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CONNECTIONS WITH BOHMIAN MECHANICS

James O. Taylor, Sheldon Goldstein

Published Jan 1, 2003• 4 citations
OpenAlex

Abstract

OF THE DISSERTATION Connections with Bohmian Mechanics by James Oliver Taylor Dissertation Director: Sheldon Goldstein We begin by formulating non-relativistic Bohmian mechanics for theories in which the wave functions are sections of a Hermitian vector bundle over a Riemannian manifold. We give geometric interpretations of the Pauli equation and the Dirac equation. We then explain the influence of non-trivial topology in formulating Bohmian theories and use that to explain the quantum story of identical particles. We take a break from Bohmian mechanics briefly to investigate when the covariant derivatives of a mapping between manifolds may be used in a Taylor polynomial fashion in order to find the best polynomial approximation to the map. We finish by first formulating and then giving some results about a new algorithm for computing solutions to Schrodinger’s equation.

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OF THE DISSERTATION Connections with Bohmian Mechanics by James Oliver Taylor Dissertation Director: Sheldon Goldstein We begin by formulating non-relativistic Bohmian mechanics for theories in which the wave functions are sections of a Hermitian vector bundle over a Riemannian manifold. We give geometric interpretations of the Pauli equation and the Dirac equation. We then explain the influence of non-trivial topology in formulating Bohmian theories and use that to explain the quantum story of identical particles. We take a break from Bohmian mechanics briefly to investigate when the covariant derivatives of a mapping between manifolds may be used in a Taylor polynomial fashion in order to find the best polynomial approximation to the map. We finish by first formulating and then giving some results about a new algorithm for computing solutions to Schrodinger’s equation.
We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on R3N\mathbb{R}^{3N}R3N to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.
In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis (ρ=∣ψ∣2)(ρ=|ψ|^{2})(ρ=∣ψ∣2) and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.
Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions. Key words: relativistic quantum mechanics, Klein-Gordon equation, Bohmian interpretation. 1
The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.
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)