= Solution
For the <momentum-space propagator in a uniform force>, the stated Fourier convention makes $x$ act on <momentum> <wavefunctions> as $i\hbar\partial_p$. Hence
$$
\boxed{\left[i\hbar\partial_t-\frac{p^2}{2m}-i\alpha\hbar\partial_p\right]K_p
=i\hbar\delta(T)\delta(p-p')}.
$$
For $T>0$, the characteristic has $p(s)=p'-\alpha(s-t')$. Along it the amplitude acquires <quantum phase> $-\int_{t'}^t p(s)^2ds/(2m\hbar)$. <momentum> is translated without a Jacobian, giving
$$
\boxed{K_p=\theta(T)\delta(p-p'+\alpha T)
\exp\left[\frac{i(p^3-p'^3)}{6\alpha m\hbar}\right]}.
$$
The cubic expression is evaluated on the delta-function support. There it equals $-i[p'^2T-\alpha p'T^2+\alpha^2T^3/3]/(2m\hbar)$, making both its sign and its continuous $\alpha\to0$ limit clear. The jump at $T=0$ is the identity <momentum> kernel.
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