156:
states that for every infinitesimal symmetry transformation that is local (local in the sense that the transformed value of a field at a given point only depends on the field configuration in an arbitrarily small neighborhood of that point), there is a corresponding conserved charge called the
773:
946:
420:
778:
This assumes that the state in question approaches the vacuum asymptotically at spatial infinity. The first integral is the surface integral at spatial infinity and the second integral is zero by the
308:
1072:
approaches infinity. If we interchange the limits, the directional charges change. This is related to the expanding electromagnetic waves spreading outwards at the speed of light (the soft photons).
233:
978:). The conclusion is that both electrons lie in different superselection sectors no matter how tiny the velocity is. At first sight, this might appear to be in contradiction with
822:
at finite values. This is consistent with the idea that symmetry transformations not affecting the boundaries are gauge symmetries whereas those that do are global symmetries. If
581:
974:
The directional charges are different for an electron that has always been at rest and an electron that has always been moving at a certain nonzero velocity (because of the
632:
1036:
613:
1062:
119:
99:
845:, conserved scalar charges (the electric charge) are seen as well as conserved vector charges and conserved tensor charges. This is not a violation of the
867:
959:, the conclusion that states with the same electric charge but different values for the directional charges lie in different superselection sectors.
333:
1501:
1336:
1138:
1322:
249:
1555:
55:. However, only a finite number of these photons are detectable, the remainder falling below the measurement threshold.
846:
81:
density of states like an ordinary particle, but instead the density of states rises like an inverse power at the mass
1358:
841:
This conclusion holds both in classical electrodynamics as well as in quantum electrodynamics. If Ī± is taken as the
838:, we get the electric charge. But for other functions, we also get conserved charges (which are not so well known).
178:
979:
1202:
1550:
44:
1468:
1126:
134:
32:
818:). Then, the Noether charge only depends upon the value of Ī± at spatial infinity but not upon the value of
975:
63:
987:
956:
768:{\displaystyle \epsilon _{0}\oint _{S^{2}}\alpha {\vec {E}}\cdot d{\vec {S}}+\int d^{3}x\alpha \left.}
431:
1510:
1441:
1404:
1367:
1264:
1211:
1174:
1076:
999:
963:
623:
153:
149:
78:
1130:
842:
166:
161:, which is the space integral of a Noether density (assuming the integral converges and there is a
28:
1526:
1280:
1254:
1227:
1097:
962:
Even though this result is expressed in terms of a particular spherical coordinates with a given
955:
and a conserved quantity. Using the result that states with different charges exist in different
1012:
589:
1041:
1395:
1393:
Buchholz, D.; Doplicher, S.; Longo, R (1986). "On
Noether's Theorem in Quantum Field Theory".
1332:
1165:
1134:
815:
1295:
Nachrichten von der Kƶniglicher
Gesellschaft den Wissenschaft zu Gƶttingen, Math-phys. Klasse
1518:
1449:
1412:
1375:
1272:
1219:
1182:
1118:
48:
36:
121:
consist of the particle together with low-energy excitations of the electromagnetic field.
1432:
314:
162:
145:
138:
130:
1119:
74:
description, where the
Hilbert space includes particle states with different velocities.
1514:
1445:
1408:
1371:
1268:
1215:
1178:
1328:
616:
240:
158:
104:
84:
1544:
1530:
1416:
1353:
1284:
1231:
1186:
983:
141:
67:
58:
The form of the electric field at infinity, which is determined by the velocity of a
52:
40:
994:
is really the greatest lower bound of a continuous mass spectrum and eigenstates of
941:{\displaystyle \lim _{r\rightarrow \infty }\epsilon _{0}r^{2}E_{r}(r,\theta ,\phi )}
1313:
328:
59:
1496:
327:
But what if there is a position-dependent (but not time-dependent) infinitesimal
77:
Because of their infraparticle properties, charged particles do not have a sharp
1006:
24:
415:{\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})}
1276:
71:
1453:
1430:
Coleman, S.; Mandula, J. (1967). "All
Possible Symmetries of the S Matrix".
