1496:
1212:
1491:{\displaystyle \delta {\mathcal {L}}_{RS}=\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }+{\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\delta \psi _{\rho }=\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }-\partial _{\nu }{\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\delta \psi _{\rho }+{\text{ boundary terms}}}
771:
2017:
183:
1878:
1606:
623:
1051:
2201:
901:
374:
1174:
1889:
1714:
74:
1768:
2211:
leads to equation with solutions representing wavefronts, some of which propagate faster than light. In other words, the field then suffers from acausal, superluminal propagation; consequently, the
466:
1507:
934:
766:{\displaystyle {\mathcal {L}}=-{\tfrac {1}{2}}\;{\bar {\psi }}_{\mu }\left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }}
222:
1760:
1652:
961:
253:
301:
2082:
1204:
1081:
801:
493:
2047:
2215:
in interaction with electromagnetism is essentially flawed. In extended supergravity, though, Das and
Freedman have shown that local supersymmetry solves this problem.
2102:
394:
2134:
2496:
Velo, Giorgio; Zwanzinger, Daniel (1969-12-25). "Noncausality and Other
Defects of Interaction Lagrangians for Particles with Spin One and Higher".
850:
2594:
2579:
2459:
Velo, Giorgio; Zwanziger, Daniel (1969-10-25). "Propagation and
Quantization of Rarita-Schwinger Waves in an External Electromagnetic Potential".
2012:{\displaystyle \psi _{0}=\kappa ,\quad \psi _{i}=\psi _{i}^{TT}+{\frac {\gamma ^{j}\partial _{j}}{\nabla ^{2}}}\gamma _{0}\partial _{i}\kappa ,}
306:
1089:
1660:
178:{\displaystyle \left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }=0}
567:
1873:{\displaystyle \gamma ^{\nu }{\partial }_{\nu }\psi _{\mu }=0,\quad \partial ^{\mu }\psi _{\mu }=0,\quad \gamma ^{\mu }\psi _{\mu }=0.}
1601:{\displaystyle \delta {\mathcal {L}}_{RS}=2\delta {\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho },}
399:
2604:
2584:
906:
610:
2124:
As in the case of the Dirac equation, electromagnetic interaction can be added by promoting the partial derivative to
191:
2125:
847:
The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation
1722:
1614:
614:
1501:
using the
Majorana flip properties we see that the second and first terms on the RHS are equal, concluding that
1046:{\displaystyle {\mathcal {L}}_{RS}={\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho },}
499:
with additional components compared to the four component spinor in the Dirac equation. It corresponds to the
2444:
2212:
2113:
2533:
Kobayashi, M.; Shamaly, A. (1978-04-15). "Minimal electromagnetic coupling for massive spin-two fields".
231:
2542:
2505:
2468:
2416:
2407:
Rarita, William; Schwinger, Julian (1941-07-01). "On a Theory of
Particles with Half-Integral Spin".
2378:
2350:
262:
39:
2052:
841:
31:
1182:
1059:
779:
471:
2311:
2280:
2025:
225:
1654:
we thus see that the equation of motion for a massless
Majorana Rarita–Schwinger spinor reads:
2599:
2558:
2521:
2484:
2432:
2087:
2589:
2550:
2513:
2476:
2424:
2386:
2358:
2323:
2290:
2208:
1719:
The gauge symmetry of the massless Rarita-Schwinger equation allows the choice of the gauge
62:
2112:
The current description of massive, higher spin fields through either Rarita–Schwinger or
17:
2546:
2509:
2472:
2420:
2382:
2354:
379:
256:
58:
54:
46:
42:
2573:
2390:
2369:
Freedman, D. Z.; Das, A. (1977). "Gauge internal symmetry in extended supergravity".
