2931:
2570:
2926:{\displaystyle {\begin{aligned}\langle n|n+\nu \rangle &\sim \int {\mathcal {D}}A{\mathcal {D}}\psi {\mathcal {D}}{\bar {\psi }}\exp {\bigg (}-\int d^{4}x{\frac {1}{2g^{2}}}{\text{tr}}F^{\mu \nu }F_{\mu \nu }+i{\bar {\psi }}{D\!\!\!/}\psi {\bigg )}\\&\sim \int {\mathcal {D}}A\det(i{D\!\!\!/})\exp {\bigg (}-\int d^{4}x{\frac {1}{2g^{2}}}{\text{tr}}F^{\mu \nu }F_{\mu \nu }{\bigg )}.\end{aligned}}}
508:
transformed into every other ground state configuration through a smooth gauge transformation then the theory has a single vacuum state, but if there are topologically distinct configurations then it has multiple vacua. This is because if there are two different configurations that are not smoothly
2118:
804:. Only mappings with the same winding number can be smoothly deformed into each other and are said to belong to the same homotopy class. Gauge transformations which preserve the winding number are called small gauge transformations while ones that change the winding number are called
509:
connected, then to transform one into the other one must pass through a configuration with non-vanishing field strength tensor, which will have non-zero energy. This means that there is an energy barrier between the two vacua, making them distinct.
2407:
1220:
3268:
Baker, C.A.; Doyle, D.D.; Geltenbort, P.; Green, K.; van der Grinten, M.G.D.; Harris, P.G.; Iaydjiev, P.; Ivanov, S.N.; May, D.J.R. (27 September 2006). "Improved experimental limit on the electric dipole moment of the neutron".
2534:
are present in the theory then the vacuum angle becomes unobservable because the fermions suppress the instanton tunnelling between topological vacua. This can be seen by considering a Yang–Mills theory with a single massless fermion
2318:
1981:
1624:
1709:
1502:
2465:
2940:
with massless fermions has at least one zero eigenvalue for any instanton configuration. While instantons no longer contribute to tunnelling between topological vacua, they instead play a role in violating
447:
361:
736:
211:
2575:
1872:
1424:
681:
1958:
1054:
562:
495:
1740:
2192:
904:
1546:
323:
599:
393:
2326:
1810:
1089:
986:
860:
1346:
1901:
1426:. The true vacuum has to be an eigenstate of both small and large gauge transformations. Similarly to the form that eigenstates take in periodic potentials according to
1319:
1275:
296:
240:
2562:
2496:
2520:
1644:
144:
1081:
931:
798:
771:
626:
2936:
This differs from the pure Yang–Mills result by the fermion determinant acquired after integrating over the fermionic fields. The determinant vanishes because the
1775:
2161:
2141:
1366:
1243:
1006:
951:
829:
263:
2208:
1646:
will give a different ground state on which excited states are built, leading to different physics. In other words, the
Hilbert space decomposes into
2113:{\displaystyle \lim _{T\rightarrow \infty }\langle \theta |e^{-iHT}|\theta \rangle =\int {\mathcal {D}}Ae^{iS+i\int d^{4}x{\mathcal {L}}_{\theta }}.}
1558:
3183:
1436:
2416:
1657:
3398:
3252:
3219:
3095:
3342:
2498:
is not gauge invariant. In quantum chromodynamics the presence of this term leads to the strong CP problem since it gives rise to a
3425:
2499:
398:
328:
686:
157:
1815:
1371:
631:
512:
The question of whether two gauge configurations can be smoothly deformed into each other is formally described by the
1919:-vacua physically distinct from each other. The ground state energy of the different vacua is split to take the form
1056:
can be deformed to the constant map and so there is a single connected vacuum state. For a gauge field configuration
2467:. In contrast to other total derivatives that can be added to the Lagrangian, this one has physical consequences in
1922:
1015:
2953:-term is still present, but its effects are heavily suppressed as they must be proportional to the fermion masses.
805:
519:
452:
1972:
1968:
1912:
1009:
2166:
865:
2564:. In the path integral formalism the tunnelling by an instanton between two topological vacua takes the form
1083:, one can always calculate its winding number from a volume integral which in the temporal gauge is given by
3420:
2410:
3065:
1714:
1510:
147:
62:
3239:. Cambridge Monographs in Mathematical Physics. Cambridge: Cambridge University Press. pp. 287–290.
35:
103:
66:
46:
2402:{\displaystyle {\tilde {F}}^{\mu \nu }={\tfrac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}
1215:{\displaystyle n={\frac {ig^{3}}{24\pi ^{2}}}\int d^{3}r\ {\text{Tr}}(\epsilon _{ijk}A^{i}A^{j}A^{k}),}
301:
567:
369:
3378:
3288:
3039:
3000:
54:
28:
20:
1780:
960:
834:
1746:
954:
243:
3312:
3278:
1754:
1324:
1877:
1296:
1252:
272:
216:
1960:, where the constant of proportionality will depend on how strong the instanton tunnelling is.
