2830:
1947:.) In short, gauge-fixing is required, but it breaks the overall gauge invariance of the theory. The Ward–Takahashi identities then describe exactly how all of the different fields are tied to one-another, under an infinitessimal gauge transformation. These Ward–Takahashi identities are generated by the Ward operator; in the linearized form, the Ward operator is the
1557:
1708:
2096:
340:
908:
96:. Such symmetries in quantum field theory (almost) always give rise to these generalized Ward–Takahashi identities which impose the symmetry on the level of the quantum mechanical amplitudes. This generalized sense should be distinguished when reading literature, such as
2291:. The breaking is due to quantization, that is, due to the anomalous Ward–Takahashi identity (rather than to a Higgs-style Mexican-hat potential, which results in an entirely different kind of symmetry breaking). The divergence of the axial current relates the
1869:
2279:. It is not possible to quantize both at the same time (due to the anomalous Ward–Takahashi identity); by convention, the vector symmetry is quantized so that the vector current is conserved, while the axial vector current is not conserved. The
1431:
1358:
1607:
1984:
1061:
163:
1934:
of a classical gauge theory. The path-integral (Lagrangian) formulation of quantum field theory does not entirely avoid the need for gauge-fixing, as there is still a need to compute the asymptotic states of the
2406:
2261:
571:
2211:
576:
1180:
1772:
1757:
1251:
961:
has its external electrons on-shell, then the amplitudes on the right-hand side of this identity each have one external particle off-shell, and therefore they do not contribute to
1117:
390:
2326:
555:
1924:
524:
480:
432:
1409:
959:
1552:{\displaystyle \delta _{\varepsilon }S=\int \left(\partial _{\mu }\varepsilon \right)J^{\mu }\mathrm {d} ^{d}x=-\int \varepsilon \partial _{\mu }J^{\mu }\mathrm {d} ^{d}x}
1271:
2605:
2463:
2119:
1287:
2433:
2139:
1588:
1892:
1703:{\displaystyle \langle \delta _{\varepsilon }{\mathcal {F}}\rangle -i\int \varepsilon \langle {\mathcal {F}}\partial _{\mu }J^{\mu }\rangle \mathrm {d} ^{d}x=0}
1381:
1081:
931:
2091:{\displaystyle \int \delta _{\varepsilon }\left({\mathcal {F}}e^{iS}\right){\mathcal {D}}\phi =\int \varepsilon \lambda {\mathcal {F}}e^{iS}\mathrm {d} ^{d}x}
349:
39:
335:{\displaystyle {\mathcal {M}}(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=\epsilon _{\mu }(k){\mathcal {M}}^{\mu }(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})}
988:
2598:
2571:
92:
More generally, a Ward–Takahashi identity is the quantum version of classical current conservation associated to a continuous symmetry by
2338:
2216:
2833:
903:{\displaystyle {\begin{aligned}k_{\mu }{\mathcal {M}}^{\mu }(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=e\sum _{i}\left\end{aligned}}}
2860:
2705:
2591:
2628:
2333:
133:
describes how a scalar term in a
Lagrangian transforms under infinitesimal gauge transformations. It is closely related to the
2176:
123:
66:
1129:
2638:
1864:{\displaystyle \int \delta _{\varepsilon }\left({\mathcal {F}}e^{i\left(S+S_{gf}\right)}\right){\mathcal {D}}\phi =0}
2776:
2715:
2643:
1213:
1719:
1412:
2669:
2292:
2329:
2855:
2614:
1229:
346:
54:
2791:
1218:
In the path integral formulation, the Ward–Takahashi identities are a reflection of the invariance of the
119:
86:
78:
1086:
359:
101:
2298:
1219:
1185:
Physically, what this identity means is the longitudinal polarization of the photon which arises in the
558:
529:
2268:
1897:
489:
445:
2465:. In this way, the chiral anomaly provides the canonical description of the pion-nuclean interaction.
