1642:
In weak gravitational curvature, flat spacetime often serves as a good approximation to weakly curved spacetime. For experiment, this approximation is good. The
Standard Model is defined on flat spacetime, and has produced the most accurate precision tests of physics to date.
1606:
Then the spacetime covariant derivative on tensor or spin-tensor fields is simply the partial derivative in flat coordinates. However the gauge covariant derivative may require a non-trivial connection
1224:
1185:
982:
669:
1353:
500:
888:
822:
563:
457:
426:
1593:
798:
371:
274:
94:
Many important examples of classical field theories which are of interest in quantum field theory are given below. In particular, these are the theories which make up the
84:
1567:
1065:
864:
1535:
914:
1500:
1632:
1292:
767:
1025:
692:
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1408:
1373:
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1267:
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1147:
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1002:
938:
732:
712:
633:
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593:
531:
391:
349:
321:
294:
222:
187:: coupling of spinor and gauge fields. Despite these being named quantum theories, the Lagrangians can be considered as those of a classical field theory.
144:
824:-valued 1-form on P satisfying technical conditions of 'projection' and 'right-equivariance': details found in the principal connection article.
1704:
Bocharov, A.V. "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999,
1382:
This completes the mathematical prerequisites for a large number of interesting theories, including those given in the examples section above.
98:
of particle physics. These examples will be used in the discussion of the general mathematical formulation of classical field theory.
538:
1771:
174:
1118:
of an associated vector bundle. The collection of these, together with gauge fields, is the matter content of the theory.
152:. This is the only theory in the uncoupled theory list which contains interactions: YangâMills contains self-interactions.
53:
1729:
1719:
1714:
De Leon, M., Rodrigues, P.R., "Generalized
Classical Mechanics and Field Theory", Elsevier Science Publishing, 1985,
1709:
1699:
1190:
52:
are the correct domain for such a description. The
Hamiltonian variant of covariant classical field theory is the
1800:
303:
as well as the required structure of an orientation, needed for a notion of integration over all of the manifold
428:, the automorphisms of spacetime. In this case the fields of the theory should transform in a representation of
1795:
1815:
1152:
954:
641:
1810:
1724:
Griffiths, P.A., "Exterior
Differential Systems and the Calculus of Variations", Boston: BirkhÀuser, 1983,
1421:
The simplifications come from the observation that flat spacetime is contractible: it is then a theorem in
1595:. In other words, vector bundles at different points are comparable. In addition, for flat spacetime the
1317:
573:
468:
1418:, there are many useful simplifications that make theories less conceptually difficult to deal with.
869:
803:
544:
431:
400:
112:
1572:
779:
1677:
949:
57:
354:
257:
67:
56:
where momenta correspond to derivatives of field variables with respect to all world coordinates.
1682:
1540:
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180:
1507:
893:
1805:
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1115:
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29:
25:
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149:
49:
1007:
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17:
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8:
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describing the (continuous) symmetries of internal degrees of freedom. The corresponding
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37:
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252:
196:
In order to formulate a classical field theory, the following structures are needed:
132:
463:
1657:
1694:
Saunders, D.J., "The
Geometry of Jet Bundles", Cambridge University Press, 1989,
1415:
164:
41:
231:(for emphasizing the manifold without additional structures such as a metric),
228:
139:
122:
95:
1789:
236:
127:
61:
1600:
33:
917:
566:
534:
1743:
1569:, and then we need not view fields as sections but simply as functions
45:
1757:
511:
232:
1168:
965:
652:
916:. It is this local form of the connection which is identified with
351:
may admit symmetries. For example, if it is equipped with a metric
1779:
1472:, and therefore identify the connection globally as a gauge field
247:
The spacetime often comes with additional structure. Examples are
1736:
Momentum Maps and
Classical Fields Part I: Covariant Field Theory
943:
940:
is flat, there are simplifications which remove this subtlety.
1269:
from above. Then for example, more concretely we may consider
1766:, "Advanced Classical Field Theory", World Scientific, 2009,
1748:
Echeverria-Enriquez, A., Munoz-Lecanda, M.C., Roman-Roy, M.,
1750:
Geometry of
Lagrangian First-order Classical Field Theories
505:
1734:
Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery R.,
1634:
which is considered to be the gauge field of the theory.