3394:
3338:
3304:
3248:
3215:
3179:
3137:
3091:
2967:
2946:
2538:
2474:
1246:
70:
2505:
1629:
116:
3386:
3296:
3240:
3207:
3171:
3127:
3047:
3008:
2468:
1289:
Topological vacua are not candidate vacuum states of Yang–Mills theories since they are not
743:
266:
3149:
1059:
909:
776:
749:
604:
3330:
3163:
3145:
1760:
1293:
of large gauge transformations and so aren't gauge invariant. Instead acting on the state
3069:
3382:
3292:
3043:
3004:
3199:
2942:
2937:
2313:{\displaystyle {\mathcal {L}}_{\theta }=\theta {\frac {g^{2}}{32\pi ^{2}}}{\text{Tr}},}
2146:
2126:
1904:
1647:
1351:
1228:
991:
936:
814:
739:
513:
248:
3027:
2988:
3414:
3358:
3316:
3012:
1427:
1290:
801:
363:. Since all points at spatial infinity now behave as a single new point, the spatial
107:
74:
32:
3370:
3362:
2195:
501:
151:
111:
86:
78:
3300:
3366:
82:
53:
distinct vacuum states. The dynamical effects of the vacuum are captured in the
3051:
738:. This means that every mapping has some integer associated with it called its
3111:
505:
449:
so that every pure gauge choice for the gauge field is described by a mapping
3244:
3211:
3175:
3141:
2413:. This term is a total derivative meaning that it can be written in the form
106:
vacuum structure of non-abelian Yang–Mills theories is often investigated in
2962:
1750:
1650:
since expectation values of gauge invariant operators between two different
3308:
1757:
between different topological vacua with an instanton with winding number
1619:{\displaystyle \Omega _{m}|\theta \rangle =e^{-i\theta m}|\theta \rangle }
364:
50:
1555:. They are eigenstates of both types of gauge transformations since now
1249:. The different classes of vacuum states with different winding numbers
3283:
3132:
3115:
2531:
1915:, however instantons lift the degeneracy, making the various different
3116:"An application of the Morse theory to the topology of Lie-groups"
2409:
is the dual field strength tensor and the trace is over the group
1497:{\displaystyle |\theta \rangle =\sum _{n}e^{in\theta }|n\rangle .}
1012:. This is in contrast to abelian gauge groups where every mapping
3390:
2460:{\displaystyle {\mathcal {L}}_{\theta }=\partial _{\mu }K^{\mu }}
1704:{\displaystyle \langle \theta |{\mathcal {O}}|\theta '\rangle =0}
242:
is some gauge transformation belonging to the non-abelian gauge
2502:
which has not yet been observed, requiring the fine tuning of
3337:. Vol. 2. Cambridge University Press. pp. 457–458.
1777:
being responsible for a tunnelling from a topological vacuum
1745:
Yang–Mills theories exhibit finite action solutions to their
2791:
2633:
2623:
2613:
2423:
2215:
2173:
2094:
2050:
1674:
746:, with it roughly describing to how many times the spatial
1430:, the vacuum state is a coherent sum of topological vacua
146:. Classical ground states of this theory have a vanishing
3168:
Advanced Topics in
Quantum Field Theory: A Lecture Course
2949:. If instead the theory has very light fermions then the
1967:-vacuum can be directly incorporated into the Yang–Mills
3088:
Gauge Field
Theories: An Introduction with Applications
3170:. Cambridge: Cambridge University Press. p. 178.
2356:
442:{\displaystyle S^{3}=\mathbb {R} ^{3}\cup \{\infty \}}
356:{\displaystyle |{\boldsymbol {x}}|\rightarrow \infty }
2573:
2541:
2508:
2477:
2419:
2329:
2211:
2169:
2149:
2129:
1984:
1925:
1880:
1818:
1783:
1763:
1717:
1660:
1632:
1561:
1513:
1439:
1374:
1354:
1327:
1299:
1255:
1231:
1092:
1062:
1018:
994:
963:
939:
912:
868:
837:
817:
779:
752:
731:{\displaystyle \pi _{3}({\text{SU}}(2))=\mathbb {Z} }
689:
634:
607:
570:
522:
455:
401:
372:
331:
304:
275:
251:
219:
206:{\displaystyle A_{i}=i\Omega \nabla _{i}\Omega ^{-1}}
160:
119:
1971:
by considering the vacuum-vacuum transitions in the
800:, with negative windings occurring due to a flipped
3028:"Vacuum Periodicity in a Yang–Mills Quantum Theory"
1507:This set of states indexed by the angular variable
2925:
2556:
2514:
2490:
2459:
2401:
2312:
2186:
2155:
2135:
2112:
1952:
1895:
1867:{\displaystyle |n_{+}\rangle =|n_{-}+\nu \rangle }
1866:
1804:
1769:
1734:
1703:
1638:
1618:
1540:
1496:
1419:{\displaystyle \Omega _{m}|n\rangle =|n+m\rangle }
1418:
1360:
1340:
1313:
1269:
1237:
1214:
1075:
1048:
1000:
980:
945:
925:
898:
854:
823:
792:
765:
730:
675:
620:
593:
556:
489:
441:
387:
355:
317:
290:
257:
234:
205:
138:
3377:. World Scientific Publishing. pp. 298–300.
