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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

A Noncommutative representation of classical dynamics. Connections with field quantization

J. A. Crawford

Published May 1, 1972• 0 citations
Publisher
DOI
10.1007/bf02735503

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Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
Abstract We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that the Heisenberg algebra gets some modifications in the constructed setup, from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering a Snyder-deformed Heisenberg algebra, we show that since the translation group is not deformed, it can be identified with a polymer-modified Heisenberg algebra. In the classical level, it is shown that the noncanonical Poisson brackets are related to their canonical counterparts by means of a Darboux transformation on the corresponding phase space.
In Foldy-Wouthuysen representation, this paper deduces the effective quantum mechanics for a relativistic particle confined to a curved surface in an external field by using the thin-layer quantization scheme. It is found that the quantum effects caused by introducing the confining potential can be taken as the results of relativistic correction in the nonrelativistic limit. Meanwhile, the spin connection of the curved surface can produce a Zeeman-like gap through the relativistic correction term. In addition, the confining potential can provide a curvature-independent energy shift resulting from the Zitterbewegung effect. As an example, this paper applies the effective Hamiltonian to a torus surface, in which the spin effects related to the introduced confining potential are obtained. These results demonstrate the scaling of the noncommutation of the nonrelativistic limit and the thin-layer quantization procedure.