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International Journal of Geometric Methods in Modern Physics

The classical limit in geometric quantization by group extension

Paul Bracken

Published Mar 22, 2025• 0 citations
Publisher

Abstract

Physical systems as a rule are associated with a symmetry group. The group approach to geometric quantization makes use of this to introduce a quantization by means of group extension. This procedure is discussed and applied to a physical system whose group law has its origin with the Galilean group. The main intention is to investigate the classical limit and its relationship under this approach to geometric quantization. The classical quantization conditions are obtained based on a deeper foundation.

DOI
10.1142/s0219887825501622

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Physical systems as a rule are associated with a symmetry group. The group approach to geometric quantization makes use of this to introduce a quantization by means of group extension. This procedure is discussed and applied to a physical system whose group law has its origin with the Galilean group. The main intention is to investigate the classical limit and its relationship under this approach to geometric quantization. The classical quantization conditions are obtained based on a deeper foundation.
A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free-particle dynamics and the harmonic oscillator.
An Introduction to the geometric quantization procedure which is the attempt at finding a procedure that maps classical observables to quantum observables.\n\nFrom this idea, we made a more mathematical approach to some very interesting physics.\nWe derived a mapping and while analysing it. It becomes clear that more conditions have to be imposed for this to produce physically correct answers. \nFrom geometric quantization, we produced the canonical quantization and the operator form of the Hamiltonian of the harmonic oscillator.\nAlong the way, we also found some physical consequences of the math.
Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.
This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.