1537:
which allows us to identify associated vector bundles as
462:
For example, for
Minkowski space, the symmetries are the
1229:
Suppose that the matter content is given by sections of
827:
Under a trivialization this can be written as a local
746:. In field theory this connection is also viewed as a
1613:
1575:
1543:
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70:
60:
is formulated as covariant classical field theory on
191:
36:, and their dynamics is phrased in the context of a
1626:
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216:
78:
769:whose action on various fields is defined later.
235:(when equipped with a Lorentzian metric), or the
1787:
1637:
737:
1385:
742:Here we take the view of the connection as a
1219:{\displaystyle L:E'\rightarrow \mathbb {R} }
944:Associated vector bundles and matter content
1504:Furthermore, there is a trivial connection
326:
242:
1212:
1029:For completeness, given a representation
890:-valued 1-form on a trivialization patch
72:
1448:In particular, this allows us to pick a
506:Gauge, principal bundles and connections
1180:{\displaystyle E'\xrightarrow {\pi } M}
635:-torsor. This is sometimes written as
167:: coupling of scalar and spinor fields.
1788:
977:{\displaystyle E\xrightarrow {\pi } M}
664:{\displaystyle P\xrightarrow {\pi } M}
177:: coupling of scalar and gauge fields.
101:
539:Lie groupâLie algebra correspondence
1348:{\displaystyle V\otimes T_{p}^{*}M}
984:associated to the principal bundle
920:in physics. When the base manifold
875:
809:
694:is the canonical projection map on
550:
158:
13:
1294:to be a bundle where the fibre at
785:
756:
299:Metric up to conformal equivalence
239:for a more geometrical viewpoint.
54:covariant Hamiltonian field theory
44:. Nowadays, it is well known that
14:
1827:
1599:is the trivial connection on the
495:{\displaystyle {\text{Iso}}(1,3)}
192:Requisite mathematical structures
1762:Giachetta, G., Mangiarotti, L.,
1425:that any fibre bundle over flat
359:
262:
22:covariant classical field theory
1187:, the Lagrangian is a function
883:{\displaystyle {\mathfrak {g}}}
817:{\displaystyle {\mathfrak {g}}}
558:{\displaystyle {\mathfrak {g}}}
452:{\displaystyle {\text{Aut}}(M)}
421:{\displaystyle {\text{Aut}}(M)}
227:This is variously known as the
1588:{\displaystyle M\rightarrow V}
1579:
1208:
1054:
1036:
853:
847:
793:{\displaystyle {\mathcal {A}}}
489:
477:
446:
440:
415:
409:
397:. The symmetries form a group
373:, these are the isometries of
1:
1688:
1121:
565:. This is referred to as the
1638:Accuracy as a physical model
738:Connections and gauge fields
366:{\displaystyle \mathbf {g} }
269:{\displaystyle \mathbf {g} }
199:
79:{\displaystyle \mathbb {R} }
7:
1646:
1562:{\displaystyle E=M\times V}
1060:{\displaystyle (V,G,\rho )}
859:{\displaystyle A_{\mu }(x)}
89:
10:
1832:
1530:{\displaystyle A_{0,\mu }}
1386:Theories on flat spacetime
909:{\displaystyle U\subset M}
1495:{\displaystyle A_{\mu }.}
1004:through a representation
1678:Non-autonomous mechanics
1627:{\displaystyle A_{\mu }}
950:associated vector bundle
58:Non-autonomous mechanics
1683:Higgs field (classical)
1390:When the base manifold
1149:: given a fiber bundle
762:{\displaystyle \nabla }
615:, otherwise known as a
327:Symmetries of spacetime
243:Structures on spacetime
181:Quantum electrodynamics
1801:Differential equations
1653:Classical field theory
1628:
1597:Levi-Civita connection
1589:
1563:
1531:
1496:
1466:
1439:
1404:
1369:
1349:
1308:
1288:
1263:
1243:
1220:
1181:
1143:
1101:
1081:
1061:
1021:
1020:{\displaystyle \rho .}
998:
978:
934:
910:
884:
860:
818:
794:
763:
734:is the base manifold.