399:
2801:
2526:
2491:
2288:
1714:
1591:
1223:
93:
31:
1390:
940:
2745:
2436:
1944:
1931:
1197:
1120:
393:
82:
43:
1595:
1256:
2811:
2786:
2542:
2173:, in an isospin triplet. This theory has not one, but two distinct global symmetries: the vector
1353:{\displaystyle \int \delta _{\varepsilon }\left({\mathcal {F}}e^{iS}\right){\mathcal {D}}\phi =0}
2710:
137:
and plays a central role in providing a geometric description of the consistent quantization of
2771:
2740:
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2441:
2104:
1936:
1563:
934:
115:
62:
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2689:
2534:
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1964:
1952:
1384:
2411:
2124:
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2633:
2264:
1972:
1421:
1416:
1274:
1201:
74:
58:
47:
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2146:
1968:
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916:
150:
138:
97:
2849:
2546:
2276:
2154:
1948:
134:
108:
1960:
1927:
111:
1063:
be the amplitude for some QED process involving an external photon with momentum
2730:
2720:
2684:
2659:
2284:
2150:
1956:
1273:(and this applies even in the case where the physical symmetry of the system is
17:
2796:
2725:
1186:
978:
1963:, the Ward–Takahashi identities correspond to a (global) right-action in the
1056:{\displaystyle {\mathcal {M}}(k)=\epsilon _{\mu }(k){\mathcal {M}}^{\mu }(k)}
2280:
2272:
114:
to which the Ward–Takahashi identity applies. The equivalent identities for
2503:
1978:
When the functional measure is not gauge invariant, but happens to satisfy
2583:
2679:
2664:
982:
974:
962:
439:
154:
70:
973:
The Ward identity is a specialization of the Ward–Takahashi identity to
2538:
2166:
2158:
2674:
2162:
1193:
565:
from our original amplitude. Then the Ward–Takahashi identity reads
353:
2482:
Ward, John Clive (1950). "An
Identity in Quantum Electrodynamics".
1930:
term. Gauge-fixing terms are required so as to be able to perform
2401:{\displaystyle f_{\pi }g_{\pi N{\overline {N}}}\simeq g_{A}m_{N}}
2256:{\displaystyle {\overline {\psi }}\gamma _{5}\gamma _{\mu }\psi }
2170:
149:
The Ward–Takahashi identity applies to correlation functions in
2587:
2566:. Westview Press. Section 7.4 ("The Ward-Takahashi identity").
2517:
Takahashi, Yasushi (1957). "On the generalized ward identity".
2055:
2033:
2008:
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536:
260:
169:
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698:
153:, which do not necessarily have all their external momenta
1363:
expresses the invariance of the functional measure where
104:'s textbook, from the original Ward–Takahashi identity.
2287:
of the vector symmetry, whereas the axial symmetry is
2206:{\displaystyle {\overline {\psi }}\gamma _{\mu }\psi }
561:
that is obtained by removing the photon with momentum
2444:
2414:
2341:
2301:
2219:
2179:
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1762:
If the gauge transformation corresponds to an actual
1722:
1610:
1576:
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1369:
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1259:
1232:
1132:
1123:
vector of the photon. Then the Ward identity reads:
1089:
1069:
991:
981:
processes and thus have all their external particles
943:
919:
574:
532:
492:
448:
402:
362:
166:
27:
Identity in abelian theories due to gauge invariance
2764:
2698:
2652:
2621:
1277:or even nonexistent; we are only worried about the
85:. Later uses include the extension of the proof of
2457:
2427:
2400:
2320:
2255:
2205:
2133:
2113:
2090:
1918:
1886:
1863:
1751:
1702:
1582:
1551:
1403:
1375:
1352:
1265:
1245:
1175:{\displaystyle k_{\mu }{\mathcal {M}}^{\mu }(k)=0}
1174:
1111:
1075:
1055:
953:
925:
902:
549:
518:
474:
426:
384:
334:
2560:Peskin, Michael E.; Schroeder, Daniel V. (1995).
1189:is unphysical and disappears from the S-matrix.
1419:. If the gauge transformation corresponds to a
107:The detailed discussion below concerns QED, an
2599:
2263:symmetries; equivalently, the left and right
1192:Examples of its use include constraining the
977:elements, which describe physically possible
46:of the theory, and which remains valid after
8:
1676:
1646:
1631:
1611:
1959:. When the gauge theory is formulated on a
1601:Then, the Ward–Takahashi identities become
1208:Derivation in the path integral formulation
2606:
2592:
2584:
1752:{\displaystyle \partial _{\mu }J^{\mu }=0}
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2443:
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2413:
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834:
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226:
213:
200:
187:
168:
167:
165:
396:vector of the photon and summation over
2563:An Introduction to Quantum Field Theory
2474:
2141:, the corresponding relation gives the
937:and is negative in sign. Note that if
77:, guaranteeing the cancellation of the
1246:{\displaystyle \delta _{\varepsilon }}
89:to all orders of perturbation theory.
