Derivation of Dirac, Klein-Gordon, Schrodinger, Diffusion and quantum heat transport equations from a universal quantum wave equation

TL;DR

This paper derives a universal quantum wave equation that can produce multiple fundamental equations like Dirac, Klein-Gordon, Schrödinger, and quantum heat equations through complex transformations, revealing new symmetries.

Contribution

It introduces a universal quantum wave equation capable of deriving several key quantum equations via complex transformations, highlighting underlying symmetries.

Findings

Derivation of Dirac, Klein-Gordon, Schrödinger, and quantum heat equations from a single universal equation.

Identification of new symmetries relating these equations through complex transformations.

Demonstration of how the Dirac equation can be obtained by specific complex substitutions.

Abstract

A universal quantum wave equation that yields Dirac, Klein-Gordon, Schrodinger and quantum heat equations is derived. These equations are related by complex transformation of space, time and mass. The new symmetry exhibited by these equations is investigated. The universal quantum equation yields Dirac equation in two ways: firstly by replacing the particle my $m_0$ by $im_0$, and secondly by changing space and time coordinates by $it$ and $i\vec{r}$, respectively.