40:
998:
is referred to as the 'representation space'. In physics sometimes the vector space is referred to as the representation, for example in the sentence 'we model the particle as transforming in the singlet representation', or even to refer to a quantum field which takes values in such a representation,
101:
is the state space for 'internal' degrees of freedom of a particle, that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the particle's position in space. Examples of such degrees of freedom are the
1510:
2844:
In quantum field theory different particles correspond one to one with gauged fields transforming in irreducible representations of the internal and
Lorentz group. Thus, a multiplet has also come to describe a collection of
2026:
1290:
1874:
545:
2113:
1624:
describing the dimension of the representation, or, with appropriate normalisation, the highest weight of the representation. In physics it is common convention to label these by half-integers instead. See
1357:
For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups. Many of these have well understood representation theory:
1046:
dimensional irreducible representation. Generally, a group may have multiple non-isomorphic representations of the same dimension, so this does not fully characterize the representation. An exception is
747:
1564:
800:
2167:
2698:
2630:
2501:
366:
1602:
591:
1417:
1205:
2462:
455:
2779:
2659:
2425:
1911:
1802:
1234:
1148:
2583:
640:
2810:
2345:
2280:
2245:
2196:
2036:
These groups all appear in the theory of the
Standard model. For theories which extend these symmetries, the representation theory of some other groups might be considered:
1723:
1691:
1660:
1325:
1076:
264:
1390:
1412:
1347:
684:
662:
286:
1943:
189:
1178:
2839:
2730:
2556:
905:
58:
2314:
2529:
1755:
976:
956:
929:
706:
329:
2750:
2385:
2365:
1622:
1392:: Part of the gauge group of the Standard model, and the gauge group for theories of electromagnetism. Irreps are all 1 dimensional and are indexed by integers
1116:
1096:
1044:
1020:
996:
614:
406:
386:
307:
236:
212:
1809:
1950:
2841:
is a vector valued function of spacetime, but is still referred to as a scalar field, as it transforms trivially under
Lorentz transformations.
1762:
2028:: The Poincaré group of isometries of flat spacetime. This can be understood in terms of the representation theory of the groups above. See
1239:
932:
1817:
460:
2043:
76:
711:
2940:. Where the number of unresolved lines is small, these are often referred to specifically as doublet or triplet peaks, while
2121:
Grand unified theories: Gauge groups which contain the
Standard model gauge group as a subgroup. Proposed candidates include
1808:, the linear symmetries of flat spacetime. All representations arise as representations of its corresponding spin group. See
1626:
2703:
Beware that besides the
Lorentz group, a field can transform under the action of a gauge group. For example, a scalar field
1521:
3006:
2558:(strictly, this might be more accurately labelled a covector field), which transforms as a 4-vector, and spinor fields
752:
2124:
3001:
2885:
2029:
147:
2664:
2596:
2467:
334:
1569:
2986:
554:
1505:{\displaystyle \rho _{n}:{\text{U}}(1)\rightarrow {\text{GL}}(\mathbb {C} );e^{i\theta }\mapsto e^{in\theta }}
2953:
Georgi, H. (1999). Lie
Algebras in Particle Physics: From Isospin to Unified Theories (1st ed.). CRC Press.
2504:
2900:
2893:
931:
is used for both Lie algebras and Lie groups as, at least in finite dimension, there is a well understood
1183:
2430:
411:
2755:
2635:
2393:
1879:
1770:
1210:
1124:
2981:
2862:
2561:
1180:
matrices. This explicit realisation of the rotation group is known as the fundamental representation
621:
135:
2784:
2319:
2254:
2219:
2172:
1697:
1665:
1634:
1299:
1050:
241:
1725:: Part of the gauge group of the Standard model. Irreps are indexed pairs of non-negative integers
1364:
2752:
is a spacetime point, might have an isospin state taking values in the fundamental representation
1395:
1330:
667:
645:
269:
2929:
2925:
2846:
1916:
156:
2040:
Conformal symmetry: For pseudo-Euclidean space, symmetries are described by the conformal group
1566:: Part of the gauge group of the Standard model. Irreps are indexed by non-negative integers in
1157:
2815:
2706:
2534:
805:
2285:
94:
2514:
1728:
961:
941:
914:
691:
314:
215:
2212:
In quantum physics, the mathematical notion is usually applied to representations of the
27:
This article is about multiplets in mathematics and physics. For multiplet siblings, see
2976:
2858:
2735:
2370:
2350:
1758:
1607:
1513:
1101:
1081:
1029:
1005:
981:
599:
391:
371:
292:
221:
197:
119:
103:
28:
2995:
2937:
2873:
1805:
2889:
2881:
2877:
2586:
2508:
123:
107:
2966:
2590:
2248:
2213:
143:
115:
2021:{\displaystyle {\text{E}}(1,3)\cong \mathbb {R} ^{1,3}\rtimes {\text{SO}}(1,3)}
1662:: The group of rotations of 3D space. Irreps are the odd-dimensional irreps of
1292:. Since the dimension of this representation space is 3, this is known as the
958:
as the representation, for example in the sentence 'consider a representation
139:
2632:. A right-handed Weyl spinor transforms in the fundamental representation,
17:
2869:
2954:
2971:
999:
and the physical particles which are modelled by such a quantum field.
