Post-Galilean transformations of space and time derivatives and their consequences

TL;DR

This paper introduces a new particle-wave equation derived from post-Galilean transformations and quantum mechanics, expanding the theoretical framework for spinless particles and revealing new invariance properties.

Contribution

It presents a novel particle-wave equation beyond Klein-Gordon and links it to Dirac's equation under specific conditions, highlighting new invariance properties.

Findings

Derived a new particle-wave equation for spinless particles

Connected the new equation to Dirac's equation under specific parameters

Established invariance of Biot-Savart law and continuity equations

Abstract

Using post-Galilean space and time derivatives transformations and quantum mechanics, we have found a new particle-wave equation besides the Klein-Gordon equation describing a spinless scalar particle. This new equation can also be obtained from Dirac's equation if $\beta=\gamma(1\pm\frac{v}{c})$. Biot-Savart law and additional continuity equations are obtained as a consequence of the invariance of Dirac's equation and Maxwell's equations under these transformations.