Principles of Quantum Mechanics · Problem 21.17

The phase-space path integral is the most general form—it treats position and momentum as independent variables evolving through time. Starting from the position-space path integral, you'll insert complete sets of momentum eigenstates at each time slice to "promote" momentum to a dynamical degree of freedom; this transforms the Hamiltonian constraint into an explicit integral over all momentum paths. The key insight is recognizing that the exponent $p\dot{x} - H(p,x)$ is the Legendre transform connecting phase space back to configuration space—when you Gaussian-integrate out $p$, its equation of motion $\dot{x} = \partial H/\partial p$ automatically emerges, and you recover the Lagrangian $L = p\dot{x} - H$.

The phase-space path integral is the most general form—it treats position and momentum as independent variables evolving through time. Starting from the position-space path integral, you'll insert complete sets of momentum eigenstates at each time slice to "promote" momentum to a dynamical degree of freedom; this transforms the Hamiltonian constraint into an explicit integral over all momentum paths. The key insight is recognizing that the exponent $p\dot{x} - H(p,x)$ is the Legendre transform connecting phase space back to configuration space—when you Gaussian-integrate out $p$, its equation of motion $\dot{x} = \partial H/\partial p$ automatically emerges, and you recover the Lagrangian $L = p\dot{x} - H$.