459:. A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position.
42:
1530:
In spite of these successes, Feynman, in his 1963 undergraduate lecture that discussed the spin-statistics connection, says: "We apologize for the fact that we cannot give you an elementary explanation." Neuenschwander echoed this in 1994, asking if there was any progress spurring additional proofs
477:
Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a
Euclidean version of
1596:
from their ability to take on "any" spin. He wrote that they were theoretically predicted to arise in low-dimensional systems where motion is restricted to fewer than three spatial dimensions. Wilczek described their spin statistics as "interpolating continuously between the usual boson and fermion
446:
Naively, spin, an angular momentum property intrinsic to a particle, would be unrelated to fundamental properties of a collection of such particles. However, these are indistinguishable particles: any physical prediction relating multiple indistinguishable particles must not change when the
1103:
Numerous notable proofs have been published, with different kinds of limitations and assumptions. They are all "negative proofs", meaning that they establish that integral spin fields cannot result in fermion statistics while half-integral spin fields cannot result in boson statistics.
1410:
which was later critiqued by Pauli. Pauli showed that
Feynman's proof explicitly relied on the first two postulates he used and implicitly used the third one by first allowing negative probabilities but then rejecting field theory results with probabilities greater than one.
1285:
1526:
derived the CPT theorem using the spin-statistics theorem and
Burgoyne's proof the spin-statistics theorem in 1958 required no constraints on the interactions nor on the form of the field theories. These results are among the most rigorous practical theorems.
462:
While the physical state does not change under the exchange of the particles' positions, it is possible for the state vector to change sign as a result of an exchange. Since this sign change is just an overall phase, this does not affect the physical state.
1563:
of finite dimension. Thus it seems impossible to construct a
Hilbert space in which all states have finite, non-zero spin and positive, Lorentz-invariant norm. This problem is overcome in different ways depending on particle spin–statistics.
1110:
1381:, and was rederived in a more systematic way by Pauli the following year. In a later summary, Pauli listed three postulates within relativistic quantum field theory as required for these versions of the theorem:
519:
are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the
614:
1004:
821:
400:
All known particles obey either Fermi-Dirac statistics or Bose-Einstein statistics. A particle's intrinsic spin always predicts the statistics of a collection of such particles and vice versa:
1514:. (The first two postulates of the Pauli era proofs involve the vacuum state and fields at separate locations.) The new result allowed more rigorous proofs of the spin-statistics theorems by
1539:
In 1987 Greenberg and
Mohaparra proposed that the spin statistics theorem could have small violations. With the help of very precise calculations for states of the He atom that violate the
1086:
900:
537:. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator
1476:
1314:
707:
672:
2189:
1096:
An elementary explanation for the spin-statistics theorem cannot be given despite the fact that the theorem is so simple to state. In the
Feynman Lectures on Physics,
736:
1512:
1368:
1341:
1027:
927:
844:
637:
438:, which mediate forces between matter particles, are all bosons. A spin-statistics theorem attempts explain the origin of this fundamental dichotomy.
515:
are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change.
2050:
Pauli, Wolfgang (1988). "Exclusion
Principle, Lorentz Group and Reflection of Space-Time and Charge". In Enz, Charles P.; v. Meyenn, Karl (eds.).
328:
122:
2709:
2247:
1531:
and books. Neuenschwander's 2013 popularization of the spin-statistics connection suggested that simple explanations remain elusive.
1280:{\displaystyle |\psi (\alpha _{1},\alpha _{2},\alpha _{3},...)|^{2}=|{\hat {P}}\psi (\alpha _{1},\alpha _{2},\alpha _{3},...)|^{2}}
1567:
For a state of integer spin the negative norm states (known as "unphysical polarization") are set to zero, which makes the use of
2699:
17:
2344:
Deilamian, K.; et al. (1995). "Search for small violations of the symmetrization postulate in an excited state of Helium".
709:, and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.