2362:
937:
811:
804:
1179:
To obtain the equations of motion we vary the
Lagrangian with respect to the fields
2207:
In 1969, Velo and
Zwanziger showed that the Rarita–Schwinger Lagrangian coupled to
941:
815:
2295:
2268:
2196:{\displaystyle \partial _{\mu }\rightarrow D_{\mu }=\partial _{\mu }-ieA_{\mu }}
955:
Consider a massless Rarita–Schwinger field described by the
Lagrangian density
896:{\displaystyle \psi _{\mu }\rightarrow \psi _{\mu }+\partial _{\mu }\epsilon }
2562:
2554:
2525:
2517:
2488:
2480:
2436:
2327:
947:"Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
837:
2341:
Das, A.; Freedman, D. Z. (1976). "Gauge quantization for spin-3/2 fields".
2428:
2269:"Massless Rarita-Schwinger equations: Half and three halves spin solution"
369:{\displaystyle \gamma _{5}=i\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}
1169:{\displaystyle \gamma ^{\mu \nu \rho }\equiv {\frac {1}{3!}}\gamma ^{}.}
1709:{\displaystyle \gamma ^{\mu \nu \rho }\partial _{\nu }\psi _{\rho }=0.}
50:
2248:
S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
2239:
S. Weinberg, "The quantum theory of fields", Vol. 1, Cambridge p. 232
2230:
S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
496:
2104:
satisfies the massless Dirac equation, therefore carrying spin 1/2.
2285:
57:
for spin-1/2 fermions. This equation was first introduced by
1624:
1517:
1222:
968:
629:
53:
in a four-dimensional flat spacetime. It is similar to the
936:
is an arbitrary spinor field. This is simply the local
461:{\displaystyle \sigma ^{\mu \nu }\equiv {\frac {i}{2}}}
641:
2137:
2090:
2055:
2028:
1892:
1771:
1725:
1663:
1617:
1510:
1215:
1185:
1092:
1062:
964:
909:
853:
782:
626:
474:
402:
382:
309:
265:
234:
194:
77:
2257:
Pierre Ramond - Field theory, a Modern Primer - p.40
929:{\displaystyle \epsilon \equiv \epsilon _{\alpha }}
2195:
2096:
2076:
2041:
2011:
1872:
1754:
1708:
1646:
1600:
1490:
1198:
1168:
1075:
1045:
928:
895:
795:
765:
487:
460:
388:
368:
295:
247:
216:
177:
2541:(8). American Physical Society (APS): 2179–2181.
2504:(5). American Physical Society (APS): 2218–2222.
2467:(5). American Physical Society (APS): 1337–1341.
217:{\displaystyle \epsilon ^{\mu \kappa \rho \nu }}
2116:formalisms is afflicted with several maladies.
1883:A solution with spins 1/2 and 3/2 is given by:
810:This equation controls the propagation of the
2310:Deser, S.; Kay, J. H.; Stelle, K. S. (1977).
2084:is doubly transverse, carrying spin 3/2, and
1056:where the sum over spin indices is implicit,
844:with spin 3/2 has been found experimentally.
8:
1755:{\displaystyle \gamma ^{\mu }\psi _{\mu }=0}
1647:{\displaystyle \delta {\mathcal {L}}_{RS}=0}
2415:(1). American Physical Society (APS): 61.
1611:plus unimportant boundary terms. Imposing
652:
609:This field equation can be derived as the
2312:"Hamiltonian Formulation of Supergravity"
2294:
2284:
2187:
2168:
2155:
2142:
2136:
2089:
2065:
2060:
2054:
2033:
2027:
1997:
1987:
1975:
1964:
1954:
1947:
1935:
1930:
1917:
1897:
1891:
1858:
1848:
1828:
1818:
1798:
1788:
1783:
1776:
1770:
1740:
1730:
1724:
1694:
1684:
1668:
1662:
1629:
1623:
1622:
1616:
1589:
1579:
1563:
1553:
1542:
1541:
1522:
1516:
1515:
1509:
1483:
1474:
1455:
1445:
1434:
1433:
1426:
1413:
1403:
1387:
1377:
1366:
1365:
1352:
1339:
1323:
1313:
1302:
1301:
1291:
1281:
1265:
1255:
1244:
1243:
1227:
1221:
1220:
1214:
1190:
1184:
1154:
1144:
1131:
1112:
1097:
1091:
1067:
1061:
1034:
1024:
1008:
998:
987:
986:
973:
967:
966:
963:
920:
908:
884:
871:
858:
852:
787:
781:
757:
739:
720:
710:
700:
681:
666:
655:
654:
640:
628:
627:
625:
479:
473:
449:
436:
419:
407:
401:
381:
360:
350:
340:
330:
314:
308:
264:
239:
233:
199:
193:
163:
145:
126:
116:
106:
87:
76:
68:In modern notation it can be written as:
951:Equations of motion in the massless case
2223:
613:corresponding to the Rarita–Schwinger
944:, and the field must be a gravitino.
