∂
2541:∂
2535:−
2521:∂
2515:ψ
2511:∂
2502:φ
2494:∇
2491:−
2479:∂
2473:ψ
2469:∂
2463:∇
2460:−
2451:∂
2439:∂
2433:−
2430:φ
2427:∇
2424:−
2380:×
2377:∇
2368:ψ
2365:∇
2349:×
2346:∇
2310:ψ
2307:∇
2304:×
2301:∇
2261:ψ
2258:∇
2247:→
2205:×
2202:∇
2178:∂
2166:∂
2160:−
2157:φ
2154:∇
2151:−
2114:φ
2066:Heaviside
1870:Wetterich
1855:Weisskopf
1805:Sudarshan
1755:Schwinger
1665:Nishijima
1630:Maldacena
1595:Leutwyler
1555:Kinoshita
1455:Goldstone
1445:Gell-Mann
1360:Doplicher
1127:Equations
788:Steinmetz
718:Kirchhoff
703:Jefimenko
698:Hopkinson
683:Helmholtz
678:Heaviside
540:Permeance
425:Impedance
165:Insulator
160:Gauss law
110:Conductor
87:Phenomena
82:Textbooks
62:Magnetism
7242:Concepts
7223:Positron
7208:Electron
7089:(2007).
6989:45172403
6981:11289894
6373:) is an
5789:Jacobian
5725:spin sum
5706:linearly
5668:extremum
5364:breaking
5360:a priori
4352:′
4308:′
4195:′
4151:′
4128:′
3811:such as
3772:′
3713:′
3560:′
3489:′
3249:′
3191:′
3148:′
3077:′
3042:′
2126:and the
1885:Wightman
1850:Weinberg
1840:Virasoro
1820:Tomonaga
1815:Thirring
1810:Symanzik
1770:Semenoff
1745:Schrader
1710:Polyakov
1625:Majorana
1560:Klebanov
1515:Ivanenko
1505:'t Hooft
1475:Guralnik
1420:Fröhlich
1415:Fritzsch
1410:Frampton
1325:Buchholz
1270:Bargmann
1260:Anderson
1040:Crossing
813:Wiechert
768:Poynting
658:Einstein
505:DC motor
500:AC motor
335:Lenz law
120:Electret
7040:9958379
7012:Bibcode
6961:Bibcode
6907:Bibcode
6849:Bibcode
6755:is the
6445:)) is:
5662:of the
5171:is the
4693:is the
4509:and in
3337:is the
3298:(where
2839:surface
2835:twisted
2689:) and (
1996:physics
1994:In the
1895:Wilczek
1860:Wentzel
1835:Veltman
1780:Shirkov
1775:Shifman
1765:Seiberg
1750:Schwarz
1730:Rubakov
1650:Naimark
1600:Lipatov
1590:Lehmann
1570:Kreimer
1550:Kendall
1440:Gelfand
1435:Glashow
1395:Feynman
1375:Faddeev
1370:Englert
1340:Coleman
1330:Cachazo
1315:Brodsky
1300:Bjorken
1290:Berezin
1280:Belavin
1020:Anomaly
878:History
798:Thomson
773:Ritchie
763:Poisson
748:Neumann
743:Maxwell
738:Lorentz
733:Liénard
663:Faraday
648:Coulomb
475:Voltage
450:Ohm law
72:History
7218:Photon
7116:
7097:
7061:
7038:
7030:
6987:
6979:
6925:
6867:
6771:) is:
6751:where
6708:) is:
6283:where
5955:where
5819:is an
5815:, any
5634:, the
5493:Yennie
5322:gauges
5308:gauges
5113:where
4627:where
4391:where
4025:where
3204:where
2817:, and
2046:orders
1950:Zumino
1915:Yukawa
1905:Witten
1900:Wilson
1890:Wigner
1825:Tyutin
1785:Skyrme
1735:Ruelle
1700:Plefka
1695:Peskin
1685:Parisi
1645:MĂžller
1635:Migdal
1620:Maiani
1615:LĂŒders
1580:Landau
1575:Kuraev
1545:KÀllén
1535:Jordan
1520:Jackiw
1460:Gribov
1350:DeWitt
1345:Dashen
1335:Callan
1305:Bleuer
1275:Becchi
1265:Anselm
783:Singer
