= Dirichlet fluctuation determinant of a semiclassical propagator
{c}
{title2=$\mathcal O_c=-m\,d^2/dt^2-V^{\prime\prime}(q_c(t))$}
Expanding a <configuration-space path integral> about a classical trajectory $q_c$ with fixed endpoints gives $S[q_c+f]=S[q_c]+\tfrac12\int f\mathcal O_cf\,dt+O(f^3)$, 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 $\mathcal O_0=-m\,d^2/dt^2$, 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.
Back to article page