2704:
543:
2133:
2069:
1990:
1805:
1788:
1709:
1107:
Proofs that avoid using any relativistic quantum field theory mechanism have defects. Many such proofs rely on a claim that
411:
A spin-statistics theorem shows that the mathematical logic of quantum mechanics predicts or explains this physical result.
2150:
466:
The essential ingredient in proving the spin-statistics relation is relativity, that the physical laws do not change under
935:
321:
755:
156:
2595:
2404:
1677:
1650:
1100:
said that this probably means that we do not have a complete understanding of the fundamental principle involved.
164:
84:
1937:
1598:
1316:
permutes the coordinates. However, the value on the left-hand-side represents the probability of particle 1 at
314:
422:
contains at most one fermion -- controls the formation of matter. The basic building blocks of matter such as
2694:
1032:
2684:
849:
89:
2555:
79:
1592:
published a research paper on the possibilities of possible fractional-spin particles, which he termed
94:
1440:
2689:
2665:
Animation of the Dirac belt trick with a double belt, showing that belts behave as spin 1/2 particles
1540:
1399:
Their analysis neglected particle interactions other than commutation/anti-commutation of the state.
521:
415:
414:
The statistics of indistinguishable particles is among the most fundamental of physical effects. The
2427:
1547:
of He using an atomic beam spectrometer. The search was unsuccessful with an upper limit of 5×10.
381:
148:
1574:
For a state of half-integer spin the argument can be circumvented by having fermionic statistics.
500:, and then boosts become rotations. The new "spacetime" has only spatial directions and is termed
1479:
481:
362:
210:
132:
56:
1290:
677:
642:
1560:
471:
467:
389:
478:
spacetime, in which the time direction is treated as a spatial one, as will be now explained.
715:
493:
227:
33:
2624:
2564:
2442:
2353:
2302:
2016:
1946:
1902:
1855:
1485:
1395:
Physical probabilities are positive (the metric of the
Hilbert space is positive definite).
1346:
1319:
1012:
912:
829:
622:
524:: two identical fermions cannot occupy the same state. This rule does not hold for bosons.
270:
8:
1419:
1407:
674:
a numerical function with complex values) creates a two-particle state with wavefunction
354:
265:
255:
67:
2628:
2568:
2446:
2357:
2306:
2051:
2020:
1950:
1906:
1885:
1859:
2640:
2614:
2389:
2326:
2208:
2032:
1615:
1437:
lead to a theorem that stated that the expectation value of the product of two fields,
1370:, and so on, and is thus quantum mechanically invalid for indistinguishable particles.
496:
of the correlation functions of a quantum field theory, the time coordinate may become
489:
140:
2508:
1846:(1939). "Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin".
1699:
1426:
more fully developed by Pauli in 1955. These proofs were notably difficult to follow.
2644:
2591:
2528:
2489:
2469:
2400:
2369:
2318:
2271:
2228:
2170:
2129:
2106:
2065:
2036:
1986:
1825:
1784:
1746:
1705:
1673:
1646:
358:
290:
232:
2670:
Animation of a Dirac belt trick variant showing that spin 1/2 particles are fermions
2330:
2086:
2632:
2572:
2520:
2481:
2450:
2361:
2310:
2263:
2220:
2162:
2098:
2057:
2024:
1954:
1910:
1863:
1817:
1738:
1620:
1415:
497:
350:
285:
260:
109:
2061:
1974:
1893:
1667:
1640:
1434:
1430:
1403:
1097:
738:
and the two operators take place at the same time; more generally, they may have
275:
237:
217:
2365:
2224:
2659:
2524:
2454:
2396:
1881:
1610:
1568:
1515:
1422:
in 1940 based on charge-conjugation invariance, leading to a connection to the
1378:
528:
346:
280:
187:
99:
51:
2636:
2485:
1726:
492:
with a different velocity and is mathematically like a rotation into time. By
2678:
2532:
2493:
2423:
2275:
2232:
2174:
2110:
2102:
1829:
1750:
1589:
1556:
485:
456:
419:
2605:
Jabs, Arthur (2010). "Connecting spin and statistics in quantum mechanics".
