# Stochastic quantum mechanics

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

## Contents

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

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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## References

### Notes

1. See I.Fényes ( 1946 , 1952 )
2. Davidson (1979) , p. 1
3. See E.Nelson ( 1966 , 1985 , 1986 )
4. Milonni (1994) , p. 128
5. Milonni (1994) , p. 290
6. See T. W.Marshall ( 1963 , 1965 )
7. Milonni (1994) , p. 129

### Papers

• de Broglie, L. (1967). "Le Mouvement Brownien d'une Particule Dans Son Onde". C. R. Acad. Sci. B264: 1041.CS1 maint: ref=harv (link)
• 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:. Bibcode:1979LMaPh...3..367D. doi:10.1007/BF00397209. ISSN   0377-9017. S2CID   6416365.CS1 maint: ref=harv (link)
• Fényes, I. (1946). "A Deduction of Schrödinger Equation". Acta Bolyaiana. 1 (5): ch. 2.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• Lindgren, J.; Liukkonen, J. (2019). "Quantum Mechanics can be understood through stochastic optimization on spacetimes". Scientific Reports. 9 (1): 19984. Bibcode:2019NatSR...919984L. doi:. PMC  . PMID   31882809.
• Nelson, E. (1966). Dynamical Theories of Brownian Motion. Princeton: Princeton University Press. OCLC   25799122.CS1 maint: ref=harv (link)
• Nelson, E. (1985). Quantum Fluctuations. Princeton: Princeton University Press. ISBN   0-691-08378-9. LCCN   84026449. OCLC   11549759.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)