An Introduction To Geometric Prequantization
Joseph Geraci

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 and the non-degenerate 2-form , which comes with . 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…
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Taxonomy
TopicsMathematics and Applications · Advanced Numerical Analysis Techniques · Algebraic and Geometric Analysis
