Quantization is geometry, after all
Quantization of a general canonically formulated classical theory is expressed in a geometric fashion utilizing the symplectic structure of phase space augmented by one of several acceptable Riemannian metric structures appended to the phase-space manifold. The purpose of the metric is to provide an adequate geometrical structure on phase space to support Brownian motion which is used in a path-integral-like quantization procedure. For a wide class of Hamiltonian the propagator of the quantum theory is given a rigorous, unambiguous, and completely geometric interpretation that is covariant under arbitrary canonical coordinate transformations. In this procedure different appended Riemannian geometries result in a quantization involving completely inequivalent sets of quantum kinematical variables. In particular, for Heisenberg variables this quantization procedure provides a natural geometric explanation of why the usual Schrödinger representation applies to only a very limited set of canonical coordinates and not to all such coordinates.
path-integral-like quantization procedure
scientific article; zbMATH DE number 4058371
scientific article