7:
2267:Valenzuela, M.; Zanelli, J. (2024).
27:Field equation for spin-3/2 fermions
568:representation of the Lorentz group
2165:
2139:
2030:
1994:
1972:
1961:
1815:
1784:
1681:
1576:
1423:
1400:
1336:
1278:
1021:
881:
717:
123:
25:
814:of composite objects such as the
248:{\displaystyle \gamma _{\kappa }}
1912:
1843:
1813:
296:{\displaystyle \kappa =0,1,2,3}
2595:Partial differential equations
2580:Eponymous equations of physics
2449:Particle physics and cosmology
2148:
2077:{\displaystyle \psi _{i}^{TT}}
1547:
1439:
1371:
1307:
1249:
1158:
1132:
992:
864:
660:
455:
429:
1:
2296:10.21468/SciPostPhys.16.3.065
1762:, reducing the equations to:
611:Euler–Lagrange equation
2391:10.1016/0550-3213(77)90041-4
2363:10.1016/0550-3213(76)90589-7
1199:{\displaystyle \psi _{\mu }}
1076:{\displaystyle \psi _{\mu }}
796:{\displaystyle \psi _{\mu }}
488:{\displaystyle \psi _{\nu }}
2042:{\displaystyle \nabla ^{2}}
2621:
2126:gauge covariant derivative
2049:is the spatial Laplacian,
1083:are Majorana spinors, and
836:) or for the conjectural
2108:Drawbacks of the equation
36:Rarita–Schwinger equation
18:Rarita-Schwinger equation
2555:10.1103/physrevd.17.2179
2518:10.1103/physrev.188.2218
2481:10.1103/physrev.186.1337
2328:10.1103/PhysRevD.16.2448
2120:Superluminal propagation
2097:{\displaystyle \kappa }
2197:
2098:
2078:
2043:
2013:
1874:
1756:
1710:
1648:
1602:
1492:
1200:
1170:
1077:
1047:
930:
897:
797:
767:
489:
462:
390:
370:
297:
249:
218:
179:
2429:10.1103/physrev.60.61
2198:
2099:
2079:
2044:
2014:
1875:
1757:
1711:
1649:
1603:
1493:
1201:
1171:
1078:
1048:
931:
898:
798:
768:
490:
463:
391:
371:
298:
250:
219:
180:
2605:Mathematical physics
2585:Quantum field theory
2135:
2088:
2053:
2026:
1890:
1769:
1723:
1661:
1615:
1508:
1485: boundary terms
1213:
1183:
1090:
1060:
962:
907:
851:
780:
776:where the bar above
624:
472:
400:
380:
307:
263:
232:
192:
75:
2547:1978PhRvD..17.2179K
2510:1969PhRv..188.2218V
2473:1969PhRv..186.1337V
2421:1941PhRv...60...61R
2383:1977NuPhB.120..221F
2355:1976NuPhB.114..271D
2073:
1943:
842:elementary particle
495:is a vector-valued
32:theoretical physics
2193:
2094:
2074:
2056:
2039:
2009:
1926:
1870:
1752:
1706:
1644:
1598:
1488:
1196:
1166:
1073:
1043:
926:
893:
793:
763:
650:
485:
458:
386:
366:
293:
245:
226:Levi-Civita symbol
214:
175:
2535:Physical Review D
2371:Nuclear Physics B
2343:Nuclear Physics B
1981:
1550:
1486:
1442:
1374:
1310:
1252:
1125:
995:
663:
649:
570:, or rather, its
427:
389:{\displaystyle m}
16:(Redirected from
2612:
2566:
2529:
2492:
2447:, Squires E.J.,
2443:Collins P.D.B.,
2440:
2395:
2394:
2366:
2338:
2332:
2331:
2322:(8): 2448–2455.
