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Groups and probabilities at the foundations of quantum mechanics

, ,  a
a Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation

The current state of the foundations of quantum mechanics is discussed. The analysis takes as its starting point the theory of probability amplitudes, which is intimately related to the group-theoretic approach. A detailed examination is presented of the relationships of classical and quantum theories, the transition to the classical limit, the different forms of uncertainty relations, and the properties of quantum structures determined by the Clebsch--Gordan coefficients. Possible future generalizations are examined, including those involving quantum algebras.

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Fulltext is also available at DOI: 10.1070/PU1992v035n12ABEH002281
PACS: 03.65.Ta, 03.65.Db, 02.50.Cw (all)
DOI: 10.1070/PU1992v035n12ABEH002281
URL: https://ufn.ru/en/articles/1992/12/a/
Citation: Smorodinskii Ya A, Shelepin A L, Shelepin L A "Groups and probabilities at the foundations of quantum mechanics" Sov. Phys. Usp. 35 (12) 1005–1051 (1992)
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Оригинал: Смородинский Я А, Шелепин А Л, Шелепин Л А «Групповые и вероятностные основы квантовой теории» УФН 162 (12) 1–95 (1992); DOI: 10.3367/UFNr.0162.199212a.0001

References (167) ↓ Cited by (10) Similar articles (20)

  1. Fon Neiman I. Matematicheskie Osnovy Kvantovoi Mekhaniki (M.: Nauka, 1964)
  2. Smorodinskii Ya.A. Shelepin L.A. Usp. Fiz. Nauk 106 3 (1972); Smorodinskii Ya A, Shelepin L A Sov. Phys. Usp. 15 1 (1972)
  3. Dirac P A M Fields Quanta 3 139 (1972)
  4. Feinman R., Khibbs A. Kvantovaya Mekhanika i Integraly Po Traektoriyam (M.: Mir, 1968)
  5. Shelepin A.L., Shelepin L.A. Tr. FIAN 218 3 (1992)
  6. Dirak P.A.M. Vospominaniya o Neobychainoi Epokhe: Sb. Statei (M.: Nauka, 1990)
  7. Vilenkin N.Ya. Spetsial’nye Funktsii i Teoriya Predstavlenii Grupp (M.: Nauka, 1965)
  8. Karruzers P., N’eto M. Kogerentnye Sostoyaniya v Kvantovoi Teorii (M.: Mir, 1972) p. 71
  9. Gitman D.M., Shelepin A.L. Kratk. Soobshch. Fiz. (1) 31 (1990)
  10. Shelepin A.L., Shelepin L.A. Teoretiko-gruppovye Metody v Fizike Vol. 2 (M.: Nauka, 1986) p. 99
  11. Shelepin A.L., Shelepin L.A. Tr. FIAN 191 46 (1989)
  12. Shelepin L.A. Tr. FIAN 70 3 (1973)
  13. Perelomov A.M. Obobshchennye Kogerentnye Sostoyaniya i Ikh Primeneniya (M.: Nauka, 1987)
  14. Zhelobenko D.P., Shtern A.M. Predstavleniya Grupp Li (M.: Nauka, 1983)
  15. Gitman D.M., Kharchev S.M., Shelepin A.L. Tr. FIAN 201 95 (1990)
  16. Gitman D.M., Shelepin A.L. Preprint MIT CTP 1990 (Massachusets, 1991)
  17. Fronsdel K. Teoriya Grupp i Elementarnye Chastitsy (M.: Mir, 1967) p. 324
  18. Fronsdal C Proc. R. Soc. London A 288 98 (1965)
  19. Klimyk A. U. Matrichnye Elementy i Koeffitsienty Klebsha-Gordana Predstavlenii Grupp (K.: Naukova dumka, 1979)
