The action-angle Wigner function: a discrete, finite and algebraic phase space formalism

The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discreti zation of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigne r function is shown to exist in the semi-discrete limit of this quantizatio n scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action -angle phase space distribution respects are derived. The dynamical propert ies of the action-angle Wigner function are analysed for discrete and finit e-dimensional model Hamiltonians. The limit of the discrete and finite-dime nsional formalism including a discrete analogue of the Gaussian wavefunctio n spread, viz. the binomial wavepacket, is examined and shown by examples t hat standard (continuum) quantum mechanical results carl be obtained as the dimension of the discrete phase space is extended to infinity.