2307:
2301:
2300:
2298:
2288:
2264:
2258:
2255:
2249:
2246:
2240:
2237:
2231:
2228:
2209:electromagnetism
2202:
2200:
2199:
2194:
2192:
2191:
2173:
2172:
2160:
2159:
2147:
2146:
2103:
2101:
2100:
2095:
2083:
2081:
2080:
2075:
2072:
2064:
2048:
2046:
2045:
2040:
2038:
2037:
2018:
2016:
2015:
2010:
2002:
2001:
1992:
1991:
1982:
1980:
1979:
1970:
1969:
1968:
1959:
1958:
1948:
1942:
1934:
1922:
1921:
1902:
1901:
1879:
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1876:
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1853:
1852:
1833:
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1802:
1793:
1792:
1787:
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1780:
1761:
1759:
1758:
1753:
1745:
1744:
1735:
1734:
1715:
1713:
1712:
1707:
1699:
1698:
1689:
1688:
1679:
1678:
1653:
1651:
1650:
1645:
1637:
1636:
1628:
1627:
1607:
1605:
1604:
1599:
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1584:
1583:
1574:
1573:
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1557:
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1551:
1543:
1530:
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1520:
1497:
1495:
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1484:
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1478:
1466:
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1449:
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1418:
1417:
1408:
1407:
1398:
1397:
1382:
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1376:
1375:
1367:
1357:
1356:
1344:
1343:
1334:
1333:
1318:
1317:
1312:
1311:
1303:
1296:
1295:
1286:
1285:
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1275:
1260:
1259:
1254:
1253:
1245:
1235:
1234:
1226:
1225:
1205:
1203:
1202:
1197:
1195:
1194:
1175:
1173:
1172:
1167:
1162:
1161:
1149:
1148:
1139:
1138:
1126:
1124:
1113:
1108:
1107:
1082:
1080:
1079:
1074:
1072:
1071:
1052:
1050:
1049:
1044:
1039:
1038:
1029:
1028:
1019:
1018:
1003:
1002:
997:
996:
988:
981:
980:
972:
971:
935:
933:
932:
927:
925:
924:
902:
900:
899:
894:
889:
888:
876:
875:
863:
862:
835:
834:
833:
826:
825:
802:
800:
799:
794:
792:
791:
772:
770:
769:
764:
762:
761:
752:
748:
747:
746:
725:
724:
715:
714:
705:
704:
695:
694:
671:
670:
665:
664:
656:
651:
642:
633:
632:
605:
603:
601:
600:
597:
594:
587:
585:
584:
581:
578:
566:
564:
562:
561:
558:
555:
548:
546:
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542:
539:
532:
530:
529:
526:
523:
516:
514:
513:
510:
507:
494:
492:
491:
486:
484:
483:
467:
465:
464:
459:
454:
453:
441:
440:
428:
420:
415:
414:
395:
393:
392:
387:
375:
373:
372:
367:
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364:
355:
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345:
344:
335:
334:
319:
318:
302:
300:
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294:
254:
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246:
244:
243:
223:
221:
220:
215:
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212:
184:
182:
181:
176:
168:
167:
158:
154:
153:
152:
131:
130:
121:
120:
111:
110:
101:
100:
63:Julian Schwinger
21:
2620:
2619:
2615:
2614:
2613:
2611:
2610:
2609:
2570:
2569:
2532:
2498:Physical Review
2495:
2461:Physical Review
2458:
2409:Physical Review
2406:
2403:
2398:
2368:
2340:
2339:
2335:
2309:
2308:
2304:
2266:
2265:
2261:
2256:
2252:
2247:
2243:
2238:
2234:
2229:
2225:
2221:
2183:
2164:
2151:
2138:
2133:
2132:
2122:
2110:
2086:
2085:
2051:
2050:
2029:
2024:
2023:
1993:
1983:
1971:
1960:
1950:
1949:
1913:
1893:
1888:
1887:
1854:
1844:
1824:
1814:
1794:
1782:
1772:
1767:
1766:
1736:
1726:
1721:
1720:
1690:
1680:
1664:
1659:
1658:
1621:
1613:
1612:
1585:
1575:
1559:
1540:
1514:
1506:
1505:
1470:
1451:
1432:
1422:
1409:
1399:
1383:
1364:
1348:
1335:
1319:
1300:
1287:
1277:
1261:
1242:
1219:
1211:
1210:
1186:
1181:
1180:
1150:
1140:
1127:
1117:
1093:
1088:
1087:
1063:
1058:
1057:
1030:
1020:
1004:
985:
965:
960:
959:
953:
916:
905:
904:
880:
867:
854:
849:
848:
832:
830:
829:
828:
824:
822:
821:
820:
819:
783:
778:
777:
753:
735:
716:
706:
696:
677:
676:
672:
653:
622:
621:
598:
595:
592:
591:
589:
582:
579:
576:
575:
573:
571:
559:
556:
553:
552:
550:
543:
540:
537:
536:
534:
527:
524:
521:
520:
518:
511:
508:
505:
504:
502:
500:
475:
470:
469:
445:
432:
403:
398:
397:
378:
377:
356:
346:
336:
326:
310:
305:
304:
261:
260:
235:
230:
229:
195:
190:
189:
159:
141:
122:
112:
102:
83:
82:
78:
73:
72:
28:
23:
22:
15:
12:
11:
5:
2618:
2616:
2608:
2607:
2602:
2597:
2592:
2587:
2582:
2572:
2571:
2568:
2567:
2530:
2493:
2456:
2451:(1989) Wiley,
2441:
2402:
2399:
2397:
2396:
2333:
2302:
2259:
2250:
2241:
2232:
2222:
2220:
2217:
2205:
2204:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2163:
2158:
2154:
2150:
2145:
2141:
2121:
2118:
2109:
2106:
2093:
2071:
2068:
2063:
2059:
2036:
2032:
2020:
2019:
2008:
2005:
2000:
1996:
1990:
1986:
1978:
1974:
1967:
1963:
1957:
1953:
1946:
1941:
1938:
1933:
1929:
1925:
1920:
1916:
1911:
1908:
1905:
1900:
1896:
1881:
1880:
1869:
1866:
1861:
1857:
1851:
1847:
1842:
1839:
1836:
1831:
1827:
1821:
1817:
1812:
1809:
1806:
1801:
1797:
1791:
1786:
1779:
1775:
1751:
1748:
1743:
1739:
1733:
1729:
1717:
1716:
1705:
1702:
1697:
1693:
1687:
1683:
1677:
1674:
1671:
1667:
1643:
1640:
1635:
1632:
1626:
1620:
1609:
1608:
1597:
1592:
1588:
1582:
1578:
1572:
1569:
1566:
1562:
1556:
1549:
1546:
1539:
1536:
1533:
1528:
1525:
1519:
1513:
1499:
1498:
1482:
1477:
1473:
1469:
1464:
1461:
1458:
1454:
1448:
1441:
1438:
1429:
1425:
1421:
1416:
1412:
1406:
1402:
1396:
1393:
1390:
1386:
1380:
1373:
1370:
1363:
1360:
1355:
1351:
1347:
1342:
1338:
1332:
1329:
1326:
1322:
