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

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

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

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

Noncommutative symmetric functions and Laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).Comment: 25 page

The twistor geometry of three-qubit entanglement

A geometrical description of three qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric ${\cal Q}$ a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of ${\cal Q}$. An invariant vanishing for the $GHZ$ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.Comment: 4 pages RevTeX

Energetic Consistency and Momentum Conservation in the Gyrokinetic Description of Tokamak Plasmas

Gyrokinetic field theory is addressed in the context of a general Hamiltonian. The background magnetic geometry is static and axisymmetric, and all dependence of the Lagrangian upon dynamical variables is in the Hamiltonian or in free field terms. Equations for the fields are given by functional derivatives. The symmetry through the Hamiltonian with time and toroidal angle invariance of the geometry lead to energy and toroidal momentum conservation. In various levels of ordering against fluctuation amplitude, energetic consistency is exact. The role of this in underpinning of conservation laws is emphasised. Local transport equations for the vorticity, toroidal momentum, and energy are derived. In particular, the momentum equation is shown for any form of Hamiltonian to be well behaved and to relax to its magnetohydrodynamic (MHD) form when long wavelength approximations are taken in the Hamiltonian. Several currently used forms, those which form the basis of most global simulations, are shown to be well defined within the gyrokinetic field theory and energetic consistency.Comment: RevTeX 4, 47 pages, no figures, revised version updated following referee comments (discussion more strictly correct/consistent, 4 references added, results unchanged as they depend on consistency of the theory), resubmitted to Physics of Plasma

Quantum Sturm-Liouville Equation, Quantum Resolvent, Quantum Integrals, and Quantum KdV : the Fast Decrease Case

We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field $O(k)$ and the regularization .Comment: minor correction