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arXiv:quant-ph/0202104 · Journal of Physics A Mathematical and General

Relativistically invariant extension of the de Broglie Bohm theory of quantum mechanics

Christopher Dewdney, George Horton

Published Nov 13, 2002• 13 citations
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Abstract

We show that quantum mechanics can be given a Lorentz-invariant realistic interpretation by applying our recently proposed relativistic extension of the de Broglie–Bohm theory to deduce non-locally correlated, Lorentz-invariant individual particle motions for the Einstein–Podolsky–Rosen experiment and the double-interferometer experiment proposed by Horne, Shimony and Zeilinger.

DOI
10.1088/0305-4470/35/47/311

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We show that quantum mechanics can be given a Lorentz-invariant realistic interpretation by applying our recently proposed relativistic extension of the de Broglie–Bohm theory to deduce non-locally correlated, Lorentz-invariant individual particle motions for the Einstein–Podolsky–Rosen experiment and the double-interferometer experiment proposed by Horne, Shimony and Zeilinger.
We demonstrate how to construct a lorentz-invariant, hidden-variable interpretation of relativistic quantum mechanics based on particle trajectories. The covariant theory that we propose employs a multi-time formalism and a lorentz-invariant rule for the coordination of the space-time points on the individual particle trajectories. In this way we show that there is no contradiction between nonlocality and lorentz invariance in quantum mechanics. The approach is illustrated for relativistic bosons, using a simple model to discuss the individual non-locally correlated particle motion which ensues when the wavefunction is entangled. A simple example of measurement is described.
We demonstrate how to construct a Lorentz-invariant, hidden-variable interpretation of relativistic quantum mechanics based on particle trajectories. The covariant theory that we propose employs a multi-time formalism and a Lorentz-invariant rule for the coordination of the spacetime points on the individual particle trajectories. In this way we show that there is no contradiction between non-locality and Lorentz-invariance in quantum mechanics. The approach is illustrated for relativistic bosons, using a simple model to discuss the individual non-locally correlated particle motion which ensues when the wavefunction is entangled. A simple example of measurement is described.
Abstract David Bohm published his “Suggested Interpretation of Quantum Theory in Terms of Hidden Variables” some twenty five years after Louis de Broglie first presented his similar Pilot Wave theory of quantum mechanics. In the following 30 years what became known as the de Broglie–Bohm approach to quantum theory was to a large extent ignored within the physics community. Even David Bohm himself became somewhat disillusioned with the lack of impact of his interpretation of quantum theory and he directed his interest elsewhere. But some 27 years after Bohm had published his interpretation of quantum theory, interest was rekindled in part by new, detailed calculations that demonstrated clearly and graphically, exactly how his interpretation explained quantum phenomena in terms of well defined individual particle trajectories. These computations encompassed two-slit interference, quantum tunnelling, neutron interferometry, Wheeler’s delayed choice experiment, orbital and intrinsic angular momentum, quantum measurement and Einstein–Podolsky–Rosen nonlocal correlations for orbital angular momentum, intrinsic angular momentum and correlated particle interferometry. Since then, the acceptance of the validity of de Broglie–Bohm theory has steadily grown, as has the interest in the consequences of the approach. For my contribution to the current celebratory volume I was asked to provide a personal review specifically of this novel work within its historical context of the last quarter of the twentieth century.
The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation contrasts with the Copenhagen orthodox interpretation, which conveys uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear transforms in Bohmian mechanics for the momentum observable. While these nonlinear operator-like transforms cannot be easily represented as matrices in the Hilbert space, we show that they can be utilized in the de Broglie-Bohm interpretations of quantum theory.