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Quantum maps and automorphisms

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The Breadth of Symplectic and Poisson Geometry

Part of the book series: Progress in Mathematics ((PM,volume 232))

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 U χ on a Hilbert space associated to (M, ω), such that A → U *χ AU χ defines an automorphism of the alge of observables. A yet stronger one, common in partial differential equations, is that U χ should be a Fourier integral operator associated to the graph of χ. We compare the definiti in the case where (M, ω) is a compact Kähler 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 H 1(M,ℂ) = {0}. We illustrate with examples from quantum maps.

This research was partially supported by NSF grant DMS-0071358 and by the Clay Mathematics Institute.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Steve Zelditch

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  1. Steve Zelditch
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Editors and Affiliations

  1. Department of Engineering and Applied Science Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Tudor S. Ratiu

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© 2005 Birkhäuser Boston

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Zelditch, S. (2005). Quantum maps and automorphisms. In: Marsden, J.E., Ratiu, T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol 232. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4419-9_22

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