1732:
1429:
1227:
1181:
2383:
999:
of the
Poisson bracket. Its algebraic isomorphism to the algebra of commutators bypasses the negative result of the Groenewold–van Hove theorem, which precludes such an isomorphism for the Poisson bracket, a question implicitly raised by Dirac in his 1926 doctoral thesis, the "method of
2164:
1424:{\displaystyle \{\{f,g\}\}\ ={\frac {2}{\hbar }}~f(x,p)\ \sin \left({{\tfrac {\hbar }{2}}({\overleftarrow {\partial }}_{x}{\overrightarrow {\partial }}_{p}-{\overleftarrow {\partial }}_{p}{\overrightarrow {\partial }}_{x})}\right)\ g(x,p).}
1017:
1447:
2232:
106:
2237:
1022:
1867:
1929:
1737:
Each correspondence map from phase space to
Hilbert space induces a characteristic "Moyal" bracket (such as the one illustrated here for the Weyl map). All such Moyal brackets are
1176:{\displaystyle {\begin{aligned}\{\{f,g\}\}&{\stackrel {\mathrm {def} }{=}}\ {\frac {1}{i\hbar }}(f\star g-g\star f)\\&=\{f,g\}+O(\hbar ^{2}),\\\end{aligned}}}
1727:{\displaystyle \{\{f,g\}\}(x,p)={2 \over \hbar ^{3}\pi ^{2}}\int dp'\,dp''\,dx'\,dx''f(x+x',p+p')g(x+x'',p+p'')\sin \left({\tfrac {2}{\hbar }}(x'p''-x''p')\right)~.}
2587:
1434:
The left & right arrows over the partial derivatives denote the left & right partial derivatives. Sometimes the Moyal bracket is referred to as the
2378:{\displaystyle {\begin{aligned}\{\{\{f,g\}\}\}&{\stackrel {\mathrm {def} }{=}}\ {\tfrac {1}{2}}(f\star g+g\star f)=fg+O(\hbar ^{2}).\\\end{aligned}}}
2412:
The sine and cosine brackets are, respectively, the results of antisymmetrizing and symmetrizing the star product. Thus, as the sine bracket is the
36:
372:
2433:
873:
937:. It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being the
2663:
Baker, George A. (1958-03-15). "Formulation of
Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space".
124:
580:
353:
2725:
2610:
536:
459:
2981:
2527:
119:
208:
2420:
in standard quantum mechanics. Similarly, as the Moyal bracket equals the
Poisson bracket up to higher orders of
866:
315:
295:
163:
942:
2818:
Krivoruchenko, M. I.; Raduta, A. A.; Faessler, Amand (2006-01-17). "Quantum deformation of the Dirac bracket".
555:
285:
2226:
Next to the sine bracket discussed, Groenewold further introduced the cosine bracket, elaborated by Baker,
2159:{\displaystyle =2i\sin \left({\tfrac {\hbar }{2}}(n_{1}m_{2}-n_{2}m_{1})\right)~T_{m_{1}+n_{1},m_{2}+n_{2}},~}
1783:
595:
333:
233:
2986:
2444:
2437:
531:
526:
497:
348:
129:
565:
310:
300:
2210:
involves an operation on equivalence classes of functions in phase space, which can be considered as a
859:
511:
482:
2413:
1008:
938:
516:
477:
430:
405:
328:
188:
2946:
2207:
600:
2775:
Fairlie, D. B.; Zachos, C. K. (1989). "Infinite-dimensional algebras, sine brackets, and SU(â)".
926:
769:
487:
395:
2880:
Curtright, T.; Fairlie, D.; Zachos, C. (1998). "Features of time-independent Wigner functions".
