1063:
1970:
807:
134:. Moyal actually appears not to know about the product in his celebrated article and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography. The popular naming after Moyal appears to have emerged only in the 1970s, in homage to his flat
1713:
1409:; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally, as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a torsion-free symplectic
482:
2194:
2356:. This situation is clearly different from the case where the positions are taken to be real-valued, but does offer insights into the overall algebraic structure of the Heisenberg algebra and its envelope, the Weyl algebra.
1708:
1058:{\displaystyle f\star g=fg+{\frac {i\hbar }{2}}\sum _{i,j}\Pi ^{ij}(\partial _{i}f)(\partial _{j}g)-{\frac {\hbar ^{2}}{8}}\sum _{i,j,k,m}\Pi ^{ij}\Pi ^{km}(\partial _{i}\partial _{k}f)(\partial _{j}\partial _{m}g)+\ldots ,}
601:
776:
1164:
284:
2036:
2445:
making the cyclicity of the phase-space trace manifest. This is a unique property of the above specific Moyal product, and does not hold for other correspondence rules' star products, such as Husimi's, etc.
1265:
1355:
2443:
359:
1965:{\displaystyle e^{-\tanh(a){\frac {q^{2}+p^{2}}{\hbar }}}\star e^{-\tanh(b){\frac {q^{2}+p^{2}}{\hbar }}}={\frac {\tanh(a+b)}{\tanh(a)+\tanh(b)}}e^{-\tanh(a+b){\frac {q^{2}+p^{2}}{\hbar }}}}
2206:
2354:
2543:
2301:
535:
698:
187:
92:
366:
2041:
1455:
703:
1091:
192:
1191:
1279:
542:
2371:
1975:
1081:
2640:
307:
2368:
star product of the Moyal type may be dropped, resulting in plain multiplication, as evident by integration by parts,
2751:
2665:
2729:
1425:
1401:
On any symplectic manifold, one can, at least locally, choose coordinates so as to make the symplectic structure
2756:
2306:
2258:
1382:
1077:
801:
107:
2464:
493:
2603:
671:
160:
65:
1445:
1374:
are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl-algebra case).
640:
477:{\displaystyle f\star g-g\star f=i\hbar \{f,g\}+{\mathcal {O}}(\hbar ^{3})\equiv i\hbar \{\{f,g\}\},}
131:
2189:{\displaystyle e^{-\alpha (q^{2}+p^{2})}e^{-\beta (q^{2}+p^{2})}=e^{-(\alpha +\beta )(q^{2}+p^{2})}}
2216:
1410:
1070:
135:
2681:
Lee, H. W. (1995). "Theory and application of the quantum phase-space distribution functions".
127:
51:
2635:. World Scientific Series in 20th Century Physics. Vol. 34. Singapore: World Scientific.
294:
119:
35:
2690:
2498:
2220:
1406:
19:
This article is about the product on functions on phase space. Not to be confused with the
8:
2609:(Lecture notes). Université Libre du Bruxelles, Institut des Hautes Études Scientifiques.
99:
2694:
2502:
2552:
1390:
1085:
2725:
2702:
2661:
2636:
648:
604:
2721:
2698:
2628:
2562:
2506:
2489:
Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory".
2255:
for the
Heisenberg group), so that the position and momenta operators are given by
2252:
2224:
1414:
662:
611:
2525:
1703:{\displaystyle \exp \left\star \exp \left={\frac {1}{1+\hbar ^{2}ab}}\exp \left.}
1080:
on the algebra of symbols and can be given a closed form (which follows from the
629:
If one restricts to polynomial functions, the above algebra is isomorphic to the
147:
95:
301:
characterized by the following properties (see below for an explicit formula):
2566:
2510:
2745:
2624:
2620:
2580:
485:
2541:
Curtright, T. L.; Zachos, C. K. (2012). "Quantum
Mechanics in Phase Space".