2314:
2290:
2373:
2190:"The Spin-Statistics Theorem and Identical Particle Distribution Functions"
2028:
1914:
1843:
1519:
1386:
1374:
532:
385:
373:
300:
2322:
1959:
1932:
1779:
Duck, Ian; Sudarshan, Ennackel Chandy George; Sudarshan, E. C. G. (1998).
1392:
Spatially separated measurements do not disturb each other (they commute),
2209:"Local Quantum Field Theory of Possible Violation of the Pauli Principle"
1982:
1867:
1550:
1423:
295:
197:
192:
182:
357:
of collections of such particles is a consequence of the mathematics of
1742:
474:
according to the spin of the particle that they create, by definition.
1821:
1804:
Curceanu, Catalina; Gillaspy, J. D.; Hilborn, Robert C. (2012-07-01).
1697:
41:
1727:"The fundamental theorem on the relation between spin and statistics"
739:
407:
Half-integral spin particle are fermions with Fermi-Dirac statistics.
2577:
2550:
2267:
2166:
2007:
Julian
Schwinger (June 15, 1951). "The Quantum Theory of Fields I".
1523:
527:
In quantum field theory, a state or a wavefunction is described by
431:
369:
222:
2619:
2056:(in German). Wiesbaden: Vieweg+Teubner Verlag. pp. 459–479.
516:
450:
427:
377:
2252:
Dwight E. Neuenschwander, Am. J. Phys. 62 (11), 972 (1994)]"
404:
Integral spin particles are bosons with Bose-Einstein statistics
1698:
Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1965).
435:
423:
1597:
cases". The effect has become the basis for understanding the
1418:
in 1950 based on time-reversal invariance followed a proof by
1385:
Any state with particle occupation has higher energy than the
2669:
2664:
1593:
1583:
512:
104:
2509:"Hamiltonian theories of the fractional quantum Hall effect"
507:
1731:
Proceedings of the Indian
Academy of Sciences - Section A
2551:"Toward an understanding of the spin–statistics theorem"
2126:
Inward bound: of matter and forces in the physical world
1543:, Deilamian, Gillaspy and Kelleher looked for the 1s2s S
609:{\displaystyle \iint \psi (x,y)\phi (x)\phi (y)\,dx\,dy}
455:
In a quantum system, a physical state is described by a
2590:(5th ed.). Princeton: Princeton University Press.
1803:
1577:
1806:"Resource Letter SS–1: The Spin-Statistics Connection"
1797:
1778:
1551:
Relation to representation theory of the Lorentz group
1488:
1443:
1349:
1322:
1293:
1113:
1035:
1015:
938:
915:
852:
832:
758:
718:
680:
645:
625:
546:
1997:
A reprint of Feynman's 1949 paper in Physical Review
1783:(1. reprint ed.). Singapore: World Scientific.
434:
are all fermions. Conversely, particles such as the
1774:
1772:
1770:
1768:
1766:
1764:
1762:
1760:
1406:gave a completely different type of proof based on
999:{\displaystyle \phi (x)\phi (y)=-\phi (y)\phi (x),}
2388:
2206:
1506:
1470:
1362:
1335:
1308:
1279:
1080:
1021:
998:
921:
894:
838:
816:{\displaystyle \phi (x)\phi (y)=\phi (y)\phi (x),}
815:
730:
701:
666:
631:
608:
345:proves that the observed relationship between the
2387:Peskin, Michael E.; Schroeder, Daniel V. (1995).
2207:Greenberg, O. W.; Mohapatra, R. N. (1987-11-30).
2676:
2585:
2428:"Quantum Mechanics of Fractional-Spin Particles"
2418:
2416:
2006:
1926:
1924:
1757:
2386:
1973:
1933:"On the Connection Between Spin and Statistics"
902:, and the field will create bosonic particles.
353:not due to the orbital motion) and the quantum
2586:Streater, Ray F.; Wightman, Arthur S. (2000).