42:that follows from the global or gauge
2267:. The corresponding currents are the
1253:represents a gauge transformation by
7:
1279:invariance of the functional measure
2149:. This example is prominent in the
1112:{\displaystyle \epsilon _{\mu }(k)}
486:final-state electrons with momenta
385:{\displaystyle \epsilon _{\mu }(k)}
2321:{\displaystyle g_{A}\approx 1.267}
2145:. The conventional example is the
2075:
1910:
1907:
1894:is the gauge invariant action and
1724:
1681:
1657:
1536:
1515:
1489:
1460:
550:{\displaystyle {\mathcal {M}}_{0}}
25:
2834:Template:Quantum mechanics topics
2169:doublet, feel forces mediated by
2143:anomalous Ward–Takahashi identity
2121:is some functional of the fields
1919:{\displaystyle S_{\mathrm {gf} }}
519:{\displaystyle q_{1}\cdots q_{n}}
475:{\displaystyle p_{1}\cdots p_{n}}
2829:
2828:
2706:Anomalous magnetic dipole moment
427:{\displaystyle \mu =0,\ldots ,3}
1570:(as a functional of the fields
53:The Ward–Takahashi identity of
1713:This is the QFT analog of the
1425:symmetry of the theory, then,
1404:{\displaystyle {\mathcal {F}}}
1163:
1157:
1106:
1100:
1050:
1044:
1027:
1021:
1005:
999:
954:{\displaystyle {\mathcal {M}}}
888:
846:
827:
811:
782:
766:
747:
705:
661:
603:
379:
373:
329:
271:
254:
248:
232:
174:
1:
75:vertex renormalization factor
67:wave function renormalization
57:(QED) was originally used by
2368:
2225:
2185:
1967:: they are generated by the
1266:{\displaystyle \varepsilon }
2629:Euler–Heisenberg Lagrangian
2334:Goldberger–Treiman relation
2283:is then interpreted as the
1715:Noether continuity equation
2877:
1211:
2820:
2644:Path integral formulation
1926:is a non-gauge-invariant
1214:Path integral formulation
124:Slavnov–Taylor identities
2812:Photon-photon scattering
2458:{\displaystyle f_{\pi }}
2293:pion-nucleon interaction
2114:{\displaystyle \lambda }
2861:Quantum electrodynamics
2756:Ward–Takahashi identity
2639:Gupta–Bleuler formalism
2615:Quantum electrodynamics
2330:axial coupling constant
356:with momentum k (where
145:Ward–Takahashi identity
55:quantum electrodynamics
38:is an identity between
36:Ward–Takahashi identity
2504:10.1103/PhysRev.78.182
2459:
2429:
2402:
2322:
2295:to pion decay, fixing
2257:
2207:
2157:. In this theory, the
2135:
2115:
2092:
1920:
1888:
1865:
1753:
1704:
1594:and assuming that the
1584:
1553:
1405:
1377:
1354:
1267:
1247:
1176:
1113:
1077:
1057:
955:
935:charge of the electron
927:
904:
551:
520:
476:
428:
386:
352:involving an external
336:
120:quantum chromodynamics
79:ultraviolet divergence
2777:Breit–Wheeler process
2716:Klein–Nishina formula
2460:
2430:
2428:{\displaystyle g_{A}}
2403:
2323:
2258:
2213:and the axial vector
2208:
2136:
2134:{\displaystyle \phi }
2116:
2093:
1921:
1889:
1866:
1754:
1705:
1585:
1583:{\displaystyle \phi }
1554:
1406:
1378:
1355:
1268:
1248:
1226:. More precisely, if
1177:
1114:
1078:
1058:
956:
928:
905:
552:
521:
477:
429:
387:
337:
40:correlation functions
2442:
2412:
2339:
2299:
2289:spontaneously broken
2277:axial vector current
2217:
2177:
2125:
2105:
1985:
1951:. The corresponding
1898:
1878:
1773:
1720:
1608:
1592:integrating by parts
1574:
1432:
1391:
1367:
1288:
1257:
1230:
1224:gauge transformation
1200:and of the electron
1130:
1087:
1067:
989:
941:
917:
572:
530:
490:
446:
400:
360:
350:correlation function
164:
32:quantum field theory
2792:Delbrück scattering
2746:Vacuum polarization
2670:Faddeev–Popov ghost
2531:1957NCim....6..371T
2496:1950PhRv...78..182W
2437:pion decay constant
1945:interaction picture
1932:second quantization
1198:vacuum polarization
87:Goldstone's theorem
83:perturbation theory
2787:Compton scattering
2539:10.1007/BF02832514
2455:
2425:
2398:
2318:
2253:
2203:
2131:
2111:
2088:
1916:
1884:
1861:
1749:
1700:
1598:can be neglected.