779:
1245:
Noether, E.; Tavel, M.A. (transl.) (2005). "Invariant
Variation Problems".
23:
is an electrically charged particle together with its surrounding cloud of
952:
850:
1259:
1002:. The electron, and other particles like it is called an infraparticle.
1522:
1223:
324:
is conserved. This is nothing other than the familiar electric charge.
313:
at the boundary at spatial infinity is zero, which is satisfied if the
1379:
1079:
besides QED. The name "infraparticle" still applies in those cases.
966:, translations changing the origin do not affect spatial infinity.
1102:
1163:
Buchholz, D. (1986). "Gauss' law and the infraparticle problem".
1096:
Schroer, B. (2008). "A note on infraparticles and unparticles".
1075:
More generally, there might exist a similar situation in other
101:
of the particle. These states which are very close in mass to
172:
If this is applied to the global U(1) symmetry, the result
303:{\displaystyle \oint _{S^{2}}{\vec {J}}\cdot d{\vec {S}}}
27:āof which there are an infinite number, by virtue of the
1068:
goes to infinity first and only then take the limit as
1005:
The existence of the directional charges is related to
1497:"The Physical State Space of Quantum Electrodynamics"
1293:
1044:
1015:
870:
635:
592:
434:
336:
252:
181:
107:
87:
1354:"Noether's theorem for Local Gauge Transformations"
1056:
1030:
940:
767:
607:
575:
414:
302:
227:
113:
93:
124:
43:. Whenever electric charges accelerate they emit
872:
1200:Weyl, H. (1929). "Elektron und Gravitation I".
1158:
1156:
1154:
1152:
1150:
228:{\displaystyle Q=\int d^{3}x\rho ({\vec {x}})}
1312:is the integral of the time component of the
853:. In particular, for each direction (a fixed
8:
982:, which implies that the whole one-particle
1121:Quantum Field Theory: A Modern Introduction
125:Noether's theorem for gauge transformations
320:falls off sufficiently fast, the quantity
1258:
1101:
1043:
1014:
911:
901:
891:
875:
869:
746:
745:
733:
706:
685:
684:
667:
666:
655:
650:
640:
634:
594:
593:
591:
554:
553:
527:
526:
512:
511:
505:
484:
483:
463:
462:
442:
433:
398:
397:
377:
376:
347:
346:
335:
289:
288:
271:
270:
262:
257:
251:
211:
210:
195:
180:
106:
86:
1247:Transport Theory and Statistical Physics
1088:
422:where Ī± is some function of position?
239:is the conserved charge where Ļ is the
1502:Communications in Mathematical Physics
1352:Karatas, D.L.; Kowalski, K.L. (1990).
1064:are the same if we take the limit as
7:
148:, there are also position dependent
47:, whereby an infinite number of the
1051:
1025:
882:
739:
544:
243:. As long as the surface integral
14:
970:Implication for particle behavior
576:{\displaystyle \int d^{3}x\left}
1324:The Feynman Lectures on Physics
1327:. Vol. 2 (2nd ed.).
935:
917:
879:
751:
690:
672:
599:
565:
559:
550:
538:
532:
523:
517:
495:
489:
480:
474:
468:
459:
409:
403:
394:
388:
382:
373:
358:
352:
343:
294:
276:
222:
216:
207:
1:
1417:10.1016/0003-4916(86)90086-2
1187:10.1016/0370-2693(86)91110-X
1009:. The directional charge at
1359:American Journal of Physics
70:. This is unlike the usual
1572:
1031:{\displaystyle t=-\infty }
608:{\displaystyle {\vec {E}}}
425:The Noether charge is now
1277:10.1080/00411457108231446
1057:{\displaystyle t=\infty }
1476:philsci-archive.pitt.edu
1454:10.1103/PhysRev.159.1251
990:, but it is not because
814:approaches infinity (in
144:symmetry related to the
45:Bremsstrahlung radiation
16:Type of dressed particle
1127:Oxford University Press
980:Wigner's classification
976:Lorentz transformations
847:ColemanāMandula theorem
135:quantum electrodynamics
33:quantum electrodynamics
1469:"Superselection Rules"
1321:Feynman, R.P. (2005).
