Abstract
We show that the mathematical formalism of the quantum statistical model can be interpreted as a method for approximation of classical (measure-theoretic) averages on the infinite-dimensional phase space. The technique of approximation is based on the Taylor expansion of functionals of classical fields. To find the order of the deviation of quantum statistical predictions from the classical predictions, we use the time-scaling arguments. We show that quantum randomness might be considered as the result of random fluctuations at the Planck time-scale.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boyer, T.H.: A brief survey of stochastic electrodynamics. In: Barut, A.O. (ed.) Foundations of Radiation Theory and Quantum Electrodynamics, pp. 124–148. Plenum, New York (1980)
Daletski, A.L., Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Spaces. Kluwer, Dordrecht (1991)
Davidson, M.: Model for the stochastic origins of Schrödinger equation. J. Math. Phys. 20, 1865–1879 (1979)
Davidson, M.: A dynamical theory of Markovian diffusion. Physica A 96, 465–483 (1979)
Dirac, P.A.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1930)
De la Pena, L., Cetto, A.M.: The Quantum Dice: An Introduction to Stochastic Electrodynamics. Kluwer, Dordrecht (1996)
Einstein, A.: The Collected Papers of Albert Einstein. Princeton University Press, Princeton (1993)
Holevo, A.S.: Statistical Structure of Quantum Theory. Springer, Berlin (2001)
Khrennikov, A.Yu.: Stochastic integrals in locally convex spaces. Usp. Mat. Nauk 37, 161–162 (1982)
Khrennikov, A.Yu.: Ito’s formula in nuclear Frechet space. Mosc. Univ. Math. Bull. 39, 85–89 (1984)
Khrennikov, A.Yu.: An existence theorem for the solution of a stochastic differential equation in nuclear Frechet space. Theory Probab. Appl. 1, 85–89 (1985)
Khrennikov, A.Yu.: A pre-quantum classical statistical model with infinite-dimensional phase space. J. Phys. A: Math. Gen. 38, 9051–9073 (2005)
Khrennikov, A.Yu.: Generalizations of quantum mechanics induced by classical statistical field theory. Found. Phys. Lett. 18, 637–650 (2006)
Khrennikov, A.Yu.: Nonlinear Schrödinger equations from prequantum classical statistical field theory. Phys. Lett. A 357(3), 171–176 (2006)
Khrennikov, A.Yu.: Quantum mechanics as an asymptotic projection of statistical mechanics of classical fields. In: Adenier, G., Khrennikov, A.Yu., Nieuwenhuizen, Th.M. (eds.) Quantum Theory: Reconsideration of Foundations—3. AIP Conference Proceedings, vol. 810, pp. 179–197. AIP, New York (2006)
Nelson, E.: Quantum Fluctuation. Princeton University Press, Princeton (1985)
Planck, M.: Über irreversible Strahlungsvorgänge. Sitzungsber. Preuß. Akad. Wiss. 5, 479 (1899)
Shiryaev, A.N.: Essentials of Stochastic Finance: Facts, Models, Theory. WSP, Singapore (1999)
’t Hooft, G.: Quantum mechanics and determinism. hep-th/0105105 (2001)
’t Hooft, G.: Determinism beneath quantum mechanics. quant-ph/0212095 (2002)
Ventzel, E.: Theory of Probability. Fizmatlit, Moscow (1958)
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khrennikov, A. Quantum Randomness as a Result of Random Fluctuations at the Planck Time Scale?. Int J Theor Phys 47, 114–124 (2008). https://doi.org/10.1007/s10773-007-9528-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-007-9528-6