= Quantum-mechanical propagator in a constant force
{title2=$K_F(q_1,q_0;T)$}
In units $\hbar=1$, a particle of mass $m$ with potential $V(q)=-Fq$ has classical path $q_c(t)=q_0+[(q_1-q_0)/T-FT/(2m)]t+Ft^2/(2m)$. Its action and exact <quantum-mechanical propagator> are
$$
S_c=\frac{m(q_1-q_0)^2}{2T}+\frac{FT(q_1+q_0)}2-\frac{F^2T^3}{24m},\qquad K_F=\sqrt{\frac{m}{2\pi iT}}e^{iS_c}.
$$
The 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>.
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