445:
343:
109:
910:
774:
492:
320:
290:
253:
400:
2507:
965:
950:
243:
228:
902:
2899:
2837:
2784:
2672:
2561:
2472:
1218:
560:
472:
198:
155:
2509:
Maverick
Mathematician: The Life and Science of J.E. Moyal (Chap. 3: Battle With A Legend)
804:
8:
946:
659:
467:
385:
213:
193:
145:
2903:
2841:
2788:
2676:
2565:
2476:
2915:
2889:
2861:
2827:
2800:
2757:
2739:
2621:
Conversely, the
Poisson bracket is formally expressible in terms of the star product,
2488:
2211:
957:
905:, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with
380:
305:
238:
150:
2573:
2865:
2853:
2804:
2796:
2761:
2721:
2688:
2606:
2523:
2492:
976:
930:
814:
789:
729:
724:
624:
590:
570:
168:
27:
2907:
2845:
2792:
2749:
2713:
2680:
2569:
2513:
2480:
2463:
Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory".
819:
809:
799:
699:
679:
664:
634:
502:
390:
2919:
1441:
A popular (Fourier) integral representation for it, introduced by George Baker is
2429:
1757:
986:
969:
961:
844:
714:
694:
440:
280:
2911:
2849:
2552:
Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics".
2417:
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
2753:
2484:
2975:
2857:
2709:
2705:
2692:
2684:
2215:
1191:
895:
839:
834:
764:
734:
704:
575:
521:
248:
223:
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
2730:
Curtright, T. L.; Zachos, C. K. (2012). "Quantum
Mechanics in Phase Space".
2598:
829:
824:
759:
744:
709:
203:
2941:: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.):
2963:
2938:
2924:
1761:
1745:
1004:
989:
934:
794:
749:
684:
639:
2518:
2424:, the cosine bracket equals the ordinary product up to higher orders of
2966:(submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011)
2894:
2832:
922:
906:
784:
754:
674:
649:
644:
629:
16:
Suitably normalized antisymmetrization of the phase-space star product
2934:
2932:
909:. In the meantime this idea was independently introduced in 1946 by
275:
2443:
The sine and cosine bracket also stand in relation to equations of
654:
2929:
2744:
2416:
of the commutator, the cosine bracket is the Wigner image of the
894:
is the suitably normalized antisymmetrization of the phase-space
887:
1744:
The Moyal bracket specifies the eponymous infinite-dimensional
2465:
Mathematical
Proceedings of the Cambridge Philosophical Society
2434:
Liouville equation (formulated in terms of the
Poisson bracket)
2206:
Generalization of the Moyal bracket for quantum systems with
2817:
2945:, pp. 267-286, VÀxjö University Press, Sweden, 2003 (
1741:
among themselves, in accordance with a systematic theory.
2879:
2299:
2026:
1661:
1304:
2235:
1932:
1786:
1450:
1230:
1020:
933:
when these observables are described as functions on
39:
968:
of it), the deformation parameter being the reduced
995:Up to formal equivalence, the Moyal Bracket is the
2586:P. A. M. Dirac (1926) Cambridge University Thesis
2377:
2221:
2158:
1861:
1726:
1423:
1175:
949:, thereby providing the quantum generalization of
100:
2943:Quantum Theory: Re-consideration of Foundationsâ2
2671:(6). American Physical Society (APS): 2198â2206.
1190:is the star-product operator in phase space (cf.
901:The Moyal bracket was developed in about 1940 by
2973:
2952:
2729:
2436:, as the cosine bracket leads to the classical
2826:(2). American Physical Society (APS): 025008.
2400:are differentiable phase-space functions, and
1202:are differentiable phase-space functions, and
997:unique one-parameter Lie-algebraic deformation
2774:
2462:
2392:is the star-product operator in phase space,
921:The Moyal bracket is a way of describing the
867:
2960:Schrodinger revisited: an algebraic approach
2658:
2656:
2547:
2545:
2264:
2261:
2258:
2246:
2243:
2240:
1851:
1848:
1836:
1833:
1748:—it is antisymmetric in its arguments
1469:
1466:
1454:
1451:
1249:
1246:
1234:
1231:
1141:
1129:
1043:
1040:
1028:
1025:
95:
69:
2432:, the Moyal bracket helps reduction to the
2878:See also the citation of Baker (1958) in:
2551:
874:
860:
2893:
2831:
2743:
2653:
2601:, "The Principles of Quantum Mechanics" (
2542:
2517:
1557:
1545:
1533:
1862:{\displaystyle =T_{i\hbar \{\{f,g\}\}}.}
1003:For instance, in a two-dimensional flat
2199:(â), with deformation parameter 1/
947:Heisenbergâs quantum equation of motion
2974:
2716:, "Quantum Mechanics in Phase Space" (
2662:
2505:
1886:, each in , and integer mode indices
1000:classical analogy" for quantization.
