Quantum Maps and automorphisms

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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.
Original languageEnglish
Pages (from-to)623
JournalThe Breadth of Symplectic and Poisson Geometry, Progress in Mathematics
StatePublished - 2005

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