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International Journal of Theoretical Physics

Comments on transition from classical to quantum mechanics in generalized coordinates via the covariant derivative

Gary R. Gruber

Published Jan 1, 1973• 0 citations
Publisher
DOI
10.1007/bf00670980

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Classically general covariance is found from the idea that a vector is a physical quantity which exists independently of choice of coordinate system and is unchanged by a change of coordinate system. It is often assumed that there exists some form of absolute mathematical space or space-time, and that in a flat space approximation vectors can be imagined between defined points in this space-time, much as we can imagine an arrowed line drawn on a piece of paper. However, while classical vector quantities can be represented on paper, in the quantum domain physical quantities do not in general exist with precise values except in measurement; a change of apparatus, for example by rotating it, may affect the outcome of the measurement, so the condition for general covariance does not apply. The purpose of this paper is to re-examine covariance within the context of an orthodox, Dirac-Von Neumann interpretation of quantum mechanics, to replace it with a new condition, here called quantum covariance, and to show that quantum covariance is the required condition for the unification of general relativity with quantum mechanics for non-interacting particles.
We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental “phase-spacetime”. We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.