Nonrelativistic Fundamental Quantum and Classical Wave Equations

TL;DR

This paper derives a variety of symmetric and asymmetric wave equations from Galilean group representations, including known and new equations, and discusses their physical implications in quantum and classical contexts.

Contribution

It introduces a systematic derivation of fundamental and non-fundamental wave equations from symmetry principles, expanding the theoretical framework of wave mechanics.

Findings

Derived new asymmetric wave equations invariant under Galilean transformations.

Connected specific equations to quantum and classical physical settings.

Discussed physical implications of fundamental and non-fundamental wave theories.

Abstract

The irreducible representations of the extended Galilean group are used to derive infinite sets of symmetric and asymmetric second-order differential equations with constant coeffcients. All derived equations are local and their Lagrangians exist. It is shown that the asymmetric equations are Galilean invariant but the symmetric ones are not. By specifying quantum and classical physical settings, the constants in the equations are determined and the fundamental wave equations, including the Schr\"odinger, Schr\"odinger-like and new asymmetric equations, are obtained; the derived wave equation is non-fundamental. Formulation of wave theories based on the fundamental and non-fundamental wave equaions is considered, and physical implications of these theories on the wave description are discussed.