1316:
1309:
1306:
1299:
1294:
1290:
1284:
1280:
1274:
1271:
1268:
1264:
1258:
1251:
1248:
1241:
1238:
1233:
1230:
1224:
1218:
1193:
1189:
1177:
1176:
1165:
1160:
1157:
1153:
1147:
1143:
1137:
1134:
1130:
1123:
1120:
1116:
1111:
1106:
1103:
1100:
1096:
1070:
1066:
1054:
1053:
1042:
1037:
1033:
1027:
1023:
1017:
1014:
1011:
1007:
1001:
994:
991:
984:
979:
976:
970:
952:
949:
923:
919:
915:
912:
892:
887:
883:
879:
874:
870:
866:
861:
857:
831:
823:
790:
786:
774:
773:
760:
756:
751:
745:
742:
738:
734:
731:
728:
723:
719:
713:
709:
703:
699:
693:
690:
687:
684:
680:
675:
669:
662:
659:
648:
645:
639:
636:
631:
482:
478:
457:
452:
448:
444:
439:
435:
431:
426:
423:
418:
413:
410:
406:
385:
363:
359:
353:
349:
343:
339:
333:
329:
325:
322:
317:
313:
292:
289:
286:
283:
280:
277:
274:
271:
268:
257:Dirac matrices
242:
238:
211:
208:
205:
202:
198:
186:
185:
174:
171:
166:
162:
157:
151:
148:
144:
140:
137:
134:
129:
125:
119:
115:
109:
105:
99:
96:
93:
90:
86:
81:
59:William Rarita
55:Dirac equation
43:field equation
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2617:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2577:
2575:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2457:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2410:
2405:
2404:
2400:
2392:
2388:
2384:
2380:
2376:
2372:
2364:
2360:
2356:
2352:
2348:
2344:
2337:
2334:
2329:
2325:
2321:
2317:
2313:
2306:
2303:
2297:
2292:
2287:
2282:
2278:
2274:
2270:
2263:
2260:
2254:
2251:
2245:
2242:
2236:
2233:
2227:
2224:
2218:
2216:
2214:
2210:
2188:
2184:
2180:
2177:
2174:
2169:
2161:
2156:
2152:
2143:
2131:
2130:
2129:
2127:
2119:
2117:
2115:
2107:
2105:
2091:
2069:
2066:
2061:
2057:
2034:
2006:
2003:
1998:
1988:
1984:
1976:
1965:
1955:
1951:
1944:
1939:
1936:
1931:
1927:
1923:
1918:
1914:
1909:
1906:
1903:
1898:
1894:
1886:
1885:
1884:
1867:
1864:
1859:
1855:
1849:
1845:
1840:
1837:
1834:
1829:
1825:
1819:
1810:
1807:
1804:
1799:
1795:
1789:
1777:
1773:
1765:
1764:
1763:
1749:
1746:
1741:
1737:
1731:
1727:
1703:
1700:
1695:
1691:
1685:
1675:
1672:
1669:
1665:
1657:
1656:
1655:
1641:
1638:
1633:
1630:
1618:
1595:
1590:
1586:
1580:
1570:
1567:
1564:
1560:
1554:
1544:
1537:
1534:
1531:
1526:
1523:
1511:
1504:
1503:
1502:
1480:
1475:
1471:
1467:
1462:
1459:
1456:
1452:
1446:
1436:
1427:
1419:
1414:
1410:
1404:
1394:
1391:
1388:
1384:
1378:
1368:
1361:
1358:
1353:
1349:
1345:
1340:
1330:
1327:
1324:
1320:
1314:
1304:
1297:
1292:
1288:
1282:
1272:
1269:
1266:
1262:
1256:
1246:
1239:
1236:
1231:
1228:
1216:
1209:
1208:
1207:
1206:, obtaining:
1191:
1187:
1163:
1155:
1151:
1145:
1141:
1135:
1128:
1121:
1118:
1114:
1109:
1104:
1101:
1098:
1094:
1086:
1085:
1084:
1068:
1064:
1040:
1035:
1031:
1025:
1015:
1012:
1009:
1005:
999:
989:
982:
977:
974:
958:
957:
956:
950:
948:
