On the biparametric quantum deformation of GL(2) x GL(1)

We study the biparametric quantum deformation of GL(2) x GL(1) and exhibit its cross-product structure. We derive explictly the associated dual algebra, i.e., the quantised universal enveloping algebra employing the R-matrix procedure. This facilitates construction of a bicovariant differential calculus which is also shown to have a cross-product structure. Finally, a Jordanian analogue of the deformation is presented as a cross-product algebra.Comment: 16 pages LaTeX, published in JM

Contraction of the G_r,s Quantum Group to its Nonstandard analogue and corresponding Coloured Quantum Groups

The quantum group G_r,s provides a realisation of the two parameter quantum GL_p,q(2) which is known to be related to the two parameter nonstandard GL_hh'(2) group via a contraction method. We apply the contraction procedure to G_r,s and obtain a new Jordanian quantum group G_m,k. Furthermore, we provide a realisation of GL_h,h'(2) in terms of G_m,k. The contraction procedure is then extended to the coloured quantum group GL_r{\lambda,\mu}(2) to yield a new Jordanian quantum group GL_m{\lambda,\mu}(2). Both G_r,s and G_m,k are then generalised to their coloured versions which inturn provide similar realisations of GL_r{\lambda,\mu}(2) and GL_m{\lambda,\mu}(2).Comment: 22 pages LaTeX, to be published in J. Math. Phy

Comment on "Groverian Entanglement Measure and Evolution of Entanglement in Search Algorithm for n(= 3, 5)-Qubit Systems with Real Coefficients" (Volume 6, Number 4, August 2007), by Arti Chamoli and C. M. Bhandari

We point out that the main results-the analytic expressions for the Groverian Measure of Entanglement, in the above mentioned paper are erroneous. The technical mistake of the paper is discussed. It is shown by an explicit example that the formula for calculating the Groverian measure yields G(|\psi>) = 0 for some entangled states.Comment: 4 pages, published online in Quantum Info. Process. on 24 July 200

Two-Parameter Differential Calculus on the h-Exterior Plane

We construct a two-parameter covariant differential calculus on the quantum $h$-exterior plane. We also give a deformation of the two-dimensional fermionic phase space.Comment: 7 page

How does Casimir energy fall? IV. Gravitational interaction of regularized quantum vacuum energy

Several years ago we demonstrated that the Casimir energy for perfectly reflecting and imperfectly reflecting parallel plates gravitated normally, that is, obeyed the equivalence principle. At that time the divergences in the theory were treated only formally, without proper regularization, and the coupling to gravity was limited to the canonical energy-momentum-stress tensor. Here we strengthen the result by removing both of those limitations. We consider, as a toy model, massless scalar fields interacting with semitransparent ($\delta$-function) potentials defining parallel plates, which become Dirichlet plates for strong coupling. We insert space and time point-split regulation parameters, and obtain well-defined contributions to the self- energy of each plate, and the interaction energy between the plates. (This self-energy does not vanish even in the conformally-coupled, strong-coupled limit.) We also compute the local energy density, which requires regularization near the plates. In general, the energy density includes a surface energy that resides precisely on the boundaries. This energy is also regulated. The gravitational interaction of this well-defined system is then investigated, and it is verified that the equivalence principle is satisfied.Comment: 14 pages, 4 figure