Klein-Gordon equation in $q$-deformed Euclidean space

TL;DR

This paper develops a $q$-deformed version of the Klein-Gordon equation in three-dimensional Euclidean space, finding solutions, propagators, and conservation laws within this quantum-deformed framework.

Contribution

It introduces the $q$-deformed Klein-Gordon equation, constructs plane wave solutions, and explores their properties and physical implications in a quantum-deformed setting.

Findings

Plane wave solutions form a complete orthogonal system

Derived propagators for the $q$-deformed equations

Established continuity equations for conserved quantities

Abstract

We introduce $q$-versions of the Klein-Gordon equation in the three-dimensional $q$-deformed Euclidean space. We determine plane wave solutions to our $q$-deformed Klein-Gordon equations. We show that these plane wave solutions form a complete orthogonal system. We discuss the propagators of our $q$-deformed Klein-Gordon equations. We derive continuity equations for the charge density, the energy density, and the momentum density of a $q$-deformed spin-zero particle.