On the pre-metric foundations of wave mechanics I: massless waves

TL;DR

This paper explores the foundational geometric and physical principles underlying wave mechanics, proposing a new perspective on the transition from wave to particle-like behavior through extended matter motion and dispersion laws.

Contribution

It introduces a novel framework linking wave motion, dispersion laws, and geodesic structures, extending the understanding of wave-particle duality and quantum fluctuations.

Findings

Wave mechanics can be grounded in conservation laws and constitutive medium laws.

Degenerate Lorentzian structures emerge only in specific dispersion law cases.

Non-quadratic dispersion laws may contribute to quantum fluctuations.

Abstract

The mechanics of wave motion in a medium are founded in conservation laws for the physical quantities that the waves carry, combined with the constitutive laws of the medium, and define Lorentzian structures only in degenerate cases of the dispersion laws that follow from the field equations. It is suggested that the transition from wave motion to point motion is best factored into an intermediate step of extended matter motion, which then makes the dimension-codimension duality of waves and trajectories a natural consequence of the bicharacteristic (geodesic) foliation associated with the dispersion law. This process is illustrated in the conventional case of quadratic dispersion, as well as quartic ones, which include the Heisenberg-Euler dispersion law. It is suggested that the contributions to geodesic motion from the non-quadratic nature of a dispersion law might represent another source of quantum fluctuations about classical extremals, in addition to the diffraction effects that are left out by the geometrical optics approximation.