Biswajit Chakraborty

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.

Dual families of noncommutative quantum systems

We demonstrate how a one parameter family of interacting noncommuting Hamiltonians, which are physically equivalent, can be constructed in noncommutative quantum mechanics. This construction is carried out exactly (to all orders in the noncommutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low-energy sector, can be constructed between the interacting commutative and a noninteracting noncommutative Hamiltonian. This approximation holds to order 1/B and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.

Twisted Galilean symmetry and the Pauli principle at low energies

We show the twisted Galilean invariance of the noncommutative parameter, even in the presence of spacetime noncommutativity. We then obtain the deformed algebra of the Schrodinger field in configuration and momentum space by studying the action of the twisted Galilean group on the non-relativistic limit of the Klein–Gordon field. Using this deformed algebra we compute the two-particle correlation function to study the possible extent to which the previously proposed violation of the Pauli principle may impact at low energies. It is concluded that any possible effect is probably well beyond detection at current energies.

Seiberg-Witten map and Galilean symmetry violation in a noncommutative planar system

An effective U(1) gauge invariant theory is constructed for a noncommutative Schroedinger field coupled to a background U(1){sub *} gauge field in 2+1-dimensions using first order Seiberg-Witten map. We show that this effective theory can be cast in the form of usual Schroedinger action with interaction terms of noncommutative origin provided the gauge field is of background type with constant magnetic field. The Galilean symmetry is investigated and a violation is found in the boost sector. We also consider the problem of Hall conductivity in this framework.

Interactions and non-commutativity in quantum Hall systems

We discuss the role that interactions play in the non-commutative structure that arises when the relative coordinates of two interacting particles are projected onto the lowest Landau level. It is shown that the interactions in general renormalize the non-commutative parameter away from the non-interacting value

Spectrum of the non-commutative spherical well

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A unifying perspective on the Moyal and Voros products and their physical meanings

. The effective non-commutative parameter is in general also angular momentum dependent. An heuristic argument, based on the non-commutative coordinates, is given to find the filling fractions at incompressibilty, which are in general renormalized by the interactions, and the results are consistent with known results in the case of singular magnetic fields. Comment: 8 pages, 2 figures

Noncommutative gauge theories and Lorentz symmetry

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular, we discuss the infinite and finite non-commutative spherical wells in two dimensions. Using this, bound states and scattering can be discussed unambiguously. Here we focus on the infinite well and solution for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.

Noncommutativity in open string: new results in a gauge-independent analysis

The Moyal and Voros formulations of non-commutative quantum field theory have been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular, we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective, the two formulations are therefore completely equivalent, but we also argue that only the Voros formulation admits a consistent physical interpretation. These considerations are elucidated by considering the free-particle transition amplitude in the two representations.

Wigner oscillators, twisted Hopf algebras, and second quantization

We explicitly derive, following a Noether-like approach, the criteria for preserving Poincare invariance in noncommutative gauge theories. Using these criteria we discuss the various spacetime symmetries in such theories. It is shown that, interpreted appropriately, Poincare invariance holds. The analysis is performed in both the commutative as well as noncommutative descriptions and a compatibility between the two is also established.