596:{\displaystyle {\overline {f\star g}}={\overline {g}}\star {\overline {f}},}
1386:
771:{\displaystyle \Pi =\sum _{i,j}\Pi ^{ij}\partial _{i}\wedge \partial _{j},}
630:
619:
123:
47:
20:
1441:
1159:{\displaystyle f\star g=m\circ e^{{\frac {i\hbar }{2}}\Pi }(f\otimes g),}
27:
537:
The 1 of the undeformed algebra is also the identity in the new algebra.
2583:(1967). "Some remarks about the associated envelope of a Lie algebra".
608:
279:{\displaystyle f\star g=fg+\sum _{n=1}^{\infty }\hbar ^{n}C_{n}(f,g),}
130:
in his 1946 doctoral dissertation, in a trenchant appreciation of the
361:
Deformation of the pointwise product — implicit in the formula above.
2633:
Quantum
Mechanics in Phase Space: An Overview with Selected Papers
2557:
2031:{\displaystyle \hbar \to 0,a/\hbar \to \alpha ,b/\hbar \to \beta }
1436:
A simple explicit example of the construction and utility of the
1381:-product used in the definition of the "algebra of symbols" of a
1363:
above, and the formulas then restrict naturally to real numbers.
2604:"Universal enveloping algebras and some applications in physics"
1260:{\displaystyle e^{A}=\sum _{n=0}^{\infty }{\frac {1}{n!}}A^{n}.}
2491:
Mathematical
Proceedings of the Cambridge Philosophical Society
1440:-product (for the simplest case of a two-dimensional euclidean
2251:
are understood to act on the complex plane (respectively, the
1350:{\displaystyle C_{n}={\frac {i^{n}}{2^{n}n!}}m\circ \Pi ^{n}.}
626:
in the second condition and eliminates the fourth condition.
1424:(where the Darboux theorem does not apply) are given by the
2527:
1377:
The relationship of the Moyal product to the generalized
618:
Note that, if one wishes to take functions valued in the
2716:
Curtright, T. L.; Fairlie, D. B.; Zachos, C. K. (2014).
2202:
between phase space and
Hilbert space, however, induces
16:
An example of a phase-space star product in mathematics
2718:
A Concise
Treatise on Quantum Mechanics in Phase Space
2715:
2619:
2438:{\displaystyle \int dx\,dp\;f\star g=\int dx\,dp~f~g,}
1359:
As indicated, often one eliminates all occurrences of
2374:
2309:
2261:
2044:
1978:
1716:
1458:
1282:
1194:
1094:
810:
706:
674:
545:
496:
369:
310:
195:
163:
68:
1188:, and the exponential is treated as a power series,
661:
To provide an explicit formula, consider a constant
639:, and the two offer alternative realizations of the
2465:"On the Principles of elementary quantum mechanics"
354:{\displaystyle f\star g=fg+{\mathcal {O}}(\hbar ),}
2437:
2348:
2295:
2188:
2030:
1964:
1702:
1349:
1259:
1158:
1057:
770:
692:
595:
529:
476:
353:
278:
181:
86:
58:. It is an associative, non-commutative product,
102:, described below). It is a special case of the
2743:
2227:, where the creation and annihilation operators
1452:-product according to a hyperbolic tangent law:
1084:). The closed form can be obtained by using the
2540:
2359:
1076:This is a special case of what is known as the
2488:
622:, then an alternative version eliminates the
468:
465:
453:
450:
412:
400:
126:–Groenewold product as it was introduced by
1389:is the universal enveloping algebra of the
484:Deformation of the Poisson bracket, called
2462:
2391:
1393:(modulo that the center equals the unit).
106:-product of the "algebra of symbols" of a
2556:
2413:
2384:
677:
166:
71:
2585:Functional Analysis and its Applications
2601:
2579:
2349:{\displaystyle p={\frac {a-a^{*}}{2i}}}
1073:, treated as a formal parameter here.