111:
90:
2367:
th symmetric power of the fundamental representation, every field has
1285:{\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})}
642:
is a Lie algebra. It is often a finite-dimensional Lie algebra over
1869:{\displaystyle {\text{SL}}(2,\mathbb {C} )\cong {\text{Spin}}(1,3)}
2908:
2904:
2866:
1236:
is a representation space. The full data of the representation is
1151:
2911:, specifically the three-dimensional fundamental representation.
2390:
Fields also transform under representations of the
Lorentz group
122:
of particle physics. Formally, we describe this state space by a
540:{\displaystyle \rho (g_{1}\cdot g_{2})=\rho (g_{1})\rho (g_{2})}
2531:, which transform in the trivial representation, vector fields
2108:{\displaystyle {\text{Conf}}(p,q)\cong O(p,q)/\mathbb {Z} _{2}}
1078:
which has exactly one irreducible representation of dimension
126:
which carries the action of a group of continuous symmetries.
33:
749:. This is a linear map which preserves the Lie bracket: for
938:
In mathematics, it is common to refer to the homomorphism
742:{\displaystyle {\mathfrak {g}}\rightarrow {\text{End}}(V)}
1913:. Irreps are indexed by pairs of non-negative integers
54:
2818:
2787:
2758:
2738:
2709:
2667:
2638:
2599:
2564:
2537:
2517:
2470:
2433:
2396:
2373:
2353:
2322:
2288:
2257:
2222:
2175:
2127:
2046:
1953:
1919:
1882:
1820:
1773:
1731:
1700:
1668:
1637:
1610:
1572:
1524:
1420:
1398:
1367:
1333:
1302:
1242:
1213:
1186:
1160:
1127:
1104:
1084:
1053:
1032:
1008:
984:
964:
944:
917:
808:
755:
714:
694:
670:
648:
624:
602:
557:
463:
414:
394:
374:
337:
317:
295:
272:
244:
224:
200:
159:
1559:{\displaystyle {\text{SU}}(2)\cong {\text{Spin}}(3)}
1121:
For example, consider real three-dimensional space,
2944:is used to describe groups of peaks in any number.
49:
may be too technical for most readers to understand
2833:
2804:
2773:
2744:
2724:
2692:
2653:
2624:
2577:
2550:
2523:
2495:
2456:
2419:
2379:
2359:
2339:
2308:
2274:
2239:
2190:
2161:
2118:Supersymmetry: Symmetry described by a supergroup.
2107:
2020:
1937:
1905:
1868:
1796:
1749:
1717:
1685:
1654:
1616:
1596:
1558:
1504:
1406:
1384:
1341:
1319:
1284:
1228:
1199:
1172:
1142:
1110:
1090:
1070:
1038:
1014:
990:
970:
950:
923:
899:
794:
741:
700:
678:
656:
634:
608:
585:
539:
449:
408:. This map must preserve the group structure: for
400:
380:
360:
323:
309:is a Lie group. This is often a compact Lie group.
301:
280:
258:
230:
206:
183:
2282:is determined by the single half-integer number
2876:, which is used to define spin quantization. A
1945:, indexing the dimension of the representation.
