---
abstract: 'The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.'
author:
- 'William F. Braasch Jr.'
- 'William K. Wootters'
bibliography:
- 'references.bib'
title: Transition probabilities and transition rates in discrete phase space
---

Introduction
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The Wigner function is a real function on phase space representing the quantum state of a system of particles. In Wigner’s original paper, he points out that the equation of evolution of this function can be written in a form that makes the deterministic dynamics look like a probabilistic process [@Wigner1932]. Specifically, for each set of positions of the particles, and for each set of possible shifts in the particles’ momenta, there is a certain probability per unit time 