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arXiv:quant-ph/0603207

Many-fingered time Bohmian mechanics

Hrvoje Nikolić

Published Mar 23, 2006• 2 citations
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Abstract

The many-fingered time (MFT) formulation of many-particle quantum mechanics and quantum field theory is a natural framework that overcomes the problem of "instantaneous collapse" in entangled systems that exhibit nonlocalities. The corresponding Bohmian interpretation can also be formulated in terms of MFT beables, which alleviates the problem of instantaneous action at a distance by using an ontology that differs from that in the standard Bohmian interpretation. The appearance of usual single-time particle-positions and fields is recovered by quantum measurements.

DOI
10.48550/arxiv.quant-ph/0603207

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The many-fingered time (MFT) formulation of many-particle quantum mechanics and quantum field theory is a natural framework that overcomes the problem of "instantaneous collapse" in entangled systems that exhibit nonlocalities. The corresponding Bohmian interpretation can also be formulated in terms of MFT beables, which alleviates the problem of instantaneous action at a distance by using an ontology that differs from that in the standard Bohmian interpretation. The appearance of usual single-time particle-positions and fields is recovered by quantum measurements.
We present a three-dimensional field ontology for Bohmian mechanics in which the single-particle wave functions defined in ordinary physical space are taken as ontological elements alongside particle positions. In this view, each particle is associated with a wave function in three-dimensional space conditioned on the actual positions of the remaining particles. The universal wave function on configuration space is retained within the formalism, but its role is interpreted as nomological: it determines the coupled evolution of the three-dimensional fields without itself being treated as a fundamental physical field. The corresponding evolution equations for the conditioned three-dimensional wave functions follow from the standard Bohmian formalism and, together with the guidance equation, yield dynamics fully equivalent to those of the conventional theory. The empirical and predictive content of Bohmian mechanics is thereby preserved, while its ontological structure is expressed in explicitly three-dimensional terms. Within a ψ-ontic perspective, this proposal makes explicit an ontological organization of Bohmian mechanics in which the dynamical fields are defined on physical space. While fully equivalent to the conventional formulation, expressing the theory in these terms may facilitate interpretation, visualization, and pedagogical presentation.
We formulate Bohmian mechanics (BM) such that the main objects of concern are macroscopic phenomena, while microscopic particle trajectories only play an auxiliary role. Such a formulation makes it easy to understand why BM always makes the same measurable predictions as standard quantum mechanics (QM), irrespective of the details of microscopic trajectories. Relativistic quantum field theory (QFT) is interpreted as an effective long-distance theory that at smaller distances must be replaced by some more fundamental theory. Analogy with condensed-matter physics suggests that this more fundamental theory could have a form of nonrelativistic QM, offering a simple generic resolution of an apparent conflict between BM and relativistic QFT.
The article describes an interpretation of the mathematical formalism of standard quantum mechanics in terms of relations. In particular, the wave function ψ(x) is interpreted as a complex-valued relation between an entity (often called “particle”) and a second entity x (often called “spatial point”). Such complex-valued relations can also be formulated for classical physical systems. Entanglement is interpreted as a relation between two entities (particles or properties of particles). Such relations define the concept of “being next to each other”, which implies that entangled entities are close to each other, even though they might appear to be far away with respect to a classical background space. However, when space is also considered to be a network of relations (of which the classical background space is a large-scale continuum limit), such nearest neighbor configurations are possible. The measurement problem is discussed from the perspective of this interpretation. It should be emphasized that this interpretation is not meant to be a serious attempt to describe the ontology of our world, but its purpose is to make it obvious that, besides Bohmian mechanics, presumably many other ontological interpretations of quantum theory exist.
It is argued that in Bohmian mechanics the effective wave function of a subsystem of the universe does not encode the influences of other par-ticles on the subsystem. This suggests that the ontology of Bohmian mechanics does not consist only in Bohmian particles and their posi-tions. It is nonetheless pointed out that since the wave function in configuration space may represent the state of ergodic motion of non-Bohmian particles in three-dimensional space, the ontology of Bohmian mechanics may still consist only in particles. Bohmian mechanics, which is also called the de Broglie-Bohm theory, is a realistic alternative to standard quantum mechanics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete realistic description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. It has been a hot topic of debate how to interpret the wave function in the theory. Recently Esfeld et al (2013) further developed the nomological interpretation of the wave function, as suggested by Dürr, Goldstein and Zangh̀ı (1997), Goldstein and Teufel (2001) and Goldstein and Zangh̀ı (2012). They argued that the ontology of Bohmian mechanics consists only in Bohmian particles and their positions, and in particular, the effective wave function of a subsystem of the universe encodes the influences of other particles in the universe on the subsystem. In this note, we will argue that this interesting view may be problematic. The Bohmian law of motion is expressed by two equations, a guiding equation for the configuration of particles in three-dimensional space and the Schrödinger equation, describing the time evolution of the wave function, which enters the guiding equation. The law can be formulated as follows: dQ(t) dt = vΨ(t)(Q(t)). (1)
This chapter provides a comprehensive overview of the Bohmian formulation of quantum mechanics. It starts with a historical review of the difficulties found by Louis de Broglie, David Bohm, and John S. Bell to convince the scientific community about the validity and utility of Bohmian mechanics. Then, a formal explanation of Bohmian mechanics for nonrelativistic, single-particle quantum systems is presented. The generalization to many-particle systems, where the exchange interaction and the spin play an important role, is also presented. After that, the measurement process in Bohmian mechanics is discussed. It is emphasized that Bohmian mechanics exactly reproduces the mean value and temporal and spatial correlations obtained from the standard, that is the Copenhagen or orthodox, formulation. The ontological characteristics of Bohmian mechanics provide a description of measurements as another type of interaction without the need for introducing the wave function collapse. Several solved problems are presented at the end of the chapter, giving additional mathematical support to some particular issues. A detailed description of computational algorithms to obtain Bohmian trajectories from the numerical solution of the Schrodinger or the Hamilton-Jacobi equations are presented in an appendix. The motivation of this chapter is twofold: first, as a didactic introduction to Bohmian formalism, which is used in the subsequent chapters, and second, as a self-contained summary for any newcomer interested in using Bohmian mechanics in his or her daily research activity.