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International Journal of Geometric Methods in Modern Physics

HIGHER-DIMENSIONAL GEOMETRIC DESCRIPTION OF THE QUANTUM KLEIN–GORDON EQUATION

Benjamin Koch

Published May 6, 2013• 3 citations
Publisher

Abstract

It is shown that the equations of relativistic Bohmian mechanics for multiple bosonic particles have a dual description in terms of a classical theory of conformally "curved" space-time. This shows that it is possible to formulate quantum mechanics as a purely classical geometrical theory. The results are further generalized to interactions with an external electromagnetic field.

DOI
10.1142/s0219887813200144

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It is shown that the equations of relativistic Bohmian mechanics for multiple bosonic particles have a dual description in terms of a classical theory of conformally "curved" space-time. This shows that it is possible to formulate quantum mechanics as a purely classical geometrical theory. The results are further generalized to interactions with an external electromagnetic field.
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Positive-energy solutions of the Klein-Gordon equation form a Hilbert space of holomorphic functions on the future tube. This domain is interpreted as an extended phase space for the associated classical particle, the extra dimensions being the time and a new variable related to the inverse equilibrium temperature in an associated canonical ensemble. This gives a compelling formulation of relativistic Bohmian mechanics where the "hidden variables" are simply the classical trajectories of the particle in phase space. Interactions may be included through "holomorphic gauge theory."