945:
943:
939:
938:supersymmetry
921:
917:
913:
910:
890:
885:
877:
872:
868:
859:
855:
845:
843:
840:. So far, no
839:
817:
816:delta baryons
813:
812:wave function
808:
806:
805:Dirac adjoint
788:
784:
758:
754:
749:
743:
740:
736:
732:
729:
726:
721:
711:
707:
701:
697:
691:
688:
685:
682:
678:
673:
667:
657:
646:
643:
637:
634:
620:
619:
618:
616:
612:
607:
569:
498:
480:
476:
450:
446:
442:
437:
433:
424:
421:
416:
411:
408:
404:
396:is the mass,
383:
361:
357:
351:
347:
341:
337:
331:
327:
323:
320:
315:
311:
290:
287:
284:
281:
278:
275:
272:
269:
266:
258:
240:
236:
227:
209:
206:
203:
200:
196:
172:
169:
164:
160:
155:
149:
146:
142:
138:
135:
132:
127:
117:
113:
107:
103:
97:
94:
91:
88:
84:
79:
71:
70:
69:
66:
64:
60:
56:
52:
48:
44:
41:
37:
33:
19:
2538:
2534:
2501:
2497:
2464:
2460:
2452:
2448:
2412:
2408:
2374:
2370:
2346:
2342:
2336:
2319:
2316:Phys. Rev. D
2315:
2305:
2276:
2273:SciPost Phys
2272:
2262:
2253:
2244:
2235:
2226:
2213:quantization
2206:
2123:
2111:
2021:
1882:
1718:
1610:
1500:
1178:
1055:
954:
946:
942:supergravity
846:
809:
803:denotes the
775:
608:
187:
67:
40:relativistic
35:
29:
2453:Section 1.6
2445:Martin A.D.
2114:Fierz–Pauli
549:, 0) ⊕ (0,
2574:Categories
2377:(2): 221.
2349:(2): 271.
2286:2305.00106
2279:(3): 065.
2219:References
615:Lagrangian
2563:0556-2821
2526:0031-899X
2489:0031-899X
2437:0031-899X
2189:μ
2175:−
2170:μ
2166:∂
2157:μ
2149:→
2144:μ
2140:∂
2092:κ
2058:ψ
2031:∇
2004:κ
1995:∂
1985:γ
1973:∇
1962:∂
1952:γ
1928:ψ
1915:ψ
1907:κ
1895:ψ
1860:μ
1856:ψ
1850:μ
1846:γ
1830:μ
1826:ψ
1820:μ
1816:∂
1800:μ
1796:ψ
1790:ν
1785:∂
1778:ν
1774:γ
1742:μ
1738:ψ
1732:μ
1728:γ
1696:ρ
1692:ψ
1686:ν
1682:∂
1676:ρ
1673:ν
1670:μ
1666:γ
1619:δ
1591:ρ
1587:ψ
1581:ν
1577:∂
1571:ρ
1568:ν
1565:μ
1561:γ
1555:μ
1548:¯
1545:ψ
1538:δ
1512:δ
1476:ρ
1472:ψ
1468:δ
1463:ρ
1460:ν
1457:μ
1453:γ
1447:μ
1440:¯
1437:ψ
1428:ν
1424:∂
1420:−
1415:ρ
1411:ψ
1405:ν
1401:∂
1395:ρ
1392:ν
1389:μ
1385:γ
1379:μ
1372:¯
1369:ψ
1362:δ
1354:ρ
1350:ψ
1346:δ
1341:ν
1337:∂
1331:ρ
1328:ν
1325:μ
1321:γ
1315:μ
1308:¯
1305:ψ
1293:ρ
1289:ψ
1283:ν
1279:∂
1273:ρ
1270:ν
1267:μ
1263:γ
1257:μ
1250:¯
1247:ψ
1240:δ
1217:δ
1192:μ
1188:ψ
1156:ρ
1152:γ
1146:ν
1142:γ
1136:μ
1129:γ
1110:≡
1105:ρ
1102:ν
1099:μ
1095:γ
1069:μ
1065:ψ
1036:ρ
1032:ψ
1026:ν
1022:∂
1016:ρ
1013:ν
1010:μ
1006:γ
1000:μ
993:¯
990:ψ
922:α
918:ϵ
914:≡
911:ϵ
891:ϵ
886:μ
882:∂
873:μ
869:ψ
865:→
860:μ
856:ψ
838:gravitino
789:μ
785:ψ
759:ν
755:ψ
744:ν
741:μ
737:σ
727:−
722:ρ
718:∂
712:κ
708:γ
698:γ
692:ν
689:ρ
686:κ
683:μ
679:ϵ
668:μ
661:¯
658:ψ
638:−
481:ν
477:ψ
451:ν
447:γ
438:μ
434:γ
417:≡
412:ν
409:μ
405:σ
358:γ
348:γ
338:γ
328:γ
312:γ
267:κ
241:κ
237:γ
210:ν
207:ρ
204:κ
201:μ
197:ϵ
165:ν
161:ψ
150:ν
147:μ
143:σ
133:−
128:ρ
124:∂
118:κ
114:γ
104:γ
98:ν
95:ρ
92:κ
89:μ
85:ϵ
65:in 1941.
