Gaussian momentum integration in a phase-space path integral ($\int dp{e}^{-\Delta t{p}^2/(2m)+ip\Delta q}=\sqrt{2\pi m/\Delta t}{e}^{-m\Delta q^2/(2\Delta t)}$)

Integrating each momentum in a quadratic Hamiltonian yields the configuration-space kinetic action and its time-slice normalization. A raw $dp$ measure gives $\sqrt{2\pi m/\Delta t}$; the normalized Fourier measure $dp/(2\pi )$ gives $\sqrt{m/(2\pi \Delta t)}$. Dropping the time dependence changes the quantum-mechanical propagator.