728:
708:
688:
665:
629:
609:
589:
559:
527:
496:
453:
422:
387:
367:
345:
317:
290:
270:
218:
171:Scalar electrodynamics
80:
1796:Differential topology
1668:Variational bicomplex
1629:
1590:
1564:
1532:
1497:
1467:
1450:global trivialization
1440:
1405:
1370:
1350:
1309:
1289:
1264:
1244:
1221:
1182:
1144:
1114:or matter field is a
1102:
1082:
1062:
1022:
999:
979:
935:
911:
885:
861:
819:
795:
764:
729:
709:
689:
666:
630:
610:
590:
560:
528:
497:
454:
423:
395:Killing vector fields
388:
368:
346:
318:
291:
271:
219:
81:
50:variational bicomplex
1816:Lagrangian mechanics
1673:Quantum field theory
1611:
1573:
1541:
1508:
1476:
1456:
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1394:
1359:
1318:
1298:
1273:
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1233:
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1153:
1133:
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1033:
1008:
988:
955:
924:
894:
870:
834:
804:
780:
774:principal connection
753:
748:covariant derivative
744:principal connection
718:
698:
687:{\displaystyle \pi }
678:
642:
619:
599:
579:
545:
517:
469:
432:
401:
377:
355:
335:
307:
280:
258:
208:
68:
18:mathematical physics
1811:Theoretical physics
1410:is flat, that is, (
1355:. This then allows
1341:
1172:
969:
656:
393:, generated by the
251:Metric: a (pseudo-)
113:KleinâGordon theory
108:Scalar field theory
64:over the time axis
1624:
1585:
1559:
1527:
1492:
1462:
1435:
1423:algebraic topology
1400:
1375:to be viewed as a
1365:
1345:
1327:
1304:
1287:{\displaystyle E'}
1284:
1259:
1239:
1216:
1177:
1139:
1097:
1077:
1057:
1017:
994:
974:
930:
906:
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625:
605:
585:
555:
523:
492:
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418:
383:
363:
341:
313:
286:
266:
214:
204:A smooth manifold
102:Uncoupled theories
76:
38:finite-dimensional
1772:978-981-283-895-7
1764:Sardanashvily, G.
1663:Lagrangian system
1465:{\displaystyle P}
1438:{\displaystyle M}
1403:{\displaystyle M}
1368:{\displaystyle L}
1307:{\displaystyle p}
1262:{\displaystyle V}
1242:{\displaystyle E}
1173:
1142:{\displaystyle L}
1100:{\displaystyle V}
1080:{\displaystyle E}
997:{\displaystyle P}
970:
933:{\displaystyle M}
727:{\displaystyle M}
707:{\displaystyle P}
657:
628:{\displaystyle G}
608:{\displaystyle P}
588:{\displaystyle G}
526:{\displaystyle G}
475:
438:
407:
386:{\displaystyle M}
344:{\displaystyle M}
316:{\displaystyle M}
289:{\displaystyle M}
253:Riemannian metric
217:{\displaystyle M}
150:YangâMills theory
1823:
1738:, November 2003
1658:Exterior algebra
1633:
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1623:
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223:
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159:Coupled theories
119:Spinor theories
85:
83:
82:
77:
75:
26:classical fields
1831:
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1744:physics/9801019
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1416:Euclidean space
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1067:, the fiber of
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165:Yukawa coupling
161:
133:Majorana theory
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92:
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65:
12:
11:
5:
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929:
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902:
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811:
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758:
739:
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703:
683:
672:
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660:
655:
651:
647:
624:
604:
584:
552:
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507:
504:
491:
488:
485:
482:
479:
464:Poincaré group
448:
445:
442:
417:
414:
411:
382:
361:
340:
331:The spacetime
328:
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244:
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229:world manifold
