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arXiv:1603.02541 · Journal of Physics A Mathematical and Theoretical

Bohmian mechanics, collapse models and the emergence of classicality

Marko Toroš, Sandro Donadi, Angelo Bassi

Published Aug 8, 2016• 8 citations
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Abstract

Abstract We discuss the emergence of classical trajectories in Bohmian mechanics, when a macroscopic object interacts with an external environment. We show that in such a case the conditional wave function of the system follows a dynamics which, under reasonable assumptions, corresponds to that of the Ghirardi–Rimini–Weber (GRW) collapse model. As a consequence, Bohmian trajectories evolve classically. Our analysis also shows how the GRW (istantaneous) collapse process can be derived by an underlying continuous interaction of a quantum system with an external agent, thus throwing a light on how collapses can emerge from a deeper level theory.

DOI
10.1088/1751-8113/49/35/355302

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•Aug 3, 2019•0 citations
Abstract We discuss the emergence of classical trajectories in Bohmian mechanics, when a macroscopic object interacts with an external environment. We show that in such a case the conditional wave function of the system follows a dynamics which, under reasonable assumptions, corresponds to that of the Ghirardi–Rimini–Weber (GRW) collapse model. As a consequence, Bohmian trajectories evolve classically. Our analysis also shows how the GRW (istantaneous) collapse process can be derived by an underlying continuous interaction of a quantum system with an external agent, thus throwing a light on how collapses can emerge from a deeper level theory.
We discuss a model of spontaneous collapse of the quantum state that does not require adding any stochastic processes to the standard dynamics. The additional ingredient with respect to the wave function is a position in the configuration space which drives the collapse in a completely deterministic way. This new variable is equivalent to a set of positions of all the particles, i.e., a set of Bohmian positions, which obey the usual guiding equation of Bohmian theory. Any superposition of quantum states of a macroscopic object occupying different regions of space is projected by a localization process onto the region occupied by the positions. Since the Bohmian positions are well defined in a single realization of the experiment, a space localization into one region is produced. The mechanism is based on the correlations between these positions arising from the cohesive forces inside macroscopic objects. The model introduces two collapse parameters which play a role very similar to those of the Ghirardi-Rimini-Weber (GRW) [Ghirardi et al., Phys. Rev. D 34, 470 (1986)] and continuous spontaneous localization (CSL) [Pearle, Phys. Rev. A 39, 2277 (1989); Ghirardi et al., Phys. Rev. A 42, 78 (1990)] theories. With appropriate values of these parameters, we check that the corresponding dynamics rapidly projects superpositions of macroscopic states localized in different regions of space into a single region, while it maintains a negligible effect in all situations where the predictions of standard quantum dynamics are known to be correct. We briefly speculate about the possible relations with gravity. We then study the evolution of the density operator and a mean-field approximation of the dynamical equations of this model, as well as the change of the evolution of the momentum introduced by the localization process. Possible theoretical interpretations are discussed. Generally speaking, this model introduces a sharper border between the quantum and classical worlds than the GRW and CSL theories and leaves a broader range of acceptable values for the parameters.
Bohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ‘matter’ moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The role of the wave function then is to govern the motion of the matter. 1. Introduction2. Bohmian Mechanics3. Ghirardi, Rimini, and Weber3.1. GRWm3.2. GRWf3.3. Empirical equivalence between GRWm and GRWf4. Primitive Ontology4.1. Primitive ontology and physical equivalence4.2. Primitive ontology and symmetry4.3. Without primitive ontology4.4. Primitive ontology and quantum state5. Differences between BM and GRW5.1. Primitive ontology and quadratic functionals5.2. Primitive ontology and equivariance6. A Plethora of Theories6.1. Particles, fields, and flashes6.2. Schrödinger wave functions and many-worlds7. The Flexible Wave Function7.1. GRWf without collapse7.2. Bohmian mechanics with collapse7.3. Empirical equivalence and equivariance8. What is a Quantum Theory without Observers?
One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini & Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focusing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way.
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.
In this paper, I describe a general strategy for deriving Newtonian mechanics from Bohm’s theory in the classical macroscopic regime. The strategy is based on two main steps: (i) the use of open systems, leading to the formation of independent subsystems; (ii) the analysis of the quantum potential for the emergent subsystems. The first step leads to and explains (within the Bohmian framework) the well-known phenomenon of decoherence. The second one provides a precise criterion for the emergence of Newtonian trajectories. In particular, I will show that, under reasonable classicality conditions, the quantum potential of the emergent subsystems becomes negligible. As a consequence, these systems will follow an approximately Newtonian dynamics.