Complex Covariance

TL;DR

This paper explores extending special relativity and quantum mechanics to complex-valued space-time and phase space, deriving Lorentz transformations for complex velocities and formulating non-Hermitian Klein-Gordon and Dirac equations.

Contribution

It introduces a framework for complex-valued space-time in relativity and quantum theory, including new Lorentz transformations and non-Hermitian wave equations.

Findings

Derived Lorentz transformations for complex velocities

Formulated non-Hermitian Klein-Gordon and Dirac equations

Proposed foundations for non-Hermitian quantum theory

Abstract

According to some generalized correspondence principle the classical limit of a non-Hermitian Quantum theory describing quantum degrees of freedom is expected to be well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity has been developed by Einstein merely for real-valued space-time and four-momentum we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, yet also to the non-Hermitian Klein-Gordon and Dirac equations which are to lay the foundations of a non-Hermitian quantum theory.