795:{\displaystyle X_{1},X_{2}\in {\mathfrak {g}},}
2593:spinors which transform in representations of
2162:{\displaystyle {\text{SU}}(5),{\text{SO}}(10)}
134:Mathematically, multiplets are described via
8:
2247:gauge theory will have multiplets which are
2693:{\displaystyle {\text{SL}}(2,\mathbb {C} )}
2625:{\displaystyle {\text{SL}}(2,\mathbb {C} )}
2496:{\displaystyle {\text{SL}}(2,\mathbb {C} )}
1154:acts naturally on this space as a group of
361:{\displaystyle G\rightarrow {\text{GL}}(V)}
106:of a particle in quantum mechanics, or the
1810:Representation theory of the Lorentz group
1597:{\displaystyle n\in \mathbb {N} _{\geq 0}}
388:to the space of invertible linear maps on
2817:
2788:
2786:
2765:
2761:
2760:
2757:
2737:
2708:
2683:
2682:
2668:
2666:
2645:
2641:
2640:
2637:
2615:
2614:
2600:
2598:
2569:
2563:
2542:
2536:
2516:
2486:
2485:
2471:
2469:
2434:
2432:
2397:
2395:
2372:
2352:
2323:
2321:
2298:
2287:
2258:
2256:
2223:
2221:
2182:
2177:
2174:
2145:
2128:
2126:
2099:
2095:
2094:
2088:
2047:
2045:
1998:
1983:
1979:
1978:
1954:
1952:
1918:
1883:
1881:
1846:
1836:
1835:
1821:
1819:
1774:
1772:
1730:
1701:
1699:
1669:
1667:
1638:
1636:
1609:
1585:
1581:
1580:
1571:
1542:
1525:
1523:
1490:
1474:
1460:
1459:
1451:
1434:
1425:
1419:
1400:
1399:
1397:
1368:
1366:
1334:
1332:
1303:
1301:
1273:
1261:
1252:
1248:
1247:
1241:
1220:
1216:
1215:
1212:
1191:
1185:
1159:
1134:
1130:
1129:
1126:
1103:
1083:
1054:
1052:
1031:
1007:
983:
963:
943:
916:
885:
863:
835:
822:
807:
783:
782:
773:
760:
754:
725:
716:
715:
713:
693:
672:
671:
669:
650:
649:
647:
626:
625:
623:
601:
586:{\displaystyle (V,{\mathfrak {g}},\rho )}
568:
567:
556:
528:
509:
487:
474:
462:
432:
419:
413:
393:
373:
344:
336:
316:
294:
274:
273:
271:
252:
251:
243:
223:
199:
158:
77:Learn how and when to remove this message
61:, without removing the technical details.
551:At the algebra level, this is a triplet
2936:is a group of related or unresolvable
2347:representations are isomorphic to the
1002:For an irreducible representation, an
153:At the group level, this is a triplet
2955:https://doi.org/10.1201/9780429499210
1763:Clebsch-Gordan coefficients for SU(3)
1512:. The index can be understood as the
1327:, and it is common to denote this as
935:between Lie groups and Lie algebras.
59:make it understandable to non-experts
7:
2849:described by these representations.
2892:is in the vector representation or
2427:, or more generally its spin group
1200:{\displaystyle \rho _{\text{fund}}}
784:
717:
627:
569:
2861:, which describes symmetries of a
2457:{\displaystyle {\text{Spin}}(1,3)}
1353:Application to theoretical physics
146:, and is usually used to refer to
25:
2316:, the isospin. Since irreducible
450:{\displaystyle g_{1},g_{2}\in G,}
2774:{\displaystyle \mathbb {C} ^{2}}
2654:{\displaystyle \mathbb {C} ^{2}}
2420:{\displaystyle {\text{SO}}(1,3)}
1906:{\displaystyle {\text{SO}}(1,3)}
1797:{\displaystyle {\text{SO}}(1,3)}
1335:
1229:{\displaystyle \mathbb {R} ^{3}}
1143:{\displaystyle \mathbb {R} ^{3}}
368:, that is, a map from the group
38:
2880:is a trivial representation, a
2578:{\displaystyle \psi _{\alpha }}
708:is an Lie algebra homomorphism
635:{\displaystyle {\mathfrak {g}}}
2924:In spectroscopy, particularly
2828:
2822:
2805:{\displaystyle {\text{SU}}(2)}
2799:
2793:
2719:
2713:
2687:
2673:
2619:
2605:
2490:
2476:
2451:
2439:
2414:
2402:
2387:symmetrized internal indices.