1580:
1549:
1401:
1373:
1350:
1263:
1243:
1220:functional measure
1172:
1109:
1073:
1053:
951:
923:
900:
898:
679:
557:to be the simpler
547:
516:
472:
424:
382:
332:
2843:
2842:
2802:Møller scattering
2772:Bhabha scattering
2741:Uehling potential
2690:Virtual particles
2573:978-0-201-50397-5
2371:
2269:isovector current
2265:chiral symmetries
2228:
2188:
1937:scattering matrix
1887:{\displaystyle S}
1376:{\displaystyle S}
1196:structure of the
1076:{\displaystyle k}
926:{\displaystyle e}
670:
118:theories such as
94:Noether's theorem
81:to all orders of
63:Yasushi Takahashi
16:(Redirected from
2868:
2832:
2831:
2807:Schwinger effect
2608:
2601:
2594:
2585:
2578:
2577:
2557:
2551:
2550:
2519:Il Nuovo Cimento
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2012:
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1999:
1965:principle bundle
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102:Daniel Schroeder
21:
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2867:
2866:
2865:
2846:
2845:
2844:
2839:
2838:
2816:
2760:
2751:Vertex function
2736:Schwinger limit
2711:Furry's theorem
2694:
2648:
2634:Feynman diagram
2617:
2612:
2582:
2581:
2574:
2559:
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2554:
2516:
2515:
2511:
2484:Physical Review
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2174:
2123:
2122:
2103:
2102:
2073:
2060:
2013:
2005:
2001:
1991:
1983:
1982:
1973:vertical bundle
1901:
1896:
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1364:
1316:
1308:
1304:
1294:
1286:
1285:
1255:
1254:
1233:
1228:
1227:
1216:
1210:
1202:vertex function
1143:
1133:
1128:
1127:
1090:
1085:
1084:
1065:
1064:
1030:
1011:
987:
986:
971:
939:
938:
915:
914:
897:
896:
878:
865:
852:
830:
814:
797:
793:
790:
786:
785:
772:
750:
734:
721:
708:
701:
684:
680:
651:
638:
625:
612:
589:
579:
570:
569:
533:
528:
527:
526:. Also define
506:
493:
488:
487:
462:
449:
444:
443:
398:
397:
363:
358:
357:
319:
306:
293:
280:
257:
238:
222:
209:
196:
183:
162:
161:
147:
59:John Clive Ward
48:renormalization
28:
23:
22:
18:Ward identities
15:
12:
11:
5:
2874:
2872:
2864:
2863:
2858:
2856:Gauge theories
2848:
2847:
2841:
2840:
2837:
2836:
2822:
2821:
2818:
2817:
2815:
2814:
2809:
2804:
2799:
2794:
2789:
2784:
2782:Bremsstrahlung
2779:
2774:
2768:
2766:
2762:
2761:
2759:
2758:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2702:
2700:
2696:
2695:
2693:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2656:
2654:
2650:
2649:
2647:
2646:
2641:
2636:
2631:
2625:
2623:
2619:
2618:
2613:
2611:
2610:
2603:
2596:
2588:
2580:
2579:
2572:
2552:
2525:(2): 371–375.