3335:The Quantum Theory of Fields: Modern Applications
2911:
2832:
2814:
2813:
2812:
2771:
2759:
2758:
2757:
2658:
676:{\displaystyle \Omega (x):S^{3}\rightarrow S^{3}}
3206:. Cambridge University Press. pp. 265–350.
2987:Callan, C.G.; Dashen, R.F.; Gross, D.J. (1976).
2799:
1986:
1953:{\displaystyle E(\theta )\propto \cos \theta }
1368:will map it to a different topological vacuum
1049:{\displaystyle S^{3}\rightarrow {\text{U}}(1)}
3120:Bulletin de la Société Mathématique de France
557:{\displaystyle \Omega (x):S^{3}\rightarrow G}
490:{\displaystyle \Omega (x):S^{3}\rightarrow G}
8:
2598:
2578:
2039:
2001:
1861:
1834:
1799:
1692:
1661:
1613:
1580:
1488:
1448:
1413:
1393:
1308:
1264:
436:
430:
45:that arises when the state is written as a
2989:"The structure of the gauge theory vacuum"
2198:contribution to the Lagrangian called the
831:it is sufficient to focus on one of their
3282:
3131:
2910:
2909:
2900:
2887:
2878:
2869:
2856:
2847:
2831:
2830:
2815:
2808:
2790:
2789:
2770:
2769:
2760:
2753:
2742:
2741:
2726:
2713:
2704:
2695:
2682:
2673:
2657:
2656:
2639:
2638:
2632:
2631:
2622:
2621:
2612:
2611:
2584:
2574:
2572:
2540:
2507:
2482:
2476:
2451:
2441:
2428:
2422:
2421:
2418:
2390:
2371:
2355:
2343:
2332:
2331:
2328:
2295:
2284:
2283:
2273:
2261:
2252:
2238:
2232:
2220:
2214:
2213:
2210:
2178:
2172:
2171:
2168:
2148:
2128:
2099:
2093:
2092:
2082:
2062:
2049:
2048:
2031:
2016:
2007:
1989:
1983:
1924:
1879:
1849:
1840:
1828:
1819:
1817:
1793:
1784:
1782:
1762:
1716:
1679:
1673:
1672:
1667:
1659:
1631:
1605:
1590:
1572:
1566:
1560:
1512:
1480:
1468:
1458:
1440:
1438:
1399:
1385:
1379:
1373:
1353:
1332:
1326:
1300:
1298:
1256:
1254:
1230:
1200:
1190:
1180:
1164:
1152:
1140:
1124:
1109:
1099:
1091:
1067:
1061:
1032:
1023:
1017:
993:
964:
962:
938:
917:
911:
892:
891:
873:
867:
838:
836:
816:
784:
778:
757:
751:
724:
723:
703:
694:
688:
667:
654:
633:
612:
606:
577:
569:
542:
521:
475:
454:
421:
417:
416:
406:
400:
379:
375:
374:
371:
342:
337:
332:
330:
309:
303:
274:
250:
218:
194:
184:
165:
159:
124:
118:
2187:{\displaystyle {\mathcal {L}}_{\theta }}
899:{\displaystyle \pi _{3}(G)=\mathbb {Z} }
2979:
1907:. Without any tunnelling the different
338:
7:
1735:{\displaystyle \theta \neq \theta '}
1626:. In pure Yang–Mills, each value of
1541:{\displaystyle \theta \in [0,2\pi )}
906:. This is because every mapping of
811:For other non-abelian gauge groups
213:, where at each point in spacetime
2438:
1996:
1563:
1376:
1329:
1321:with a large gauge transformation
635:
523:
456:
433:
350:
310:
306:
276:
220:
191:
181:
177:
14:
1963:The complicated structure of the
318:{\displaystyle \Omega _{\infty }}
957:deformed into a mapping onto an
683:, which has a homotopy group of
594:{\displaystyle G={\text{SU}}(2)}
388:{\displaystyle \mathbb {R} ^{3}}
110:in some fixed gauge such as the
564:. For example, the gauge group
73:. It was discovered in 1976 by
3026:Jackiw, R.; Rebbi, C. (1976).
2821:
2802:
2747:
2644:
2585:
2551:
2545:
2500:neutron electric dipole moment
2337:
2304:
2289:
2266:
2032:
2008:
1993:
1935:
1929:
1841:
1820:
1805:{\displaystyle |n_{-}\rangle }
1785:
1680:
1668:
1606:
1573:
1535:
1520:
1481:
1441:
1400:
1386:
1301:
1257:
1206:
1157:
1043:
1037:
1029:
981:{\displaystyle {\text{SU}}(2)}
975:
969:
885:
879:
855:{\displaystyle {\text{SU}}(2)}
849:
843:
717:
714:
708:
700:
660:
644:
638:
601:has an underlying manifold of
588:
582:
548:
532:
526:
481:
465:
459:
347:
343:
333:
285:
279:
229:
223:
16:Yang–Mills theory vacuum state
1:
3375:Current Algebra And Anomalies
3301:10.1103/PhysRevLett.97.131801
3074:Lecture Notes on Gauge Theory
1008:, a result that follows from
3086:Guidry, M. W. (1991). "13".
