Quantum maps and automorphisms

Abstract

What does it mean to quantize a symplectic map $\chi$? In deformation quantization, it means to construct an automorphism of the $*$ algebra associated to $\chi$. In quantum chaos it means to construct unitary operators $U_{\chi}$ such that $A \to U_{\chi} A U_{\chi}^*$ defines an automorphism of the algebra of observables. In geometric quantization and in PDE it means to construct a unitary Fourier integral (or Toeplitz) operator associated to the graph of $\chi$. We compare the definitions in the setting of Kahler manifolds $(M, g)$. The main result is a Toeplitz analogue of the Duistermaat-Singer theorem on automorphisms of the pseudo-differential algebra, and its extension to non-simply connected phase spaces, which often occur in applications (quantized symplectic torus automorphisms.