2291:"Predicted energy shifts for "paronic" Helium"
2187:
2148:
1930:
1880:
1693:
1691:
1689:
451:Quantum states and indistinguishable particles
2548:
2506:
2413:
2128:(Reprint ed.). Oxford: Clarendon Press
2085:Lüders, Gerhart; Zumino, Bruno (1958-06-15).
1921:
1431:mathematical foundations of quantum mechanics
395:
322:
2507:Murthy, Ganpathy; Shankar, R. (2003-10-03).
2084:
1886:"The Connection Between Spin and Statistics"
1842:
1686:
2151:"Question ♯7. The spin-statistics theorem"
2000:
329:
315:
40:
2618:
2576:
2343:
1958:
1724:
1639:Dirac, Paul Adrien Maurice (1981-01-01).
599:
592:
531:operating on some basic state called the
508:Exchange symmetry or permutation symmetry
2588:PCT, Spin & Statistics, and All That
2470:"Nobel Lecture: Fractional quantization"
2467:
2188:Neuenschwander, Dwight E. (2015-07-28).
2149:Neuenschwander, Dwight E. (1994-11-01).
2087:"Connection between Spin and Statistics"
2549:Duck, Ian; Sudarshan, E. C. G. (1998).
2422:
2391:An Introduction to Quantum Field Theory
2245:
1701:The Feynman Lectures on Physics, Vol. 3
1669:General principles of quantum mechanics
1373:The first proof was formulated in 1939
1088:, and the particles will be fermionic.
742:separation, as is explained hereafter.
14:
2677:
1081:{\displaystyle \psi (x,y)=-\psi (y,x)}
470:. The field operators transform under
2288:
2049:
1781:Pauli and the spin-statistics theorem
1665:
1638:
1534:
895:{\displaystyle \psi (x,y)=\psi (y,x)}
2604:
2123:
1578:Quasiparticle anyons in 2 dimensions
1009:then only the antisymmetric part of
749:, meaning that the following holds:
484:include 3-dimensional rotations and
238:Grand potential / Landau free energy
1977:(1961). "The theory of positrons".
1642:The Principles of Quantum Mechanics
24:
2542:
25:
2721:
2652:
2246:Hilborn, Robert C. (1995-04-01).
905:On the other hand, if the fields
2710:Theorems in mathematical physics
1725:Sudarshan, E. C. G. (May 1968).
1645:. Clarendon Press. p. 149.
1471:{\displaystyle \phi (x)\phi (y)}
826:then only the symmetric part of
2500:
2461:
2380:
2337:
2282:
2239:
2200:
2181:
2142:
2117:
2078:
2043:
1967:
1938:Progress of Theoretical Physics
1704:. Addison-Wesley. p. 4.1.
2700:Statistical mechanics theorems
2468:Laughlin, R. B. (1999-07-01).
1874:
1836:
1718:
1666:Pauli, Wolfgang (1980-01-01).
1659:
1632:
1599:fractional quantum hall effect
1501:
1489:
1465:
1459:
1453:
1447:
1300:
1267:
1262:
1211:
1202:
1192:
1178:
1173:
1122:
1115:
1075:
1063:
1051:
1039:
990:
984:
978:
972:
960:
954:
948:
942:
889:
877:
868:
856:
807:
801:
795:
789:
780:
774:
768:
762:
696:
684:
661:
649:
589:
583:
577:
571:
565:
553:
13:
1:
2705:Theorems in quantum mechanics
1626:
441:
2250:The spin-statistics theorem,
2248:"Answer to Question ♯7 [
2062:10.1007/978-3-322-90270-2_41
27:Theorem in quantum mechanics
7:
2556:American Journal of Physics
2366:10.1103/PhysRevLett.74.4787
2256:American Journal of Physics
2225:10.1103/PhysRevLett.59.2507
2155:American Journal of Physics
1810:American Journal of Physics
1604:
80:Indistinguishable particles
10:
2726:
2525:10.1103/RevModPhys.75.1101
2455:10.1103/PhysRevLett.49.957
1581:
1309:{\displaystyle {\hat {P}}}
702:{\displaystyle \psi (x,y)}
667:{\displaystyle \psi (x,y)}
447:particles are exchanged.
