Quantum maps and automorphisms

Steve Zelditch*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapter

9 Scopus citations

Abstract

There are several inequivalent definitions of what it means to quantize a symplectic map on a symplectic manifold (M, ω). One definition is that the quantization is an automorphism of a *-algebra associated to (M, ω). Another is that it is unitary operator Uχ on a Hilbert space associated to (M, ω), such that A → U*χAUχ defines an automorphism of the algebra of observables. A yet stronger one, common in partial differential equations, is that Uχ should be a Fourier integral operator associated to the graph of χ.We compare the definitions in the case where (M, ω) is a compact Kahler manifold. The main result is a Toeplitz analogue of the Duistermaat-Singer theorem on automorphisms of pseudodifferential algebras, and an extension which does not assume H1(M,ℂ) = {0}. We illustrate with examples from quantum maps.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Pages623-654
Number of pages32
DOIs
StatePublished - Jan 1 2005

Publication series

NameProgress in Mathematics
Volume232
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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