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 Incoming links: Momentum-space propagator in a uniform force

For the momentum-space propagator in a uniform force, the stated Fourier convention makes

x

i ℏ \partial_{p}

. Hence

[i ℏ \partial_{t} - \frac{p ^{2}}{2 m} - i α ℏ \partial_{p}] K_{p} = i ℏ δ (T) δ (p - p^{'}) .

For

T > 0

, the characteristic has

p (s) = p^{'} - α (s - t^{'})

. Along it the amplitude acquires quantum phase

- \int_{t^{'}}^{t} p (s)^{2} d s / (2 m ℏ)

. momentum is translated without a Jacobian, giving

K_{p} = θ (T) δ (p - p^{'} + α T) exp [\frac{i ( p ^{3} - p ^{'3} )}{6 α m ℏ}] .

The cubic expression is evaluated on the delta-function support. There it equals

- i [p^{'2} T - α p^{'} T^{2} + α^{2} T^{3} /3] / (2 m ℏ)

, making both its sign and its continuous

α \to 0

limit clear. The jump at

T = 0

is the identity momentum kernel.