3013:10.1016/0370-2693(76)90277-X
2526:Modification due to fermions
298:approaches some fixed value
2163:the Yang–Mills action, and
1753:. They are responsible for
1341:{\displaystyle \Omega _{m}}
806:large gauge transformations
3442:
3235:Pokorski, S. (2000). "8".
3090:. Wiley VCH. p. 447.
3052:10.1103/PhysRevLett.37.172
2945:and thus give rise to the
1896:{\displaystyle \nu =\pm 1}
1314:{\displaystyle |n\rangle }
1270:{\displaystyle |n\rangle }
291:{\displaystyle \Omega (x)}
235:{\displaystyle \Omega (x)}
57:through the presence of a
862:subgroups, ensuring that
773:is mapped onto the group
3245:10.1017/CBO9780511612343
3212:10.1017/CBO9780511565045
3176:10.1017/CBO9781139013352
2557:{\displaystyle \psi (x)}
2491:{\displaystyle K^{\mu }}
3271:Physical Review Letters
3032:Physical Review Letters
2515:{\displaystyle \theta }
1639:{\displaystyle \theta }
628:so that the mapping is
139:{\displaystyle A_{0}=0}
85:, and independently by
3426:Quantum chromodynamics
2927:
2558:
2516:
2492:
2461:
2403:
2314:
2188:
2157:
2137:
2114:
1954:
1897:
1868:
1806:
1771:
1736:
1705:
1648:superselection sectors
1640:
1620:
1542:
1498:
1420:
1362:
1342:
1315:
1271:
1239:
1216:
1077:
1050:
1002:
982:
947:
927:
900:
856:
825:
794:
767:
732:
677:
622:
595:
558:
491:
443:
395:behaves as a 3-sphere
389:
357:
319:
292:
259:
236:
207:
140:
63:quantum chromodynamics
49:of an infinite set of
27:is the semi-classical
2928:
2559:
2517:
2493:
2462:
2404:
2315:
2189:
2158:
2138:
2115:
1955:
1898:
1869:
1807:
1772:
1737:
1706:
1641:
1621:
1543:
1499:
1421:
1363:
1343:
1316:
1272:
1240:
1217:
1078:
1076:{\displaystyle A^{i}}
1051:
1003:
983:
948:
928:
926:{\displaystyle S^{3}}
901:
857:
826:
795:
793:{\displaystyle S^{3}}
768:
766:{\displaystyle S^{3}}
733:
678:
623:
621:{\displaystyle S^{3}}
596:
559:
504:configuration can be
492:
444:
390:
358:
320:
293:
265:. To ensure that the
260:
237:
208:
150:which corresponds to
148:field strength tensor
141:
69:problem known as the
3237:Gauge Field Theories
2571:
2539:
2506:
2475:
2417:
2327:
2209:
2167:
2147:
2143:is the Hamiltonian,
2127:
1982:
1923:
1878:
1816:
1781:
1770:{\displaystyle \nu }
1761:
1715:
1658:
1630:
1559:
1511:
1437:
1372:
1352:
1348:with winding number
1325:
1297:
1253:
1229:
1090:
1060:
1016:
992:
961:
937:
910:
866:
835:
815:
777:
750:
742:, also known as its
687:
632:
605:
568:
520:
453:
399:
370:
329:
302:
273:
249:
217:
158:
117:
55:Lagrangian formalism
21:quantum field theory
3383:1985caa..book.....J
3293:2006PhRvL..97m1801B
3204:Aspects of Symmetry
3044:1976PhRvL..37..172J
3005:1976PhLB...63..334C
1747:equations of motion
1277:are referred to as
108:Euclidean spacetime
89:and Claudio Rebbi.
36:Yang–Mills theories
3133:10.24033/bsmf.1472
2923:
2921:
2554:
2522:to be very small.