213:
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185:chromodynamics
178:
175:chromodynamics
168:
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145:Maxwell theory
140:Gauge theories
137:
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96:Standard model
91:
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9:
6:
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3:
2:
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1806:Fiber bundles
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1758:dg-ga/9505004
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1730:3-7643-3103-8
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1720:0-444-87753-3
1717:
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1710:0-8218-0958-X
1707:
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1700:0-521-36948-7
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1424:
1419:
1417:
1413:
1397:
1383:
1380:
1378:
1362:
1342:
1337:
1332:
1328:
1324:
1321:
1301:
1280:
1277:
1256:
1236:
1227:
1204:
1201:
1197:
1194:
1174:
1169:
1165:
1160:
1157:
1136:
1129:
1119:
1117:
1113:
1108:
1094:
1074:
1051:
1048:
1045:
1042:
1039:
1027:
1014:
1011:
991:
971:
966:
962:
958:
951:
941:
927:
919:
903:
900:
897:
850:
842:
838:
830:
825:
775:
770:
749:
745:
735:
721:
701:
681:
658:
653:
649:
645:
638:
637:
636:
622:
602:
582:
575:
570:
568:
540:
536:
520:
513:
503:
486:
483:
480:
465:
460:
443:
412:
396:
380:
338:
324:
310:
298:
283:
254:
250:
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237:base manifold
234:
230:
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197:
186:
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162:
151:
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138:
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126:
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121:
120:
118:
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111:
110:
109:
106:
105:
99:
97:
87:
63:
62:fiber bundles
59:
55:
51:
47:
43:
39:
35:
34:fiber bundles
31:
27:
23:
19:
1749:
1735:
1641:
1605:
1601:frame bundle
1503:
1449:
1447:
1445:is trivial.
1420:
1389:
1381:
1379:of a field.
1376:
1228:
1127:
1125:
1111:
1109:
1028:
947:
918:gauge fields
828:
826:
773:
771:
741:
673:
571:
537:through the
509:
461:
330:
302:
246:
226:
203:
195:
123:Dirac theory
93:
21:
15:
1752:, May 1995
1249:with fibre
829:gauge field
567:gauge group
541:is denoted
535:Lie algebra
128:Weyl theory
46:jet bundles
24:represents
1790:Categories
1689:References
1377:functional
1128:Lagrangian
1122:Lagrangian
1780:0811.0331
1620:μ
1580:→
1554:×
1523:μ
1485:μ
1338:∗
1325:⊗
1209:→
1170:π
1052:ρ
1012:ρ
967:π
901:⊂
843:μ
757:∇
682:π
654:π
574:principal
512:Lie group
233:spacetime
200:Spacetime
40:space of
1647:See also
1281:′
1205:′
1166:→
1161:′
963:→
776:denoted
650:→
595:-bundle
90:Examples
48:and the
30:sections
1116:section
1770:
1728:
1718:
1708:
1698:
1412:Pseudo
674:where
42:fields
1776:arXiv
1754:arXiv
1740:arXiv
1112:field
800:is a
1768:ISBN
1726:ISBN
1716:ISBN
1706:ISBN
1696:ISBN
866:, a
714:and
1452:of
1314:is
1087:is
948:An
474:Iso
437:Aut
406:Aut
276:on
32:of
28:by
16:In
1792::
1603:.
1414:-)
1226:.
1126:A
1110:A
1107:.
772:A
572:A
569:.
510:A
502:.
459:.
323:.
224:.
86:.
20:,
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1778::
1774:(
1756::
1742::
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1548:=
1545:E
1520:,
1517:0
1513:A
1490:.
1481:A
1460:P
1433:M
1398:M
1363:L
1343:M
1333:p
1329:T
1322:V
1302:p
1278:E
1257:V
1237:E
1213:R
1202:E
1198::
1195:L
1175:M
1158:E
1137:L
1095:V
1075:E
1055:)
1049:,
1046:G
1043:,
1040:V
1037:(
1015:.
992:P
972:M
959:E
928:M
904:M
898:U
876:g
854:)
851:x
848:(
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810:g
786:A
722:M
702:P
659:M
646:P
623:G
603:P
583:G
551:g
521:G
490:)
487:3
484:,
481:1
478:(
447:)
444:M
441:(
416:)
413:M
410:(
381:M
360:g
339:M
311:M
296:.
284:M
263:g
212:M
183:/
173:/
73:R
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