We construct a special class of semiclassical Fourier integral operators whose wave
fronts are the symplectic micromorphisms of our previous work (J. SymplecticGeom.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.
Keywords
quantization, Fourier integral operators, symplectic
micromorphisms
