On computing quantum waves exactly from classical and relativistic action

Abstract

We show that the Schr\"odinger equation can be solved exactly based only on classical least action. Fundamental postulates of quantum mechanics can in turn be derived directly from this construction. The results extend to the relativistic Klein-Gordon, Pauli, Dirac, and Maxwell equations, and suggest a smooth transition between physics across scales. Most quantum mechanics problems have classical versions which involve multiple least action solutions. The associated classical multipaths stem either from the initial position or momentum distribution, or from branch points, generated, e.g., by a multiply connected manifold (double slit experiment), by spatial inequality constraints (particle in a box), or by a singularity (Coulomb potential). 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, where a key point is that $\rho$ can be easily computed from $\phi$ along each extremal action path. The construction is general and does not involve any quasi-classical approximation. Examples illustrate how the quantum wave functions for the double-slit experiment or e.g., the hydrogen atom can be computed exactly from their classical least action counterparts. In a quantum measurement process, randomness originates from the determined forward mapping of an initial classical density distribution. In the Einstein-Podolsky-Rosen experiment, while Bell's inequalities are violated, from this perspective there is indeed a hidden variable in the form of a complex spinor. These results also provide a simpler computational alternative to Feynman path integrals, as they use only a minimal subset of classical paths and avoid zig-zag paths and time-slicing altogether.