2509:
2473:
2472:
2470:
2467:
2452:
2448:
2422:
2418:
2395:
2391:
2385:
2381:
2377:
2370:
2367:
2362:
2359:
2355:
2349:
2345:
2317:
2314:
2309:
2305:
2252:
2247:
2243:
2237:
2233:
2227:
2224:
2202:
2197:
2193:
2187:
2184:
2155:nuclear forces
2147:chiral anomaly
2130:
2110:
2099:
2098:
2087:
2082:
2077:
2070:
2067:
2063:
2057:
2052:
2049:
2046:
2043:
2040:
2035:
2029:
2023:
2020:
2016:
2010:
2004:
1998:
1994:
1990:
1969:Lie derivative
1912:
1909:
1904:
1883:
1872:
1871:
1860:
1857:
1854:
1849:
1843:
1836:
1830:
1827:
1823:
1819:
1816:
1812:
1808:
1804:
1798:
1792:
1786:
1782:
1778:
1764:gauge symmetry
1748:
1745:
1740:
1736:
1730:
1726:
1711:
1710:
1699:
1696:
1693:
1688:
1683:
1678:
1673:
1669:
1663:
1659:
1653:
1648:
1645:
1642:
1639:
1636:
1633:
1628:
1621:
1617:
1613:
1579:
1560:
1559:
1548:
1543:
1538:
1531:
1527:
1521:
1517:
1513:
1510:
1507:
1504:
1501:
1496:
1491:
1484:
1480:
1475:
1471:
1466:
1462:
1457:
1453:
1450:
1447:
1442:
1438:
1398:
1372:
1361:
1360:
1349:
1346:
1343:
1338:
1332:
1326:
1323:
1319:
1313:
1307:
1301:
1297:
1293:
1262:
1240:
1236:
1209:
1206:
1183:
1182:
1171:
1168:
1165:
1162:
1159:
1154:
1148:
1140:
1136:
1108:
1105:
1102:
1097:
1093:
1072:
1052:
1049:
1046:
1041:
1035:
1029:
1026:
1023:
1018:
1014:
1010:
1007:
1004:
1001:
996:
970:
967:
948:
922:
911:
910:
894:
890:
885:
881:
877:
872:
868:
864:
859:
855:
851:
848:
845:
842:
837:
833:
829:
826:
821:
817:
813:
808:
802:
796:
792:
789:
787:
784:
779:
775:
771:
768:
765:
762:
757:
753:
749:
746:
741:
737:
733:
728:
724:
720:
715:
711:
707:
704:
702:
699:
695:
689:
683:
677:
673:
669:
666:
663:
658:
654:
650:
645:
641:
637:
632:
628:
624:
619:
615:
611:
608:
605:
600:
594:
586:
582:
578:
577:
544:
538:
513:
509:
505:
500:
496:
469:
465:
461:
456:
452:
438:initial-state
423:
420:
417:
414:
411:
408:
405:
381:
378:
375:
370:
366:
343:
342:
331:
326:
322:
318:
313:
309:
305:
300:
296:
292:
287:
283:
279:
276:
273:
268:
262:
256:
253:
250:
245:
241:
237:
234:
229:
225:
221:
216:
212:
208:
203:
199:
195:
190:
186:
182:
179:
176:
171:
151:momentum space
146:
143:
139:gauge theories
122:(QCD) are the
98:Michael Peskin
65:to relate the
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2873:
2862:
2859:
2857:
2854:
2853:
2851:
2835:
2827:
2824:
2823:
2819:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2769:
2767:
2763:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2703:
2701:
2697:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2657:
2655:
2651:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2626:
2624:
2620:
2616:
2609:
2604:
2602:
2597:
2595:
2590:
2589:
2586:
2575:
2569:
2565:
2564:
2556:
2553:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2513:
2510:
2505:
2501:
2497:
2493:
2489:
2485:
2478:
2475:
2468:
2466:
2450:
2446:
2438:
2420:
2416:
2393:
2389:
2383:
2379:
2375:
2365:
2360:
2357:
2353:
2347:
2343:
2335:
2331:
2315:
2312:
2307:
2303:
2294:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2250:
2245:
2241:
2235:
2231:
2222:
2200:
2195:
2191:
2182:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2128:
2108:
2085:
2080:
2068:
2065:
2061:
2050:
2047:
2044:
2041:
2038:
2027:
2021:
2018:
2014:
2002:
1996:
1992:
1988:
1981:
1980:
1979:
1976:
1974:
1970:
1966:
1962:
1958:
1954:
1950:
1949:BRST operator
1946:
1942:
1938:
1933:
1929:
