Momentum-space propagator in a uniform force ($K_p=\theta (T)\delta (p-p^{\prime }+\alpha T)e^{i(p^3-p^{\prime 3})/(6\alpha m\hslash )}$)

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.