2600:Fermions
903:, where
51:fermions
2590:Spinors
2543:Bibcode
2506:Bibcode
2469:Bibcode
2417:Bibcode
2401:Sources
2379:Bibcode
2351:Bibcode
2022:where
602:
590:
586:
574:
563:
551:
547:
535:
531:
519:
515:
503:
224:is the
38:is the
2561:
2524:
2487:
2435:
827:Δ
606:part.
533:) ⊗ ((
497:spinor
468:, and
303:) and
259:(with
188:where
34:, the
2281:arXiv
588:) ⊕ (
49:-3/2
2559:ISSN
2522:ISSN
2485:ISSN
2433:ISSN
604:, 1)
572:(1,
255:are
61:and
47:spin
2551:doi
2514:doi
2502:188
2477:doi
2465:186
2425:doi
2387:doi
2375:120
2359:doi
2347:114
2324:doi
2291:doi
940:of
45:of
30:In
2576::
2557:.
2549:.
2539:17
2537:.
2520:.
2512:.
2500:.
2483:.
2475:.
2463:.
2431:.
2423:.
2413:60
2411:.
2385:.
2373:.
2367:;
2357:.
2345:.
2320:16
2318:.
2314:.
2289:.
2277:16
2275:.
2271:.
2128::
1868:0.
1704:0.
807:.
617::
565:))
517:,
376:,
228:,
2565:.
2553::
2545::
2528:.
2516::
2508::
2491:.
2479::
2471::
2455:.
2439:.
2427::
2419::
2393:.
2389::
2381::
2365:.
2361::
2353::
2330:.
2326::
2299:.
2293::
2283::
2203:.
2185:A
2181:e
2178:i
2162:=
2153:D
2070:T
2067:T
2062:i
2035:2
2007:,
1999:i
1989:0
1977:2
1966:j
1956:j
1945:+
1940:T
1937:T
1932:i
1924:=
1919:i
1910:,
1904:=
1899:0
1865:=
1841:,
1838:0
1835:=
1811:,
1808:0
1805:=
1750:0
1747:=
1701:=
1642:0
1639:=
1634:S
1631:R
1625:L
1596:,
1535:2
1532:=
1527:S
1524:R
1518:L
1481:+
1359:=
1298:+
1237:=
1232:S
1229:R
1223:L
1164:.
1159:]
1133:[
1122:!
1119:3
1115:1
1041:,
983:=
978:S
975:R
969:L
878:+
818:(
750:)
733:m
730:i
702:5
674:(
647:2
644:1
635:=
630:L
599:2
596:/
593:1
583:2
580:/
577:1
560:2
557:/
554:1
544:2
541:/
538:1
528:2
525:/
522:1
512:2
509:/
506:1
501:(
456:]
443:,
430:[
425:2
422:i
384:m
362:3
352:2
342:1
332:0
324:i
321:=
316:5
291:3
288:,
285:2
282:,
279:1
276:,
273:0
270:=
173:0
170:=
156:)
139:m
136:i
108:5
80:(
20:)
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