Symplectic microgeometry, IV: Quantization

Abstract

We construct a special class of semiclassical Fourier integral operators whose wave fronts are the symplectic micromorphisms of our previous work (J. Symplectic Geom. 8 (2010), 205–223). These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semiclassical pseudodifferential calculus and offers a functorial framework for the semiclassical analysis of the Schrödinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.