| In this thesis, a quantization method for classical maps on the torus is presented. The quantum algebra of observables is defined as the quantization of measurable functions on the torus with generators exp {dollar}(2pi ix){dollar} and exp {dollar}(2pi ip).{dollar} The Hilbert space we use remains the infinite-dimensional {dollar}Lsp2 (IR, dx).{dollar} The dynamics is given by a unitary quantum propagator such that as {dollar}hbar to 0,{dollar} the classical dynamics is returned. We construct such a quantization for the Kronecker map, the cat map, the baker's map, the kick map, and the Harper map. For the cat map, we find the same for the propagator on the plane the same integral kernel conjectured in (HB) using semiclassical methods.; We also define a quantum "integral over phase space" as a trace over the quantum algebra. Using this definition, we proceed to define quantum ergodicity and mixing for maps on the torus. We prove that the quantum cat map and Kronecker map are both ergodic, but only the cat map is mixing, true to its classical origins.; For Planck's constant satisfying the integrality condition {dollar}h = 1/N,{dollar} with {dollar}Nindoubzsp+,{dollar} we construct an explicit isomorphism between {dollar}Lsp2 (IR, dx){dollar} and the Hilbert space of sections of an N-dimensional vector bundle over a {dollar}theta{dollar}-torus T{dollar}sp2{dollar} of boundary conditions. The basis functions are distributions in {dollar}Lsp2 (IR, dx),{dollar} given by an infinite comb of Dirac {dollar}delta{dollar}-functions. In Bargmann space these distributions take on the form of Jacobi {dollar}vartheta{dollar}-functions. Transformations from position to momentum representation can be implemented via a finite N-dimensional discrete Fourier transform. With the {dollar}theta{dollar}-torus, we provide a connection between the finite-dimensional quantum maps given in the physics literature and the canonical quantization presented here and found in the language of pseudo-differential operators elsewhere in mathematics circles. Specifically, at a fixed point of the dynamics on the {dollar}theta{dollar}-torus, we return a finite-dimensional matrix propagator. We present this connection explicitly for several examples. |