2195:) then emerges as a deformation of
13:
2287:
2284:
2281:
1375:
1358:
1338:
1321:
1066:
1063:
1060:
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
2998:
1830:
1666:
1263:
1086:
43:
2872:
2811:
2732:Asia Pacific Physics Newsletter
2222:Sine bracket and cosine bracket
945:, an equivalent formulation of
2768:
2699:
2615:
2592:
2580:
2499:
2456:
2445:a purely algebraic description
2365:
2352:
2334:
2310:
2083:
2037:
2002:
1933:
1816:
1787:
1710:
1672:
1646:
1612:
1606:
1572:
1484:
1472:
1415:
1403:
1389:
1315:
1286:
1274:
1163:
1150:
1116:
1092:
556:Relativistic quantum mechanics
88:
81:
62:
1:
2574:10.1016/S0031-8914(46)80059-4
2450:
1878:, with periodic coordinates
1760:. The corresponding abstract
596:Quantum statistical mechanics
2797:10.1016/0370-2693(89)91057-5
7:
1011:, the Moyal bracket reads,
916:
566:Quantum information science
10:
3003:
2958:M. R. Brown, B. J. Hiley:
2912:10.1103/PhysRevD.58.025002
2850:10.1103/physrevd.73.025008
1923:, this Lie algebra reads,
1872:On a 2-torus phase space,
1214:is their Poisson bracket.
943:Moyalâs dynamical equation
2982:Mathematical quantization
2754:10.1142/S2251158X12000069
2485:10.1017/S0305004100000487
2409:is the ordinary product.
2685:10.1103/physrev.109.2198
2438:HamiltonâJacobi equation
2208:second-class constraints
956:Mathematically, it is a
601:Quantum machine learning
354:Wheeler's delayed-choice
1009:Weyl-map correspondence
927:phase space formulation
311:LeggettâGarg inequality
2964:arXiv:quant-ph/0005026
2925:arXiv:hep-th/9711183v3
2603:Clarendon Press Oxford
2447:of quantum mechanics.
2379:
2160:
1895:, for basis functions
1863:
1728:
1425:
1221:language, this equals
1217:More specifically, in
1177:
925:of observables in the
102:
2380:
2161:
1864:
1729:
1426:
1178:
939:WignerâWeyl transform
296:Elitzur–Vaidman
286:Davisson–Germer
103:
2233:
1930:
1784:
1756:, and satisfies the
1448:
1228:
1219:operational calculus
1018:
951:Hamiltonâs equations
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
2987:Symplectic geometry
2904:1998PhRvD..58b5002C
2842:2006PhRvD..73b5008K
2789:1989PhLB..224..101F
2720:, Singapore, 2005)
2677:1958PhRv..109.2198B
2588:"Quantum Mechanics"
2566:1946Phy....12..405G
2519:10.22459/MM.08.2006
2506:Moyal, Ann (2006).
2477:1949PCPS...45...99M
2212:quantum deformation
1739:formally equivalent
960:of the phase-space
349:Stern–Gerlach
146:Classical mechanics
2375:
2373:
2308:
2156:
2035:
1859:
1724:
1670:
1421:
1313:
1173:
1171:
941:. It underlies
903:José Enrique Moyal
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
2882:Physical Review D
2820:Physical Review D
2777:Physics Letters B
2726:978-981-238-384-6
2644:
2611:978-0-19-852011-5
2391:
2307:
2297:
2292:
2169:which reduces to
2155:
2093:
2034:
1968:
1802:
1776:
1720:
1669:
1517:
1399:
1381:
1364:
1344:
1327:
1312:
1291:
1270:
1266:
1254:
1189:
1090:
1076:
1071:
977:group contraction
931:quantum mechanics
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
2994:
2967:
2956:
2950:
2936:
2927:
2923:
2897:
2876:
2870:
2869:
2835:
2815:
2809:
2808:
2783:(1â2): 101â107.
