Quantum probability and Hilbert’s sixth problem. (English) Zbl 1470.81008
Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376, No. 2118, Article ID 20180030, 32 p. (2018).
Summary: With the birth of quantum mechanics, the two disciplines that Hilbert proposed to axiomatize, probability and mechanics, became entangled and a new probabilistic model arose in addition to the classical one. Thus, to meet Hilbert’s challenge, an axiomatization should account deductively for the basic features of all three disciplines. This goal was achieved within the framework of quantum probability. The present paper surveys the quantum probabilistic axiomatization.
MSC:
| 81P05 | General and philosophical questions in quantum theory |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 60J99 | Markov processes |
References:
| [1] | 1Browder FE (ed.). 1976 Mathematical developments arising from Hilbert problems, Proc. Symp. Pure Mathematics, vol. XXVIII, part 2. Providence, RI: American Mathematical Society. · Zbl 0326.00002 |
| [2] | 2Gibbs JW. 1902 Elementary principles in statistical mechanics. New Haven, CT: Yale University Press. (Reprinted 2014, Dover, New York.) |
| [3] | 3Einstein A. 1906 Zur Theorie der Brownschen Bewegung. Ann. Physik 324 (2), 371-381. · JFM 37.0814.02 |
| [4] | 4Jordan P. 1927 Über eine neue Begründung der Quantenmechanik. Z. Physik 40, 809-838. (doi:10.1007/BF01390903) · JFM 53.0849.04 · doi:10.1007/BF01390903 |
| [5] | 5Hilbert D, von Neumann J, Nordheim L. 1927 Uber die Grundlagen der Quantenmechanik. Math. Ann. 98, 1-30. (doi:10.1007/BF01451579) · JFM 53.0849.03 · doi:10.1007/BF01451579 |
| [6] | 6Dirac PAM. 1947 The principles of quantum mechanics. Oxford, UK: Oxford University Press. (First published 1930, Clarendon Press, Oxford.) · Zbl 0030.04801 |
| [7] | 7von Neumann J. 1955 Mathematical foundations of quantum mechanics. Princeton, NJ: Princeton University Press. (Originally published in German as Mathematische Grundlagen der Quantenmechanik, 1932, Springer, Berlin.) · Zbl 0064.21503 |
| [8] | 8Kolmogorov AN. 1933 Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin, Germany: Springer. (Translated and reprinted in English as Foundations of the theory of probability, 1956, Chelsea, New York.) · JFM 59.1152.03 |
| [9] | 9Hilbert D. 1930 Grundlagen der Geometrie. Leipzig, Germany: Teubner. (First appeared in press in 1899.) · JFM 56.0483.01 |
| [10] | 10Feynman R. 1948 Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367-385. (doi:10.1103/RevModPhys.20.367) · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367 |
| [11] | 11Segal IE. 1947 Postulates for general quantum mechanics. Ann. Math. 48, 930-948. (doi:10.2307/1969387) · Zbl 0034.06602 · doi:10.2307/1969387 |
| [12] | 12von Neumann J. 1981 Continuous geometries with a transition probability, Memoirs of the American Mathematical Society, no. 252 (ed. I Halperin). Providence, RI: American Mathematical Society. · Zbl 0479.46050 |
| [13] | 13Accardi L, Cecchini C. 1982 Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal. 45, 245-273. (doi:10.1016/0022-1236(82)90022-2) · Zbl 0483.46043 · doi:10.1016/0022-1236(82)90022-2 |
| [14] | 14Accardi L. 1999 Quantum probability: an historical survey. Invited talk to the Annual meeting of the Japan Mathematical Society, Tokyo, 26 March 1999. Japanese translation published by the Japan Mathematical Society. 2000 Invited contribution in Probability on algebraic structures (eds G Budzban, Ph Feinsilver, A Mukherjea). Contemporary Mathematics, vol. 261, pp. 145-159. Providence, RI: American Mathematical Society. |
