Vol. 312, No. 2, 2021

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Symplectic microgeometry, IV: Quantization

Alberto S. Cattaneo, Benoit Dherin and Alan Weinstein

Vol. 312 (2021), No. 2, 355–399
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.

Keywords
quantization, Fourier integral operators, symplectic micromorphisms
Mathematical Subject Classification
Primary: 58J40, 81S10
Milestones
Received: 3 August 2020
Revised: 30 January 2021
Accepted: 1 May 2021
Published: 31 August 2021
Authors
Alberto S. Cattaneo
Universität Zürich
Zürich
Switzerland
Benoit Dherin
Google, Inc.
Dublin
Ireland
Alan Weinstein
University of California
Berkeley, CA
United States