2772:
2766:
2765:
2747:
2718:World Scientific
2703:
2697:
2696:
2660:
2651:
2649:
2642:
2635:
2619:
2613:
2596:
2590:
2584:
2578:
2577:
2549:
2540:
2539:
2537:
2536:
2521:
2503:
2497:
2496:
2460:
2427:
2423:
2408:
2399:
2395:
2389:
2384:
2382:
2381:
2376:
2374:
2364:
2363:
2309:
2300:
2295:
2294:
2293:
2291:
2290:
2278:
2273:
2186:
2165:
2163:
2162:
2157:
2153:
2149:
2148:
2147:
2146:
2134:
2133:
2121:
2120:
2108:
2107:
2091:
2090:
2086:
2082:
2081:
2072:
2071:
2059:
2058:
2049:
2048:
2036:
2027:
2001:
2000:
1999:
1998:
1986:
1985:
1966:
1965:
1964:
1963:
1962:
1950:
1949:
1922:
1894:
1885:
1881:
1877:
1868:
1866:
1865:
1860:
1855:
1854:
1815:
1814:
1800:
1799:
1798:
1774:
1773:
1755:
1751:
1733:
1731:
1730:
1725:
1718:
1717:
1713:
1709:
1701:
1690:
1682:
1671:
1662:
1645:
1628:
1605:
1588:
1568:
1556:
1544:
1532:
1518:
1516:
1515:
1514:
1505:
1504:
1491:
1430:
1428:
1427:
1422:
1397:
1396:
1392:
1388:
1387:
1382:
1374:
1371:
1370:
1365:
1357:
1351:
1350:
1345:
1337:
1334:
1333:
1328:
1320:
1314:
1305:
1289:
1268:
1267:
1259:
1252:
1213:
1201:
1197:
1187:
1182:
1180:
1179:
1174:
1172:
1162:
1161:
1122:
1091:
1089:
1078:
1074:
1073:
1072:
1070:
1069:
1057:
1052:
984:
974:
964:(essentially an
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
3002:
3001:
2997:
2996:
2995:
2993:
2992:
2991:
2972:
2971:
2970:
2957:
2953:
2937:
2930:
2877:
2873:
2816:
2812:
2773:
2769:
2704:
2700:
2665:Physical Review
2661:
2654:
2637:
2622:
2620:
2616:
2597:
2593:
2585:
2581:
2550:
2543:
2534:
2532:
2530:
2504:
2500:
2461:
2457:
2453:
2430:classical limit
2425:
2421:
2401:
2397:
2393:
2372:
2371:
2359:
2355:
2298:
2280:
2279:
2274:
2272:
2271:
2267:
2236:
2234:
2231:
2230:
2224:
2178:
2142:
2138:
2129:
2125:
2116:
2112:
2103:
2099:
2098:
2094:
2077:
2073:
2067:
2063:
2054:
2050:
2044:
2040:
2025:
2024:
2020:
1994:
1990:
1981:
1977:
1976:
1972:
1958:
1954:
1945:
1941:
1940:
1936:
1931:
1928:
1927:
1917:
1907:
1896:
1892:
1887:
1883:
1879:
1873:
1826:
1822:
1810:
1806:
1794:
1790:
1785:
1782:
1781:
1770:
1765:
1764:is realized by
1758:Jacobi identity
1753:
1749:
1702:
1694:
1683:
1675:
1660:
1659:
1655:
1638:
1621:
1598:
1581:
1561:
1549:
1537:
1525:
1510:
1506:
1500:
1496:
1495:
1490:
1449:
1446:
1445:
1432:
1383:
1373:
1372:
1366:
1356:
1355:
1346:
1336:
1335:
1329:
1319:
1318:
1303:
1302:
1298:
1258:
1229:
1226:
1225:
1203:
1199:
1195:
1170:
1169:
1157:
1153:
1120:
1119:
1082:
1077:
1059:
1058:
1053:
1051:
1050:
1046:
1021:
1019:
1016:
1015:
987:Poisson bracket
979:
972:
970:Planck constant
962:Poisson bracket
919:
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de BroglieâBohm
462:
460:Interpretations
452:
451:
450:
420:
412:
411:
410:
368:
360:
359:
358:
325:
281:CHSH inequality
270:
262:
261:
260:
189:Complementarity
183:
175:
174:
173:
141:
112:
87:
76:
75:
61:
51:
46:
38:
35:
34:
17:
12:
11:
5:
3000:
2990:
2989:
2984:
2969:
2968:
2951:
2928:
2895:hep-th/9711183
2871:
2833:hep-th/0507049
2810:
2767:
2698:
2652:
2614:
2591:
2579:
2560:(7): 405â460.