2744:
2296:{\displaystyle q={\frac {a+a^{*}}{2}}}
792:. The star product of two functions
113:
2655:
2523:
2364:Inside a phase-space integral, just
530:{\displaystyle f\star 1=1\star f=f,}
2680:
122:, but is also sometimes called the
13:
1448:: two Gaussians compose with this
1335:
1224:
1133:
1031:
1021:
1002:
992:
976:
963:
899:
880:
864:
756:
743:
730:
707:
420:
334:
233:
14:
2768:
2200:Every correspondence prescription
2019:
1999:
1955:
1827:
1767:
1444:) is given in the article on the
1124:
838:
693:{\displaystyle \mathbb {R} ^{2n}}
447:
397:
342:
182:{\displaystyle \mathbb {R} ^{2n}}
118:The Moyal product is named after
87:{\displaystyle \mathbb {R} ^{2n}}
2215:Similar results are seen in the
1082:Baker–Campbell–Hausdorff formula
2709:
2544:Asia Pacific Physics Newsletter
1426:Kontsevich quantization formula
1396:
1385:follows from the fact that the
651:of a vector space of dimension
643:of the space of polynomials in
2674:
2649:
2613:
2595:
2573:
2534:
2517:
2482:
2456:
2181:
2155:
2152:
2140:
2124:
2098:
2082:
2056:
2022:
2002:
1982:
1925:
1913:
1893:
1887:
1875:
1869:
1858:
1846:
1797:
1791:
1737:
1731:
1150:
1138:
1043:
1017:
1014:
988:
911:
895:
892:
876:
438:
425:
345:
339:
270:
258:
1:
2602:Bekaert, Xavier (June 2005).
2449:
141:
2703:10.1016/0370-1573(95)00007-4
2360:Inside phase-space integrals
1383:universal enveloping algebra
802:pseudo-differential operator
585:
572:
559:
108:universal enveloping algebra
7:
2660:. New York: Prentice-Hall.
1431:
1422:arbitrary Poisson manifolds
1366:Note that if the functions
1170:is the multiplication map,
800:can then be defined as the
10:
2773:
2463:Groenewold, H. J. (1946).
782:is a real number for each
98:(with a generalization to
18:
2752:Mathematical quantization
2567:10.1142/S2251158X12000069
2511:10.1017/S0305004100000487
1420:More general results for
1269:That is, the formula for
804:acting on both of them,
136:phase-space quantization
56:phase-space star product
2658:Time-Frequency Analysis
1972:The classical limit at
1071:reduced Planck constant
44:Weyl–Groenewold product
2439:
2350:
2297:
2190:
2032:
1966:
1704:
1351:
1261:
1228:
1160:
1059:
772:
694:
597:
531:
478:
355:
280:
237:
183:
88:
62:, on the functions on
52:Hilbrand J. Groenewold
2440:
2351:
2298:
2191:
2033:
1967:
1705:
1446:Wigner–Weyl transform
1352:
1262:
1208:
1161:
1060:
773:
695:
598:
532:
479:
356:
295:differential operator
281:
217:
184:
89:
54:) is an example of a
2757:Mathematical physics
2372:
2307:
2259:
2221:theta representation
2217:Segal–Bargmann space
2042:
1976:
1714:
1456:
1280:
1192:
1092:
808:
704:
672:
543:
494:
367:
308:
193:
161:
100:symplectic manifolds
94:, equipped with its
66:
2695:1995PhR...259..147L
2524:Moyal, Ann (2006).
2503:1949PCPS...45...99M
132:Weyl correspondence
114:Historical comments
2435:
2346:
2293:
2186:
2028:
1962:
1700:
1413:. This makes it a
1391:Heisenberg algebra
1347:
1257:
1156:
1055:
961:
862:
768:
728:
690:
647:variables (or the
593:
527:
474:
351:
276:
179:
120:José Enrique Moyal
84:
38:; also called the
36:José Enrique Moyal
2656:Cohen, L (1995).
