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arXiv:1405.3822

Relativistic Bohmain Mechanics

Jafar Khodagholizadeh, Javad Kazemi, Alireza Babazadeh

Published May 15, 2014• 1 citation
arXivHTMLPDF

Abstract

In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis $(ρ=|ψ|^{2})$ and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.

DOI
10.48550/arxiv.1405.3822

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In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis (ρ=∣ψ∣2)(ρ=|ψ|^{2})(ρ=∣ψ∣2) and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
Bohmian mechanics also known as de Broglie-Bohm theory is the most popular alternative approach to quantum mechanics. Whereas the standard interpretation of quantum mechanics is based on the complementarity principle Bohmian mechanics assumes that both particle and wave are concrete physical objects. In 1993 Peter Holland has written an ardent account on the plausibility of the de Broglie-Bohm theory. He proved that it fully reproduces quantum mechanics if the initial particle distribution is consistent with a solution of the Schrödinger equation. Which may be the reasons that Bohmian mechanics has not yet found global acceptance? In this article it will be shown that predicted properties of atoms and molecules are in conflict with experimental findings. Moreover it will be demonstrated that repeatedly published ensembles of trajectories illustrating double slit diffraction processes do not agree with quantum mechanics. The credibility of a theory is undermined when recognizably wrong data presented frequently over years are finally not declared obsolete.
Abstract. I present a relativistic covariant version of the Bohmian interpretation of quantum mechanics and discuss the corresponding measurable predictions. The covariance is incoded in the fact that the nonlocal quantum potential transforms as a scalar, which is a consequence of the fact that the nonlocal wave function transforms as a scalar. The measurable predictions that can be obtained with the deterministic Bohmian interpretation cannot be obtained with the conventional interpretation simply because the conventional probabilistic interpretation does not work in the case of relativistic quantum mechanics.
I present a relativistic covariant version of the Bohmian interpretation of quantum mechanics and discuss the corresponding measurable predictions. The covariance is incoded in the fact that the nonlocal quantum potential transforms as a scalar, which is a consequence of the fact that the nonlocal wave function transforms as a scalar. The measurable predictions that can be obtained with the deterministic Bohmian interpretation cannot be obtained with the conventional interpretation simply because the conventional probabilistic interpretation does not work in the case of relativistic quantum mechanics.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a "quantum theory without observers," alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schrödinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
In this article, we investigate Bohm's view of quantum theory, especially Bohm's quantum potential, from a new perspective. We develop a quasi-Newtonian approach to Bohmian mechanics. We show that to arrive at Bohmian formulation of quantum mechanics, there is no necessity to start from the Schrödinger equation. We also obtain an equation that restricts the possible forms of quantum potential and determines the functional form of it without appealing to the wave function and the Schrödinger equation. Finally, we discuss about the significance of quantum potential in the conceptual structure of quantum theory.