Semiclassical Propagation and the Dynamics of Configuration Space

Abstract

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.