Quantum maps and automorphisms

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*_χAU_χ defines an automorphism of the algebra 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 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.