778:Savart
758:Ărsted
723:Larmor
713:Kelvin
668:Fizeau
638:AmpĂšre
560:Stator
67:Optics
6985:S2CID
6951:arXiv
6923:S2CID
6897:arXiv
6865:S2CID
6411:ghost
6022:SU(3)
5833:SU(2)
5781:SU(3)
5495:gauge
5328:with
3956:and
3601:) = 0
2700:gauge
2070:Gibbs
2024:shear
2016:field
1945:Zuber
1795:Stora
1760:Segal
1740:Salam
1725:Proca
1720:Popov
1690:Pauli
1670:Oehme
1660:Neveu
1655:Nambu
1640:Mills
1525:Jaffe
1500:Hagen
1495:Higgs
1470:Gupta
1465:Gross
1450:Glimm
1430:Furry
1400:Fierz
1390:Fermi
1385:Fayet
1380:Fadin
1365:Dyson
1355:Dirac
1320:Brout
1295:Bethe
1255:Adler
1014:Tools
808:Weber
803:Volta
793:Tesla
708:Joule
693:Hertz
688:Henry
673:Gauss
555:Rotor
7114:ISBN
7095:ISBN
7059:ISBN
7036:PMID
7028:ISSN
6977:PMID
6700:and
6361:The
6034:and
6020:For
5831:For
5749:and
5312:The
4706:the
4513:by:
3330:and
3262:and
2886:and
2874:The
2852:U(1)
2805:and
2731:U(1)
2679:and
2656:and
2087:and
1910:Yang
1880:Wick
1875:Weyl
1865:Wess
1845:Ward
1540:Jost
1490:Hepp
1480:Haag
1405:Fock
1285:Bell
728:Lenz
653:Davy
643:Biot
7020:doi
6969:doi
6915:doi
6857:doi
6437:or
6369:or
5785:QCD
5783:of
5708:or
5501:= 3
5481:= 1
4729:.)
3826:as
3586:= 0
3341:at
2748:of
2695:).
2094:of
2014:in
1998:of
1930:Zee
1610:Low
1585:Lee
753:Ohm
7395::
7085:;
7034:.
7026:.
7018:.
7008:36
7006:.
6983:.
6975:.
6967:.
6959:.
6947:86
6921:.
6913:.
6905:.
6893:70
6891:.
6877:^
6863:.
6855:.
6845:24
6843:.
6759:.
6469:0.
6433:,
5757:.
5721:ΌΜ
5698:ΌΜ
5686:.
5503:.
5278:0.
5219:0.
5175:.
4710:.
4697:,
4615:0.
4570:0.
4431:SI
4400:,
3794:.
3597:,
3364:,
3345:.
3213:,
2965:)
2866:.
2821:.
2813:,
2723:,
2669:,
2328:)
2002:,
7150:e
7143:t
7136:v
7122:.
7103:.
7067:.
7042:.
7022::
7014::
6991:.
6971::
6963::
6953::
6929:.
6917::
6909::
6899::
6871:.
6859::
6851::
6807:2
6803:k
6799:=
6790:A
6780:A
6753:x
6739:0
6736:=
6727:A
6717:x
6673:.
6670:u
6667:d
6664:)
6661:t
6658:,
6654:r
6650:u
6647:(
6643:E
6637:1
6632:0
6620:r
6613:=
6610:)
6607:t
6604:,
6600:r
6596:(
6574:u
6571:d
6567:u
6564:)
6561:t
6558:,
6554:r
6550:u
6547:(
6543:B
6537:1
6532:0
6520:r
6513:=
6510:)
6507:t
6504:,
6500:r
6496:(
6492:A
6466:=
6462:A
6454:r
6391:0
6388:=
6328:a
6318:a
6309:A
6305:=
6294:A
6271:,
6266:]
6260:2
6255:)
6250:7
6241:A
6237:(
6232:+
6227:2
6222:)
6217:6
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6199:+
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6184:5
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6095:2
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6085:1
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6066:[
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6057:D
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6039:8
6036:λ
6032:3
6029:λ
6006:.
