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arXiv:2605.25858

Geometric Quantization on Orbifolds

Peiyuan Teng

quant-phSubmitted May 25, 2026• Announced May 26, 2026
arXivHTMLPDF

Abstract

This text introduces geometric quantization on orbifolds. After reviewing the necessary background, it develops new treatments of prequantization, polarizations, and metaplectic correction for symplectic orbifolds.

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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 MMM and the non-degenerate 2-form ωωω, which comes with MMM. 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 (R2n,∑i=1ndpi∧dqi\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_iR2n,∑i=1n​dpi​∧dqi​) 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.
Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric quantization is applicable to other symplectic manifolds, not only cotangent spaces. The resulting formalism provides a way of looking at quantum theory that is distinct from conventional approaches to the subject, e.g., the Dirac bra-ket formalism. In particular, such familiar features as the quantization of spin, the canonical quantization of position and momentum, and the Schr\{o}dinger equation all emerge from geometric quantization. This paper serves as a review of the subject written in an informal style, often taking an example-based approach to exposition, and attempts to present the material without assuming the reader is an expert in differential geometry.
Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric quantization is applicable to other symplectic manifolds, not only cotangent spaces. The resulting formalism provides a way of looking at quantum theory that is distinct from conventional approaches to the subject, e.g., the Dirac bra-ket formalism. In particular, such familiar features as the quantization of spin, the canonical quantization of position and momentum, and the Schrödinger equation all emerge from geometric quantization. This paper serves as a review of the subject written in an informal style, often taking an example-based approach to exposition, and attempts to present the material without assuming the reader is an expert in differential geometry.