Symplectic Microgeometry IV: Quantization

TL;DR

This paper develops a new category of semiclassical Fourier integral operators with symplectic micromorphisms, providing a functorial framework for semi-classical analysis and applications to quantum mechanics and Poisson manifold quantization.

Contribution

It introduces a novel class of operators with symplectic micromorphisms, extending semi-classical calculus and enabling a functorial approach to quantization.

Findings

Operators form a category with a functor to cotangent microbundles

Total symbol calculus developed for these operators

Framework applicable to Schrödinger equation and Poisson manifolds

Abstract

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. 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 semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schr\"odinger 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.