2642:978-981-238-384-6
2629:Curtright, Thomas
2581:Berezin, Felix A.
2428:
2422:
2344:
2291:
1958:
1897:
1830:
1770:
1657:
1604:
1407:Darboux's theorem
1326:
1242:
1131:
934:
932:
847:
845:
713:
649:symmetric algebra
605:complex conjugate
588:
575:
562:
23:on graded posets.
2764:
2736:
2735:
2722:World Scientific
2713:
2707:
2706:
2678:
2672:
2671:
2653:
2647:
2646:
2617:
2611:
2610:
2608:
2599:
2593:
2592:
2577:
2571:
2570:
2560:
2538:
2532:
2531:
2521:
2515:
2514:
2486:
2480:
2479:
2469:
2460:
2444:
2442:
2441:
2436:
2426:
2420:
2355:
2353:
2352:
2347:
2345:
2343:
2335:
2334:
2333:
2317:
2302:
2300:
2299:
2294:
2292:
2287:
2286:
2285:
2269:
2253:upper half-plane
2250:
2236:
2225:Heisenberg group
2211:
2195:
2193:
2192:
2187:
2185:
2184:
2180:
2179:
2167:
2166:
2128:
2127:
2123:
2122:
2110:
2109:
2086:
2085:
2081:
2080:
2068:
2067:
2037:
2035:
2034:
2029:
2018:
1998:
1971:
1969:
1968:
1963:
1961:
1960:
1959:
1954:
1953:
1952:
1940:
1939:
1929:
1898:
1896:
1861:
1838:
1833:
1832:
1831:
1826:
1825:
1824:
1812:
1811:
1801:
1773:
1772:
1771:
1766:
1765:
1764:
1752:
1751:
1741:
1709:
1707:
1706:
1701:
1696:
1692:
1691:
1687:
1686:
1685:
1673:
1672:
1658:
1656:
1649:
1648:
1632:
1621:
1605:
1603:
1596:
1595:
1576:
1571:
1567:
1566:
1562:
1561:
1560:
1548:
1547:
1513:
1509:
1508:
1504:
1503:
1502:
1490:
1489:
1451:
1439:
1415:Fedosov manifold
1380:
1373:
1369:
1362:
1356:
1354:
1353:
1348:
1343:
1342:
1327:
1325:
1318:
1317:
1307:
1306:
1297:
1292:
1291:
1275:
1266:
1264:
1263:
1258:
1253:
1252:
1243:
1241:
1230:
1227:
1222:
1204:
1203:
1187:
1169:
1165:
1163:
1162:
1157:
1137:
1136:
1132:
1127:
1119:
1068:
1064:
1062:
1061:
1056:
1039:
1038:
1029:
1028:
1010:
1009:
1000:
999:
987:
986:
974:
973:
960:
933:
928:
927:
918:
907:
906:
888:
887:
875:
874:
861:
846:
841:
833:
799:
795:
791:
781:
777:
775:
774:
769:
764:
763:
751:
750:
741:
740:
727:
699:
697:
696:
691:
689:
688:
680:
667:
663:Poisson bivector
657:
646:
638:
625:
612:antiautomorphism
602:
600:
599:
594:
589:
581:
576:
568:
563:
558:
547:
536:
534:
533:
528:
483:
481:
480:
475:
437:
436:
424:
423:
360:
358:
357:
352:
338:
337:
300:
292:
285:
283:
282:
277:
257:
256:
247:
246:
236:
231:
188:
186:
185:
180:
178:
177:
169:
156:
152:
148:smooth functions
146:The product for
128:H. J. Groenewold
105:
93:
91:
90:
85:
83:
82:
74:
61:
2772:
2771:
2767:
2766:
2765:
2763:
2762:
2761:
2742:
2741:
2740:
2739:
2732:
2714:
2710:
2683:Physics Reports
2679:
2675:
2668:
2654:
2650:
2643:
2631:, eds. (2005).