  20. Gruber B., Klimyk A.V. J. Math. Phys. 19 2009 (1978)
  21. Varshalovich D.M., Moskalev A.N., Khersonskii V.K. Kvantovaya Teoriya Uglovogo Momenta (L.: Nauka, 1975)
  22. Bidenkharn L., Lauk Dzh. Uglovoi Moment v Kvantovoi Mekhanike Vol. 1, 2 (M.: Mir, 1984)
  23. Yutsis A.P., Levinson I.B., Vanagas V.V. Matematicheskii Apparat Momenta Kolichestva Dvizheniya (Vil’nyus: GIPNL, 1960)
  24. Landau L.D., Lifshits E.M. Kvantovaya Mekhanika. Nerelyativistskaya Teoriya (M.: Nauka, 1989)
  25. Lyubarskii G.Ya. Teoriya Grupp i Ee Primenenie v Fizike (M.: Gostekhizdat, 1957)
  26. Karasev V.P. Tr. FIAN 70 147 (1973)
  27. Chacon E., Levi P., Moshinsky M. J. Math. Phys. 16 1876 (1975)
  28. Barut A., Ronchka R. Teoriya Predstavlenii Grupp i Ee Prilozheniya (M.: Mir, 1980)
  29. Dicke R H Phys. Rev. 93 99 (1954)
  30. Allen L., Eberli Dzh. Opticheskii Rezonans i Dvukhurovnevye Atomy (M.: Mir, 1978)
  31. Khilton P., Uaili S. Teoriya Gomologii. Vvedenie v Algebraicheskuyu Topologiyu (M.: Mir, 1966)
  32. Racah G Phys. Rev. 62 438 (1942); Racah G Phys. Rev. 63 367 (1949); Racah G Phys. Rev. 76 1352 (1949)
  33. Nikitin A.A., Rudzikas Z.B. Osnovy Teorii Spektrov Atomov i Ionov (M.: Nauka, 1983)
  34. Sobel’man I.I. Vvedenie v Teoriyu Atomnykh Spektrov (M.: Fizmatgiz, 1963)
  35. Fedorov F.I. Optika Aktivnykh Sred (Minsk: Izd-vo AN BSSR, 1958)
  36. Perina Ya. Kogerentnost’ Sveta (M.: Mir, 1974)
  37. O’Neil E. Vvedenie v Statisticheskuyu Optiku (M.: Mir, 1966)
  38. Dzherrard A., Berch Dzh.M. Vvedenie v Matrichnuyu Optiku (M.: Mir, 1978)
  39. Meitlend A., Dann M. Vvedenie v Fiziku Lazerov (M.: Nauka, 1978)
  40. Vasgu N., Cadilhac M. Phys. Rev. A 23 2533 (1981)
  41. Kogelnic H Bell. Syst. Tech. J. 44 455 (1965)
  42. Sudarshan E.C.G., Simon R., Mucunda N. Phys. Rev. A 28 2921 and 2933 (1983)
  43. Vol’f K.B., Man’ko V.I. Tr. FIAN 176 96 (1986)
  44. Roman P Nuovo Cimento 13 974 (1959)
  45. Vul’fson K.S. Usp. Fiz. Nauk 152 667 (1987); Vul’fson K S Sov. Phys. Usp. 30 724 (1985)
  46. Baracat R J. Opt. Soc. Am. 53 317 (1963)
  47. Potekhin V.K., Shelepin L.A. Tr. FIAN 124 114 (1980); Pomekhin V K and L A Shelepin Tr. FIAN 173 173 (1986)
  48. Andreev A.V., Emel’yanov V.I., Il’inskii Yu.A. Kooperativnye Yavleniya v Optike (M.: Nauka, 1988)
  49. Shelepin L.A. Zh. Eksp. Teor. Fiz. 54 1463 (1968)
  50. Makhviladze T.M., Shelepin L.A. Tr. FIAN 70 120 (1973)
  51. Klauder Dzh., Sudarshan E. Osnovy Kvantovoi Optiki (M.: Mir, 1970)
  52. Akhmanov S.A., D’yakov Yu.B., Chirkin A.S. Vvedenie v Statisticheskuyu Radiofiziku i Optiku (M.: Nauka, 1981)
  53. Hanbury Brown R., Twiss R.Q. Nature 177 27 (1956); Hanbury Brown R., Twiss R.Q. Nature 178 1046; Hanbury Brown R., Twiss R.Q. Proc. R. Soc. London A 242 300 (1957); Hanbury Brown R., Twiss R.Q. Proc. R. Soc. London A 243 291 (1957)
  54. Arekki F., Skalli M., Khaken G., Veidlikh V. Kvantovye Fluktuatsii Izlucheniya Lazera (M.: Mir, 1974)
