Quantum Mechanics on a Torus

Abstract

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},τ_\hbar) $ obtained as the completion of the algebra of observables $A_{\hslash }$ in the norm induced by the following inner product $(A,B)={\tau }_{\hslash }(A^{†}B)$, where ${\tau }_{\hslash }$ 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}.

27 pages, 2 figures