2340:{\displaystyle {\text{SU}}(2)}
2334:
2328:
2275:{\displaystyle {\text{SU}}(2)}
2269:
2263:
2240:{\displaystyle {\text{SU}}(2)}
2234:
2228:
2191:{\displaystyle {\text{E}}_{6}}
2156:
2150:
2139:
2133:
2085:
2073:
2064:
2052:
2015:
2003:
1971:
1959:
1932:
1920:
1900:
1888:
1863:
1851:
1840:
1826:
1791:
1779:
1744:
1732:
1718:{\displaystyle {\text{SU}}(3)}
1712:
1706:
1686:{\displaystyle {\text{SU}}(2)}
1680:
1674:
1655:{\displaystyle {\text{SO}}(3)}
1649:
1643:
1627:Representation theory of SU(2)
1553:
1547:
1536:
1530:
1483:
1464:
1456:
1448:
1445:
1439:
1379:
1373:
1320:{\displaystyle {\text{SO}}(3)}
1314:
1308:
1279:
1243:
1098:for each non-negative integer
1071:{\displaystyle {\text{SU}}(2)}
1065:
1059:
894:
891:
878:
869:
856:
850:
844:
841:
815:
812:
736:
730:
722:
580:
558:
534:
521:
515:
502:
493:
467:
355:
349:
341:
259:{\displaystyle K=\mathbb {R} }
178:
160:
1:
2464:which can be identified with
1385:{\displaystyle {\text{U}}(1)}
2857:The best known example is a
1407:{\displaystyle \mathbb {Z} }
1342:{\displaystyle \mathbf {3} }
1150:. The group of 3D rotations
679:{\displaystyle \mathbb {C} }
657:{\displaystyle \mathbb {R} }
281:{\displaystyle \mathbb {C} }
1938:{\displaystyle (\mu ,\nu )}
1761:of the representation. See
184:{\displaystyle (V,G,\rho )}
148:irreducible representations
3023:
2886:fundamental representation
118:state of particles in the
26:
1173:{\displaystyle 3\times 3}
214:is a vector space over a
2987:Multiplicity (chemistry)
2834:{\displaystyle \phi (x)}
2725:{\displaystyle \phi (x)}
2551:{\displaystyle A_{\mu }}
2251:whose representation of
978:', and the vector space
900:{\displaystyle \rho ()=}
331:is a group homomorphism
238:, generally taken to be
130:Mathematical formulation
2505:exceptional isomorphism
2030:Wigner's classification
218:(in the algebra sense)
2907:are in a multiplet of
2894:adjoint representation
2835:
2806:
2775:
2746:
2726:
2694:
2655:
2626:
2579:
2552:
2525:
2497:
2458:
2421:
2381:
2361:
2341:
2310:
2309:{\displaystyle s=:n/2}
2276:
2241:
2192:
2163:
2109:
2022:
1939:
1907:
1870:
1798:
1751:
1719:
1687:
1656:
1618:
1598:
1560:
1506:
1414:, given explicitly by
1408:
1386:
1343:
1321:
1286:
1230:
1201:
1174:
1144:
1112:
1092:
1072:
1040:
1016:
992:
972:
952:
925:
901:
796:
743:
702:
680:
658:
636:
610:
587:
541:
451:
402:
382:
362:
325:
303:
282:
260:
232:
208:
185:
2836:
2807:
2776:
2747:
2727:
2695:
2656:
2627:
2580:
2553:
2526:
2524:{\displaystyle \phi }
2498:
2459:
2422:
2382:
2362:
2342:
2311:
2277:
2242:
2193:
2164:
2110:
2023:
1940:
1908:
1871:
1799:
1752:
1750:{\displaystyle (m,n)}
1720:
1688:
1657:
1619:
1599:
1561:
1507:
1409:
1387:
1344:
1322:
1287:
1231:
1202:
1175:
1145:
1113:
1093:
1073:
1041:
1017:
993:
973:
971:{\displaystyle \rho }
953:
951:{\displaystyle \rho }
926:
924:{\displaystyle \rho }
902:
797:
744:
703:
701:{\displaystyle \rho }
681:
659:
637:
611:
588:
542:
452:
403:
383:
363:
326:
324:{\displaystyle \rho }
304:
283:
261:
233:
209:
186:
150:(irreps, for short).
142:or its corresponding
2982:Group representation
2863:group representation
2816:
2785:
2756:
2736:
2707:
2665:
2636:
2597:
2562:
2535:
2515:
2468:
2431:
2394:
2371:
2351:
2320:
2286:
2255:
2220:
2208:Quantum field theory
2173:
2125:
2044:
1951:
1917:
1880:
1876:: The spin group of
1818:
1771:
1729:
1698:
1666:
1635:
1608:
1570:
1522:
1418:
1396:
1365:
1331:
1300:
1240:
1211:
1184:
1158:
1125:
1102:
1082:
1051:
1030:
1006:
982:
962:
942:
915:
806:
753:
712:
692:
668:
646:
622:
600:
555:
461:
412:
392:
372:
335:
315:
293:
270:
242:
222:
198:
157:
93:and particularly in
3007:Rotational symmetry
2847:subatomic particles
2511:, commonly denoted
2507:. Examples include
1296:representation for
2930:X-ray spectroscopy
2926:Gamma spectroscopy
2831:
2802:
2771:
2742:
2722:
2690:
2651:
2622:
2575:
2548:
2521:
2493:
2454:
2417:
2377:
2357:
2337:
2306:
2272:
2237:
2216:. For example, an
2188:
2159:
2105:
2018:
1935:
1903:
1866:
1794:
1747:
1715:
1683:
1652:
1614:
1594:
1556:
1502:
1404:
1382:
1339:
1317:
1282:
1226:
1197:
1170:
1140:
1108:
1088:
1068:
1036:
1012:
988:
968:
948:
921:
897:
792:
739:
698:
676:
654:
632:
606:
583:
537:
447:
398:
378:
358:
321:
299:
278:
256:
228:
204:
181:
3002:Quantum mechanics
2791:
2745:{\displaystyle x}
2671:
2603:
2474:
2437:
2400:
2380:{\displaystyle n}
2360:{\displaystyle n}
2326:
2261:
2226:
2180:
2148:
2131:
2050:
2001:
1957:
1886:
1849:
1824:
1777:
1757:, describing the
1704:
1672:
1641:
1617:{\displaystyle n}
1545:
1528:
1454:
1437:
1371:
1306:
1276:
1264:
1194:
1111:{\displaystyle n}
1091:{\displaystyle n}
1057:
1039:{\displaystyle n}
1015:{\displaystyle n}
991:{\displaystyle V}
728:
609:{\displaystyle V}
401:{\displaystyle V}
381:{\displaystyle G}
347:
302:{\displaystyle G}
231:{\displaystyle K}
207:{\displaystyle V}
87:
86:
79:
16:(Redirected from
3014:
2840:
2838:
2837:
2832:
2811:
2809:
2808:
2803:
2792:
2789:
2780:
2778:
2777:
2772:
2770:
2769:
2764:
2751:
2749:
2748:
2743:
2731:
2729:
2728:
2723:
2699:
2697:
2696:
2691:
2686:
2672:
2669:
2660:
2658:
2657:
2652:
2650:
2649:
2644:
2631:
2629:
2628:
2623:
2618:
2604:
2601:
2584:
2582:
2581:
2576:
2574:
2573:
2557:
2555:
2554:
2549:
2547:
2546:
2530:
2528:
2527:
2522:
2502:
2500:
2499:
2494:
2489:
2475:
2472:
2463:
2461:
2460:
2455:
2438:
2435:
2426:
2424:
2423:
2418:
2401:
2398:
2386:
2384:
2383:
2378:
2366:
2364:
2363:
2358:
2346:
2344:
2343:
2338:
2327:
2324:
2315:
2313:
2312:
2307:
2302:
2281:
2279:
2278:
2273:
2262:
2259:
2246:
2244:
2243:
2238:
2227:
2224:
2197:
2195:
2194:
2189:
2187:
2186:
2181:
2178:
2168:
2166:
2165:
2160:
2149:
2146:
2132:
2129:
2114:
2112:
2111:
2106:
2104:
2103:
2098:
2092:
2051:
2048:
2027:
2025:
2024:
2019:
2002:
1999:
1994:
1993:
1982:
1958:
1955:
1944:
1942:
1941:
1936:
1912:
1910:
1909:
1904:
1887:
1884:
1875:
1873:
1872:
1867:
1850:
1847:
1839:
1825:
1822:
1803:
1801:
1800:
1795:
1778:
1775:
1756:
1754:
1753:
1748:
1724:
1722:
1721:
1716:
1705:
1702:
1692:
1690:
1689:
1684:
1673:
1670:
1661:
1659:
1658:
1653:
1642:
1639:
1623:
1621:
1620:
1615:
1603:
1601:
1600:
1595:
1593:
1592:
1584:
1565:
1563:
1562:
1557:
1546:
1543:
1529:
1526:
1511:
1509:
1508:
1503:
1501:
1500:
1482:
1481:
1463:
1455:
1452:
1438:
1435:
1430:
1429:
1413:
1411:
1410:
1405:
1403:
1391:
1389:
1388:
1383:
1372:
1369:
1348:
1346:
1345:
1340:
1338:
1326:
1324:
1323:
1318:
1307:
1304:
1291:
1289:
1288:
1283:
1278:
1277:
1274:
1265:
1262:
1257:
1256:
1251:
1235:
1233:
1232:
1227:
1225:
1224:
1219:
1206:
1204:
1203:
1198:
1196:
1195:
1192:
1179:
1177:
1176:
1171:
1149:
1147:
1146:
1141:
1139:
1138:
1133:
1117:
1115:
1114:
1109:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1058:
1055:
1045:
1043:
1042:
1037:
1021:
1019:
1018:
1013:
997:
995:
994:
989:
977:
975:
974:
969:
957:
955:
954:
949:
930:
928:
927:
922:
906:
904:
903:
898:
890:
889:
868:
867:
840:
839:
827:
826:
801:
799:
798:
793:
788:
787:
778:
777:
765:
764:
748:
746:
745:
740:
729:
726:
721:
720:
707:
705:
704:
699:
685:
683:
682:
677:
675:
663:
661:
660:
655:
653:
641:
639:
638:
633:
631:
630:
615:
613:
612:
607:
592:
590:
589:
584:
573:
572:
546:
544:
543:
538:
533:
532:
514:
513:
492:
491:
479:
478:
456:
454:
453:
448:
437:
436:
424:
423:
407:
405:
404:
399:
387:
385:
384:
379:
367:
365:
364:
359:
348:
345:
330:
328:
327:
322:
308:
306:
305:
300:
287:
285:
284:
279:
277:
265:
263:
262:
257:
255:
237:
235:
234:
229:
213:
211:
210:
205:
190:
188:
187:
182:
95:particle physics
82:
75:
71:
68:
62:
42:
41:
34:
21:
3022:
3021:
3017:
3016:
3015:
3013:
3012:
3011:
2992:
2991:
2963:
2950:
2922:
2917:
2874:Lorentz algebra
2855:
2814:
2813:
2783:
2782:
2759:
2754:
2753:
2734:
2733:
2705:
2704:
2663:
2662:
2639:
2634:
2633:
2595:
2594:
2565:
2560:
2559:
2538:
2533:
2532:
2513:
2512:
2466:
2465:
2429:
2428:
2392:
2391:
2369:
2368:
2349:
2348:
2318:
2317:
2284:
2283:
2253:
2252:
2218:
2217:
2210:
2205:
2176:
2171:
2170:
2123:
2122:
2093:
2042:
2041:
1977:
1949:
1948:
1915:
1914:
1878:
1877:
1816:
1815:
1769:
1768:
1727:
1726:
1696:
1695:
1664:
1663:
1633:
1632:
1606:
1605:
1579:
1568:
1567:
1520:
1519:
1486:
1470:
1421:
1416:
1415:
1394:
1393:
1363:
1362:
1355:
1329:
1328:
1298:
1297:
1269:
1246:
1238:
1237:
1214:
1209:
1208:
1187:
1182:
1181:
1156:
1155:
1128:
1123:
1122:
1100:
1099:
1080:
1079:
1049:
1048:
1028:
1027:
1004:
1003:
980:
979:
960:
959:
940:
939:
913:
912:
881:
859:
831:
818:
804:
803:
769:
756:
751:
750:
710:
709:
690:
689:
666:
665:
644:
643:
620:
619:
598:
597:
553:
552:
524:
505:
483:
470:
459:
458:
428:
415:
410:
409:
390:
389:
370:
369:
333:
332:
313:
312:
291:
290:
268:
267:
240:
239:
220:
219:
196:
195:
155:
154:
136:representations
132:
83:
72:
66:
63:
55:help improve it
52:
43:
39:
32:
23:
22:
15:
12:
11:
5:
3020:
3018:
3010:
3009:
3004:
2994:
2993:
2990:
2989:
2984:
2979:
2977:Spin (physics)
2974:
2969:
2962:
2959:
2958:
2957:
2949:
2946:
2938:spectral lines
2921:
2918:
2916:
2913:
2859:spin multiplet
2854:
2851:
2830:
2827:
2824:
2821:
2801:
2798:
2795:
2768:
2763:
2741:
2721:
2718:
2715:
2712:
2689:
2685:
2681:
2678:
2675:
2648:
2643:
2621:
2617:
2613:
2610:
2607:
2572:
2568:
2545:
2541:
2520:
2492:
2488:
2484:
2481:
2478:
2453:
2450:
2447:
2444:
2441:
2416:
2413:
2410:
2407:
2404:
2376:
2356:
2336:
2333:
2330:
2305:
2301:
2297:
2294:
2291:
2271:
2268:
2265:
2236:
2233:
2230:
2209:
2206:
2204:
2201:
2200:
2199:
2185:
2158:
2155:
2152:
2144:
2141:
2138:
2135:
2119:
2116:
2102:
2097:
2091:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2034:
2033:
2017:
2014:
2011:
2008:
2005:
1997:
1992:
1989:
1986:
1981:
1976:
1973:
1970:
1967:
1964:
1961:
1946:
1934:
1931:
1928:
1925:
1922:
1902:
1899:
1896:
1893:
1890:
1865:
1862:
1859:
1856:
1853:
1845:
1842:
1838:
1834:
1831:
1828:
1813:
1793:
1790:
1787:
1784:
1781:
1766:
1759:highest weight
1746:
1743:
1740:
1737:
1734:
1714:
1711:
1708:
1693:
1682:
1679:
1676:
1651:
1648:
1645:
1630:
1613:
1591:
1588:
1583:
1578:
1575:
1555:
1552:
1549:
1541:
1538:
1535:
1532:
1517:
1514:winding number
1499:
1496:
1493:
1489:
1485:
1480:
1477:
1473:
1469:
1466:
1462:
1458:
1450:
1447:
1444:
1441:
1433:
1428:
1424:
1402:
1381:
1378:
1375:
1354:
1351:
1337:
1316:
1313:
1310:
1281:
1272:
1268:
1260:
1255:
1250:
1245:
1223:
1218:
1190:
1169:
1166:
1163:
1137:
1132:
1107:
1087:
1067:
1064:
1061:
1035:
1011:
987:
967:
947:
933:correspondence
920:
909:
908:
896:
893:
888:
884:
880:
877:
874:
871:
866:
862:
858:
855:
852:
849:
846:
843:
838:
834:
830:
825:
821:
817:
814:
811:
791:
786:
781:
776:
772:
768:
763:
759:
738:
735:
732:
724:
719:
697:
687:
674:
652:
629:
617:
605:
582:
579:
576:
571:
566:
563:
560:
549:
548:
536:
531:
527:
523:
520:
517:
512:
508:
504:
501:
498:
495:
490:
486:
482:
477:
473:
469:
466:
446:
443:
440:
435:
431:
427:
422:
418:
397:
377:
357:
354:
351:
343:
340:
320:
310:
298:
288:
276:
254:
250:
247:
227:
203:
180:
177:
174:
171:
168:
165:
162:
131:
128:
120:Standard model
85:
84:
46:
44:
37:
29:Multiple birth
24:
14:
13:
10:
9:
6:
4:
3:
2:
3019:
3008:
3005:
3003:
3000:
2999:
2997:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2964:
2960:
2956:
2952:
2951:
2947:
2945:
2943:
2939:
2935:
2931:
2927:
2919:
2914:
2912:
2910:
2906:
2902:
2897:
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2868:
2864:
2860:
2852:
2850:
2848:
2842:
2825:
2819:
2796:
2766:
2739:
2716:
2710:
2701:
2679:
2676:
2646:
2611:
2608:
2592:
2588:
2570:
2566:
2543:
2539:
2518:
2510:
2509:scalar fields
2506:
2482:
2479:
2448:
2445:
2442:
2411:
2408:
2405:
2388:
2374:
2354:
2331:
2303:
2299:
2295:
2292:
2289:
2266:
2250:
2231:
2215:
2207:
2202:
2183:
2153:
2142:
2136:
2120:
2117:
2100:
2089:
2082:
2079:
2076:
2070:
2067:
2061:
2058:
2055:
2039:
2038:
2037:
2031:
2012:
2009:
2006:
1995:
1990:
1987:
1984:
1974:
1968:
1965:
1962:
1947:
1929:
1926:
1923:
1897:
1894:
1891:
1860:
1857:
1854:
1843:
1832:
1829:
1814:
1811:
1807:
1806:Lorentz group
1788:
1785:
1782:
1767:
1764:
1760:
1741:
1738:
1735:
1709:
1694:
1677:
1646:
1631:
1628:
1611:
1589:
1586:
1576:
1573:
1550:
1539:
1533:
1518:
1515:
1497:
1494:
1491:
1487:
1478:
1475:
1471:
1467:
1442:
1431:
1426:
1422:
1376:
1361:
1360:
1359:
1352:
1350:
1311:
1295:
1270:
1266:
1258:
1253:
1221:
1188:
1167:
1164:
1161:
1153:
1135:
1119:
1105:
1085:
1062:
1033:
1026:refers to an
1025:
1009:
1000:
985:
965:
945:
936:
934:
918:
886:
882:
875:
872:
864:
860:
853:
847:
836:
832:
828:
823:
819:
809:
789:
779:
774:
770:
766:
761:
757:
733:
695:
688:
618:
616:is as before.
603:
596:
595:
594:
577:
574:
564:
561:
529:
525:
518:
510:
506:
499:
496:
488:
484:
480:
475:
471:
464:
444:
441:
438:
433:
429:
425:
420:
416:
395:
375:
352:
338:
318:
311:
296:
289:
248:
245:
225:
217:
201:
194:
193:
192:
175:
172:
169:
166:
163:
151:
149:
145:
141:
137:
129:
127:
125:
121:
117:
113:
109:
105:
100:
96:
92:
81:
78:
70:
60:
56:
50:
47:This article
45:
36:
35:
30:
19:
2941:
2933:
2923:
2920:Spectroscopy
2898:
2890:spin triplet
2882:spin doublet
2878:spin singlet
2856:
2843:
2702:
2389:
2211:
2035:
1356:
1293:
1120:
1023:
1001:
937:
910:
550:
152:
133:
124:vector space
98:
88:
73:
64:
48:
2967:Hypercharge
2214:gauge group
1516:of the map.
