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Geometric quantization

Published Jan 1, 2005• 3 citations
Publisher

Abstract

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.

DOI
10.1142/9789812701268_0006

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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.
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.
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations we find a compatibility condition restricting the set of admissible connections. In the special case when phase space is a cotangent bundle this compatibility condition has many solutions, and the resulting quantum theory not only reproduces the well-known geometric quantization scheme, but also allows to quantize all interesting observables. For Kähler manifolds there is no compatibility condition, but a canonical choice for the parameters. The explicit form of the observables however remains undetermined.
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.
Quantization of R2R^2R2 and S1×S1S^1 \times S^1S1×S1 phase spaces are explicitly carried out tweaking the techniques of geometric quantization. Crucial is a combined use of left and right invariant vector fields. Canonical bases, operators and their actions are explicitly presented. Arguments of Dirac and also Wu and Yang for monopoles are applied for obtaining the quantization of the phase space area. Equivalence of states in the infinite dimensional prequantum Hilbert space resulting in a physical Hilbert space of dimension equal to the phase space area in units of the Planck constant is demonstrated. These techniques can be applied to any manifold with a symplectic structure.