COVARIANT FORMULATION OF NOETHER'S THEOREM FOR κ-MINKOWSKI TRANSLATIONS

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 space–times. In this paper, we formulate Noether's theorem for translations of κ-Minkowski noncommutative space–time on the basis of the five-dimensional κ-Poincaré 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 space–time, which is necessary for the Noether analysis.