2618:
2614:
2606:
2600:
2596:
2578:
2574:
2539:
2535:
2522:
2518:
2487:
2483:
2467:
2461:
2457:
2452:
2373:
2370:
2369:
2362:
2336:
2329:
2325:
2318:
2316:
2308:
2305:
2304:
2281:
2277:
2270:
2268:
2260:
2257:
2256:
2238:
2228:
2209:
2196:, as expected.
2175:
2171:
2162:
2158:
2136:
2132:
2118:
2114:
2105:
2101:
2091:
2087:
2076:
2072:
2063:
2059:
2049:
2045:
2043:
2040:
2039:
2014:
1994:
1977:
1974:
1973:
1948:
1944:
1935:
1931:
1930:
1928:
1903:
1899:
1862:
1839:
1837:
1820:
1816:
1807:
1803:
1802:
1800:
1781:
1777:
1760:
1756:
1747:
1743:
1742:
1740:
1721:
1717:
1715:
1712:
1711:
1681:
1677:
1668:
1664:
1663:
1659:
1644:
1640:
1633:
1622:
1620:
1616:
1612:
1591:
1587:
1580:
1575:
1556:
1552:
1543:
1539:
1538:
1534:
1527:
1523:
1498:
1494:
1485:
1481:
1480:
1476:
1469:
1465:
1457:
1454:
1453:
1449:
1437:
1434:
1399:
1378:
1371:
1367:
1360:
1338:
1334:
1313:
1309:
1308:
1302:
1298:
1296:
1287:
1283:
1281:
1278:
1277:
1274:
1270:
1248:
1244:
1234:
1229:
1223:
1212:
1199:
1195:
1193:
1190:
1189:
1171:
1167:
1120:
1118:
1117:
1113:
1093:
1090:
1089:
1078:Berezin formula
1066:
1034:
1030:
1024:
1020:
1005:
1001:
995:
991:
979:
975:
966:
962:
938:
923:
919:
917:
902:
898:
883:
879:
867:
863:
851:
834:
832:
809:
806:
805:
797:
793:
783:
779:
759:
755:
746:
742:
733:
729:
717:
705:
702:
701:
681:
676:
675:
673:
670:
669:
665:
652:
644:
637:
633:
623:
580:
567:
548:
546:
544:
541:
540:
495:
492:
491:
432:
428:
419:
418:
368:
365:
364:
333:
332:
309:
306:
305:
298:
293:is a certain bi
291:
287:
252:
248:
242:
238:
232:
221:
194:
191:
190:
189:takes the form
170:
165:
164:
162:
159:
158:
154:
150:
144:
116:
103:
96:Poisson bracket
75:
70:
69:
67:
64:
63:
59:
24:
17:
12:
11:
5:
2770:
2760:
2759:
2754:
2738:
2737:
2730:
2708:
2673:
2667:978-0135945322
2666:
2648:
2641:
2625:Fairlie, David
2621:Zachos, Cosmas
2612:
2594:
2572:
2533:
2530:. ANU E-press.
