On computing quantum waves exactly from classical action (2405.06328v9)

Abstract: We show that the Schr\"odinger equation in quantum mechanics can be solved exactly based only on classical least action and its associated classical density. Most quantum mechanics problems have classical versions which involve multiple least action solutions. These extremal action paths may stem from spatial inequality constraints (as in the double slit experiment), from singularities in the Hamiltonian (as in a Coulomb potential), from a closed configuration manifold (as for a spinning particle), or just from the initial action distribution. We show that the exact Schr\"odinger wave function $\Psi$ of the original quantum problem can be constructed by combining this classical multi-valued action $\Phi$ with the density $\rho$ of the classical position dynamics, which can be computed from $\Phi$ along each extremal action path. This construction is general and does not involve any quasi-classical approximation. Quantum wave collapse corresponds to transitioning between multi-valued classical action branches at a branch point (e.g., position measurement), or to measuring the classical eigendensity field (e.g., momentum or energy measurement). Entanglement corresponds to a sum of classical individual particle actions mapping to a tensor product of spinors. Examples illustrate how the quantum wave functions for the double-slit experiment, the hydrogen atom, or EPR can be computed exactly from their classical least action counterparts. These coordinate-invariant results provide a simpler computing alternative to Feynman path integrals, as they use only a discrete set of classical paths and avoid zig-zag paths and time-slicing altogether. They extend to the relativistic Klein-Gordon and Dirac equation, and suggest a smooth transition between physics across scales.