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Zeitschrift für Naturforschung A

An Approach to a Discrete Quantum Mechanics in Closed Riemannian Spaces

Léopold Halpern

Published Oct 1, 1973• 2 citations
Publisher

Abstract

Quantum mechanics is contemplated in closed Riemannian spaces where the momentum spectrum is discrete. The theory formulated in this case is equivalent to an exclusively discrete scheme; continuity occurs only statistically. The operators corresponding to the dynamical variables are determined and the uncertainity relation is examined. Agreement is obtained in the local limit with quantum mechanics as formulated in flat space within the experimental accuracy.

DOI
10.1515/zna-1973-1008

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Quantum mechanics is contemplated in closed Riemannian spaces where the momentum spectrum is discrete. The theory formulated in this case is equivalent to an exclusively discrete scheme; continuity occurs only statistically. The operators corresponding to the dynamical variables are determined and the uncertainity relation is examined. Agreement is obtained in the local limit with quantum mechanics as formulated in flat space within the experimental accuracy.
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In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.
The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\ensuremath{\tau}$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\ensuremath{\tau}$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\ensuremath{\tau}$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system's initial condition and dynamics. Such sets of histories are said to decohere. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained. We describe how a quasiclassical domain described by averages of densities of approximately conserved quantities could be an emergent feature of an initial condition of the universe that implies the approximate classical behavior of spacetime on accessible scales.