On computing quantum waves and spin from classical and relativistic action

• URL: http://arxiv.org/abs/2405.06328v4
• Date: Tue, 05 Nov 2024 19:48:13 GMT
• Title: On computing quantum waves and spin from classical and relativistic action
• Authors: Winfried Lohmiller, Jean-Jacques Slotine,
• Abstract summary: We show that the Schroedinger equation of quantum physics can be solved using a generalized form of the classical Hamilton-Jacobi least action equation. The results, which extend to the relativistic setting, build on two developments. They suggest a smooth transition between physics across scales, with the Hamilton-Jacobi formalism extending to general relativity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show that the Schroedinger equation of quantum physics can be solved using a generalized form of the classical Hamilton-Jacobi least action equation, extending a key result of Feynman applicable only to quadratic actions. The results, which extend to the relativistic setting, build on two developments. The first is incorporating geometric constraints directly in the classical least action problem. This leads to multi-valued least action solutions where each local action is its own set element. The multiple solutions replace in part the probabilistic setting by the non-uniqueness of solutions of the constrained problem. For instance, in the double slit experiment or for a particle in a box, spatial inequality constraints create impulsive constraint forces, which lead to multiple path solutions. Second, an approximate mapping $ \ \Psi \approx e^{\frac{i }{\hbar} \Phi } \ $ between action $\Phi$ and wave function $\Psi$ has been known since Dirac and even Schroedinger. We show that this mapping can be made exact by introducing a compression ratio of the proposed multi-valued action, which can in turn be interpreted as a probability density on classical trajectories of a fluid flow field. Branch points of the multi-valued least action imply a quantum wave collapse. These developments leave the results of associated Feynman path integrals unchanged, but the computation can be greatly simplified as only multi-valued least actions are used, avoiding time-slicing and zig-zag trajectories altogether. They also suggest a smooth transition between physics across scales, with the Hamilton-Jacobi formalism extending to general relativity, in a coordinate-invariant framework. In particular, the Klein-Gordon equation may have a natural extension to general relativity.

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