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The Wave-Function Is a Multi-Field

Mario Hubert, Davide Romano

Published Oct 9, 2017• 13 citations
OpenAlex

Abstract

It is generally argued that if the wave-function in the de Broglie--Bohm theory is a physical field, it must be a field in configuration space. Nevertheless, it is possible to interpret the wave-function as a multi-field in three-dimensional space. This approach hasn't received the attention yet it really deserves. The aim of this paper is threefold: first, we show that the wave-function is naturally and straightforwardly construed as a multi-field; second, we show why this interpretation is superior to other field interpretations; third, we clarify common misconceptions.

Related papers

Configuration-space dynamics, velocity fields, and the origin of nonlocality in de Broglie–Bohm quantum theory

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In many-particle quantum mechanics the wave function is irreducibly defined on configuration space. In the de Broglie–Bohm theory this fact is made explicit: both the wave function and the actual motion are naturally described as evolving in configuration space, while the familiar three-dimensional motions of particles arise only as projections of this higher-dimensional dynamics. In this paper we argue that configuration space should be regarded as the fundamental arena of quantum dynamics. On this view, the nonlocal correlations observed in ordinary three-dimensional space are not mysterious dynamical influences but direct consequences of local evolution in configuration space. This perspective is developed and illustrated through a detailed analysis of momentum measurement and which-way detection in the iconic two slit experiment, in the two-particle interferometry experiment of Horne–Shimony–Zeilinger and in the Einstein–Podolsky–Rosen experiment.

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One of the major concerns of Schr\"odinger, Lorentz, Einstein, and many others about the wave function is that it is defined on the $3\mathbf{N}$-dimensional configuration space, rather than on the three-dimensional (3D) physical space. This gives the impression that quantum mechanics cannot have a $3\mathrm{D}$ space or space-time ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wave function as a physical entity, in particular, the many-worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory. Here, a representation of the many-particle states is given, as multilayered fields defined on the three-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wave functions as defined on the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the nonexplicit and informal intuition of many working quantum physicists, who, by considering the wave function sometimes to be defined on the configuration space and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schr\"odinger cats just like the usual one. But, it may help various interpretations to solve these problems, through inclusion of the wave function as (part of) their primitive ontology. In appendices, it is shown how the multilayered field representation can be extended to quantum field theory.

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We try to give a physical meaning to the wave function or quantum state of a system, apart from being a very efficient tool for predicting results of measurements on that system. In other words, we ask what does it mean for a system outside the laboratories to have a wave function? We first explain why two possible, and probably common, answers to that question do not really answer it. Then, we explain how the de Broglie – Bohm theory does give a satisfactory meaning to the quantum state outside the laboratories, while avoiding the problems faced by the other answers.

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There are three possible interpretations of the wave function in the de Broglie-Bohm theory: taking the wave function as corresponding to a physical entity or a property of the Bohmian particles or a law. In this paper, we argue that the first interpretation is favored by an analysis of protective measurements. The de Broglie-Bohm theory is an alternative to standard quantum mechan-ics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. The wave function follows the linear Schrödinger equation and never collapses. The motion of the particles, which are usually called Bohmian particles, follows the so-called guiding equation. Although the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics, there is no clear consensus with regard to its physical interpretation. In particular, the interpre-

