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

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.Comment: 15 page

Our Trysts with `Bal' and Noncommutative Geometry

This contributory article begins with our fond and sincere reminiscences about our beloved Prof. A.P. Balachandran. In the main part, we discuss our recent formulation of quantum mechanics on (1+1)D noncommutative space-time using Hilbert-Schmidt operators. As an application, we demonstrate how geometrical phase in a system of time-dependent forced harmonic oscillator living in the Moyal space-time can emerge.Comment: In Honor of A.P.Balachandran on the Occasion of His 85th Birthday (To be published in the Festschrift volume: Particles, Fields and Topology by World Scientific, Singapore