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Translations and reflections on the torus: Identities for discrete Wigner functions and transforms
Alfredo Miguel Ozorio de Almeida (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil)
A finite Hilbert space can be associated to a periodic phase space, that is, a torus. A finite subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner functions and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translation. Connections between products of Wigner functions and mixed (transition) Wigner functions also arise. Finally, generalizations of translations and reflections to a doubled phase space connect the Weyl representation of the evolution operator to the propagator of Wigner functions.