We ask and answer the question: How do we derive the approximation of Heller (that the Wigner function propagates along classical paths) from the path integral formulation of the Wigner kernel of Kitahara?.
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 on Zd×Zd (d an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
We present the path-integral solutions to the distributions in classical (Gibbs) and quantum (Wigner) statistical mechanics. The kernel of the distributions are derived in two ways - one by time slicing and defining the appropriate short-time interval phase space matrix element and second by making use of the kernel in the path-integral approach to quantum mechanics. We show that the two approaches are perturbatively identical. We also present another computation for the Wigner kernel, which is also the Liouville kernel, for the harmonic oscillator and free particle. These kernels may be used as the starting point in the perturbative expansion of the Wigner kernel for any potential. With the kernel solved, we essentially solve also the distributions in classical and quantum statistical mechanics.
The structure of the semiclassical Feynman kernel in the coherent state representation is studied.The analysis is focused on classically realizable (i.e.real-valued) trajectories, whose contributions to the semiclassical kernel persist in the limit → 0. A smooth deformation of the real-valued guiding trajectory that starts from the center of the "initial" coherent state and obeys the Hamilton equation induces complex-valued trajectories that contribute to the semiclassical kernel "linearized" around the guiding trajectory.The linearized semiclassical kernel describes Heller's Gaussian wavepacket dynamics (GWD).Two mechanisms that break the GWD approximation are studied in terms of the semiclassical coherent state path integral.The rate at which a caustic approaches the guiding trajectory is estimated.In addition, the deviation of the semiclassical kernel from a Gaussian shape is estimated, with the assumption that the influence of caustics is negligible.These results provide an estimation of departure times between classical and quantum evolutions.In particular, for classically integrable systems, the present argument elucidates that the departure time due to caustics is much larger than that due to the deviation from a Gaussian shape.Furthermore, a disagreement between the estimations of the departure times obtained from the Husimi representation and the Wigner representation is pointed out.