396:Spin-statistics connection
2637:10.1007/s10701-009-9351-4
2513:Reviews of Modern Physics
2486:10.1103/RevModPhys.71.863
2474:Reviews of Modern Physics
1541:Pauli exclusion principle
1522:and by Burgoyne. In 1957
1091:
522:Pauli exclusion principle
488:. A boost transfers to a
416:Pauli exclusion principle
2103:10.1103/PhysRev.110.1450
382:Bose-Einstein statistics
2658:A nice nearly-proof at
2435:Physical Review Letters
2315:10.1103/PhysRevA.39.897
2213:Physical Review Letters
1979:Quantum Electrodynamics
1931:Wolfgang Pauli (1950).
1561:unitary representations
731:{\displaystyle x\neq y}
482:Lorentz transformations
472:Lorentz transformations
468:Lorentz transformations
418:-- that every occupied
363:reduced Planck constant
343:spin–statistics theorem
123:Thermodynamic ensembles
75:Spin–statistics theorem
18:Spin-statistics theorem
2607:Foundations of Physics
2289:Drake, G.W.F. (1989).
2124:Pais, Abraham (2002).
2029:10.1103/PhysRev.82.914
1915:10.1103/PhysRev.58.716
1848:Helvetica Physica Acta
1508:
1480:analytically continued
1472:
1364:
1337:
1310:
1281:
1082:
1023:
1000:
929:has the property that
923:
896:
840:
817:
732:
703:
668:
633:
610:
390:Fermi-Dirac statistics
2660:John Baez's home page
1509:
1507:{\displaystyle (x-y)}
1473:
1365:
1363:{\displaystyle r_{2}}
1338:
1336:{\displaystyle r_{1}}
1311:
1282:
1083:
1029:contributes, so that
1024:
1022:{\displaystyle \psi }
1001:
924:
922:{\displaystyle \phi }
897:
846:contributes, so that
841:
839:{\displaystyle \psi }
818:
733:
704:
669:
634:
632:{\displaystyle \phi }
611:
494:analytic continuation
228:Helmholtz free energy
157:Isoenthalpic–isobaric
34:Statistical mechanics
2695:Quantum field theory
1868:10.5169/seals-110930
1486:
1441:
1347:
1320:
1291:
1111:
1033:
1013:
936:
913:
850:
830:
756:
716:
678:
643:
623:
544:
2685:Particle statistics
2629:2010FoPh...40..776J
2569:1998AmJPh..66..284D
2447:1982PhRvL..49..957W
2358:1995PhRvL..74.4787D
2307:1989PhRvA..39..897D
2021:1951PhRv...82..914S
1960:10.1143/ptp/5.4.526
1951:1950PThPh...5..526P
1907:1940PhRv...58..716P
1884:(15 October 1940).
1860:1939AcHPh..12....3F
1672:. Springer-Verlag.
1588:In 1982, physicist
1559:has no non-trivial
1482:to all separations
1420:Frederik Belinfante
1408:vacuum polarization
1287:where the operator
712:Let us assume that
355:particle statistics
165:Isothermal–isobaric
68:Particle statistics
2426:(4 October 1982).
1743:10.1007/BF03049366
1616:Anyonic statistics
1535:Experimental tests
1504:
1468:
1360:
1333:
1306:
1277:
1078:
1019:
996:
919:
892:
836:
813:
728:
699:
664:
629:
606:
490:frame of reference
361:. In units of the
105:Anyonic statistics
2352:(24): 4787–4790.
2219:(22): 2507–2510.
2135:978-0-19-851997-3
2071:978-3-322-90271-9
1992:978-0-201-36075-2
1822:10.1119/1.4704899
1790:978-981-02-3114-9
1711:978-0-201-02118-9
1303:
1205:
359:quantum mechanics
339:
338:
233:Gibbs free energy
85:Maxwell–Boltzmann
16:(Redirected from
2717:
2690:Physics theorems
2648:
2622:
2601:
2582:
2580:
2537:
2536:
2519:(4): 1101–1158.
