The Scretching–Planck Action as a Path Integral: Density and Phase from the Sum over Histories

This paper formulates the Schrödinger field path integral in Scretching–Planck (SP) density–phase variables, \(\psi=\sqrt{P}\,e^{iS/\hbar}\). Substituting into the Schrödinger Lagrangian gives \(\mathcal L_{\mathrm{Sch}}=-\mathcal L_{\mathrm{SP}}\), so the real-time action weight in these variables is \(e^{-iS_{\mathrm{SP}}/\hbar}\). In the particle propagator, stationary phase recovers Hamilton’s principal function and the Van Vleck amplitude. The SP quantum potential is determined by the curvature of the propagator’s amplitude wherever that amplitude is regular and nonzero. In the time-sliced example, the error decreases quadratically with the time step. The SP Hamilton–Jacobi equation has an RMS residual of \(1.6\times10^{-4}\), compared with \(1.365\) when the quantum term is omitted. In imaginary time, the transfer matrix extracts the ground-state energy and density. The resulting density satisfies the stationary balance \(V+Q_{\mathrm{SP}}[P_0]=E_0\) in the benchmark calculations. For the double well, path-integral Monte Carlo gives \(E_0=1.1458\pm0.0023\), compared with the reference value \(1.1439\). A ground-state transformation converts the Euclidean evolution into a diffusion whose drift is the osmotic velocity \(u=(\hbar/2m)\partial_x\ln P_0\). The simulated stationary density differs from \(P_0\) by \(0.0033\) in \(L^1\). For a ring threaded by magnetic flux, the SP circulation sum and the path-winding sum with Aharonov–Bohm phases agree to \(2.5\times10^{-16}\) in the numerical check. Eight solved problems are included.