2512:
2488:
2457:
2399:
2365:
2310:
2184:
2153:
2133:
2110:
2000:
1950:
1893:
1874:. Instantons with
1864:
1802:
1767:
1732:
1701:
1636:
1616:
1538:
1494:
1463:
1416:
1358:
1338:
1311:
1267:
1235:
1212:
1073:
1046:
998:
978:
943:
923:
896:
852:
821:
790:
763:
728:
673:
618:
591:
554:
487:
439:
385:
353:
315:
288:
255:
232:
203:
136:
3185:978-0-521-19084-8
2993:Physics Letters B
2968:Strong CP problem
2947:chiral condensate
2881:
2876:
2750:
2707:
2702:
2647:
2364:
2340:
2292:
2264:
2259:
2156:{\displaystyle S}
2136:{\displaystyle H}
1985:
1454:
1361:{\displaystyle m}
1279:topological vacua
1247:coupling constant
1238:{\displaystyle g}
1155:
1151:
1131:
1035:
1001:{\displaystyle G}
967:
946:{\displaystyle G}
841:
824:{\displaystyle G}
706:
580:
258:{\displaystyle G}
98:Topological vacua
93:Yang–Mills vacuum
71:strong CP problem
38:specified by the
3433:
3405:
3404:
3355:
3349:
3348:
3327:
3321:
3320:
3286:
3265:
3259:
3258:
3232:
3226:
3225:
3196:
3190:
3189:
3160:
3154:
3153:
3135:
3108:
3102:
3101:
3083:
3077:
3076:
3062:
3056:
3055:
3023:
3017:
3016:
2984:
2932:
2930:
2929:
2924:
2922:
2915:
2914:
2908:
2907:
2895:
2894:
2882:
2879:
2877:
2875:
2874:
2873:
2857:
2852:
2851:
2836:
2835:
2820:
2819:
2795:
2794:
2779:
2775:
2774:
2765:
2764:
2752:
2751:
2743:
2734:
2733:
2721:
2720:
2708:
2705:
2703:
2701:
2700:
2699:
2683:
2678:
2677:
2662:
2661:
2649:
2648:
2640:
2637:
2636:
2627:
2626:
2617:
2616:
2588:
2563:
2561:
2560:
2555:
2521:
2519:
2518:
2513:
2497:
2495:
2494:
2489:
2487:
2486:
2471:physics because
2469:non-perturbative
2466:
2464:
2463:
2458:
2456:
2455:
2446:
2445:
2433:
2432:
2427:
2426:
2408:
2406:
2405:
2400:
2398:
2397:
2385:
2384:
2366:
2357:
2351:
2350:
2342:
2341:
2333:
2319:
2317:
2316:
2311:
2303:
2302:
2294:
2293:
2285:
2281:
2280:
2265:
2262:
2260:
2258:
2257:
2256:
2243:
2242:
2233:
2225:
2224:
2219:
2218:
2193:
2191:
2190:
2185:
2183:
2182:
2177:
2176:
2162:
2160:
2159:
2154:
2142:
2140:
2139:
2134:
2119:
2117:
2116:
2111:
2106:
2105:
2104:
2103:
2098:
2097:
2087:
2086:
2054:
2053:
2035:
2030:
2029:
2011:
1999:
1959:
1957:
1956:
1951:
1911:-vacua would be
1902:
1900:
1899:
1894:
1873:
1871:
1870:
1865:
1854:
1853:
1844:
1833:
1832:
1823:
1811:
1809:
1808:
1803:
1798:
1797:
1788:
1776:
1774:
1773:
1768:
1741:
1739:
1738:
1733:
1731:
1710:
1708:
1707:
1702:
1691:
1683:
1678:
1677:
1671:
1645:
1643:
1642:
1637:
1625:
1623:
1622:
1617:
1609:
1604:
1603:
1576:
1571:
1570:
1547:
1545:
1544:
1539:
1503:
1501:
1500:
1495:
1484:
1479:
1478:
1462:
1444:
1425:
1423:
1422:
1417:
1403:
1389:
1384:
1383:
1367:
1365:
1364:
1359:
1347:
1345:
1344:
1339:
1337:
1336:
1320:
1318:
1317:
1312:
1304:
1276:
1274:
1273:
1268:
1260:
1244:
1242:
1241:
1236:
1221:
1219:
1218:
1213:
1205:
1204:
1195:
1194:
1185:
1184:
1175:
1174:
1156:
1153:
1149:
1145:
1144:
1132:
1130:
1129:
1128:
1115:
1114:
1113:
1100:
1082:
1080:
1079:
1074:
1072:
1071:
1055:
1053:
1052:
1047:
1036:
1033:
1028:
1027:
1007:
1005:
1004:
999:
987:
985:
984:
979:
968:
965:
952:
950:
949:
944:
932:
930:
929:
924:
922:
921:
905:
903:
902:
897:
895:
878:
877:
861:
859:
858:
853:
842:
839:
830:
828:
827:
822:
799:
797:
796:
791:
789:
788:
772:
770:
769:
764:
762:
761:
744:Pontryagin index
737:
735:
734:
729:
727:
707:
704:
699:
698:
682:
680:
679:
674:
672:
671:
659:
658:
627:
625:
624:
619:
617:
616:
600:
598:
597:
592:
581:
578:
563:
561:
560:
555:
547:
546:
496:
494:
493:
488:
480:
479:
448:
446:
445:
440:
426:
425:
420:
411:
410:
394:
392:
391:
386:
384:
383:
378:
362:
360:
359:
354:
346:
341:
336:
324:
322:
321:
316:
314:
313:
297:
295:
294:
289:
264:
262:
261:
256:
241:
239:
238:
233:
212:
210:
209:
204:
202:
201:
189:
188:
170:
169:
145:
143:
142:
137:
129:
128:
3441:
3440:
3436:
3435:
3434:
3432:
3431:
3430:
3411:
3410:
3409:
3408:
3401:
3357:
3356:
3352:
3345:
3329:
3328:
3324:
3267:
3266:
3262:
3255:
3234:
3233:
3229:
3222:
3198:
3197:
3193:
3186:
3162:
3161:
3157:
3110:
3109:
3105:
3098:
3085:
3084:
3080:
3064:
3063:
3059:
3025:
3024:
3020:
2986:
2985:
2981:
2976:
2959:
2920:
2919:
2896:
2883:
2865:
2861:
2843:
2777:
2776:
2722:
2709:
2691:
2687:
2669:
2601:
2569:
2568:
2537:
2536:
2528:
2504:
2503:
2478:
2473:
2472:
2447:
2437:
2420:
2415:
2414:
2386:
2367:
2330:
2325:
2324:
2282:
2269:
2248:
2244:
2234:
2212:
2207:
2206:
2170:
2165:
2164:
2145:
2144:
2125:
2124:
2091:
2078:
2058:
2012:
1980:
1979:
1921:
1920:
1905:BPST instantons
1876:
1875:
1845:
1824:
1814:
1813:
1789:
1779:
1778:
1759:
1758:
1724:
1713:
1712:
1684:
1656:
1655:
1628:
1627:
1586:
1562:
1557:
1556:
1509:
1508:
1464:
1435:
1434:
1428:Bloch's theorem
1375:
1370:
1369:
1350:
1349:
1328:
1323:
1322:
1295:
1294:
1287:
1251:
1250:
1227:
1226:
1196:
1186:
1176:
1160:
1136:
1120:
1116:
1105:
1101:
1088:
1087:
1063:
1058:
1057:
1019:
1014:
1013:
990:
989:
959:
958:
935:
934:
913:
908:
907:
869:
864:
863:
833:
832:
813:
812:
780:
775:
774:
753:
748:
747:
690:
685:
684:
663:
650:
630:
629:
608:
603:
602:
566:
565:
538:
518:
517:
516:of the mapping
471:
451:
450:
415:
402:
397:
396:
373:
368:
367:
327:
326:
305:
300:
299:
271:
270:
247:
246:
215:
214:
190:
180:
161:
156:
155:
154:configurations
120:
115:
114:
100:
95:
61:-term which in
17:
12:
11:
5:
3439:
3437:
3429:
3428:
3423:
3421:Gauge theories
3413:
3412:
3407:
3406:
3400:978-9971966966
3399:
3350:
3343:
3333:(1995). "23".
3322:
3284:hep-ex/0602020
3277:(13): 131801.
3260:
3254:978-0537478169
3253:
3227:
3221:978-0521318273
3220:
3191:
3184:
3155:
3103:
3097:978-0471631170
3096:
3078:
3057:
3038:(3): 172–175.
3018:
2999:(3): 334–340.
2978:
2977:
2975:
2972:
2971:
2970:
2965:
2958:
2955:
2938:Dirac operator
2934:
2933:
2918:
2913:
2906:
2903:
2899:
2893:
2890:
2886:
2872:
2868:
2864:
2860:
2855:
2850:
2846:
2842:
2839:
2834:
2829:
2826:
2823:
2818:
2811:
2807:
2804:
2801:
2798:
2793:
2788:
2785:
2782:
2780:
2778:
2773:
2768:
2763:
2756:
2749:
2746:
2740:
2737:
2732:
2729:
2725:
2719:
2716:
2712:
2698:
2694:
2690:
2686:
2681:
2676:
2672:
2668:
2665:
2660:
2655:
2652:
2646:
2643:
2635:
2630:
2625:
2620:
2615:
2610:
2607:
2604:
2602:
2600:
2597:
2594:
2591:
2587:
2583:
2580:
2577:
2576:
2553:
2550:
2547:
2544:
2527:
2524:
2511:
2485:
2481:
2454:
2450:
2444:
2440:
2436:
2431:
2425:
2396:
2393:
2389:
2383:
2380:
2377:
2374:
2370:
2363:
2360:
2354:
2349:
2346:
2339:
2336:
2321:
2320:
2309:
2306:
2301:
2298:
2291:
2288:
2279:
2276:
2272:
2268:
2255:
2251:
2247:
2241:
2237:
2231:
2228:
2223:
2217:
2181:
2175:
2152:
2132:
2121:
2120:
2109:
2102:
2096:
2090:
2085:
2081:
2077:
2074:
2071:
2068:
2065:
2061:
2057:
2052:
2047:
2044:
2041:
2038:
2034:
2028:
2025:
2022:
2019:
2015:
2010:
2006:
2003:
1998:
1995:
1992:
1988:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1892:
1889:
1886:
1883:
1863:
1860:
1857:
1852:
1848:
1843:
1839:
1836:
1831:
1827:
1822:
1801:
1796:
1792:
1787:
1766:
1730:
1727:
1723:
1720:
1700:
1697:
1694:
1690:
1687:
1682:
1676:
1670:
1666:
1663:
1654:-vacua vanish
1635:
1615:
1612:
1608:
1602:
1599:
1596:
1593:
1589:
1585:
1582:
1579:
1575:
1569:
1565:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1505:
1504:
1493:
1490:
1487:
1483:
1477:
1474:
1471:
1467:
1461:
1457:
1453:
1450:
1447:
1443:
1415:
1412:
1409:
1406:
1402:
1398:
1395:
1392:
1388:
1382:
1378:
1357:
1335:
1331:
1310:
1307:
1303:
1286:
1283:
1266:
1263:
1259:
1234:
1223:
1222:
1211:
1208:
1203:
1199:
1193:
1189:
1183:
1179:
1173:
1170:
1167:
1163:
1159:
1148:
1143:
1139:
1135:
1127:
1123:
1119:
1112:
1108:
1104:
1098:
1095:
1070:
1066:
1045:
1042:
1039:
1031:
1026:
1022:
997:
977:
974:
971:
942:
920:
916:
894:
890:
887:
884:
881:
876:
872:
851:
848:
845:
820:
787:
783:
760:
756:
740:winding number
726:
722:
719:
716:
713:
710:
702:
697:
693:
670:
666:
662:
657:
653:
649:
646:
643:
640:
637:
615:
611:
590:
587:
584:
576:
573:
553:
550:
545:
541:
537:
534:
531:
528:
525:
514:homotopy group
486:
483:
478:
474:
470:
467:
464:
461:
458:
438:
435:
432:
429:
424:
419:
414:
409:
405:
382:
377:
352:
349:
345:
340:
335:
312:
308:
287:
284:
281:
278:
254:
231:
228:
225:
222:
200:
197:
193:
187:
183:
179:
176:
173:
168:
164:
135:
132:
127:
123:
112:temporal gauge
104:semi-classical
99:
96:
94:
91:
15:
13:
10:
9:
6:
4:
3:
2:
3438:
3427:
3424:
3422:
3419:
3418:
3416:
3402:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3354:
3351:
3346:
3344:9780521670548
3340:
3336:
3332:
3326:
3323:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3285:
3280:
3276:
3272:
3264:
3261:
3256:
3250:
3246:
3242:
3238:
3231:
3228:
3223:
3217:
3213:
3209:
3205:
3202:(1985). "7".
3201:
3195:
3192:
3187:
3181:
3177:
3173:
3169:
3166:(2012). "5".
3165:
3159:
3156:
3151:
3147:
3143:
3139:
3134:
3129:
3125:
3121:
3117:
3113:
3107:
3104:
3099:
3093:
3089:
3082:
3079:
3075:
3071:
3067:
3061:
3058:
3053:
3049:
3045:
3041:
3037:
3033:
3029:
3022:
3019:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2983:
2980:
2973:
2969:
2966:
2964:
2961:
2960:
2956:
2954:
2952:
2948:
2944:
2939:
2916:
2904:
2901:
2897:
2891:
2888:
2884:
2870:
2866:
2862:
2858:
2853:
2848:
2844:
2840:
2837:
2827:
2824:
2816:
2809:
2805:
2796:
2786:
2783:
2781:
2766:
2761:
2754:
2744:
2738:
2735:
2730:
2727:
2723:
2717:
2714:
2710:
2696:
2692:
2688:
2684:
2679:
2674:
2670:
2666:
2663:
2653:
2650:
2641:
2628:
2618:
2608:
2605:
2603:
2595:
2592:
2589:
2581:
2567:
2566:
2565:
2548:
2542:
2533:
2525:
2523:
2509:
2501:
2483:
2479:
2470:
2452:
2448:
2442:
2434:
2429:
2412:
2394:
2391:
2387:
2381:
2378:
2375:
2372:
2368:
2361:
2358:
2352:
2347:
2344:
2334:
2307:
2299:
2296:
2286:
2277:
2274:
2270:
2253:
2249:
2245:
2239:
2235:
2229:
2226:
2221:
2205:
2204:
2203:
2201:
2197:
2179:
2150:
2130:
2107:
2100:
2088:
2083:
2079:
2075:
2072:
2069:
2066:
2063:
2059:
2055:
2045:
2042:
2036:
2026:
2023:
2020:
2017:
2013:
2004:
1990:
1978:
1977:
1976:
1974:
1973:path integral
1970:
1966:
1961:
1947:
1944:
1941:
1938:
1932:
1926:
1918:
1914:
1910:
1906:
1903:are known as
1890:
1887:
1884:
1881:
1858:
1855:
1850:
1846:
1837:
1829:
1825:
1794:
1790:
1764:
1756:
1752:
1748:
1743:
1728:
1725:
1721:
1718:
1698:
1695:
1688:
1685:
1664:
1653:
1649:
1633:
1610:
1600:
1597:
1594:
1591:
1587:
1583:
1577:
1567:
1554:
1552:
1548:are known as
1532:
1529:
1526:
1523:
1517:
1514:
1491:
1485:
1475:
1472:
1469:
1465:
1459:
1455:
1451:
1445:
1433:
1432:
1431:
1429:
1410:
1407:
1404:
1396:
1390:
1380:
1355:
1333:
1305:
1292:
1284:
1282:
1280:
1261:
1248:
1232:
1209:
1201:
1197:
1191:
1187:
1181:
1177:
1171:
1168:
1165:
1161:
1146:
1141:
1137:
1133:
1125:
1121:
1117:
1110:
1106:
1102:
1096:
1093:
1086:
1085:
1084:
1068:
1064:
1040:
1024:
1020:
1011:
1010:Botts theorem
995:
972:
956:
940:
918:
914:
888:
882:
874:
870:
846:
818:
809:
807:
803:
785:
781:
758:
754:
745:
741:
720:
711:
695:
691:
668:
664:
655:
651:
647:
641:
613:
609:
585:
574:
571:
551:
543:
539:
535:
529:
515:
510:
507:
503:
498:
484:
476:
472:
468:
462:
427:
422:
412:
407:
403:
380:
366:
282:
268:
252:
245:
226:
198:
195:
185:
174:
171:
166:
162:
153:
149:
133:
130:
125:
121:
113:
109:
105:
97:
92:
90:
88:
84:
80:
76:
75:Curtis Callan
72:
68:
65:leads to the
64:
60:
56:
52:
51:topologically
48:
47:superposition
44:
41:
37:
34:
30:
26:
22:
3391:10.1142/0131
3374:
3353:
3334:
3331:Weinberg, S.
