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On a Symmetry Argument for the Guidance Equation

Bradford Skow

Published Jan 1, 2010• 1 citation
OpenAlex

Abstract

Bohmian mechanics faces an underdetermination problem: when it comes to solving the measurement problem, alternatives to the Bohmian guidance equa- tion work just as well as the official guidance equation. Dürr, Goldstein, and Zanghì have argued that of the candidate guidance equations, the official guid- ance equation is the simplest Galilean-invariant candidate. This symmetry argument—if it worked—would solve the underdetermination problem. But the argument does not work. It fails because it rests on assumptions about how Galilean transformations (especially boosts) act on the wavefunction that are (in this context) unwarranted. My discussion has larger morals about the phys- ical significance of certain mathematical results (like, for example, Wigner’s theorem) in non-orthodox interpretations of quantum mechanics.

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Abstract Michael Esfeld’s minimalist ontology is committed to two axioms relating to (1) distance relations that identify simple objects (permanent matter points) while (2) the distances between them change. This article scrutinizes such a conceptual strategy to determine whether it can successfully be applied to all levels of physical reality, as Esfeld contends. To do so, it explores one of his paradigmatic sources, that is, Bohmian mechanics. Two arguments are proposed. First, while Bohm’s original formulation of Bohmian mechanics and the interpretation advocated by Dürr, Goldstein & Zanghì are typically taken as mathematically equivalent, this article argues that Esfeld’s minimalist ontology does not cover the former’s ontological richness. To secure its achievement, the minimalist ontology framework needs to (i) break the equivalence between the two versions via a commitment to the nomological interpretation of the wavefunction (ii) yet attribute some kind of physical efficacy to the wavefunction as a guiding parameter for the evolution of particles living in three-dimensional space. Both requirements will be critically addressed. Second, the article shows that Esfeld’s metaphysical program is not only forced to rely on a theoretically suspicious formulation of quantum mechanics, but that more fundamental, under-development approaches in theoretical physics are way less reconcilable with its axioms, thus questioning its alleged universality.

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Bohmian mechanics faces an underdetermination problem: when it comes to solving the measurement problem, alternatives to the Bohmian guidance equa- tion work just as well as the official guidance equation. Dürr, Goldstein, and Zanghì have argued that of the candidate guidance equations, the official guid- ance equation is the simplest Galilean-invariant candidate. This symmetry argument—if it worked—would solve the underdetermination problem. But the argument does not work. It fails because it rests on assumptions about how Galilean transformations (especially boosts) act on the wavefunction that are (in this context) unwarranted. My discussion has larger morals about the phys- ical significance of certain mathematical results (like, for example, Wigner’s theorem) in non-orthodox interpretations of quantum mechanics.
Bohmian mechanics faces an underdetermination problem: when it comes to solving the measurement problem, alternatives to the Bohmian guidance equation work just as well as the official guidance equation. Dürr, Goldstein, and Zanghì have argued that of the candidate guidance equations, the official guidance equation is the simplest Galilean‐invariant candidate. This symmetry argument—if it worked—would solve the underdetermination problem. But the argument does not work. It fails because it rests on assumptions about how Galilean transformations (especially boosts) act on the wavefunction that are (in this context) unwarranted. My discussion has larger morals about the physical significance of certain mathematical results (like, for example, Wigner’s theorem) in non‐orthodox interpretations of quantum mechanics.
There is a recurring line of argument in the literature to the effect that Bohm's theory fails to solve the measurement problem. I show that this argument fails in all its variants. Hence Bohm's theory, whatever its drawbacks, at least succeeds in solving the measurement problem. I briefly discuss a similar argument that has been raised against the GRW theory. 1. Introduction2. Three Theories3. The Basic Argument4. Branches and Worlds5. Theory and Interpretation6. Size Matters7. Conclusion
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)
Abstract Michael Esfeld’s minimalist ontology is committed to two axioms relating to (1) distance relations that identify simple objects (permanent matter points) while (2) the distances between them change. This article scrutinizes such a conceptual strategy to determine whether it can successfully be applied to all levels of physical reality, as Esfeld contends. To do so, it explores one of his paradigmatic sources, that is, Bohmian mechanics. Two arguments are proposed. First, while Bohm’s original formulation of Bohmian mechanics and the interpretation advocated by Dürr, Goldstein & Zanghì are typically taken as mathematically equivalent, this article argues that Esfeld’s minimalist ontology does not cover the former’s ontological richness. To secure its achievement, the minimalist ontology framework needs to (i) break the equivalence between the two versions via a commitment to the nomological interpretation of the wavefunction (ii) yet attribute some kind of physical efficacy to the wavefunction as a guiding parameter for the evolution of particles living in three-dimensional space. Both requirements will be critically addressed. Second, the article shows that Esfeld’s metaphysical program is not only forced to rely on a theoretically suspicious formulation of quantum mechanics, but that more fundamental, under-development approaches in theoretical physics are way less reconcilable with its axioms, thus questioning its alleged universality.