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...

Source: http://arxiv.org/abs/math/0307175v2