1203:Zeitschrift fĆ¼r Physik
1077:quantum field theories
1058:
1032:
957:superselection sectors
942:
769:
609:
577:
416:
304:
229:
115:
95:
64:superselection sectors
1495:Buchholz, D. (1982).
1059:
1033:
988:superselection sector
943:
770:
610:
578:
417:
305:
230:
150:gauge transformations
137:, in addition to the
116:
96:
1556:Quantum field theory
1467:Giulini, D. (2007).
1042:
1013:
1000:rigged Hilbert space
868:
633:
624:integration by parts
590:
432:
334:
329:gauge transformation
250:
179:
105:
85:
51:soft photons become
1515:1982CMaPh..85...49B
1446:1967PhRv..159.1251C
1409:1986AnPhy.170....1B
1372:1990AmJPh..58..123K
1319:by definition. See
1269:1971TTSP....1..186N
1216:1929ZPhy...56..330W
1179:1986PhLB..174..331B
1133:ā184, Appendix A6.
843:spherical harmonics
834:) = 1 all over the
782:. Also assume that
235:(over all of space)
167:continuity equation
66:for the particle's
35:. That is, it is a
29:infrared divergence
1523:10.1007/BF02029133
1224:10.1007/BF01339504
1054:
1028:
938:
886:
765:
605:
573:
412:
300:
225:
111:
91:
1396:Annals of Physics
1338:978-0-8053-9065-0
1166:Physics Letters B
1140:978-0-19-507652-3
1117:Kaku, M. (1993).
986:lies in a single
871:
816:polar coordinates
754:
693:
675:
602:
562:
535:
520:
492:
471:
406:
385:
355:
297:
279:
219:
154:Noether's theorem
114:{\displaystyle m}
94:{\displaystyle m}
1563:
1535:
1534:
1492:
1486:
1485:
1483:
1482:
1473:
1464:
1458:
1457:
1440:(5): 1251ā1256.
1427:
1421:
1420:
1390:
1384:
1383:
1349:
1343:
1342:
1307:
1301:
1298:
1288:
1262:
1242:
1236:
1235:
1210:(5ā6): 330ā352.
1197:
1191:
1190:
1160:
1145:
1144:
1124:
1114:
1108:
1107:
1105:
1093:
1063:
1061:
1060:
1055:
1037:
1035:
1034:
1029:
998:only exist in a
947:
945:
944:
939:
916:
915:
906:
905:
896:
895:
885:
861:), the quantity
774:
772:
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747:
738:
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92:
37:dressed particle
1571:
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1561:
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1551:Electrodynamics
1541:
1540:
1539:
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1471:
1466:
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1461:
1433:Physical Review
1429:
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1424:
1392:
1391:
1387:
1380:10.1119/1.16219
1351:
1350:
1346:
1339:
1320:
1308:
1304:
1292:
1291:Translation of
1260:physics/0503066
1244:
1243:
1239:
1199:
1198:
1194:
1162:
1161:
1148:
1141:
1116:
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1111:
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1090:
1085:
1040:
1039:
1011:
1010:
972:
907:
897:
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866:
865:
849:as there is no
729:
722:
718:
702:
651:
646:
636:
631:
630:
588:
587:
501:
455:
451:
438:
430:
429:
332:
331:
315:current density
258:
253:
248:
247:
191:
177:
176:
165:satisfying the
163:Noether current
146:electric charge
131:electrodynamics
127:
103:
102:
83:
82:
17:
12:
11:
5:
1569:
1567:
1559:
1558:
1553:
1543:
1542:
1537:
1536:
1487:
1459:
1422:
1385:
1366:(2): 123ā131.
