I've been reading a little about the usual, continuous Wigner functions and phase space quasi-distributions in general, and I believe I understand the idea behind them. The Wigner function arises when you decompose a general operator in terms of a reflection (Fourier transform of translation) basis. As we're talking about phase space distributions, we're actually translating not only position or momentum, but both at the same time. That's why the Wigner function has a very strict connection with the operator $\alpha \hat{q} + \beta \hat{p}$. See http://arxiv.org/abs/quant--ph/0401155.
Wigner functions have been defined also in discrete quantum systems (the article I cited does this), such as the ones arising in Nuclear Magnetic Resonance. The fact that there are many different definitions and some of them are in conflict doesn't matter now.
My question regards the nature of the phase space whose lines we're assigning observables to. What is a discrete phase space, mathematically?
A continuous phase space is a fibre of a cotangent bundle, and it is a differentiable manifold. A discrete phase space is not a manifold, but admits triangulations (simplexes). Is is a cotangent bundle of something? Can we consider operators of position or momentum over it (I believe we can't)? To make things worse, this discrete space is actually$\mod N$ (with $N$ prime or power of a prime), so it's somehow associated to a discrete torus. This only makes things worse for me.
P.S.: Notice I'm not talking about the phase space of a discrete dynamical system, but about a discretisation of a phase space itself. I believe those are not related.
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