3325:
3274:
3270:
3263:
3236:
3230:
3203:
3194:
3167:
3158:
3123:
3119:
3106:
3087:
3081:
3073:
3060:
3035:
3031:
3021:
2996:
2992:
2982:
2950:
2943:axial charge
2935:
2530:If massless
2529:
2322:
2199:
2196:CP violating
2122:
1964:
1962:
1916:
1908:
1744:
1651:
1550:
1549:
1506:
1288:
1278:
1224:
988:subgroup of
955:continuously
810:
511:
502:ground state
499:
101:
87:Roman Jackiw
79:Roger Dashen
58:
42:
40:vacuum angle
39:
29:vacuum state
25:theta vacuum
24:
18:
3367:Treiman, S.
3200:Coleman, S.
3164:Shifman, M.
3126:: 251–281.
1291:eigenstates
1285:Theta vacua
802:orientation
500:When every
269:is finite,
83:David Gross
67:fine tuning
3415:Categories
3371:Zumino, B.
3363:Jackiw, R.
3359:Witten, E.
2974:References
2411:generators
1975:formalism
1969:Lagrangian
1913:degenerate
1755:tunnelling
1751:instantons
152:pure gauge
3317:119431442
3142:0037-9484
2963:Instanton
2905:ν
2902:μ
2892:ν
2889:μ
2841:∫
2838:−
2828:
2787:∫
2784:∼
2767:ψ
2748:¯
2745:ψ
2731:ν
2728:μ
2718:ν
2715:μ
2667:∫
2664:−
2654:
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2642:ψ
2629:ψ
2609:∫
2606:∼
2599:⟩
2596:ν
2579:⟨
2543:ψ
2510:θ
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2453:μ
2443:μ
2439:∂
2430:θ
2395:σ
2392:ρ
2382:σ
2379:ρ
2376:ν
2373:μ
2369:ϵ
2348:ν
2345:μ
2338:~
2300:ν
2297:μ
2290:~
2278:ν
2275:μ
2250:π
2230:θ
2222:θ
2194:is a new
2180:θ
2101:θ
2076:∫
2046:∫
2040:⟩
2037:θ
2018:−
2005:θ
2002:⟨
1997:∞
1994:→
1948:θ
1945:
1939:∝
1933:θ
1888:±
1882:ν
1862:⟩
1859:ν
1851:−
1835:⟩
1800:⟩
1795:−
1765:ν
1726:θ
1722:≠
1719:θ
1693:⟩
1686:θ
1665:θ
1662:⟨
1634:θ
1614:⟩
1611:θ
1598:θ
1592:−
1581:⟩
1578:θ
1564:Ω
1533:π
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1515:θ
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1476:θ
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1449:⟩
1446:θ
1414:⟩
1394:⟩
1377:Ω
1330:Ω
1309:⟩
1265:⟩
1162:ϵ
1134:∫
1122:π
1030:→
871:π
692:π
661:→
636:Ω
549:→
524:Ω
482:→
457:Ω
434:∞
428:∪
351:∞
348:→
311:∞
307:Ω
277:Ω
221:Ω
196:−
192:Ω
182:∇
178:Ω
3373:(1985).
3309:17026025
3114:(1956).
3112:Bott, R.
3068:(2018),
3066:Tong, D.
2957:See also
2532:fermions
1729:′
1689:′
506:smoothly
365:manifold
3379:Bibcode
3289:Bibcode
3150:0087035
3040:Bibcode
3001:Bibcode
1749:called
1245:is the
953:can be
33:abelian
31:of non-
3397:
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3307:
3251:
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3182:
3148:
3140:
3094:
2323:where
2202:-term
1553:-vacua
1225:where
1150:
267:action
81:, and
23:, the
3313:S2CID
3279:arXiv
2123:Here
933:onto
244:group
3395:ISBN
3339:ISBN
3305:PMID
3249:ISBN
3216:ISBN
3180:ISBN
3138:ISSN
3092:ISBN
102:The
3387:doi
3297:doi
3241:doi
3208:doi
3172:doi
3128:doi
3070:"3"
3048:doi
3009:doi
2825:exp
2800:det
2651:exp
1987:lim
1942:cos
1812:to
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325:as
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