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Cited by 226Quantum Mechanics and ApplicationsQuantum chaos and dynamical systemsStatistical Mechanics and Entropy

Dynamical origin of quantum probabilities

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Valentini, Antony, Westman, Hans·2005·Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
DOI 10.1098/rspa.2004.1394
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Abstract

We study the origin of the Born probability rule ρ = | ψ | 2 in the de Broglie–Bohm pilot–wave formulation of quantum theory. It is argued that quantum probabilities arise dynamically, and have a status similar to thermal probabilities in ordinary statistical mechanics. This is illustrated by numerical simulations for a two–dimensional system. We show that a simple initial ensemble, with a non–standard distribution ρ ≠ | ψ | 2 of particle positions, evolves towards the quantum distribution to high accuracy. The relaxation process ρ →| ψ | 2 is quantified in terms of a coarse–grained H –function (equal to minus the relative entropy of ρ with respect to | ψ | 2 ), which is found to decrease approximately exponentially over time, with a time constant that accords with a simple theoretical estimate.

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  1. [10][10] A. Valentini, Pilot-Wave Theory of Physics and Cosmology (Cambridge University Press, Cambridge, forthcoming). -
  2. [42][42] [A. Valentini, quant-ph/0309107.](http://arxiv.org/abs/quant-ph/0309107) - · arXiv
  3. [44][44] D. Deutsch, Proc. Roy. Soc. Lond. A 455 , 3129 (1999). -
  4. [45][45] D. Wallace, quant-ph/0211104; quant-ph/0303050; quant-ph/0312157. - · arXiv
  5. [46][46] H. Barnum, C.M. Caves, J. Finkelstein, C.A. Fuchs and R. Schack, Proc. Roy. Soc. Lond. A 456 , 1175 (2000). -
  6. [50][50] S. Saunders, Proc. Roy. Soc. Lond. A 460 , 1 (2004). -
  7. [52][52] L. Hardy, quant-ph/0101012; L. Hardy, in: Non-locality and Modality , eds. T. Placek and J. Butterfield (Kluwer, Dordrecht, 2002). - · arXiv

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Authors
Valentini, A., & Westman, H.
Affiliations
  • Augustus College, 14 Augustus Road, London SW19 6LN, UK
  • Chalmers University of Technology (SE)
Corresponding author
Valentini, Antony
ORCID
Valentini, Antony
Venue
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Publisher
The Royal Society
Published
2005-01-08
Volume
461
Issue
2053
Pages
253-272
ISSN
  • 1364-5021 (ISSN-L)
  • 1471-2946
DOI
10.1098/rspa.2004.1394
OpenAlex
W2130195092
Cited by
226
References
62
Citations by year
2026: 3 · 2025: 8 · 2024: 4 · 2023: 14 · 2022: 10 · 2021: 11 · 2020: 12 · 2019: 36 · 2018: 9 · 2017: 7 · 2016: 8 · 2015: 12 · 2014: 10 · 2013: 4 · 2012: 18
FWCI
3.48
Type
journal article
Language
en
Open access
Licence
royalsociety.org/journals/ethics-policies/data-sharing-mining/ (tdm)
Locations
  • Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
Keywords
Born rule, Bohmian mechanics, quantum probability, H-function, Kullback-Leibler divergence, coarse-graining, two-dimensional quantum systems

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