Pilot-wave theory and quantum fields

Pilot-wave theories provide possible solutions to the measurement problem. In such theories, quantum systems are not only described by the state vector, but also by some additional variables. These additional variables, also called beables, can be particle positions, field configurations, strings, etc. In this paper we focus our attention on pilot-wave theories in which the additional variables are field configurations. The first such theory was proposed by Bohm for the free electromagnetic field. Since Bohm, similar pilot-wave theories have been proposed for other quantum fields. The purpose of this paper is to present an overview and further development of these proposals. We discuss various bosonic quantum field theories such as the Schroedinger field, the free electromagnetic field, scalar quantum electrodynamics and the Abelian Higgs model. In particular, we compare the pilot-wave theories proposed by Bohm and by Valentini for the electromagnetic field, finding that they are equivalent. We further discuss the proposals for fermionic fields by Holland and Valentini. In the case of Holland's model we indicate that further work is required in order to show that the model is capable of reproducing the standard quantum predictions. We also consider a similar model, which does not seem to reproduce the standard quantum predictions. In the case of Valentini's model we point out a problem that seems hard to overcome.Comment: 65 pages, no figures, LaTex; v2 minor changes, some extensions; v3 minor improvements; v4 some typos correcte

Bohmian Trajectories For a Time Foliation with Kinks

This paper concerns the hypersurface Bohm-Dirac model, i.e., the version of Bohmian mechanics in a relativistic space-time proposed by D\"urr et al. [1], which assumes a preferred foliation of space-time into spacelike hypersurfaces (called the time foliation) as given. We show that the leaves of the time foliation do not have to be smooth manifolds but can be allowed to have kinks. More precisely, we show that, also for leaves with kinks, the trajectories are still well defined and the appropriate $|\psi|^2$ distribution is still equivariant, so that the theory is still empirically equivalent to standard quantum mechanics. This result applies to the case where the time foliation is determined by the previously proposed law $dn=0$, since such a foliation generically has kinks.Comment: 14 pages, 4 figures, LaTe

On quantum potential dynamics

Non-relativistic de Broglie-Bohm theory describes particles moving under the guidance of the wave function. In de Broglie's original formulation, the particle dynamics is given by a first-order differential equation. In Bohm's reformulation, it is given by Newton's law of motion with an extra potential that depends on the wave function--the quantum potential--together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohm's formulation of the de Broglie-Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint.Comment: 10 pages, 4 figures, LaTex; v2 minor additions; v3 figures adde

Scope of the action principle

Laws of motion given in terms of differential equations can not always be derived from an action principle, at least not without introducing auxiliary variables. By allowing auxiliary variables, e.g. in the form of Lagrange multipliers, an action is immediately obtained. Here, we consider some ways how this can be done, with illustrations from the literature, and apply this to Bohmian mechanics. We also discuss the possible metaphysical status of these auxiliary variables. A particularly interesting approach brings the theory in the form of a gauge theory, with the auxiliary variables as gauge degrees of freedom

Bohmian quantum gravity and cosmology

Quantum gravity aims to describe gravity in quantum mechanical terms. How exactly this needs to be done remains an open question. Various proposals have been put on the table, such as canonical quantum gravity, loop quantum gravity, string theory, etc. These proposals often encounter technical and conceptual problems. In this chapter, we focus on canonical quantum gravity and discuss how many conceptual problems, such as the measurement problem and the problem of time, can be overcome by adopting a Bohmian point of view. In a Bohmian theory (also called pilot-wave theory or de Broglie-Bohm theory, after its originators de Broglie and Bohm), a system is described by certain variables in space-time such as particles or fields or something else, whose dynamics depends on the wave function. In the context of quantum gravity, these variables are a space-time metric and suitable variable for the matter fields (e.g., particles or fields). In addition to solving the conceptual problems, the Bohmian approach yields new applications and predictions in quantum cosmology. These include space-time singularity resolution, new types of semi-classical approximations to quantum gravity, and approximations for quantum perturbations moving in a quantum background.Comment: 45 pages, 6 figures, PDFLaTeX; written for "Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology", edited by Xavier Oriols Pladevall and Jordi Mompart; v2 typos correcte