Subir Ghosh

Topics in Noncommutative Geometry Inspired Physics

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.

Quantisation of O(N) invariant nonlinear sigma model in the Batalin-Tyutin formalism

We quantise the O(N) nonlinear sigma model using the Batalin-Tyutin (BT) approach of converting a second class system into a first class system. It is a nontrivial application of the BT method since the quantisation of this model by the conventional Dirac procedure suffers from operator ordering ambiguities. The first class constraints, the BRST hamiltonian and the BRST charge are explicitly computed. The partition function is constructed and evaluated in the unitary gauge and a multiplier (ghost) dependent gauge.

Spinning particles in 2 + 1 dimensions

Abstract Lagrangian and Hamiltonian formulations of a free spinning particle in 2 + 1-dimensions or anyon are established, following closely the analysis of Hanson and Regge. Two viable (and inequivalent) Lagrangians are derived. It is also argued that one of them is more favourable. In the Hamiltonian analysis non-trivial Dirac Brackets of the fundamental variables are computed for both the models. Important qualitative differences with a recently proposed model for anyons are pointed out.

Bosonization in the noncommutative plane

Abstract In this Letter, we study bosonization of the noncommutative massive Thirring model in 2+1 dimensions. We show that, contrary to the duality between massive Thirring model and Maxwell–Chern–Simons model in ordinary spacetime, in the low energy (or large fermion mass) limit, their noncommutative versions are not equivalent, in the same approximation.

The chiral oscillator and its applications in quantum theory

The fundamental importance of the chiral oscillator is elaborated. Its quantum invariants are computed. As an application the Zeeman effect is analysed. We also show that the chiral oscillator is the most basic example of a duality invariant model, simulating the effect of the familiar electric - magnetic duality.The fundamental importance of the chiral oscillator is elaborated. Its quantum invariants are computed. As an application the Zeeman effect is analysed. We also show that the chiral oscillator is the most basic example of a duality invariant model, simulating the effect of the familiar electric-magnetic duality.

Non-abelian bosonisation in three-dimensional field theory

We develop a method based on the generalised Stuckelberg prescription for discussing bosonisation in the low energy regime of the SU(2) massive Thirring model in 2+1 dimensions. For arbitrary values of the coupling parameter the bosonised theory is found to be a nonabelian gauge theory whose physical sector is explicitly obtained. In the case of vanishing coupling this gauge theory can be identified with the SU(2) Yang-Mills Chern-Simons theory in the limit when the Yang-Mills term vanishes. Bosonisation identities for the fermionic current are derived.Abstract We develop a method based on the generalised Stuckelberg prescription for discussing bosonisation in the low energy regime of the SU(2) massive Thirring model in 2+1 dimensions. For arbitrary values of the coupling parameter the bosonised theory is found to be a non-abelian gauge theory whose physical sector is explicitly obtained. In the case of vanishing coupling this gauge theory can be identified with the SU(2) Yang-Mills-Chern-Simons theory in the limit when the Yang-Mills term vanishes. Bosonisation identities for the fermionic current are derived.

Noncommutative geometry and geometric phases

We have studied particle motion in generalized forms of noncommutative phase space, that simulate monopole and other forms of Berry curvature, that can be identified as effective internal magnetic fields, in coordinate and momentum space. The Ahranov-Bohm effect has been considered in this form of phase space, with operatorial structures of noncommutativity. The physical significance of our results is also discussed.

GAUGE-INVARIANT EXTENSION OF LINEARIZED HORAVA GRAVITY

In the present paper we have constructed a gauge invariant extension of a generic Horava Gravity (HG) model (with quadratic curvature terms) in linearized version in a systematic procedure. No additional fields are introduced. The linearized HG model is explicitly shown to be a gauge fixed version of the Einstein Gravity (EG) thus proving the Bellorin-Restuccia conjecture in a robust way. In the process we have explicitly computed the correct Hamiltonian dynamics using Dirac Brackets appearing from the Second Class Constraints present in the HG model. We comment on applying this scheme to the full non-linear HG.

A novel 'magnetic' field and its dual non-commutative phase space

In this Letter we have studied a new form of non-commutative (NC) phase space with an operatorial form of noncommutativity. A point particle in this space feels the effect of an interaction with an “internal” magnetic field, that is singular at a specific position θ−1. By “internal” we mean that the effective magnetic fields depends essentially on the particle properties and modifies the symplectic structure. Here θ is the NC parameter and induces the coupling between the particle and the “internal” magnetic field. The magnetic moment of the particle is computed. Interaction with an external physical magnetic field reveals interesting features induced by the inherent fuzziness of the NC phase space: introduction of non-trivial structures into the charge and mass of the particle and possibility of the particle dynamics collapsing to a Hall type of motion. The dynamics is studied both from Lagrangian and symplectic (Hamiltonian) points of view. The canonical (Darboux) variables are also identified. We briefly comment, that the model presented here, can play interesting role in the context of (recently observed) real space Berry curvature in material systems.

Noncommutative Chern-Simons soliton

We have studied noncommutative extension of the relativistic Chern-Simons-Higgs model, in the first nontrivial order in {theta}, with only spatial noncommutativity. Both Lagrangian and Hamiltonian formulations of the problem have been discussed, with the focus being on the canonical and symmetric forms of the energy-momentum tensor. In the Hamiltonian scheme, constraint analysis and the induced Dirac brackets have been provided. The space-time translation generators and their actions on the fields are discussed in detail. The effects of noncommutativity on the soliton solutions have been analyzed thoroughly and we have come up with some interesting observations. Considering the relative strength of the noncommutative effects, we have shown that there is a universal character in the noncommutative correction to the magnetic field--it depends only on {theta}. On the other hand, in the cases of all other observables of physical interest, such as the potential profile, soliton mass, or the electric field, {theta} as well as {tau} (the latter comprised solely of commutative Chern-Simons-Higgs model parameters) appear on similar footings. Lastly, we have shown that noncommutativity imposes a further restriction on the form of the Higgs field so that the Bogomolny-Prasad-Sommerfeld equations are compatible with the full variational equations of motion.