International Journal of Geometric Methods in Modern Physics

A modification of the standard product used in local field theory by means of an associative deformed product is proposed. We present a class of deformed products, one for every spin S = 0, 1/2, 1, that induces a nonlocal theory, displaying different form for different fields. This type of deformed product is naturally supersymmetric and it has an intriguing duality.

Deformation quantization of Poisson manifolds is studied within the framework of an expansion in powers of derivatives of Poisson structures. We construct the Lie group associated with a Poisson bracket algebra which defines a second order deformation in the derivative expansion.

In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.

A 2-dimensional strongly pseudoconvex Hartogs domain D_F can be equipped with a natural Kähler metric g_F. In this paper we prove that if the second term of Engliš's expansion of Rawnsley's epsilon function is constant then (D_F, g_F) is holomorphically isometric to an open subset of the 2-dimensional complex hyperbolic space.

We study abelian groups graded (or color) Koszul–Vinberg algebras and their modules. Koszul–Vinberg cohomology and homology of these algebras are studied. As applications, we investigate some extensions of graded algebras and graded modules.

We study lightlike surfaces in Minkowski 3-space.

The unconstrained frame-like formulation of an infinite tower of completely symmetric tensor gauge fields is reviewed and examined in the limit where the cosmological constant goes to zero. By partially fixing the gauge and solving the torsion constraints, the form of the gauge transformations in the unconstrained metric-like formulation are obtained till first order in a weak field expansion. The algebra of the corresponding gauge symmetries is shown to be equivalent, at this order and modulo (unphysical) gauge parameter redefinitions, to the Lie algebra of Hermitian differential operators on ℝⁿ, the restriction of which to the spin-two sector is the Lie algebra of infinitesimal diffeomorphisms.

We propose an approach based on a generalized quantum mechanics to deal with the basic features of the intrinsic spin Hall effect. This can be done by considering two decoupled harmonic oscillators on the noncommutative plane and evaluating the spin Hall conductivity. Focusing on the high frequency regime, we obtain a diagonalized Hamiltonian. After getting the corresponding spectrum, we show that there is a Hall conductivity without an external magnetic field, which is noncommutativity parameter θ-dependent. This allows us to make contact with the spin Hall effect and also give different interpretations. Fixing θ, one can recover three different approaches dealing with the phenomenon.

We shall prove here that conservation laws from Holst's Lagrangian, often used in LQG, do not agree with the corresponding conservation laws in standard GR. Nevertheless, these differences vanish on-shell, i.e. along solutions, so that they eventually define the same classical conserved quantities. Accordingly, they define in particular the same entropy of solutions, and the standard law is reproduced for systems described by Holst's Lagrangian.

This provides the classical support to the computation usually done in LQG for the entropy of black holes which is in turn used to fix the Barbero–Immirzi parameter.