Covariant formulation of Noether's Theorem for translations on kappa-Minkowski spacetime

TL;DR

This paper develops a covariant formulation of Noether's theorem for translations in kappa-Minkowski noncommutative spacetime, deriving conserved currents and charges for a free scalar field and establishing their relation to relativistic dispersion.

Contribution

It introduces a covariant approach to Noether's theorem in noncommutative spacetime using kappa-Poincare' calculus, deriving conserved quantities for scalar fields.

Findings

Five conserved Noether currents for scalar theory.

Energy-momentum charges satisfy a relativistic dispersion relation.

Derived equations of motion from Hamilton's principle in noncommutative spacetime.

Abstract

The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative spacetimes. In this paper, we formulate Noether's theorem for translations of kappa-Minkowski noncommutative spacetime on the basis of the 5-dimensional kappa-Poincare' covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy-momentum charges. By applying our result to plane waves it follows that the energy-momentum charges satisfy a special-relativity dispersion relation with a generalized mass given by the fifth charge. In this paper we provide also a rigorous derivation of the equation of motion from Hamilton's principle in noncommutative spacetime, which is necessary for the Noether analysis.