Revisiting quantum relativistic effects from phase transition by catastrophe theory

TL;DR

This paper explores quantum relativistic effects as phase transitions using catastrophe theory, deriving relativistic equations from a modified Schrödinger framework to provide a new perspective on high-speed particle behavior.

Contribution

It introduces a novel approach by modeling relativistic effects as phase transitions via catastrophe theory, deriving Klein-Gordon and Dirac equations from a modified Schrödinger equation.

Findings

Relativistic effects can be modeled as phase transitions.

Derived Klein-Gordon and Dirac equations from a modified Schrödinger framework.

Quantum relativistic effects are describable using catastrophe models.

Abstract

In this paper we start from the Schr\"odinger equation to revisit some classical quantum mechanics from the perspective of phase transition process. Here the relativistic effect of particles moving at high speed can be regarded as the phase transition process when the velocity variable increases. Considering that the catastrophe models could describe qualitatively any phase transition process, we adopt the simplest folding catastrophe type as the potential function in the Schr\"odinger equation to obtain a revised Schr\"odinger relativistic equation through the dimensionless analysis first, and then further to derive out the steady-state Klein-Gordon equation and Dirac relativistic equation gradually. These results reveal that the quantum relativistic effect could be considered as the phase transition process, which could be described by adopting the catastrophe models as the potential function in the classical Schr\"odinger equation.