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The Metaplectic Correction

N M J Woodhouse

Published Mar 26, 1992• 0 citations
Publisher

Abstract

Abstract The half-form bundles are also defined for complex polarizations. By including them in the quantization construction, we arrive at a more useful pairing, which can be applied to a much wider class of nonnegative polarizations. It seems to give the correct physical relationship between the wave functions in the Hilbert spaces of different polarizations. In the case of two Fock spaces, the new pairing is unitary. When P is complex, δp may be nontrivial, and its inclusion in the construction can modify quantization conditions and alter the form of the quantum observables. In the case of the harmonic oscillator, the modification is precisely that needed to get the correct zero-point energy (see §9.2).

DOI
10.1093/oso/9780198536734.003.0010

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Abstract The half-form bundles are also defined for complex polarizations. By including them in the quantization construction, we arrive at a more useful pairing, which can be applied to a much wider class of nonnegative polarizations. It seems to give the correct physical relationship between the wave functions in the Hilbert spaces of different polarizations. In the case of two Fock spaces, the new pairing is unitary. When P is complex, δp may be nontrivial, and its inclusion in the construction can modify quantization conditions and alter the form of the quantum observables. In the case of the harmonic oscillator, the modification is precisely that needed to get the correct zero-point energy (see §9.2).
Abstract The problem with prequantization is that the Hilbert space it constructs is too large to represent the phase space of a physically reasonable quantum system. In the first example in §8.4 (a cotangent bundle), the Hilbert space contained wave functions that depended on momentum as well as position; and in the third (a Coadjoint orbit), the group representation was reducible. We have already touched on a possible solution: we should admit only sections of B that are parallel along a polarization. We shall now look in detail at how this works. It should be stressed, however, that the physical justification is not based on general mathematical results (such as the Borel-Weil theorem), but on the way in which the construction works in particular examples. It generalizes and unifies a number of quantization techniques that, in the past, have not appeared to have any obvious connection with each other and that have sometimes seemed overspecialized with applications only to particular physical systems. Also it should be noted that not all syrnplectic manifolds admit polarizations (Gotay 1987).
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Abstract Real polarizations play an important part in linking classical and quantum mechanics. We shall see that picking a polarization in a classical phase space amounts to choosing a representation in the underlying quantum theory-’representation’ in the sense of a position or momentum space representation. We shall also come across complex representations, which in many ways are easier to handle. An obvious example is the Fock space treatment of the harmonic oscillator, in which the quantum Hilbert space is realized as a space of holomorphic functions Complex representations arise from complex polarizations, which are natural generalizations of real polarizations: one simply relaxes the condition that the Hamiltonian vector fields spanning P should be real.
We study the Maslov correction to semiclassical states by using a Kähler regularized BKS pairing map from the energy representation to the Schrödinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of Kähler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of Kähler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration space are correctly reproduced.
Abstract The problem of quantizing a particle on a two-sphere has been treated by numerous approaches, including Isham’s global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham’s scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.