1902:
1881:
1858:
1855:
1852:
1841:
1834:
1828:
1825:
1821:
1817:
1814:
1810:
1806:
1802:
1790:
1784:
1780:
1776:
1769:
1768:
1767:
1765:
1760:
1746:
1743:
1738:
1734:
1728:
1716:
1697:
1694:
1691:
1686:
1671:
1667:
1661:
1643:
1640:
1637:
1634:
1619:
1615:
1604:
1603:
1602:
1599:
1597:
1596:surface terms
1593:
1577:
1569:
1565:
1546:
1541:
1529:
1525:
1519:
1511:
1508:
1505:
1502:
1499:
1494:
1482:
1478:
1473:
1469:
1464:
1455:
1451:
1448:
1445:
1440:
1436:
1428:
1427:
1426:
1424:
1423:
1418:
1414:
1386:
1370:
1347:
1344:
1341:
1330:
1324:
1321:
1317:
1305:
1299:
1295:
1291:
1284:
1283:
1282:
1280:
1276:
1260:
1238:
1234:
1225:
1221:
1215:
1207:
1205:
1203:
1199:
1195:
1190:
1188:
1169:
1166:
1160:
1152:
1138:
1134:
1126:
1125:
1124:
1122:
1103:
1095:
1091:
1070:
1047:
1039:
1024:
1016:
1012:
1008:
1002:
985:. Again let
984:
980:
976:
969:Ward identity
968:
966:
964:
936:
920:
892:
883:
879:
875:
870:
866:
862:
857:
853:
849:
843:
840:
835:
831:
824:
819:
815:
806:
794:
777:
773:
769:
763:
760:
755:
751:
744:
739:
735:
731:
726:
722:
718:
713:
709:
703:
693:
681:
675:
671:
667:
664:
656:
652:
648:
643:
639:
635:
630:
626:
622:
617:
613:
609:
606:
598:
584:
580:
568:
567:
566:
564:
560:
542:
511:
507:
503:
498:
494:
485:
467:
463:
459:
454:
450:
442:with momenta
441:
437:
434:is implied),
421:
418:
415:
412:
409:
406:
403:
395:
376:
368:
364:
355:
351:
348:
324:
320:
316:
311:
307:
303:
298:
294:
290:
285:
281:
277:
274:
266:
251:
243:
239:
235:
227:
223:
219:
214:
210:
206:
201:
197:
193:
188:
184:
180:
177:
160:
159:
158:
156:
152:
144:
142:
140:
136:
135:BRST operator
132:
131:Ward operator
127:
125:
121:
117:
113:
110:
105:
103:
99:
95:
90:
88:
84:
80:
76:
72:
68:
64:
60:
56:
51:
49:
45:
41:
37:
33:
19:
2825:
2755:
2562:
2555:
2522:
2518:
2512:
2487:
2483:
2477:
2142:
2100:
1977:
1961:fiber bundle
1940:
1928:gauge fixing
1873:
1761:
1712:
1600:
1567:
1561:
1420:
1362:
1281:here), then
1278:
1217:
1191:
1187:ξ gauge
1184:
1121:polarization
972:
912:
562:
483:
435:
394:polarization
344:
148:
130:
128:
106:
91:
52:
35:
29:
2731:Self-energy
2721:Landau pole
2685:Positronium
2660:Dual photon
2285:gauge boson
2151:sigma model
1957:BRST charge
116:non-abelian
2850:Categories
2797:Lamb shift
2726:QED vacuum
2490:(2): 182.
2469:References
2275:) and the
2153:theory of
1562:for some "
1413:functional
1212:See also:
979:scattering
965:elements.
44:symmetries
2826:See also:
2765:Processes
2653:Particles
2622:Formalism
2547:121528462
2451:π
2376:≃
2369:¯
2358:π
2348:π
2313:≈
2281:rho meson
2273:rho meson
2251:ψ
2246:μ
2242:γ
2232:γ
2226:¯
2223:ψ
2201:ψ
2196:μ
2192:γ
2186:¯
2183:ψ
2129:ϕ
2109:λ
2051:λ
2048:ε
2045:∫
2039:ϕ
1997:ε
1993:δ
1989:∫
1853:ϕ
1785:ε
1781:δ
1777:∫
1739:μ
1729:μ
1725:∂
1677:⟩
1672:μ
1662:μ
1658:∂
1647:⟨
1644:ε
1641:∫
1635:−
1632:⟩
1620:ε
1616:δ
1612:⟨
1578:ϕ
1530:μ
1520:μ
1516:∂
1512:ε
1509:∫
1506:−
1483:μ
1470:ε
1465:μ
1461:∂
1452:∫
1441:ε
1437:δ
1342:ϕ
1300:ε
1296:δ
1292:∫
1261:ε
1239:ε
1235:δ
1153:μ
1139:μ
1096:μ
1092:ϵ
1040:μ
1017:μ
1013:ϵ
876:⋯
850:⋯
825:⋯
795:−
770:⋯
761:−
745:⋯
719:⋯
672:∑
649:⋯
623:⋯
599:μ
585:μ
559:amplitude
504:⋯
460:⋯
440:electrons
416:…
404:μ
369:μ
365:ϵ
317:⋯
291:⋯
267:μ
244:μ
240:ϵ
220:⋯
194:⋯
2699:Concepts
2680:Positron
2665:Electron
2408:relates
2165:, in an
1590:) after
1222:under a
1204:in QED.
