Path integral formulation of finite-dimensional quantum mechanics in discrete phase space

Abstract: We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function defined on $\mathbb{Z}_d \times \mathbb{Z}_d$ (with $d$ an odd prime), and the associated Weyl transform built from generalized displacement operators, we derive an exact evolution kernel that propagates the discrete Wigner function in time. By exploiting the composition law of the kernel and iterating the short-time approximation, we obtain a sum-over-paths expression for the propagator weighted by a discrete phase-space action that is the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates, we show that the fluctuation sum factorizes and, at times strictly commensurate with the lattice (the Clifford-covariant regime), collapses to a deterministic shift realizing the discrete analog of classical Hamiltonian flow. The formulation is applied to a single qutrit ($d=3$) under a diagonal Hamiltonian, and to two interacting qutrits, where we show explicitly that the full entanglement dynamics -- captured by a closed-form expression for the linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors of the action. The $\tildeμ= 0$ sector alone is non-real at finite time step and collapses to a trivial (uniform) kernel in the continuum limit, failing to reproduce the entanglement dynamics in either regime. We discuss the relevance of this construction for semiclassical simulation of many-body spin systems and the characterization of non-classicality through Wigner negativity.