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Stochastic quantum mechanics (or the stochastic interpretation) is an interpretation of quantum mechanics.

The modern application of stochastics to quantum mechanics involves the assumption of spacetime stochasticity, the idea that the small-scale structure of spacetime is undergoing both metric and topological fluctuations (John Archibald Wheeler's "quantum foam"), and that the averaged result of these fluctuations recreates a more conventional-looking metric at larger scales that can be described using classical physics, along with an element of nonlocality that can be described using quantum mechanics. A stochastic interpretation of quantum mechanics is due to persistent vacuum fluctuation. The main idea is that vacuum or spacetime fluctuations are the reason for quantum mechanics and not a result of it as it is usually considered.

Stochastic mechanics

The first relatively coherent stochastic theory of quantum mechanics was put forward by Hungarian physicist Imre Fényes[1] who was able to show the Schrödinger equation could be understood as a kind of diffusion equation for a Markov process.[2][3]

Louis de Broglie[4] felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another.[5] Perhaps the most widely known theory where quantum mechanics is assumed to describe an inherently stochastic process was put forward by Edward Nelson[6] and is called stochastic mechanics. This was also developed by Davidson, Guerra, Ruggiero and others.[7]
Stochastic electrodynamics
Main article: Stochastic electrodynamics

Stochastic quantum mechanics can be applied to the field of electrodynamics and is called stochastic electrodynamics (SED).[8] SED differs profoundly from quantum electrodynamics (QED) but is nevertheless able to account for some vacuum-electrodynamical effects within a fully classical framework.[9] In classical electrodynamics it is assumed there are no fields in the absence of any sources, while SED assumes that there is always a constantly fluctuating classical field due to zero-point energy. As long as the field satisfies the Maxwell equations there is no a priori inconsistency with this assumption.[10] Since Trevor W. Marshall[11] originally proposed the idea it has been of considerable interest to a small but active group of researchers.[12]

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Bell's theorem
De Broglie–Bohm theory
Diffusion equation
Hidden variable theory
Quantum foam
Quantum nonlocality
Interpretations of quantum mechanics
Scale relativity
Stochastic process
Stochastic electrodynamics
Zero-point energy

References
Notes

See I. Fényes (1946, 1952)
Davidson (1979), p. 1
de la Peña & Cetto (1996), p. 36
de Broglie (1967)
de la Peña & Cetto (1996), p. 36
See E. Nelson (1966, 1985, 1986)
de la Peña & Cetto (1996), p. 36
de la Peña & Cetto (1996), p. 65
Milonni (1994), p. 128
Milonni (1994), p. 290
See T. W. Marshall (1963, 1965)

Milonni (1994), p. 129

Papers

de Broglie, L. (1967). "Le Mouvement Brownien d'une Particule Dans Son Onde". C. R. Acad. Sci. B264: 1041.
Davidson, M. P. (1979). "The Origin of the Algebra of Quantum Operators in the Stochastic Formulation of Quantum Mechanics". Letters in Mathematical Physics. 3 (5): 367–376. arXiv:quant-ph/0112099. Bibcode:1979LMaPh...3..367D. doi:10.1007/BF00397209. ISSN 0377-9017. S2CID 6416365.
Fényes, I. (1946). "A Deduction of Schrödinger Equation". Acta Bolyaiana. 1 (5): ch. 2.
Fényes, I. (1952). "Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik". Zeitschrift für Physik. 132 (1): 81–106. Bibcode:1952ZPhy..132...81F. doi:10.1007/BF01338578. ISSN 1434-6001. S2CID 119581427.
Marshall, T. W. (1963). "Random Electrodynamics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 276 (1367): 475–491. Bibcode:1963RSPSA.276..475M. doi:10.1098/rspa.1963.0220. ISSN 1364-5021. S2CID 202575160.
Marshall, T. W. (1965). "Statistical Electrodynamics". Mathematical Proceedings of the Cambridge Philosophical Society. 61 (2): 537–546. Bibcode:1965PCPS...61..537M. doi:10.1017/S0305004100004114. ISSN 0305-0041.
Lindgren, J.; Liukkonen, J. (2019). "Quantum Mechanics can be understood through stochastic optimization on spacetimes". Scientific Reports. 9 (1): 19984. Bibcode:2019NatSR...919984L. doi:10.1038/s41598-019-56357-3. PMC 6934697. PMID 31882809.
Nelson, E. (1966). Dynamical Theories of Brownian Motion. Princeton: Princeton University Press. OCLC 25799122.
Nelson, E. (1985). Quantum Fluctuations. Princeton: Princeton University Press. ISBN 0-691-08378-9. LCCN 84026449. OCLC 11549759.
Nelson, E. (1986). "Field Theory and the Future of Stochastic Mechanics". In Albeverio, S.; Casati, G.; Merlini, D. (eds.). Stochastic Processes in Classical and Quantum Systems. Lecture Notes in Physics. 262. Berlin: Springer-Verlag. pp. 438–469. doi:10.1007/3-540-17166-5. ISBN 978-3-662-13589-1. OCLC 864657129.

Books

de la Peña, Luis; Cetto, Ana María (1996). van der Merwe, Alwyn (ed.). The Quantum Dice: An Introduction to Stochastic Electrodynamics. Dordrecht; Boston; London: Kluwer Academic Publishers. ISBN 0-7923-3818-9. LCCN 95040168. OCLC 832537438.
Jammer, M. (1974). The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective. New York: Wiley. ISBN 0-471-43958-4. LCCN 74013030. OCLC 613797751.
Namsrai, K. (1985). Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics. Dordrecht; Boston: D. Reidel Publishing Co. doi:10.1007/978-94-009-4518-0. ISBN 90-277-2001-0. LCCN 85025617. OCLC 12809936.
Milonni, Peter W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Boston: Academic Press. ISBN 0-12-498080-5. LCCN 93029780. OCLC 422797902.

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