2504:
2498:
2497:
2465:
2459:
2458:
2432:
2420:
2411:
2410:
2394:
2384:
2378:
2377:
2341:
2335:
2334:
2286:
2280:
2279:
2243:
2237:
2236:
2204:
2198:
2197:
2185:
2179:
2178:
2146:
2140:
2139:
2121:
2115:
2114:
2097:(6): 1450–1453.
2082:
2076:
2075:
2047:
2041:
2040:
2004:
1998:
1996:
1971:
1965:
1964:
1962:
1928:
1919:
1918:
1890:
1878:
1872:
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1840:
1834:
1833:
1801:
1795:
1794:
1776:
1755:
1754:
1722:
1716:
1715:
1695:
1684:
1683:
1663:
1657:
1656:
1636:
1621:Braid statistics
1513:
1511:
1510:
1505:
1477:
1475:
1474:
1469:
1416:Julian Schwinger
1369:
1367:
1366:
1361:
1359:
1358:
1343:, particle 2 at
1342:
1340:
1339:
1334:
1332:
1331:
1315:
1313:
1312:
1307:
1305:
1304:
1296:
1286:
1284:
1283:
1278:
1276:
1275:
1270:
1249:
1248:
1236:
1235:
1223:
1222:
1207:
1206:
1198:
1195:
1187:
1186:
1181:
1160:
1159:
1147:
1146:
1134:
1133:
1118:
1087:
1085:
1084:
1079:
1028:
1026:
1025:
1020:
1005:
1003:
1002:
997:
928:
926:
925:
920:
901:
899:
898:
893:
845:
843:
842:
837:
822:
820:
819:
814:
737:
735:
734:
729:
708:
706:
705:
700:
673:
671:
670:
665:
639:an operator and
638:
636:
635:
630:
615:
613:
612:
607:
351:angular momentum
331:
324:
317:
110:Braid statistics
44:
30:
29:
21:
2725:
2724:
2720:
2719:
2718:
2716:
2715:
2714:
2675:
2674:
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2578:10.1119/1.18860
2545:
2543:Further reading
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2462:
2441:(14): 957–959.
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2346:Phys. Rev. Lett
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2268:10.1119/1.17953
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2167:10.1119/1.17652
2161:(11): 972–972.
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2118:
2091:Physical Review
2083:
2079:
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2009:Physical Review
2005:
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1975:Richard Feynman
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1968:
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1894:Physical Review
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1438:
1435:Arthur Wightman
1404:Richard Feynman
1377:, a student of
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1098:Richard Feynman
1094:
1034:
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218:Internal energy
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149:Grand canonical
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2653:External links
2651:
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2424:Wilczek, Frank
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2397:Addison-Wesley
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2281:
2262:(4): 298–299.
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2053:Wolfgang Pauli
2042:
2015:(6): 914–917.
1999:
1991:
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1945:(4): 526–543.
1920:
1901:(8): 716–722.
1882:Wolfgang Pauli
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1816:(7): 561–577.
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1569:gauge symmetry
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1516:Gerhart Luders
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2295:Phys. Rev. A
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1844:Markus Fierz
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1529:
1520:Bruno Zumino
1429:Work on the
1428:
1413:
1401:
1398:
1387:vacuum state
1375:Markus Fierz
1372:
1106:
1102:
1095:
1008:
907:anti-commute
906:
904:
825:
746:
744:
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618:
533:
526:
511:
501:
480:
476:
465:
461:
457:state vector
454:
445:
413:
410:
399:
386:half-integer
376:have either
374:3 dimensions
365:
342:
340:
161:
153:
145:
137:
129:
74:
1983:Basic Books
1854:(1): 3–37.
1571:necessary.