| [15] | 15Mackey GW. 1976 Mathematical foundations of quantum mechanics. Reading, MA: Addison-Wesley. |
| [16] | 16Varadarajan VS. 1968 Geometry of quantum theory. Princeton, NJ: Van Nostrand. · Zbl 0155.56802 |
| [17] | 17Piron C. 1976 Foundations of quantum physics. Reading, MA: Addison-Wesley. · Zbl 0333.46050 |
| [18] | 18Accardi L. 1985 Non Kolmogorovian probabilistic models and quantum theory. Invited talk at 45th Int. Statistical Institute Conf., Amsterdam, August, with discussion by JS Watson. Abstract in Bull. l’Inst. Int. Stat. 51(4). Re-elaborated in Accardi L, Reviglio E. 1994 Quantum probability and the interpretation of quantum theory. Volterra Preprint no. 166. |
| [19] | 19Feynmann RP. 1967 The character of physical law. Cambridge, MA: MIT Press. |
| [20] | 20Bell JS. 1964 On the Einstein Podolsky Rosen paradox. Physics 1, 195-200. |
| [21] | 21Accardi L, Fedullo A. 1982 On the statistical meaning of complex numbers in quantum mechanics. Lett. Nuovo Cimento 34, 161-172. (doi:10.1007/BF02817051) · doi:10.1007/BF02817051 |
| [22] | 22Accardi L. 1984 Some trends and problems in quantum probability. In Quantum probability and applications to the quantum theory of irreversible processes (eds L Accardi, A Frigerio, V Gorini). Lecture Notes in Mathematics, vol. 1055, pp. 1-19. Berlin, Germany: Springer. · Zbl 0544.60003 |
| [23] | 23Accardi L. 1984 The probabilistic roots of the quantum mechanical paradoxes. In The wave-particle dualism (eds S Diner, D Fargue, G Lochak, F Selleri), pp. 297-330. Dordrecht, The Netherlands: Reidel. |
| [24] | 24Accardi L. 1988 Foundations of quantum mechanics: a quantum probabilistic approach. In The nature of quantum paradoxes (eds G Tarozzi, A van der Merwe), pp. 257-323. Dordrecht, The Netherlands: Reidel. |
| [25] | 25Accardi L. 1993 On the axioms of probability. In Proc. Conf. on Foundations of Quantum Mechanics, Lecce, Italy. Volterra Preprint no. 194 (1994). |
| [26] | 26Accardi L. 1994 Einstein-Bohr: one all. Acta Enciclopedica, Istituto dell’Enciclopedia Italiana. Volterra Preprint no. 174 (1993). |
| [27] | 27Accardi L, Khrennikov A, Ohya M. 2009 Quantum Markov model for data from Shafir-Tversky experiments in cognitive psychology. Open Syst. Info. Dyn. 16, 371-385. (doi:10.1142/S123016120900027X) · Zbl 1182.81051 · doi:10.1142/S123016120900027X |
| [28] | 28Asano M, Basieva I, Khrennikov A, Ohya M, Tanaka Y, Yamato I. 2012 Quantum-like model for the adaptive dynamics of the genetic regulation of E. coli’s metabolism of glucose/lactose. Syst. Synth. Biol. 6, 1-7. (doi:10.1007/s11693-012-9091-1) · doi:10.1007/s11693-012-9091-1 |
| [29] | 29Gnedenko BV. 1950 Theory of probability. Moscow, Russia: Nauka. |
| [30] | 30Schwinger J 1970 Quantum kinematics and dynamics. New York, NY: Academic Press. · Zbl 1094.81500 |
| [31] | 31Accardi L. 1995 Can mathematics help solving the interpretational problems of quantum theory? Nuovo Cimento B 110, 685-721. (doi:10.1007/BF02741475) · doi:10.1007/BF02741475 |
| [32] | 32Feynmann RP, Leighton RP, Sands M. 1966 Lectures on Physics, vol. III. Reading, MA: Addison-Wesley. |
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