Quantum-mechanical propagator in a constant force ($K_F(q_1,q_0;T)$)

In units $\hslash =1$, a particle of mass $m$ with potential $V(q)=-Fq$ has classical path $q_c(t)=q_0+[(q_1-q_0)/T-FT/(2m)]t+F{t}^2/(2m)$. Its action and exact quantum-mechanical propagator areThe fluctuation operator is the free one because the potential is linear. This is a special case of classical-path factorization of a quadratic path integral.

For potential $V(x)=\alpha x$, momentum follows a characteristic with $\dot{p}=-\alpha$. The retarded quantum-mechanical propagator translates its momentum argument and accumulates the kinetic action along that characteristic. Its apparent division by $\alpha$ is removable on the delta-function support; at zero force it becomes the free diagonal momentum kernel. Fourier transform gives the usual constant-force position kernel with a cubic-time action term.