911:The symbol
144:Lie algebra
116:hypercharge
2996:Categories
2948:References
2915:Other uses
2503:due to an
104:spin state
67:March 2012
18:Multiplets
2942:multiplet
2934:multiplet
2820:ϕ
2711:ϕ
2571:α
2567:ψ
2544:μ
2519:ϕ
2068:≅
1996:⋊
1975:≅
1930:ν
1924:μ
1844:≅
1587:≥
1577:∈
1540:≅
1498:θ
1484:↦
1479:θ
1449:→
1423:ρ
1271:ρ
1189:ρ
1165:×
966:ρ
946:ρ
919:ρ
876:ρ
854:ρ
810:ρ
780:∈
723:→
696:ρ
593:, where
578:ρ
519:ρ
500:ρ
481:⋅
465:ρ
439:∈
342:→
319:ρ
176:ρ
140:Lie group
99:multiplet
2961:See also
2870:subgroup
2853:Examples
2732:, where
2585:such as
802:we have
457:we have
2972:Isospin
2872:of the
2812:. Then
2203:Physics
1604:, with
1294:triplet
191:where
112:isospin
91:physics
53:Please
2905:quarks
2888:and a
2865:of an
2249:fields
1804:: The
2909:SU(3)
2884:is a
2867:SU(2)
2661:, of
2587:Dirac
1263:SO(3)
1207:, so
1152:SO(3)
216:field
138:of a
108:color
2932:, a
2928:and
2591:Weyl
2436:Spin
2169:and
2049:Conf
1848:Spin
1544:Spin
1275:fund
1193:fund
1024:plet
114:and
97:, a
2901:QCD
2899:In
2781:of
2589:or
727:End
664:or
266:or
89:In
57:to
2998::
2903:,
2896:.
2790:SU
2700:.
2670:SL
2602:SL
2473:SL
2399:SO
2325:SU
2293:=:
2260:SU
2225:SU
2154:10
2147:SO
2130:SU
2000:SO
1885:SO
1823:SL
1776:SO
1703:SU
1671:SU
1640:SO
1527:SU
1453:GL
1349:.
1305:SO
1118:.
1056:SU
346:GL
110:,
2829:)
2826:x
2823:(
2800:)
2797:2
2794:(
2767:2
2762:C
2740:x
2720:)
2717:x
2714:(
2688:)
2684:C
2680:,
2677:2
2674:(
2647:2
2642:C
2620:)
2616:C
2612:,
2609:2
2606:(
2540:A
2491:)
2487:C
2483:,
2480:2
2477:(
2452:)
2449:3
2446:,
2443:1
2440:(
2415:)
2412:3
2409:,
2406:1
2403:(
2375:n
2355:n
2335:)
2332:2
2329:(
2304:2
2300:/
2296:n
2290:s
2270:)
2267:2
2264:(
2235:)
2232:2
2229:(
2198:.
2184:6
2179:E
2157:)
2151:(
2143:,
2140:)
2137:5
2134:(
2115:.
2101:2
2096:Z
2090:/
2086:)
2083:q
2080:,
2077:p
2074:(
2071:O
2065:)
2062:q
2059:,
2056:p
2053:(
2032:.
2016:)
2013:3
2010:,
2007:1
2004:(
1991:3
1988:,
1985:1
1980:R
1972:)
1969:3
1966:,
1963:1
1960:(
1956:E
1933:)
1927:,
1921:(
1901:)
1898:3
1895:,
1892:1
1889:(
1864:)
1861:3
1858:,
1855:1
1852:(
1841:)
1837:C
1833:,
1830:2
1827:(
1812:.
1792:)
1789:3
1786:,
1783:1
1780:(
1765:.
1745:)
1742:n
1739:,
1736:m
1733:(
1713:)
1710:3
1707:(
1681:)
1678:2
1675:(
1650:)
1647:3
1644:(
1629:.
1612:n
1590:0
1582:N
1574:n
1554:)
1551:3
1548:(
1537:)
1534:2
1531:(
1495:n
1492:i
1488:e
1476:i
1472:e
1468:;
1465:)
1461:C
1457:(
1446:)
1443:1
1440:(
1436:U
1432::
1427:n
1401:Z
1380:)
1377:1
1374:(
1370:U
1336:3
1315:)
1312:3
1309:(
1280:)
1267:,
1259:,
1254:3
1249:R
1244:(
1222:3
1217:R
1168:3
1162:3
1136:3
1131:R
1106:n
1086:n
1066:)
1063:2
1060:(
1034:n
1022:-
1010:n
986:V
907:.
895:]
892:)
887:2
883:X
879:(
873:,
870:)
865:1
861:X
857:(
851:[
848:=
845:)
842:]
837:2
833:X
829:,
824:1
820:X
816:[
813:(
790:,
785:g
775:2
771:X
767:,
762:1
758:X
737:)
734:V
731:(
718:g
686:.
673:C
651:R
628:g
604:V
581:)
575:,
570:g
565:,
562:V
559:(
547:.
535:)
530:2
526:g
522:(
516:)
511:1
507:g
503:(
497:=
494:)
489:2
485:g
476:1
472:g
468:(
445:,
442:G
434:2
430:g
426:,
421:1
417:g
396:V
376:G
356:)
353:V
350:(
339:G
297:G
275:C
253:R
249:=
246:K
226:K
202:V
179:)
173:,
170:G
167:,
164:V
161:(
80:)
74:(
69:)
65:(
51:.
31:.
20:)
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