Topics in Noncommutative Gauge Theories and Deformed Relativistic Theories

TL;DR

This thesis explores noncommutative gauge theories, discovering new solitons and instantons using generalized Bose operators, and investigates the implications of noncommutativity in quantum optics and DSR theories.

Contribution

It introduces a novel approach to static solutions in noncommutative gauge theories using generalized Bose operators and finds new instanton solutions distinct from previous constructions.

Findings

New static solitons and instantons in noncommutative gauge theories

Alternative descriptions of known magnetic flux tube solutions

Implications of noncommutativity in quantum optics and DSR thermodynamics

Abstract

This is my PhD thesis. In this thesis we study the gauge theories on noncommutative Moyal space. We find new static solitons and instantons in terms of the so called generalized Bose operators. Generalized Bose operators are constructed to describe reducible representation of the oscillator algebra. They create/annihilate $k$-quanta, $k$ being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of generalized Bose operators. The Nielsen-Olesen vortex solutions found in terms of these operators reduce to the already found ones. On the contrary we find a class of new instaton solutions which are unitarily inequivalant to the the ones found from ADHM construction on noncommutative space. The charge of the instaton has a description in terms of the index representing the reducibility of the Fock space, i.e., $k$. After studying the static solitonic solutions in noncommutative Minkowski space and the instaton solutions in noncommutative Euclidean space we go on to study the implications of the time-space noncommutativity in Minkowski space. To understand it properly we study the time-dependent transitions of a forced harmonic oscillator in noncommutative 1+1 dimensional spacetime. We also try to understand the implications of the found results in the context of quantum optics. We then shift to the so called DSR theories which are related to a different kind of noncommutative ($\kappa$-Minkowski) space. DSR (Doubly/Deformed Special Relativity) aims to search for an alternate relativistic theory which keeps a length/energy scale (the Planck scale) and a velocity scale (the speed of light scale) invariant. We study thermodynamics of an ideal gas in such a scenario.