Quantum Mechanics on a Torus
  We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space $ L^2 (A_\hbar},\tau_\hbar) $ obtained as the completion of the algebra of observables $ A_\hbar $ in the norm induced by the following inner product $(A,B) =\tau_{\hbar}(A^{\dagger}B) $, where $\tau_{\hbar}$ is a linear functional on the algebra analogous to the classical ``integral over phase space.'' We also derive explicit formulas connecting this formulation to the $\theta $-torus decomposition of Bargmann space introduced in ref. \QCITE{cite}{}{KLMR}. 
Ron Rubin and Andrew Lesniewski