Pradip Mukherjee

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.

Note on the noncommutative correction to gravity

An apparent contradiction in the leading order correction to noncommutative (NC) gravity reported in the literature has been pointed out. We show by direct computation that actually there is no such controvarsy and all perturbative NC corrections start from the second order in the NC parameter. The role of symmetries in the vanishing of the first order correction is manifest in our calculation.

Lie algebraic noncommutative gravity

We exploit the Seiberg-Witten map technique to formulate the theory of gravity defined on a Lie algebraic noncommutative space-time. Detailed expressions of the Seiberg-Witten maps for the gauge parameters, gauge potentials, and the field strengths have been worked out. Our results demonstrate that notwithstanding the introduction of more general noncommutative structure there is no first order correction, exactly as happens for a canonical (i.e. constant) noncommutativity.

A canonical approach to the quantization of the damped harmonic oscillator

We provide a new canonical approach for studying the quantum mechanical damped harmonic oscillator based on the doubling of degrees of freedom approach. Explicit expressions for Lagrangians of the elementary modes of the problem, characterizing both forward and backward time propagation, are given. A Hamiltonian analysis, showing the equivalence with the Lagrangian approach, is also done. Based on this Hamiltonian analysis, the quantization of the model is discussed.

Deformed Reissner-Nordstrom solutions in noncommutative gravity

The leading order corrections to Reissner-Nordstrom solutions of the Einsteins equations on noncommutative spacetime have been worked out based on a noncommutative gauge theory of gravity. From the corrected metric the horizons have been derived and the curvature scalar is also computed. The introduction of noncommutativity leads to the removal of the coordinate singularities.

Symmetries of topological gravity with torsion in the hamiltonian and lagrangian formalisms

A systematic analysis of the symmetries of topological 3D gravity with torsion and a cosmological term, in the first order formalism, has been performed in details - both in the hamiltonian and lagrangian formalisms. This illuminates the connection between the symmetries of curved spacetime (diffeomorphisms plus local Lorentz transformations) with the Poincare gauge transformations. The Poincare gauge symmetries of the action are shown to be inequivalent to its gauge symmetries. Finally, the complete analysis is compared with the metric formulation where the diffeomorphism symmetry is shown to be equivalent to the gauge symmetry.

Gauge symmetry and W-algebra in higher derivative systems

The problem of gauge symmetry in higher derivative Lagrangian systems is discussed from a Hamiltonian point of view. The number of independent gauge parameters is shown to be in general less than the number of independent primary first classc onstraints, thereby distinguishing it from conventional first order systems. Different models have been considered as illustrative examples. In particular we show a direct connection between the gauge symmetry and the W-algebra for the rigid relativistic particle.

GAUGE INVARIANCES VIS-À-VIS DIFFEOMORPHISMS IN SECOND-ORDER METRIC GRAVITY: A NEW HAMILTONIAN APPROACH

A new analysis of the gauge invariances and their unity with diffeomorphism invariances in second-order metric gravity is presented which strictly follows Diracs constrained Hamiltonian approach.

General algorithm for nonrelativistic diffeomorphism invariance

An algorithmic approach towards the formulation of non-relativistic diffeomorphism invariance has been developed which involves both matter and gauge fields. A step by step procedure has been provided which can accommodate all types of (abelian) gauge interaction. The algorithm is applied to the problem of a two dimensional electron moving under an external field and also under the Chern-Simons dynamics.

New Hamiltonian analysis of Regge Teitelboim minisuperspace cosmology

A new Hamiltonian formulation of the minisuperspace cosmology following from the geodetic brane gravity model introduced by Regge and Teitelboim is presented. The model is considered in the framework of higher derivative theories which facilitates Hamiltonian formulation. The analysis is done using the equivalent first order approach. The gauge generator containing the exact number of gauge parameters is constructed. Equivalence between the gauge and reparametrization symmetries has been demonstrated. Complete gauge fixed computations have been provided and formal quantization is done indicating the Wheeler de Witt equation. Compatibility with existing results is shown.