•May 24, 2013•0 citations
It is generally argued that if the wave-function in the de Broglie--Bohm theory is a physical field, it must be a field in configuration space. Nevertheless, it is possible to interpret the wave-function as a multi-field in three-dimensional space. This approach hasn't received the attention yet it really deserves. The aim of this paper is threefold: first, we show that the wave-function is naturally and straightforwardly construed as a multi-field; second, we show why this interpretation is superior to other field interpretations; third, we clarify common misconceptions.
In many-particle quantum mechanics the wave function is irreducibly defined on configuration space. In the de Broglie–Bohm theory this fact is made explicit: both the wave function and the actual motion are naturally described as evolving in configuration space, while the familiar three-dimensional motions of particles arise only as projections of this higher-dimensional dynamics. In this paper we argue that configuration space should be regarded as the fundamental arena of quantum dynamics. On this view, the nonlocal correlations observed in ordinary three-dimensional space are not mysterious dynamical influences but direct consequences of local evolution in configuration space. This perspective is developed and illustrated through a detailed analysis of momentum measurement and which-way detection in the iconic two slit experiment, in the two-particle interferometry experiment of Horne–Shimony–Zeilinger and in the Einstein–Podolsky–Rosen experiment.
One of the major concerns of Schr\"odinger, Lorentz, Einstein, and many others about the wave function is that it is defined on the 3N3\mathbf{N}3N-dimensional configuration space, rather than on the three-dimensional (3D) physical space. This gives the impression that quantum mechanics cannot have a 3D3\mathrm{D}3D space or space-time ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wave function as a physical entity, in particular, the many-worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory. Here, a representation of the many-particle states is given, as multilayered fields defined on the three-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wave functions as defined on the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the nonexplicit and informal intuition of many working quantum physicists, who, by considering the wave function sometimes to be defined on the configuration space and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schr\"odinger cats just like the usual one. But, it may help various interpretations to solve these problems, through inclusion of the wave function as (part of) their primitive ontology. In appendices, it is shown how the multilayered field representation can be extended to quantum field theory.
We try to give a physical meaning to the wave function or quantum state of a system, apart from being a very efficient tool for predicting results of measurements on that system. In other words, we ask what does it mean for a system outside the laboratories to have a wave function? We first explain why two possible, and probably common, answers to that question do not really answer it. Then, we explain how the de Broglie – Bohm theory does give a satisfactory meaning to the quantum state outside the laboratories, while avoiding the problems faced by the other answers.
This paper provides a detailed and unified historical account of early debates over wave-function realism, the modern term for the view that the wave function of quantum theory is physically real. As this paper will show, the idea of physical waves associated with particles had its roots in work by Einstein and de Broglie, who both originally thought of these waves as propagating in three-dimensional physical space. De Broglie quickly turned this wave--particle duality into an early pilot-wave theory, on which a particle's associated ``phase wave'' piloted or guided the particle along its trajectory. Schrödinger built on de Broglie's phase-wave hypothesis to provide a comprehensive account of the nascent quantum theory. However, Schrödinger's new ``undulatory mechanics'' came at the cost of replacing de Broglie's phase waves propagating in physical space with a wave function propagating in a system's abstract configuration space. The present work will argue that this move from three-dimensional physical space to a many-dimensional configuration space was a decisive reason why the founders of quantum theory uniformly abandoned the physical reality of the wave function. This paper will further clarify that de Broglie introduced two distinct pilot-wave theories, and will then argue that it was Bohm's rediscovery of the second of these two pilot-wave theories over two decades later, as well as Bohm's vociferous defense of wave-function realism, that were responsible for resurrecting the idea of an ontological wave function. Finally, this paper will describe how the influence of Schrödinger and Bohmended up playing a central role in Everett's development of the ``many worlds'' interpretation.
Abstract The three major interpretations of quantum mechanics—Bohm, GRW, and many worlds—all treat the wave function as a physical entity. But this entity apparently resides in a high-dimensional configuration space, with three coordinates for each particle in the universe. Some conclude on this basis that the three-dimensionality of the world of experience is an illusion. Others respond by adding three-dimensional ontology to the theory or by reinterpreting the wave function either as a law or as an indirect representation of properties in three-dimensional space. It is argued that the latter possibility makes it unnecessary to postulate either additional ontology or a fundamentally high-dimensional world.
There are three possible interpretations of the wave function in the de Broglie-Bohm theory: taking the wave function as corresponding to a physical entity or a property of the Bohmian particles or a law. In this paper, we argue that the first interpretation is favored by an analysis of protective measurements. The de Broglie-Bohm theory is an alternative to standard quantum mechan-ics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. The wave function follows the linear Schrödinger equation and never collapses. The motion of the particles, which are usually called Bohmian particles, follows the so-called guiding equation. Although the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics, there is no clear consensus with regard to its physical interpretation. In particular, the interpre-