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arXiv:math-ph/0003029

Comparison between Geometric Quantisation and Covariant Quantum Mechanics

Marco Modugno, Carlos Prieto, Raffaele F. Vitolo

Published Mar 24, 2000• 8 citations
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Abstract

We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation.
DOI
10.48550/arxiv.math-ph/0003029

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This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for nnn particles, as the nnn-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interpretation of two-valued wavefunctions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.
The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrödinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some results fit well-known facts of other mathematical models of quantum mechanics, but are derived by innovative procedures. Other results are completely new. The most important original contributions can be summarized by the following two statements: The quantum symmetries constitute Lie algebras, which are naturally isomorphic to subalgebras of the new classical Lie algebra of quantisable functions. Let me stress that this algebra is not a Poisson subalgebra. On the other hand, by means of the quantum Lagrangian formalism, we can associate a conserved quantum current with any element of a distinguished classical subalgebra. These results might be the prequantum source of the probabilistic interpretation of quantum mechanics.