  55. Glauber R. Kvantovaya Optika i Kvantovaya Radiofizika (M.: Mir, 1966) p. 91
  56. Smirnov D.F., Troshin A.S. Usp. Fiz. Nauk 153 233 (1987); Smirnov D F, Troshin A S Sov. Phys. Usp. 30 851 (1987)
  57. Taish M.K., Sale B.E.A. Usp. Fiz. Nauk 161 (4) 171 (1989)
  58. Bykov V.P. Usp. Fiz. Nauk 161 (10) 151 (1991); Bykov V P Sov. Phys. Usp. 34 910 (1991)
  59. Yuu F.T.S. Vvedenie v Teoriyu Difraktsii, Obrabotku Informatsii i Golografiyu (M.: Sov. radio, 1979)
  60. Bell R.Dzh. Vvedenie v Fur’e-spektroskopiyu (M.: Mir, 1975)
  61. Sklyanin E K Funct. Anal. Appl. 16 262 (1982)
  62. Kulish P.P., Reshetikhin N.Y. J. Sov. Math. 23 2435 (1983)
  63. Drinfel’d V.G. Dokl. Akad. Nauk SSSR 32 254 (1985)
  64. Moore G., Seiberg N. Commun. Math. Phys. 123 177 (1989)
  65. Jimbo M Int. J. Mod. Phys. A 4 3759 (1989)
  66. Heine E Grelle’s J. F. D. M. 34 285 (1847)
  67. Jacobi C G J Grelle’s J. F. D. M. 34 197 (1846)
  68. Thomae J J. Reine Angew. Math. 70 258 (1869)
  69. Symmetries In Science V (New York: Plenum Press, 1991)
  70. Gel’fond A.O. Ischislenie Konechnykh Raznostei (M.: Nauka, 1967)
  71. Rikhtmaier R., Morton K. Raznostnye Metody Resheniya Kraevykh Zadachs (M.: Mir, 1972)
  72. Samarskii A.A., Karamzin Yu.N. Raznostnye Uravneniya (M.: Nauka, 1978)
  73. Karasev V.P., Shelepin L.A. Tr. FIAN 87 55 (1976)
  74. Thomae J Z. Math. Phys. 16 428 (1871)
  75. Suslov C.K. "K teorii raznostnykh analogov gipergeometricheskikh funktsii na ravnomernykh setkakh" Preprint IAE—4445/1 (M., 1987)
  76. Gel’fand I.M., MinlosR.A., Shapiro Z.Ya. Predstavleniya Gruppy Vrashchenii i Gruppy Lorentsa (M.: Fizmatgiz, 1958)
  77. Kharmut Kh. Primenenie Metodov Teorii Informatsii v Fizike (M.: Mir, 1989)
  78. Khemming R.V. Teoriya Informatsii i Teoriya Kodirovaniya (M.: Radio i svyaz’, 1983)
  79. Koblits I. R-adicheskie Chisla, R-adicheskii Analiz i Dzeta-funktsiya. (M.: Mir, 1982)
  80. Nomura M J. Phys. Soc. Jpn. 57 3653 (1988); Nomura M J. Phys. Soc. Jpn. 58 2694 (1989); Nomura M J. Phys. Soc. Jpn. 59 439 (1990); Nomura M J. Phys. Soc. Jpn. 59 1954 (1990); Nomura M J. Phys. Soc. Jpn. 59 2345 (1990); Nomura M J. Phys. Soc. Jpn. 60 710 (1991); Nomura M J. Phys. Soc. Jpn. 60 1906 (1991)
  81. Bidenharn L C J. Phys. A 22 873 (1989)
  82. Kachurik I.I., Klimyk A.U. J. Phys. A 23 2717 (1990)
  83. Floreanini R., Vinet L.t Preprint Université de Montreal UdeM-LPN-TH-54 (1991); Floreanini R., Vinet L. J. Phys. A 23 1019 (1990)
  84. Smirnov Yu.F., Tolstoi V.N., Kharitonov Yu.I. Yad. Fiz. 53 959 (1991)
  85. Klimyk A.U., Smirnov Y.F., Gruber V. Symmetries In Science V (New York: Plenum Press, 1991) p. 341
  86. Ross M Commun. Math. Phys. 117 581 (1988)
  87. Kachurik I.I. Preprint ITF-90-37R (Kiev, 1990)
  88. Nomura M J. Math. Phys. 30 2397 (1989)
  89. Alvares/Gamme L., Gomes C., Sierre G. Phys. Lett. B 220 142 (1989)
  90. Zhilinskii B.N., Smirnov Yu.F. Yad. Fiz. 54 17 (1991)
