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arXiv:1605.06562

How particles can emerge in a relativistic version of Bohmian Quantum Field Theory

Harder, T. Mark

Published May 20, 2016• 1 citation
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Abstract

It is shown how bosonic material particles can emerge from a covariant formulation of de Broglie-Bohm theory. The formulation is based on the work of Nikolic. Material particles are continuous fields, formed as the eigenvalue of the Schrodinger field operator, evaluated along a Bohmian trajectory.
DOI
10.48550/arxiv.1605.06562

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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.
This chapter provides a fully comprehensive overview of the Bohmian formulation of quantum phenomena. It deals with a historical review of the difficulties found by Louis de Broglie, David Bohm and John Bell to convince the scientific community about the validity and utility of Bohmian mechanics. The chapter presents a formal explanation of Bohmian mechanics for nonrelativistic single-particle quantum systems. It explains a didactic introduction of the Bohmian formalism and a self-contained summary for any newcomer interested in using Bohmian mechanics in their daily research activity. The quantum theory revolves around the notions of particles and waves. Einstein died in 1955, convinced that a correct reformulation of quantum theory would preserve local causality. Aerodynamics and hydrodynamics are examples of such continuous material systems. The initial formulation of quantum mechanics with Bohmian trajectories developed by de Broglie and Bohm was performed for dynamical systems with associated velocities much slower than the speed of light.
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 is endowed with an ontological package that supposedly allows to solve the main interpretational problems of quantum mechanics. We are concerned in this work by the emergence of classicality from the quantum mechanical substrate. We will argue that although being superficially attractive, the de Broglie-Bohm interpretation does not shed new light on the quantum-to-classical transition. This is due to nature of the dynamical law of Bohmian mechanics by which the particles follow the streamlines of the probability flow. As a consequence, Bohmian trajectories can be highly non-classical even when the wavefunction propagates along classical trajectories, as happens in semiclassical systems. In order to account for classical dynamics, Bohmian mechanics needs non-spreading and non-interfering wave packets: this is achieved for practical purposes by having recourse to decoherence and dense measurements. However one then faces the usual fundamental problems associated with the meaning of reduced density matrices. Moreover the specific assets of the de Broglie-Bohm interpretation – in particular the existence of point-like particles pursuing well-defined trajectories – would play no role in accounting for the emergence of classical dynamics.