Semiclassical Propagation and the Dynamics of Configuration Space

Florentin Langouche, D. Roekaerts, E. Tirapegui

V. S. Morales-Salgado

This work explores the quantum propagator $K(x,t)$ as a solution of the system's dynamical equation. We develop a generalized propagator framework in which the propagator is written in the form $K=\exp\!\left(R+\frac{i}{\hbar}S\right)$, where $S$ governs the semiclassical phase structure and $R$ governs the amplitude transport and weighting of configurations. Starting from nonrelativistic quantum mechanics, the classical Hamilton--Jacobi equation emerges in the semiclassical limit, while $R$ reduces to the logarithm of the Van Vleck determinant. The formalism is then extended to relativistic field theory and minisuperspace quantum cosmology using functional methods and generalized Hamiltonian constraints. Then the resulting semiclassical equations recover the corresponding classical dynamics along characteristic flows. In minisuperspace models, the formalism yields coupled equations for geometric and matter sectors analogous to semiclassical Wheeler--DeWitt systems. In semiclassical gravitational settings, the gravitational contribution may admit an entropy-like interpretation, consistent with thermodynamic weighting factors appearing in Euclidean gravitational path integrals. The resulting framework suggests a unified semiclassical description in which propagator phases encode dynamical evolution while amplitude functionals encode transport and configuration weighting. This perspective motivates viewing mechanics as a relation between the differential structure of spacetime and the differential structure of the space of dynamical configurations.

Guangcan Yang

In this paper a semiclassical propagator in a mixed position–momentum space is derived in the formalism of Maslov's multi-dimensional semiclassical theory. The corresponding mixed van Vleck determinant is also given explicitly. The propagator can be used to locally fix semiclassical divergences in singular regions of configuration space. It is shown that when a semiclassical propagator is transformed from one representation to another, its form is invariant.

M. Roncadelli

A new path integral representation of the quantum mechanical propagator is proposed, which is even more intimately connected with classical mechanics than the Feynman path integral. The emergence of the semiclassical propagator is discussed in detail. It is argued that those paths which actually contribute in the new path integral have a very simple relation with the classical dynamical trajectories in configuration space.

R. Rossi, K. M. Fonseca-Romero, Maria Carolina Nemes