1344:
1337:
1329:Addison-Wesley
1302:
1300:
1299:
1253:(3): 235ā257.
1237:
1192:
1173:(3): 331ā334.
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617:electric field
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598:
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583:
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241:charge density
237:
236:
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209:
206:
203:
198:
194:
190:
187:
184:
159:Noether charge
126:
123:
110:
90:
79:delta function
53:real particles
39:rather than a
15:
13:
10:
9:
6:
4:
3:
2:
1568:
1557:
1554:
1552:
1549:
1548:
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1067:
1048:
1045:
1022:
1019:
1016:
1008:
1003:
1001:
997:
993:
989:
985:
984:Hilbert space
981:
977:
969:
967:
965:
960:
958:
954:
932:
929:
926:
923:
920:
912:
908:
902:
898:
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888:
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848:
844:
839:
837:
833:
829:
825:
821:
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813:
809:
805:
801:
798:) approaches
797:
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789:
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781:
762:
758:
748:
742:
734:
730:
726:
723:
719:
715:
712:
707:
703:
699:
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681:
678:
669:
663:
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652:
647:
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637:
629:
628:
627:
625:
620:
618:
596:
569:
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541:
529:
514:
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502:
498:
486:
477:
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452:
448:
443:
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435:
428:
427:
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263:
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213:
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155:
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147:
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132:
122:
108:
88:
80:
75:
73:
69:
68:Hilbert space
65:
61:
56:
54:
50:
46:
42:
41:bare particle
38:
34:
30:
26:
22:
21:infraparticle
1509:(1): 49ā71.
1506:
1500:
1490:
1479:. Retrieved
1475:
1462:
1437:
1431:
1425:
1400:
1394:
1388:
1363:
1357:
1347:
1323:
1316:
1314:four-current
1309:
1305:
1294:
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1246:
1240:
1207:
1201:
1195:
1170:
1164:
1120:
1112:
1091:
1074:
1069:
1065:
1007:soft photons
1004:
995:
991:
973:
961:
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854:
840:
835:
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238:
171:
128:
76:
60:point charge
57:
25:soft photons
20:
18:
1403:(1): 1ā17.
1129:. pp.
1545:Categories
1481:2010-02-21
1297:: 235ā257.
1083:References
72:Fock space
62:, defines
1531:120467701
1285:119019843
1232:186233130
1103:0804.3563
1052:∞
1026:∞
1023:−
933:ϕ
927:θ
889:ϵ
883:∞
880:→
780:Gauss law
752:→
743:⋅
740:∇
731:ϵ
727:−
724:ρ
716:α
700:∫
691:→
679:⋅
673:→
664:α
648:∮
638:ϵ
600:→
560:→
548:α
545:∇
542:⋅
533:→
518:→
503:ϵ
490:→
478:ρ
469:→
457:α
436:∫
404:→
392:ψ
383:→
371:α
353:→
341:ψ
338:δ
295:→
283:⋅
277:→
255:∮
217:→
205:ρ
189:∫
953:c-number
851:mass gap
1511:Bibcode
1442:Bibcode
1405:Bibcode
1368:Bibcode
1265:Bibcode
1212:Bibcode
1175:Bibcode
615:is the
49:virtual
1529:
1335:
1283:
1230:
1137:
964:origin
622:Using
586:where
139:global
1527:S2CID
1472:(PDF)
1281:S2CID
1255:arXiv
1228:S2CID
1098:arXiv
951:is a
810:) as
1333:ISBN
1135:ISBN
1038:and
857:and
142:U(1)
133:and
1519:doi
1450:doi
1438:159
1413:doi
1401:170
1376:doi
1273:doi
1220:doi
1183:doi
1171:174
1131:177
873:lim
169:).
129:In
31:of
19:An
1547::
1525:.
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