2516:
2481:
2454:
2453:
2451:
2448:
2434:
2431:
2425:
2419:
2416:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2390:
2387:
2383:
2380:
2377:
2367:
2361:
2358:
2342:
2339:
2332:
2328:
2324:
2321:
2315:
2312:
2290:
2284:
2280:
2276:
2273:
2267:
2264:
2205:
2183:
2178:
2174:
2170:
2165:
2161:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2135:
2131:
2126:
2121:
2117:
2113:
2108:
2104:
2100:
2097:
2094:
2090:
2084:
2079:
2075:
2071:
2066:
2062:
2058:
2055:
2052:
2048:
2027:
2024:
2021:
2017:
2013:
2010:
2007:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1957:
1951:
1947:
1943:
1938:
1934:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1902:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1836:
1829:
1823:
1819:
1815:
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1799:
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1790:
1787:
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1755:
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1746:
1739:
1736:
1733:
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1727:
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1699:
1695:
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1684:
1680:
1676:
1671:
1667:
1662:
1655:
1652:
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1643:
1639:
1636:
1631:
1628:
1625:
1619:
1615:
1611:
1608:
1602:
1599:
1594:
1590:
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1583:
1579:
1574:
1570:
1565:
1559:
1555:
1551:
1546:
1542:
1537:
1533:
1530:
1526:
1522:
1519:
1516:
1512:
1507:
1501:
1497:
1493:
1488:
1484:
1479:
1475:
1472:
1468:
1464:
1461:
1433:
1430:
1398:
1395:
1346:
1341:
1337:
1333:
1330:
1324:
1321:
1316:
1312:
1305:
1301:
1295:
1290:
1286:
1272:
1256:
1251:
1247:
1240:
1237:
1233:
1226:
1221:
1218:
1215:
1211:
1207:
1202:
1198:
1155:
1152:
1149:
1146:
1143:
1140:
1135:
1130:
1126:
1123:
1116:
1112:
1109:
1106:
1103:
1100:
1097:
1054:
1051:
1048:
1045:
1042:
1037:
1033:
1027:
1023:
1019:
1016:
1013:
1008:
1004:
998:
994:
990:
985:
982:
978:
972:
969:
965:
959:
956:
953:
950:
947:
944:
941:
937:
931:
926:
922:
916:
913:
910:
905:
901:
897:
894:
891:
886:
882:
878:
873:
870:
866:
860:
857:
854:
850:
844:
840:
837:
831:
828:
825:
822:
819:
816:
813:
767:
762:
758:
754:
749:
745:
739:
736:
732:
726:
723:
720:
716:
712:
709:
687:
684:
679:
635:
616:
615:
592:
587:
584:
579:
574:
571:
566:
561:
557:
554:
551:
538:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
489:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
435:
431:
427:
422:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
362:
350:
347:
344:
341:
336:
331:
328:
325:
322:
319:
316:
313:
289:
275:
272:
269:
266:
263:
260:
255:
251:
245:
241:
235:
230:
227:
224:
220:
216:
213:
210:
207:
204:
201:
198:
176:
173:
168:
143:
140:
115:
112:
81:
78:
73:
15:
9:
6:
4:
3:
2:
2769:
2758:
2755:
2753:
2750:
2749:
2747:
2733:
2731:9789814520430
2727:
2723:
2719:
2712:
2704:
2700:
2696:
2692:
2688:
2684:
2677:
2669:
2663:
2659:
2652:
2644:
2638:
2634:
2630:
2626:
2622:
2616:
2605:
2598:
2590:
2586:
2582:
2576:
2568:
2564:
2559:
2554:
2550:
2546:
2545:
2537:
2529:
2528:
2520:
2512:
2508:
2504:
2500:
2496:
2492:
2485:
2477:
2473:
2466:
2459:
2455:
2447:
2432:
2429:
2423:
2417:
2414:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2388:
2385:
2381:
2378:
2375:
2365:
2357:
2340:
2337:
2330:
2326:
2322:
2319:
2313:
2310:
2288:
2282:
2278:
2274:
2271:
2265:
2262:
2254:
2249:
2245:
2241:
2235:
2231:
2226:
2222:
2218:
2213:
2208:
2203:
2201:
2197:
2176:
2172:
2168:
2163:
2159:
2149:
2146:
2143:
2137:
2133:
2129:
2119:
2115:
2111:
2106:
2102:
2095:
2092:
2088:
2077:
2073:
2069:
2064:
2060:
2053:
2050:
2046:
2025:
2015:
2011:
2008:
2005:
1995:
1991:
1988:
1985:
1979:
1949:
1945:
1941:
1936:
1932:
1922:
1919:
1916:
1910:
1907:
1904:
1900:
1890:
1884:
1881:
1878:
1872:
1866:
1863:
1855:
1852:
1849:
1843:
1840:
1834:
1821:
1817:
1813:
1808:
1804:
1794:
1788:
1785:
1782:
1778:
1774:
1761:
1757:
1753:
1748:
1744:
1734:
1728:
1725:
1722:
1718:
1710:Equivalently,
1697:
1693:
1688:
1682:
1678:
1674:
1669:
1665:
1660:
1653:
1650:
1645:
1641:
1637:
1634:
1629:
1626:
1623:
1617:
1613:
1609:
1606:
1600:
1597:
1592:
1588:
1584:
1581:
1577:
1572:
1568:
1563:
1557:
1553:
1549:
1544:
1540:
1535:
1531:
1528:
1524:
1520:
1517:
1514:
1510:
1505:
1499:
1495:
1491:
1486:
1482:
1477:
1473:
1470:
1466:
1462:
1459:
1447:
1443:
1429:
1427:
1423:
1418:
1416:
1412:
1408:
1404:
1394:
1392:
1388:
1384:
1375:
1364:
1357:
1344:
1339:
1331:
1328:
1322:
1319:
1314:
1310:
1303:
1299:
1293:
1288:
1284:
1267:
1254:
1249:
1245:
1238:
1235:
1231:
1219:
1216:
1213:
1209:
1205:
1200:
1196:
1186:
1182:
1178:
1174:
1153:
1147:
1144:
1141:
1128:
1121:
1114:
1110:
1107:
1104:
1101:
1098:
1095:
1087:
1083:
1079:
1074:
1072:
1052:
1049:
1046:
1040:
1035:
1025:
1011:
1006:
996:
983:
980:
970:
967:
957:
954:
951:
948:
945:
942:
939:
935:
929:
924:
920:
914:
908:
903:
889:
884:
871:
868:
858:
855:
852:
848:
842:
835:
829:
826:
823:
820:
817:
814:
811:
803:
790:
786:
765:
760:
752:
747:
737:
734:
724:
721:
718:
714:
710:
685:
682:
664:
659:
656:
650:
642:
632:
627:
621:
613:
610:
606:
590:
582:
577:
569:
564:
555:
552:
549:
539:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
490:
487:
486:Moyal bracket
471:
462:
459:
456:
444:
441:
433:
429:
415:
409:
406:
403:
394:
391:
388:
385:
382:
379:
376:
373:
370:
363:
348:
329:
326:
323:
320:
317:
314:
311:
304:
303:
302:
296:
273:
267:
264:
261:
253:
249:
243:
239:
228:
225:
222:
218:
214:
211:
208:
205:
202:
199:
196:
174:
171:
149:
139:
137:
133:
129:
125:
121:
111:
109:
101:
97:
79:
76:
57:
53:
49:
45:
41:
37:
33:
32:Moyal product
29:
22:
2717:
2711:
2686:
2682:
2676:
2657:
2651:
2632:
2615:
2597:
2588:
2584:
2575:
2548:
2542:
2536:
2526:
2519:
2494:
2490:
2484:
2475:
2471:
2458:
2363:
2247:
2243:
2239:
2233:
2229:
2214:
2199:
2198:
1435:
1421:
1419:
1402:
1400:
1397:On manifolds
1387:Weyl algebra
1376:
1365:
1358:
1268:
1184:
1180:
1176:
1172:
1075:
788:
784:
660:
654:
631:Weyl algebra
628:
620:real numbers
617:
145:
117:
55:
48:Hermann Weyl
43:
40:star product
39:
31:
25:
21:star product
2219:and in the
1442:phase space
1086:exponential
286:where each
28:mathematics
2746:Categories
2689:(3): 147.
