An Introduction To Geometric Prequantization

TL;DR

This paper introduces geometric prequantization, a mathematical framework that extends classical symplectic geometry to better understand the transition to quantum mechanics, focusing on explicit constructions for simple phase spaces.

Contribution

It provides an accessible introduction to geometric prequantization, illustrating the method for basic symplectic manifolds without involving complex line bundles.

Findings

Constructs prequantization for ($\

$ ext{R}^{2n}$ with standard symplectic form.

Highlights that the resulting Hilbert space is too large for physical applications.

Abstract

Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some symplectic manifold $M$ and the non-degenerate 2-form $\omega$, which comes with $M$. Geometric prequantization is a process by which one does this in a mathematically "rigorous" manner and we shall attempt to just introduce the methods here. We do this by exploring this contruction for ($\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_i$) which avoids using complex line bundles in any non-trivial way. One should note however that the Hilbert space one obtains is in fact too "big", in the sense that it has too many functions in order to correspond with actual physically significant Hilbert spaces. Geometric quantization remedies this situation but it should be noted that not all manifolds are prequantizable. We shall not discuss either of these issues however.