Time scales for dynamical relaxation to the Born rule

被引:47
作者
Towler, M. D. [1 ,2 ,3 ]
Russell, N. J. [1 ]
Valentini, Antony [4 ]
机构
[1] Univ Cambridge, Condensed Matter Theory Grp, Cavendish Lab, Cambridge CB3 OHE, England
[2] UCL, Dept Phys & Astron, London WC1E 6BT, England
[3] Apuan Alps Ctr Phys, I-55020 Tuscany, LU, Italy
[4] Clemson Univ, Dept Phys & Astron, Kinard Lab 303, Clemson, SC 29631 USA
来源
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 2012年 / 468卷 / 2140期
关键词
de Broglie-Bohm theory; relaxation to quantum equilibrium; Born rule; SIGNAL-LOCALITY; SUGGESTED INTERPRETATION; QUANTUM-THEORY; UNCERTAINTY; TERMS;
D O I
10.1098/rspa.2011.0598
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie-Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie-Bohm ontology, the box contains an objectively existing 'pilot wave' which guides the electron trajectory, and this is represented mathematically by a Schrodinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to 'quantum equilibrium'. The proximity to equilibrium is characterized by the coarse-grained subquantum H-function which is found to decrease roughly exponentially towards zero over the course of time. The time scale tau for this relaxation is calculated for various values of M and the coarse-graining length epsilon. Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, tau proportional to M (1), is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of tau on epsilon is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.
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页码:990 / 1013
页数:24
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