Phase-space path integral ($K=\int \mathcal{D}q\mathcal{D}p{e}^{i\int (p\dot{q}-H)dt}$)

A phase-space path integral represents a quantum transition kernel by integrating both coordinates and conjugate momenta with action $\int (p\dot{q}-H)dt$. It follows from inserting position and momentum completeness relations between short-time evolution operators. For a quadratic momentum dependence the momentum variables can be integrated by a Gaussian integral.

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.

The first-order term means ${\sum }_i{p}_i(q_{i+1}-q_i)$ in a finite-dimensional integral. Specify the endpoints and an ordering prescription, then take the maximum time step to zero. Typical Euclidean paths are not classically differentiable, so the discrete prescription is essential.