TY - CHAP

T1 - Quantum maps and automorphisms

AU - Zelditch, Steve

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/0-8176-4419-9_22

DO - 10.1007/0-8176-4419-9_22

M3 - Chapter

AN - SCOPUS:47749123530

T3 - Progress in Mathematics

SP - 623

EP - 654

BT - Progress in Mathematics

PB - Springer Basel

ER -