1083:, where
983:on-shell
975:S-matrix
963:S-matrix
155:on-shell
71:electron
2527:Bibcode
2492:Bibcode
2435:to the
2328:as the
2167:isospin
2159:neutron
1971:on the
1955:is the
1943:in the
1564:current
1415:of the
1383:is the
1119:is the
933:is the
392:is the
157:. Let
109:abelian
73:to its
69:of the
2675:Photon
2570:
2545:
2332:. The
2163:proton
1953:charge
1874:where
1422:global
1417:fields
1385:action
1275:global
1194:tensor
913:where
482:, and
354:photon
112:theory
2543:S2CID
2316:1.267
2271:(the
2171:pions
2101:with
1766:then
1411:is a
345:be a
2568:ISBN
2161:and
1387:and
129:The
100:and
61:and
34:, a
2535:doi
2500:doi
1941:e.g
347:QED
126:.
30:In
2852::
2541:.
2533:.
2521:.
2498:.
2488:78
2486:.
1975:.
1759:.
1566:"
141:.
50:.
2607:e
2600:t
2593:v
2576:.
2549:.
2537::
2529::
2523:6
2506:.
2502::
2494::
2447:f
2421:A
2417:g
2394:N
2390:m
2384:A
2380:g
2366:N
2361:N
2354:g
2344:f
2308:A
2304:g
2236:5
2086:x
2081:d
2076:d
2069:S
2066:i
2062:e
2056:F
2042:=
2034:D
2028:)
2022:S
2019:i
2015:e
2009:F
2003:(
1939:(
1911:f
1908:g
1903:S
1882:S
1859:0
1856:=
1848:D
1842:)
1835:)
1829:f
1826:g
1822:S
1818:+
1815:S
1811:(
1807:i
1803:e
1797:F
1791:(
1747:0
1744:=
1735:J
1698:0
1695:=
1692:x
1687:d
1682:d
1668:J
1652:F
1638:i
1627:F
1568:J
1547:x
1542:d
1537:d
1526:J
1503:=
1500:x
1495:d
1490:d
1479:J
1474:)
1456:(
1449:=
1446:S
1397:F
1371:S
1348:0
1345:=
1337:D
1331:)
1325:S
1322:i
1318:e
1312:F
1306:(
1170:0
1167:=
1164:)
1161:k
1158:(
1147:M
1135:k
1107:)
1104:k
1101:(
1071:k
1051:)
1048:k
1045:(
1034:M
1028:)
1025:k
1022:(
1009:=
1006:)
1003:k
1000:(
995:M
947:M
921:e
893:]
889:)
884:n
880:q
871:1
867:q
863:;
858:n
854:p
847:)
844:k
841:+
836:i
832:p
828:(
820:1
816:p
812:(
807:0
801:M
783:)
778:n
774:q
767:)
764:k
756:i
752:q
748:(
740:1
736:q
732:;
727:n
723:p
714:1
710:p
706:(
694:0
688:M
682:[
676:i
668:e
665:=
662:)
657:n
653:q
644:1
640:q
636:;
631:n
627:p
618:1
614:p
610:;
607:k
604:(
593:M
581:k
563:k
543:0
537:M
512:n
508:q
499:1
495:q
484:n
468:n
464:p
455:1
451:p
436:n
422:3
419:,
413:,
410:0
407:=
380:)
377:k
374:(
330:)
325:n
321:q
312:1
308:q
304:;
299:n
295:p
286:1
282:p
278:;
275:k
272:(
261:M
255:)
252:k
249:(
236:=
233:)
228:n
224:q
215:1
211:q
207:;
202:n
198:p
189:1
185:p
181:;
178:k
175:(
170:M
20:)
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