Geometric Quantization on Manifolds
Geometric quantization is a type of quantization, which is a general term for a procedure that associates a quantum system with a given classical system. In practical terms, if one is trying to deduce what sort of quantum system should model a given physi
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23.1 Introduction Geometric quantization is a type of quantization, which is a general term for a procedure that associates a quantum system with a given classical system. In practical terms, if one is trying to deduce what sort of quantum system should model a given physical phenomenon, one often begins by observing the classical limit of the system. Electromagnetic radiation, for example, is describable on a macroscopic scale by Maxwell’s equations. On a finer scale, quantum effects (photons) become important. How should one determine the correct quantum theory of electromagnetism? It seems that the only reasonable way to proceed is to “quantize” Maxwell’s equations— and then to compare the resulting quantum system to experiment. Meanwhile, not every physically interesting system has R2n as its phase space. Geometric quantization, then, is an attempt to construct a quantum Hilbert space, together with appropriate operators, starting from a physical system having an arbitrary 2n-dimensional symplectic manifold N as its phase space. To perform geometric quantization on N, one must first choose a polarization, that is, roughly, a choice of n directions on N in which the wave functions will be constant. If N = T ∗ M, then one may use the “vertical polarization,” in which the wave functions are constant along the fibers of T ∗ M. For cotangent bundles with the vertical polarization, geometric quantization reproduces the “half-density quantization” of Blattner [4]. (See Examples 23.45 and 23.48.) Even for cotangent bundles, however, it is of interest to use polarizations other than the vertical polarization, as B.C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, DOI 10.1007/978-1-4614-7116-5 23, © Springer Science+Business Media New York 2013
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23. Geometric Quantization on Manifolds
we have seen already in the Rn case. In the case of the cotangent bundle of a compact Lie group, for example, the paper [20] shows how quantization with a complex polarization gives rise to a generalized Segal–Bargmann transform. Some phase spaces, meanwhile, may not even be in the form of a cotangent bundle. In the orbit method in representation theory, for example, the relevant symplectic manifolds are “coadjoint orbits,” which typically are not cotangent bundles. [In the SU(2) case, for instance, these orbits are 2-spheres with the natural rotationally invariant symplectic form.] In quantum field theory, meanwhile, one encounters Lagrangians that are linear, rather than quadratic, in the “velocity” variables. In such cases, the initial velocity is determined by the initial position, and one cannot think of the space of initial conditions as a (co)tangent bundle. Systems of this form can still be symplectic, but they are not cotangent bundles. Furthermore, it is common to think of compact symplectic manifolds (such as S 2 with a rotationally invariant symplectic form) as classical models of internal degrees of freedom, such as spin. To quantize these more general symplectic manifolds, one needs a more g
