From Hamilton–Jacobi Theory to the Relativistic Schrödinger picture via von Neumann-like linear extension in tempered distribution spaces

Introduction

This work establishes a novel connection between the relativistic Hamilton–Jacobi equation (HJE) and the relativistic massive Schrödinger equation (RSE) in free space, achieved without relying on the semiclassical limit (ℏ → 0). The approach operates mode-by-mode on spectral fibers associating with any Minkowski momentum bra S_p = <p, . > the de Broglie wave η_p. The families S and η generate distinct subspaces in tempered distribution spaces: the former spans a real four-dimensional vector space isomorphic to Minkowski momentum space, while the latter comprises the full space of complex tempered distributions on Minkowski space.

Methods

The key families of distributions — Minkowski bras (S) and de Broglie basis (η) — span (via suitable subfamilies) the solution spaces of the HJE’s and RSE’s, respectively. Employing Schwartz linear algebra for complex tempered fields, we apply a von Neumann-style linear-continuous operator extension to lift the HJE (formulated in complex variables) from certainty momentum states |p⟩ to complex amplitude-probability states ψ. This extension mirrors procedures used in game theory and follows standard von Neumann techniques. The construction is further extended to the Maxwell–Schrödinger formalism - in complex tempered distribution 3-field - through de Broglie-Maxwell isomorphisms (Fe), which map wave distributions to corresponding electromagnetic-like fields while preserving translation representations, dispersion relations, and polarization structures.

Results

The principal finding demonstrates that the relativistic massive Schrödinger equations are von Neumann-like linear extensions of the relativistic HJE in complex form. These equations are uniquely determined spectrally by the Einstein energy-momentum relation within the tempered distribution framework.

Conclusions

This framework provides a vast, concrete (although partial) unification of classical relativistic mechanics, relativistic quantum mechanics for massive particles, and Maxwellian field theory - all within the setting of tempered distributions - offering new insights into the foundational relationships among these domains.