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Quantization

N M J Woodhouse

Published Mar 26, 1992• 0 citations
Publisher

Abstract

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).

DOI
10.1093/oso/9780198536734.003.0009

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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).
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.
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.
Abstract The formulation of geometric quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to quantum mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the physics literature.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
A method for quantizing a classical system, based upon the notion of localization on phase space, is developed. This method, to which is given the name prime quantization, is based upon the mathematical theory of positive-operator-valued measures and their relationship to reproducing kernel Hilbert spaces, and to a notion of generalized kinematical variables related to phase space. For a system with openRn{\mathrm{openR}}^{n}openRn as its configuration space, it has the advantage of being able to connect the well-known problem of ordering in quantum mechanics to the choice of a measuring apparatus in a joint measurement of position and momentum (within the limits of the uncertainty principle). Moreover, for such a system, a choice of ordering in the present context also turns out to be a choice of polarization in the method of geometric quantization.