2541:
2528:
2498:
2454:
2452:
2449:
2418:anticommutator
2386:
2385:
2370:
2367:
2362:
2358:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2306:
2303:
2289:
2286:
2283:
2277:
2270:
2268:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2238:
2223:
2220:
2182: ⥠4
2177:) for integer
2167:
2166:
2152:
2145:
2141:
2137:
2132:
2128:
2124:
2119:
2115:
2111:
2106:
2102:
2097:
2089:
2085:
2080:
2076:
2070:
2066:
2062:
2057:
2053:
2047:
2043:
2039:
2033:
2030:
2023:
2019:
2016:
2013:
2010:
2007:
2004:
1997:
1993:
1989:
1984:
1980:
1975:
1971:
1961:
1957:
1953:
1948:
1944:
1939:
1935:
1915:
1905:
1890:
1870:
1869:
1858:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1825:
1821:
1818:
1813:
1809:
1805:
1797:
1793:
1789:
1768:
1735:
1734:
1723:
1716:
1712:
1708:
1705:
1700:
1697:
1693:
1689:
1686:
1681:
1678:
1674:
1668:
1665:
1658:
1654:
1651:
1648:
1644:
1641:
1637:
1634:
1631:
1627:
1624:
1620:
1617:
1614:
1611:
1608:
1604:
1601:
1597:
1594:
1591:
1587:
1584:
1580:
1577:
1574:
1571:
1567:
1564:
1560:
1555:
1552:
1548:
1543:
1540:
1536:
1531:
1528:
1524:
1521:
1513:
1509:
1503:
1499:
1494:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1395:
1391:
1386:
1380:
1377:
1369:
1363:
1360:
1354:
1349:
1343:
1340:
1332:
1326:
1323:
1317:
1311:
1308:
1301:
1297:
1294:
1288:
1285:
1282:
1279:
1276:
1273:
1265:
1262:
1257:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1223:
1184:
1183:
1168:
1165:
1160:
1156:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1123:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1088:
1085:
1081:
1068:
1065:
1062:
1056:
1049:
1047:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1023:
1007:, and for the
918:
915:
911:Hip Groenewold
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
612:
611:
608:
607:
604:
603:
598:
593:
588:
586:Density matrix
583:
578:
573:
568:
563:
558:
552:
549:
548:
545:
544:
540:
539:
534:
529:
524:
519:
514:
509:
508:
507:
506:
505:
490:
485:
480:
475:
470:
464:
463:
458:
457:
454:
453:
449:
448:
443:
438:
433:
428:
422:
421:
418:
417:
414:
413:
409:
408:
403:
398:
393:
388:
383:
377:
376:
375:
369:
366:
365:
362:
361:
357:
356:
351:
346:
340:
339:
338:
337:
336:
334:Delayed-choice
329:Quantum eraser
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
272:
271:
268:
267:
264:
263:
259:
258:
257:
256:
246:
241:
236:
231:
226:
221:
219:Quantum number
216:
211:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
160:
159:
158:
153:
148:
142:
139:
138:
135:
134:
133:
132:
127:
122:
114:
113:
108:
97:
94:
90:
83:
80:
74:
71:
68:
64:
57:
54:
50:
45:
42:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
2999:
2988:
2985:
2983:
2980:
2979:
2977:
2965:
2961:
2955:
2948:
2944:
2940:
2935:
2933:
2926:
2921:
2917:
2913:
2909:
2905:
2901:
2896:
2891:
2888:(2): 025002.
