Light-Ray Radon Transform for Abelianin and Nonabelian Connection in 3 and 4 Dimensional Space with Minkowsky Metric

We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R).Comment: We add an important discussion part, establishing the relation of our Radon transform with the self-dual Yang-Mills, string theory, and the represntation theory of the group SL(2,R

The Associated Metric for a Particle in a Quantum Energy Level

We show that the probabilistic distribution over the space in the spectator world, can be associated via noncommutative geometry (with some modifications) to a metric in which the particle lives. According to this geometrical view, the metric in the particle world is ``contracted'' or ``stretched'' in an inverse proportion to the probability distribution.Comment: 14 pages, latex, epsf, 3 figures. Some clarifications were adde

Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis

We consider the Lie group $\mathbb{R}^D_\kappa$ generated by the Lie algebra of $\kappa$-Minkowski space. Imposing the invariance of the metric under the pull-back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with $\mathbb{R}^D_\kappa$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_\kappa$ relevant for the realization of the non-commutative $\kappa$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg algebra. Using a suitable set of generalized coherent states, we select the particular Hilbert space and realize the non-commutative $\kappa$-Minkowski space as an algebra of the Hilbert-Schmidt operators. We define dequantization map and fuzzy variant of the Laplace-Beltrami operator such that dequantization map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami operator on the half of de Sitter space-time.Comment: 21 pages, v3 differs from version published in Fortschritte der Physik by a note and references added and adjuste

SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

We show how the density-matrix numerical renormalization group (DM-NRG) method can be used in combination with non-Abelian symmetries such as SU(N), where the decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.Comment: 18 pages, 9 figure

Isotropic subbundles of TM⊕T∗MTM\oplus T^*M

We define integrable, big-isotropic structures on a manifold $M$ as subbundles $E\subseteq TM\oplus T^*M$ that are isotropic with respect to the natural, neutral metric (pairing) $g$ of $TM\oplus T^*M$ and are closed by Courant brackets (this also implies that $[E,E^{\perp_g}]\subseteq E^{\perp_g}$). We give the interpretation of such a structure by objects of $M$, we discuss the local geometry of the structure and we give a reduction theorem.Comment: LaTex, 37 pages, minimization of the defining condition

Spin-S bilayer Heisenberg models: Mean-field arguments and numerical calculations

Spin-S bilayer Heisenberg models (nearest-neighbor square lattice antiferromagnets in each layer, with antiferromagnetic interlayer couplings) are treated using dimer mean-field theory for general S and high-order expansions about the dimer limit for S=1, 3/2,...,4. We suggest that the transition between the dimer phase at weak intraplane coupling and the Neel phase at strong intraplane coupling is continuous for all S, contrary to a recent suggestion based on Schwinger boson mean-field theory. We also present results for S=1 layers based on expansions about the Ising limit: In every respect the S=1 bilayers appear to behave like S=1/2 bilayers, further supporting our picture for the nature of the order-disorder phase transition.Comment: 6 pages, Revtex 3.0, 8 figures (not embedded in text