Dirac's Equation in $R$-Minkowski Spacetime

TL;DR

This paper derives the Dirac equation within the $R$-Minkowski spacetime, revealing its equivalence to the de Sitter spacetime's Dirac equation, and introduces a novel Schrödinger picture approach for solutions.

Contribution

It establishes the Dirac equation in $R$-Minkowski spacetime as equivalent to that in de Sitter spacetime and proposes a new solution method using the Schrödinger picture.

Findings

Dirac equation in $R$-Minkowski spacetime matches de Sitter spacetime form

New Schrödinger picture approach for solving Dirac's equation

Connection between $R$-Poincaré algebra and de Sitter spacetime

Abstract

We recently constructed the $R$-Poincar\'e algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the $R$-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac's equation in the de Sitter spacetime using the Schr\"{o}dinger picture.