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 (A~, τ~) obtained as the completion of the algebra of observables A~ in the norm induced by the following inner product (A,B) = τ~ ( A†B ) , where τ~ 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 θ-torus decomposition of Bargmann space introduced in ref.. ∗26 pages, 2 Figures