2887:
2883:
2875:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2834:
2829:
2825:
2821:
2814:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2778:
2771:
2763:
2759:
2755:
2751:
2746:
2741:
2737:
2733:
2727:
2723:
2719:
2715:
2711:
2707:
2702:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2659:
2657:
2648:
2640:
2633:
2629:
2625:
2618:
2612:
2608:
2604:
2600:
2595:
2589:
2583:
2575:
2571:
2567:
2563:
2559:
2555:
2548:
2546:
2531:
2529:9781920942595
2525:
2520:
2515:
2511:
2510:
2502:
2494:
2490:
2486:
2482:
2478:
2474:
2471:(1): 99â124.
2470:
2466:
2459:
2455:
2448:
2446:
2441:
2439:
2435:
2431:
2419:
2415:
2410:
2407:
2404:
2388:Here, again,
2368:
2360:
2356:
2349:
2346:
2343:
2340:
2337:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2304:
2301:
2275:
2269:
2255:
2252:
2249:
2229:
2228:
2227:
2219:
2217:
2216:Dirac bracket
2213:
2209:
2204:
2202:
2198:
2194:
2190:
2185:
2181:
2176:
2172:
2150:
2143:
2139:
2135:
2130:
2126:
2122:
2117:
2113:
2109:
2104:
2100:
2095:
2087:
2078:
2074:
2068:
2064:
2060:
2055:
2051:
2045:
2041:
2031:
2028:
2021:
2017:
2014:
2011:
2008:
2005:
1995:
1991:
1987:
1982:
1978:
1973:
1969:
1959:
1955:
1951:
1946:
1942:
1937:
1926:
1925:
1924:
1920:
1914:
1910:
1904:
1900:
1893:
1876:
1856:
1845:
1842:
1839:
1827:
1823:
1819:
1811:
1807:
1803:
1795:
1791:
1780:
1779:
1778:
1772:
1763:
1759:
1747:
1742:
1740:
1721:
1714:
1706:
1703:
1698:
1695:
1691:
1687:
1684:
1679:
1676:
1663:
1656:
1652:
1649:
1642:
1639:
1635:
1632:
1629:
1625:
1622:
1618:
1615:
1609:
1602:
1599:
1595:
1592:
1589:
1585:
1582:
1578:
1575:
1569:
1565:
1562:
1558:
1553:
1550:
1546:
1541:
1538:
1534:
1529:
1526:
1522:
1519:
1511:
1507:
1501:
1497:
1492:
1487:
1481:
1478:
1475:
1463:
1460:
1457:
1444:
1443:
1442:
1439:
1437:
1431:
1418:
1412:
1409:
1406:
1400:
1393:
1384:
1378:
1367:
1361:
1352:
1347:
1341:
1330:
1324:
1309:
1306:
1299:
1295:
1292:
1283:
1280:
1277:
1271:
1260:
1255:
1243:
1240:
1237:
1222:
1220:
1215:
1211:
1207:
1193:
1192:Moyal product
1166:
1158:
1154:
1147:
1144:
1138:
1135:
1132:
1126:
1124:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1083:
1079:
1054:
1048:
1037:
1034:
1031:
1014:
1013:
1012:
1010:
1006:
1001:
998:
993:
991:
988:
982:
978:
971:
967:
963:
959:
954:
952:
948:
944:
940:
936:
932:
928:
924:
914:
912:
908:
904:
899:
897:
893:
892:Moyal bracket
889:
877:
872:
870:
865:
863:
858:
857:
855:
854:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
610:
609:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
576:Quantum chaos
574:
572:
569:
567:
564:
562:
559:
557:
554:
553:
547:
546:
538:
535:
533:
532:Transactional
530:
528:
525:
523:
522:Quantum logic
520:
518:
515:
513:
510:
504:
501:
500:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
461:
456:
455:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
423:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
378:
374:
371:
370:
364:
363:
355:
352:
350:
347:
345:
342:
341:
335:
332:
331:
330:
327:
326:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
266:
265:
255:
252:
251:
250:
249:Wave function
247:
245:
242:
240:
237:
235:
232:
230:
229:Superposition
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
161:
157:
154:
152:
149:
147:
144:
143:
137:
136:
131:
128:
126:
123:
121:
118:
117:
116:
115:
111:
78:
72:
55:
52:
48:
40:
33:
32:
29:
26:
25:
21:
20:
2959:
2954:
2942:
2885:
2881:
2874:
2823:
2819:
2813:
2780:
2776:
2770:
2735:
2731:
2717:
2714:T. Curtright
2701:
2668:
2664:
2646:
2638:
2631:
2627:
2623:
2617:
2602:
2599:P.A.M. Dirac
2594:
2582:
2557:
2553:
2533:. Retrieved
2508:
2501:
2468:
2464:
2458:
2442:
2411:
2405:
2402:
2387:
2225:
2205:
2200:
2196:
2192:
2188:
2183:
2179:
2174:
2170:
2168:
1918:
1912:
1908:
1902:
1898:
1888:
1874:
1871:
1766:
1743:
1738:
1736:
1440:
1436:Sine bracket
1435:
1433:
1224:
1216:
1209:
1205:
1185:
1002:
996:
994:
980:
975:. Thus, its
955:
920:
900:
896:star product
891:
885:
431:KleinâGordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
2939:B. J. Hiley
1777:, so that
1762:Lie algebra
1746:Lie algebra
1005:phase space
990:Lie algebra
985:yields the
958:deformation
935:phase space
820:von Neumann
805:Schrödinger
581:EPR paradox
512:Many-worlds
446:Schrödinger
401:Schrödinger
396:Phase-space
386:Interaction
291:Double-slit
269:Experiments
244:Uncertainty
214:Nonlocality
209:Measurement
194:Decoherence
164:Hamiltonian
2976:Categories
2710:D. Fairlie
2535:2010-05-02
2451:References
2428:. In the
2414:Wigner map
923:commutator
907:Paul Dirac
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
2866:119131374
2858:1550-7998
2805:120159881
2762:119230734
2745:1104.5269
2738:: 37â46.
2693:0031-899X
2493:124183640
2357:ℏ
2329:⋆
2317:⋆
2061:−
2029:ℏ
2018:
1831:ℏ
1771:≡ f
1692:−
1667:ℏ
1653:
1520:∫
1508:π
1498:ℏ
1379:→
1376:∂
1362:←
1359:∂
1353:−
1342:→
1339:∂
1325:←
1322:∂
1307:ℏ
1296:
1264:ℏ
1194:), while
1155:ℏ
1111:⋆
1105:−
1099:⋆
1087:ℏ
966:extension
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
2706:C.Zachos
2605:, 1958)
1707:′
1699:″
1688:″
1680:′
1643:″
1626:″
1603:′
1586:′
1566:″
1554:′
1542:″
1530:′
917:Overview
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
2900:Bibcode
2838:Bibcode
2785:Bibcode
2673:Bibcode
2562:Bibcode
2554:Physica
2473:Bibcode
2214:of the
1186:where
888:physics
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
2920:288935
2918:
2864:
2856:
2803:
2760:
2724:
2712:, and
2691:
2609:
2526:
2491:
2296:
2154:
2092:
1967:
1801:
1719:
1398:
1290:
1269:
1253:
1075:
890:, the
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
2916:S2CID
2890:arXiv
2862:S2CID
2828:arXiv
2801:S2CID
2758:S2CID
2740:arXiv
2489:S2CID
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
2854:ISSN
2722:ISBN
2689:ISSN
2641:(log
2607:ISBN
2524:ISBN
2396:and
1897:exp(
1882:and
1752:and
1198:and
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
2947:PDF
2908:doi
2846:doi
2793:doi
2781:224
2750:doi
2681:doi
2669:109
2636:= 2
2570:doi
2514:doi
2481:doi
2184:Ï/ħ
2015:sin
1650:sin
1293:sin
929:of
886:In
2978::
2962:,
2931:^
2914:.
2906:.
2898:.
2886:58
2884:.
2860:.
2852:.
2844:.
2836:.
2824:73
2822:.
2799:.
2791:.
2779:.
2756:.
2748:.
