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Prequantization

N M J Woodhouse

Published Mar 26, 1992• 0 citations
Publisher

Abstract

Abstract These are Dirac’s quantum conditions (Dirac 1925): they imply that the quantum operators form a Hilbert space representation of the classical observables. In his terms, the Poisson bracket is the classical analogue of the quantum commentator. The quantum conditions do not determine the underlying quantum system uniquely. Nor is every representation of the classical observables a physically reasonable quantization.

DOI
10.1093/oso/9780198536734.003.0008

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Abstract These are Dirac’s quantum conditions (Dirac 1925): they imply that the quantum operators form a Hilbert space representation of the classical observables. In his terms, the Poisson bracket is the classical analogue of the quantum commentator. The quantum conditions do not determine the underlying quantum system uniquely. Nor is every representation of the classical observables a physically reasonable quantization.
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.
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
In previous work the approach to stochastic quantization, originally proposed by Nelson, has been formulated in the frame of the stochastic variational principles of control theory. Then the Hamilton-Jacobi-Madelung equation is interpreted as the programming equation of the controlled problem, to be associated with the hydrodynamical continuity equation. Here we point out explicitly the canonical Hamiltonian structure of these equations, by introducing a suitable symplectic structure on the underlying phase space in various representations. One possible representation leads to the Schr\"odinger equation, which, together with its complex conjugate, can be recognized as a particular form of the Hamilton canonical equations in this frame. Then a suitably selected time-invariant subalgebra of the classical hydrodynamical algebra, closed under Poisson bracket pairing, is shown to be connected to the standard quantum observable operator algebra. In this correspondence Poisson brackets for hydrodynamical observables become averages of quantum observables in the given state. From this point of view stochastic quantization can be interpreted as giving an explanation for the standard quantization procedure of replacing the classical particle (or field) observables with operators, according to the scheme $p\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}x}$, $l\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}\ensuremath{\varphi}}$, etc. This discussion shows also the relevance of the canonical symplectic structure of the quantum state space, a feature which seems to have been overlooked in the axiomatic approaches to quantum mechanics.