Spectral distances on doubled Moyal plane using Dirac eigen-spinors

We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role throughout this paper. At the end, a toy model of "Higgs field" has been constructed by fluctuating the Dirac operator and the variation on the transverse distance has been demonstrated, through an explicit computation

Twisted Galilean symmetry and the Pauli principle at low energies

We show the twisted Galilean invariance of the noncommutative parameter, even in presence of space-time noncommutativity. We then obtain the deformed algebra of the Schr\"odinger 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.Comment: 16 pages Latex, 2 figures Some modifications made in the abstract, introduction, typographical errors correcte

Dual families of non-commutative quantum systems

We demonstrate how a one parameter family of interacting non-commuting Hamiltonians, which are physically equivalent, can be constructed in non-commutative quantum mechanics. This construction is carried out exactly (to all orders in the non-commutative 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 non-interacting non-commutative Hamiltonian. This approximation holds to order 1/B and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.Comment: 11 pages, no figure

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

An effective U(1) gauge invariant theory is constructed for a non-commutative Schrodinger field coupled to a background U(1)_{\star} 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 Schrodinger 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.Comment: REVTeX, 4 pages, Title changed, Paper shortened, Appendix removed, A new section on Galilean symmetry adde

Thermal effective potential in two- and three- dimensional non-commutative spaces

Thermal correlation functions and the associated effective statistical potential are computed in two- and three-dimensional non-commutative space using an operator formulation that makes no reference to a star product. The corresponding results for the Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. The forms of the correlation function and the effective potential are found to be the same, except that in the Voros case the thermal length undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of $\sqrt{\theta}$. It is shown that in a suitable basis (called here quasi-commutative) in the multi-particle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases, with an appropriate non-commutative correction to the thermal length in the Voros case, and that the Pauli principle is restored.Comment: 31 pages, 7 figure

On the role of twisted statistics in the noncommutative degenerate electron gas

We consider the problem of a degenerate electron gas in the background of a uniformly distributed positive charge, ensuring overall neutrality of the system, in the presence of non-commutativity. In contrast to previous calculations that did not include twisted statistics, we find corrections to the ground state energy already at first order in perturbation theory when the twisted statistics is taken into account. These corrections arise since the interaction energy is sensitive to two particle correlations, which are modified for twisted anti-commutation relations