2736:01
2734:.
2708:,
2687:.
2679:.
2667:.
2655:^
2645:)
2634:}
2630:,
2624:iħ
2568:.
2558:12
2556:.
2544:^
2522:.
2512:.
2487:.
2479:.
2469:45
2467:.
2440:.
2218:.
2203:.
2197:SU
2189:SU
2187:.
2171:SU
1921:))
1438:.
1212:}
1208:,
992:.
983:â0
953:.
913:.
898:.
2949:)
2922:.
2910::
2902::
2892::
2868:.
2848::
2840::
2830::
2807:.
2795::
2787::
2764:.
2752::
2742::
2728:.
2695:.
2683::
2675::
2650:.
2647:g
2643:â
2639:f
2632:g
2628:f
2626:{
2576:.
2572::
2564::
2538:.
2516::
2495:.
2483::
2475::
2426:ħ
2422:ħ
2406:g
2403:f
2398:g
2394:f
2390:â
2369:.
2366:)
2361:2
2353:(
2350:O
2347:+
2344:g
2341:f
2338:=
2335:)
2332:f
2326:g
2323:+
2320:g
2314:f
2311:(
2305:2
2302:1
2288:f
2285:e
2282:d
2276:=
2265:}
2262:}
2259:}
2256:g
2253:,
2250:f
2247:{
2244:{
2241:{
2201:N
2193:N
2191:(
2180:N
2175:N
2173:(
2151:,
2144:2
2140:n
2136:+
2131:2
2127:m
2123:,
2118:1
2114:n
2110:+
2105:1
2101:m
2096:T
2088:)
2084:)
2079:1
2075:m
2069:2
2065:n
2056:2
2052:m
2046:1
2042:n
2038:(
2032:2
2022:(
2012:i
2009:2
2006:=
2003:]
1996:2
1992:n
1988:,
1983:1
1979:n
1974:T
1970:,
1960:2
1956:m
1952:,
1947:1
1943:m
1938:T
1934:[
1919:p
1916:2
1913:m
1911:+
1909:x
1906:1
1903:m
1901:(
1899:i
1891:i
1889:m
1884:p
1880:x
1875:T
1857:.
1852:}
1849:}
1846:g
1843:,
1840:f
1837:{
1834:{
1828:i
1824:T
1820:=
1817:]
1812:g
1808:T
1804:,
1796:f
1792:T
1788:[
1775:â
1769:f
1767:T
1754:g
1750:f
1722:.
1715:)
1711:)
1704:p
1696:x
1685:p
1677:x
1673:(
1664:2
1657:(
1647:)
1640:p
1636:+
1633:p
1630:,
1623:x
1619:+
1616:x
1613:(
1610:g
1607:)
1600:p
1596:+
1593:p
1590:,
1583:x
1579:+
1576:x
1573:(
1570:f
1563:x
1559:d
1551:x
1547:d
1539:p
1535:d
1527:p
1523:d
1512:2
1502:3
1493:2
1488:=
1485:)
1482:p
1479:,
1476:x
1473:(
1470:}
1467:}
1464:g
1461:,
1458:f
1455:{
1452:{
1419:.
1416:)
1413:p
1410:,
1407:x
1404:(
1401:g
1394:)
1390:)
1385:x
1368:p
1348:p
1331:x
1316:(
1310:2
1300:(
1287:)
1284:p
1281:,
1278:x
1275:(
1272:f
1261:2
1256:=
1250:}
1247:}
1244:g
1241:,
1238:f
1235:{
1232:{
1210:g
1206:f
1204:{
1200:g
1196:f
1188:â
1167:,
1164:)
1159:2
1151:(
1148:O
1145:+
1142:}
1139:g
1136:,
1133:f
1130:{
1127:=
1117:)
1114:f
1108:g
1102:g
1096:f
1093:(
1084:i
1080:1
1067:f
1064:e
1061:d
1055:=
1044:}
1041:}
1038:g
1035:,
1032:f
1029:{
1026:{
981:ħ
973:ħ
875:e
868:t
861:v
89:|
79:H
73:=
63:|
56:t
53:d
49:d
41:i
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.