  91. Koh I.G., Zhong Qi Ma. Preprint IC/89/259 (Trieste, 1989); Koh I.G., Zhong Qi Ma. Phys. Lett. B 234 480 (1990)
  92. Beitmen G., Erdeii A. Vysshie Transtsendentnye Funktsii Vol. 1-3 (M.: Nauka, 1973)
  93. Endryus G. Teoriya Razbienii (M.: Nauka, 1982)
  94. Karasev V.P. Kratk. Soobshch. Fiz. (9) 31 (1991)
  95. Fateev V.A., Lukianov S.L. Int. J. Mod. Phys. A 3 507 (1988)
  96. Zamolodchikov A.B. Teor. Mat. Fiz. 65 347 (1985)
  97. Rocek M Phys. Lett. B 255 554 (1991)
  98. Basov N.G., Prokhorov A.M. Zh. Eksp. Teor. Fiz. 27 431 (1954)
  99. Gordon J. P., Zeiger H.J., Townes C.H. Phys. Rev. 95 282 (1954)
  100. Schrödinger E Naturwissenschaften 14 664 (1926)
  101. Malkin I.A., Man’ko V.I. Dinamicheskie Simmetrii i Kogerentnye Sostoyaniya Kvantovykh Sistem (M.: Nauka, 1979)
  102. Man’ko V.I. (Red.) Kogerentnye Sostoyaniya v Kvantovoi Teorii (M.: Mir, 1972)
  103. Redcliffe J M J. Phys. A 4 313 (1971)
  104. Perelemov A M Comm. Math. Phys. 26 222 (1972)
  105. Perelomov AM. Yad. Fiz. 29 1688 (1979)
  106. Dodonov V.V., Man’ko V.I.o Tr. FIAN 183 5 (1987)
  107. Delbourgo R J. Phys. A 10 1837 (1977)
  108. Delbourgo R J. Phys. A 10 233L (1977)
  109. Gitman D.M., Shelepin A.L. "Symmetries and Algebraic Structure in Physics" Proc. 18th Int. Coll. on Group-Theoretical Methods in Physics (New York: Nova Science, 1991) p. 251
  110. Berezin F.A. Metod Vtorichnogo Kvantovaniya (M.: Nauka, 1986)
  111. Klauder J.R., Skagerstam B. S. Coherent States, Application In Physics And Mathematical Physics (Singapore: World Sci. Publ. Co., 1985)
  112. Lieb E M Comm. Math. Phys. 31 2211 (1973)
  113. Arecchi F.T., Courteus E., Gilmor R., Thomas H. Phys. Rev. 6 2211 (1972)
  114. Naimark M.A. Lineinye Predstavleniya Gruppy Lorentsa (M.: Fizmatgiz, 1958)
  115. Zhelobenko D.P. Kompaktnye Gruppy Li i Ikh Predstavleniyas (M.: Nauka, 1970)
  116. Schmitt H.A., Multi A. J. Phys. A 23 861L (1990)
  117. Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i Ryady. Spetsial’nye Funktsii] (M.: Nauka, 1983); Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i Ryady. Dopolnitel’nye Glavy (M.: Nauka, 1986)
  118. Karasev V.P. Kratk. Soobshch. Fiz. (3) 32 (1976)
  119. Kurchan J, P Leboeuf, and M Saraceno Phys. Rev. A 40 6800 (1989)
  120. Ariano G.D., Rasetti M., Vadacchino M. J. Phys. A 18 1295 (1985)
  121. Berezin F.A., Shubin M.A. Uravnenie Shredingera (M.: Izd-vo MGU, 1983)
  122. Berezin F A Commun. Math. Phys. 40 153 (1975)
  123. Berezin F.A. Usp. Fiz. Nauk 132 497 (1980); Berezin F A Sov. Phys. Usp. 23 763 (1980)
  124. Simon B Commun. Math. Phys. 71 247 (1980)
  125. Yaffe L G Rev. Mod. Phys. 54 1992 (1982)
  126. Bellisard J., Holtz R. J. Math. Phys. 15 1275 (1974)
  127. Shelepin A.L., Shelepin L.A. "Symmetries and Algebraic Structure in Physics" Proc. 18th Int. Coll. on Group Theoretical Methods in Physics (New York: Nova Science, 1991) p. 279
  128. Karasev V.P. Kratk. Soobshch. Fiz. (3) 32 (1976)
