Particle Dynamics and Lie-algebraic type of Non-commutativity of space-time

TL;DR

This paper investigates particle dynamics in non-commutative space-time using Dirac's constraint analysis, revealing classical $oldsymbol{ ext{kappa-Minkowski}}$ algebra and undeformed symmetry generators.

Contribution

It introduces a novel approach to analyze non-commutative space-time effects on particle dynamics via constraint analysis and extended space formalism.

Findings

Deformed Dirac brackets similar to $oldsymbol{ ext{kappa}}$-deformed space-time.

Classical $oldsymbol{ ext{kappa-Minkowski}}$ algebra obtained from phase space analysis.

Undeformed Galilean and Poincaré algebra generators.

Abstract

In this paper, we present the results of our investigation relating particle dynamics and non-commutativity of space-time by using Dirac's constraint analysis. In this study, we re-parameterise the time $t=t(\tau)$ along with $x=x(\tau)$ and treat both as configuration space variables. Here, $\tau$ is a monotonic increasing parameter and the system evolves with this parameter. After constraint analysis, we find the deformed Dirac brackets similar to the $\kappa$-deformed space-time and also, get the deformed Hamilton's equations of motion. Moreover, we study the effect of non-commutativity on the generators of Galilean group and Poincare group and find undeformed form of the algebra. Also, we work on the extended space analysis in the Lagrangian formalism. We find the primary as well as the secondary constraints. Strikingly on calculating the Dirac brackets among the phase space variables, we obtain the classical version of $\kappa$-Minkowski algebra.