Mathematics – Quantum Algebra
Scientific paper
2003-07-12
The breadth of symplectic and Poisson geometry, 623--654, Progr. Math., 232, Birkh\~A?user Boston, Boston, MA, 2005
Mathematics
Quantum Algebra
Scientific paper
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.
Zelditch Steve
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