  129. Karasev V.P., Karasev P.P., San’ko V.A., Shelepin L.A. Tr. FIAN 106 119 (1979)
  130. Kuznetsov G.I., Smorodinskii Ya.A. Yad. Fiz. 21 1135 (1975)
  131. Regge T Nuovo Cimento 10 544 (1958)
  132. Shelepin L A Nucl. Phys. B 2 608 (1967)
  133. Ponzano G., Regge T. Spectroscopy And Group Theoretical Methods In Physics (Amsterdam: North-Holland, 1968) p. 1
  134. Maslov V.P. Dokl. Akad. Nauk SSSR 192 (2) 272 (1970)
  135. Maslov V.P. Kompleksnye Markovskie Tsepi i Kontinual’nyi Integral Feinmana (M.: Nauka, 1976)
  136. Zagidullin M.V., Shelepin L.A. Tr. FIAN 106 154 (1979)
  137. Shelepin A.L., Shelepin L.A. Tr. FIAN 173 142 (1986)
  138. Spravochnik Po Teorii Veroyatnostei i Matematicheskoi Statistike (M.: Nauka, 1985)
  139. Pawula R F Trans. IEEE IT-13 33 (1967)
  140. Tikhonov V.I., Mironov M.A. Markovskie Protsessy (M.: Sov. radio, 1977)
  141. Ohlson J E Trans. IEEE SOM-22 43 (1973)
  142. Feller V. Vvedenie v Teoriyu Veroyatnostei i Ee Prilozheniya Vol. 1, 2 (M.: Nauka, 1967)
  143. Gnedenko B.V. Kurs Teorii Veroyatnostei (M.: Gostekhizdat, 1954)
  144. Dynkin E.B. Osnovaniya Teorii Markovskikh Protsessov (M.: Fizmatgiz, 1959)
  145. Gikhman I.I., Skorokhod A.V. Vvedenie v Teoriyu Sluchainykh Protsessov (M.: Nauka, 1977)
  146. Shapiro I.S. Dokl. Akad. Nauk SSSR 106 647 (1956)
  147. Popov B.C. Zh. Eksp. Teor. Fiz. 37 1116 (1959)
  148. Fok V.A. Raboty Po Kvantovoi Teorii Polya (L.: Izd-vo LGU, 1957)
  149. Nikiforov A.F., Uvarov V.B. Spetsial’nye Funktsii Matematicheskoi Fiziki (M.: Nauka, 1986)
  150. Miller U. Simmetrii i Razdelenie Peremennykh (M.: Mir, 1981)
  151. Talman J Special Functions: A Group Theoretical Approach (New York: W.A. Benjamin, 1968)
  152. Nikiforov A.F., Suslov S.K., Uvarov V.B. Klassicheskie Ortogonal’nye Polinomy Diskretnoi Peremennoi (M.: Nauka, 1985)
  153. Smorodinskii Ya.A., Suslov S.K. Yad. Fiz. 35 192 (1982); Smorodinskii Ya.A., Suslov S.K. Yad. Fiz. 36 1066 (1982)
  154. Askey R., Wilson LA. SIAMJ. Math. Anal. 10 1008 (1979)
  155. Wilson J A SIAMJ. Math. Anal. 11 690 (1980)
  156. Hahn W Math. Nachr. 2 (1) 4 (1949); Hahn W Math. Nachr. 2 (5) 263 (1949); Hahn W Math. Nachr. 2 (6) 340 (1949)
  157. Karlin S., McGregor J.L. Scripta Math. 26 33 (1961)
  158. Suslov S.K. Teor. Mat. Fiz. 88 66 (1991)
  159. Gel’fand I.M. Dokl. Akad. Nauk SSSR 288 (1) 14 (1986)
  160. Gel’fand I.M., Zelevinskii A.V. Funkts. Analiz Ego Pril. 20 (3) 17 (1986)
  161. Rozenblyum A.V., Rozenblyum L.V. Dif. Uravn. 18 1787 (1982)
  162. Wybourn B.G., Bowick M.J. Austral. J. Phys. 30 (3) 259 (1977)
  163. Wybourne B G Austral. J. Phys. 32 (5) 417 (1979)
  164. Elliot Dzh., Dober P. Simmetriya v Fizike Vol. 1, 2 (M.: Mir, 1983)
  165. Shelepin A.L., Shelepin L.A. Kratk. Soobshch. Fiz. (7) 20 (1989)
  166. Gantmakher F.R. Mat. Sb.. 5 217 (1939)
  167. Barut A.O., Ruczka R. Proc. R. Soc. London 287 519 (1965)
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