2478:: 405–460.
2450:References
2212:-product.
1411:connection
609:antilinear
142:Definition
2558:1104.5269
2405:∫
2396:⋆
2376:∫
2331:∗
2323:−
2283:∗
2150:β
2144:α
2138:−
2096:β
2093:−
2054:α
2051:−
2026:β
2023:→
2020:ℏ
2006:α
2003:→
2000:ℏ
1983:→
1980:ℏ
1956:ℏ
1911:
1905:−
1885:
1867:
1844:
1828:ℏ
1789:
1783:−
1775:⋆
1768:ℏ
1729:
1723:−
1642:ℏ
1618:−
1610:
1589:ℏ
1529:−
1521:
1515:⋆
1471:−
1463:
1336:Π
1332:∘
1225:∞
1210:∑
1145:⊗
1134:Π
1125:ℏ
1111:∘
1099:⋆
1050:…
1032:∂
1022:∂
1003:∂
993:∂
977:Π
964:Π
936:∑
921:ℏ
915:−
900:∂
881:∂
865:Π
849:∑
839:ℏ
815:⋆
757:∂
753:∧
744:∂
731:Π
715:∑
708:Π
586:¯
578:⋆
573:¯
560:¯
553:⋆
513:⋆
501:⋆
448:ℏ
442:≡
430:ℏ
398:ℏ
386:⋆
380:−
374:⋆
343:ℏ
315:⋆
297:of order
240:ℏ
234:∞
219:∑
200:⋆
138:picture.
1432:Examples
1403:constant
641:Weyl map
46:, after
2691:Bibcode
2499:Bibcode
2472:Physica
2223:of the
2204:its own
1069:is the
34:(after
2728:
2664:
2639:
2551:: 37.
2497:: 99.
2427:
2421:
2207:proper
1166:where
1065:where
778:where
607:is an
30:, the
2607:(PDF)
2591:: 91.
2553:arXiv
2468:(PDF)
1405:, by
2726:ISBN
2662:ISBN
2637:ISBN
2303:and
2237:and
1908:tanh
1882:tanh
1864:tanh
1841:tanh
1786:tanh
1726:tanh
1370:and
1183:) =
796:and
603:The
153:and
124:Weyl
50:and
2699:doi
2687:259
2563:doi
2507:doi
2366:one
2038:is
1607:exp
1518:exp
1460:exp
1276:is
668:on
658:).
157:on
42:or
26:In
2748::
2724:.
2720:.
2697:.
2685:.
2627:;
2623:;
2587:.
2561:.
2547:.
2505:.
2495:45
2493:.
2476:12
2474:.
2470:.
2248:∂z
2242:=
2232:=
1428:.
1417:.
1185:ab
1179:⊗
1088::
787:,
700::
110:.
2734:.
2705:.
2701::
2693::
2670:.
2645:.
2589:1
2569:.
2565::
2555::
2549:1
2513:.
2509::
2501::
2433:,
2430:g
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2418:p
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2402:=
2399:g
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2341:i
2338:2
2327:a
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2314:=
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2289:2
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2272:a
2266:=
2263:q
2246:/
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2240:a
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2125:)
2120:2
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2112:+
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2078:2
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2070:+
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2057:(
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2016:/
2012:b
2009:,
1996:/
1992:a
1989:,
1986:0
1950:2
1946:p
1942:+
1937:2
1933:q
1926:)
1923:b
1920:+
1917:a
1914:(
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1891:b
1888:(
1879:+
1876:)
1873:a
1870:(
1859:)
1856:b
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1847:(
1835:=
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1573:=
1569:]
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798:g
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780:Π
766:,
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645:n
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634:A
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469:}
466:}
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299:n
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288:C
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259:(
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250:C
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229:1
226:=
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215:+
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172:2
167:R
155:g
151:f
104:★
80:n
77:2
72:R
60:★
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