6000:a
5990:a
5981:A
5977:=
5966:A
5943:,
5938:]
5932:2
5927:)
5922:2
5913:A
5909:(
5904:+
5899:2
5894:)
5889:1
5880:A
5876:(
5870:[
5866:x
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5857:d
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5840:Ï
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5638:Ο
5636:R
5620:Ο
5617:R
5584:2
5580:B
5574:2
5566:+
5557:A
5542:B
5539:=
5534:L
5519:B
5511:Ο
5508:R
5499:Ο
5488:Ο
5485:R
5479:Ο
5466:Ο
5460:Ο
5457:R
5452:Ο
5430:2
5424:2
5419:)
5409:A
5394:(
5384:=
5379:L
5341:L
5319:Ο
5315:R
5305:Ο
5301:R
5275:=
5266:j
5216:=
5207:A
5158:]
5152:j
5147:,
5140:c
5135:[
5131:=
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5095:j
5089:0
5081:=
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4999:=
4991:2
4987:t
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4932:J
4926:0
4918:=
4913:A
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4890:2
4886:t
4876:A
4870:2
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4851:c
4847:1
4822:0
4809:=
4799:2
4783:2
4779:t
4765:2
4750:2
4746:c
4742:1
4702:Ό
4699:â
4680:]
4674:A
4669:,
4660:c
4657:1
4649:[
4645:=
4636:A
4612:=
4603:A
4567:=
4561:t
4542:c
4539:1
4534:+
4529:A
4497:0
4494:=
4488:t
4467:2
4463:c
4459:1
4454:+
4449:A
4404:)
4402:t
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4396:(
4394:Ï
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4341:3
4335:d
4329:R
4323:4
4318:)
4315:t
4312:,
4304:r
4299:(
4295:B
4279:=
4276:)
4273:t
4270:,
4266:r
4262:(
4258:A
4235:t
4226:)
4223:t
4220:,
4216:r
4212:(
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4184:3
4178:d
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4166:4
4161:)
4158:t
4155:,
4147:r
4142:(
4137:E
4115:=
4112:)
4109:t
4106:,
4102:r
4098:(
4066:z
4060:,
4051:y
4045:,
4036:x
4013:,
4004:y
3998:)
3989:x
3979:r
3975:(
3972:B
3969:=
3965:A
3938:z
3932:B
3929:=
3925:B
3914:Ο
3911:R
3902:A
3898:A
3883:B
3875:r
3869:2
3866:1
3858:=
3855:)
3852:t
3849:,
3845:r
3841:(
3836:A
3820:B
3776:.
3768:r
3761:3
3757:d
3751:r
3745:3
3741:d
3734:R
3728:4
3723:)
3720:t
3717:,
3709:r
3704:(
3700:B
3693:)
3690:t
3687:,
3683:r
3679:(
3675:B
3665:=
3661:r
3655:3
3651:d
3647:)
3644:t
3641:,
3637:r
3633:(
3628:2
3623:A
3608:A
3599:t
3595:r
3593:(
3591:Ï
3584:Ï
3581:â
3564:.
3556:r
3549:3
3545:d
3540:]
3533:d
3525:3
3521:R
3514:R
3505:)
3496:t
3493:,
3485:r
3480:(
3476:J
3462:c
3458:/
3454:R
3449:0
3440:[
3421:0
3410:4
3406:1
3401:=
3398:)
3395:t
3392:,
3388:r
3384:(
3380:A
3368:)
3366:t
3362:r
3360:(
3358:J
3343:r
3335:r
3332:d
3328:r
3304:r
3300:r
3285:|
3281:R
3277:|
3273:=
3270:R
3245:r
3236:r
3232:=
3228:R
3217:)
3215:t
3211:r
3209:(
3207:Ï
3187:r
3180:3
3176:d
3169:R
3163:4
3158:)
3155:t
3152:,
3144:r
3139:(
3135:B
3119:=
3116:)
3113:t
3110:,
3106:r
3102:(
3098:A
3073:r
3066:3
3062:d
3056:R
3052:)
3049:t
3046:,
3038:r
3033:(
3014:0
3003:4
2999:1
2994:=
2991:)
2988:t
2985:,
2981:r
2977:(
2935:.
2931:0
2928:=
2925:)
2922:t
2919:,
2915:r
2911:(
2906:A
2783:.
2773:+
2769:A
2761:A
2750:A
2727:)
2725:t
2721:r
2719:(
2717:Ï
2708:Ï
2692:2
2686:1
2681:Ï
2677:A
2673:)
2671:t
2667:r
2665:(
2663:Ï
2658:B
2654:E
2650:E
2641:)
2639:2
2637:(
2617:t
2562:.
2556:t
2546:A
2531:)
2524:t
2505:+
2498:(
2488:=
2482:t
2454:t
2444:A
2421:=
2417:E
2406:E
2390:.
2385:A
2374:=
2371:)
2362:+
2357:A
2352:(
2343:=
2338:B
2316:0
2313:=
2291:B
2282:)
2280:1
2278:(
2255:+
2251:A
2243:A
2215:.
2210:A
2199:=
2194:B
2188:,
2181:t
2171:A
2148:=
2143:E
2131:A
2092:B
2085:E
2068:â
1983:e
1976:t
1969:v
842:e
835:t
828:v
20:)
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