1478:, could be
1424:CPT theorem
1414:A proof by
286:von Neumann
95:Fermi–Dirac
2679:Categories
2194:Radiations
1627:References
442:Background
250:Scientists
211:Potentials
2645:122488238
2620:0810.2399
2533:0034-6861
2494:0034-6861
2276:0002-9505
2233:0031-9007
2175:0002-9505
2111:0031-899X
2037:121971249
1830:0002-9505
1751:0370-0089
1496:−
1457:ϕ
1445:ϕ
1301:^
1242:α
1229:α
1216:α
1209:ψ
1203:^
1153:α
1140:α
1127:α
1120:ψ
1061:ψ
1058:−
1037:ψ
1017:ψ
982:ϕ
970:ϕ
967:−
952:ϕ
940:ϕ
917:ϕ
875:ψ
854:ψ
834:ψ
799:ϕ
787:ϕ
772:ϕ
760:ϕ
740:spacelike
723:≠
682:ψ
647:ψ
627:ϕ
581:ϕ
569:ϕ
551:ψ
548:∬
502:Euclidean
498:imaginary
432:electrons
370:particles
281:Ehrenfest
261:Boltzmann
141:Canonical
2374:10058599
2331:35775478
1605:See also
1524:Res Jost
1402:In 1949
517:Fermions
428:neutrons
276:Einstein
223:Enthalpy
188:Einstein
2625:Bibcode
2565:Bibcode
2443:Bibcode
2354:Bibcode
2323:9901315
2303:Bibcode
2017:Bibcode
1947:Bibcode
1903:Bibcode
1856:Bibcode
747:commute
424:protons
378:integer
256:Maxwell
2643:
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2109:
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2035:
1989:
1828:
1787:
1749:
1708:
1676:
1649:
1594:anyons
1092:Proofs
619:(with
534:vacuum
513:Bosons
486:boosts
436:photon
430:, and
368:, all
291:Tolman
177:Models
2641:S2CID
2615:arXiv
2431:(PDF)
2327:S2CID
2033:S2CID
1889:(PDF)
1584:Anyon
301:Fermi
296:Debye
271:Gibbs
198:Potts
193:Ising
183:Debye
2592:ISBN
2529:ISSN
2490:ISSN
2401:ISBN
2370:PMID
2319:PMID
2272:ISSN
2229:ISSN
2171:ISSN
2130:ISBN
2107:ISSN
2066:ISBN
1987:ISBN
1826:ISSN
1785:ISBN
1747:ISSN
1706:ISBN
1674:ISBN
1647:ISBN
1555:The
1518:and
341:The
266:Bose
2633:doi
2573:doi
2521:doi
2482:doi
2451:doi
2362:doi
2311:doi
2264:doi
2221:doi
2163:doi
2099:doi
2095:110
2058:doi
2025:doi
1955:doi
1911:doi
1864:doi
1818:doi
1739:doi
1433:by
384:or
162:NPT
154:NPH
146:µVT
138:NVT
130:NVE
2681::
2639:.
2631:.
2623:.
2611:40
2609:.
2571:.
2561:66
2559:.
2553:.
2527:.
2517:75
2515:.
2511:.
2488:.
2478:71
2476:.
2472:.
2449:.
2439:49
2437:.
2433:.
2415:^
2399:.
2395:.
2368:.
2360:.
2350:74
2348:.
2325:.
2317:.
2309:.
2299:39
2297:.
2293:.
2270:.
2260:63
2258:.
2254:.
2227:.
2217:59
2215:.
2211:.
2192:.
2169:.
2159:62
2157:.
2153:.
2105:.
2093:.
2089:.
2064:.
2031:.
2023:.
2013:82
2011:.
1985:.
1981:.
1953:.
1941:.
1935:.
1923:^
1909:.
1899:58
1897:.
1891:.
1862:.
1852:12
1850:.
1824:.
1814:80
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1808:.
1759:^
1745:.
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1688:^
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2647:.
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2581:.
2575::
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2496:.
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2409:.
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2333:.
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2278:.
2266::
2235:.
2223::
2177:.
2165::
2138:.
2113:.
2101::
2074:.
2060::
2039:.
2027::
2019::
1995:.
1963:.
1957::
1949::
1943:5
1917:.
1913::
1905::
1870:.
1866::
1858::
1832:.
1820::
1793:.
1753:.
1741::
1714:.
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