Semiclassical propagator

A semiclassical propagator sums $e^{i{S}_{\mathrm{cl}}/\hslash }$ over classical paths joining its endpoints, weighted by their quadratic fluctuation determinants and phase indices. It is exact for quadratic actions.

Expanding a configuration-space path integral about a classical trajectory $q_c$ with fixed endpoints gives $S[q_c+f]=S[q_c]+\frac{1}{2}\int f{\mathcal{O}}_{c}fdt+O(f^3)$, where $f$ obeys homogeneous Dirichlet boundary conditions. The Gaussian functional integral contributes $(\det {\mathcal{O}}_c{)}^{-1/2}$ with the phase prescribed by continuation of the quantum-mechanical propagator. Relative to the free operator ${\mathcal{O}}_0=-m{d}^2/d{t}^2$, the prefactor is $D_0(T)[\det {\mathcal{O}}_c/\det {\mathcal{O}}_0{]}^{-1/2}$. In a general potential the operator, and thus the prefactor, depends on the endpoints through $q_c$. For a quadratic potential $V^{\prime \prime }$ is constant and all higher fluctuation terms vanish, so the approximation is exact away from singular focusing times. At a zero mode the simple determinant formula requires a limiting prescription or a different treatment of the zero mode.

The Van Vleck determinant is the determinant of mixed endpoint derivatives of a classical action. It measures how nearby classical trajectories focus and supplies the fluctuation prefactor of the semiclassical propagator.