Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

SU(1, 1) Perturbations of the s-Wave Morse Potential Problem

A purely algebraic perturbation theory based on deforming the generators of the dynamical group SU(1, 1) is applied to the l = 0 Morse potential problem with . In particular, perturbations of the form and are treated explicitly.     

Zero Energy Modes of Massless Fermions in S3R Spacetime

The Dirac-type equation on topology is worked out and the complete set of solutions in the particular physical case of the zero-energy modes of the massless field quanta is derived. Unlike the Minkowskian case, the 1/2fermionic vacua on the manifold is made of nontrivial static modes of defined chirality.     

Bicross-Product Structure of Affine Quantum Groups

We show that the affine quantum group is isomorphic to a bicross-product central extension of the quantum loop group by a quantum cocycle in R-matrix form.     

Multipoint Series of Gromov–Witten Invariants of CP1

We derive explicit formulas for the multipoint series of in degree 0 from the Toda hierarchy, using the recursions of the Toda hierarchy. The Toda equation then yields inductive formulas for the higher degree multipoint series of . We also obtain explicit formulas for the Hodge integrals , in the cases i=0 and 1.     

Tunneling Across an Inhomogeneous Delta-Barrier

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy (where 0$$ " align="middle" border="0"> and 0$$ " align="middle" border="0"> are two constants). In particular, the perpendicular incidence of an electron with a given value of the wave vector is considered. The electron is forward-scattered into the region behind the barrier (region 2: 0$$ " align="middle" border="0"> ), i. e. the wave function is composed of plane waves with all wave vectors such that and \left. 0 \right)} $$ " align="middle" border="0"> ) (where ). Therefore, if 0$$ " align="middle" border="0"> , the wave function of the electron is represented as , where . An approximate formula is derived for the amplitude . The authors pay a special attention to the flow density and calculate this function in two cases: 1. for the plane and 2. for high values of is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.     

Dirac-Hestenes Lagrangian

We formulate the variational principle of theDirac equation within the noncommutative even space-timesubalgebra, the Clifford -algebra . A fundamental ingredient in ourmultivectorial algebraic formulation is a -complex geometry, . We derive the Lagrangian for theDirac-Hestenes equation and show that it must be mapped on , where denotes an -algebra of functions.     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

Poincaré-Invariant Markov Processes and Gaussian Random Fields on Relativistic Phase Space

We give a complete characterization, including a Lévy–Itô decomposition, of Poincaré-invariant Markov processes on , the relativistic phase space in 1+1 spacetime dimensions. Then, by means of such processes, we construct Poincaré-invariant Gaussian random fields, and we prove a no-go theorem for the random fields corresponding to Brownian motions on .     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

Exactly Solvable su(N) Mixed Spin Ladders

It is shown that solvable mixed spin ladder models can be constructed from su(N) permutators. Heisenberg rung interactions appear as chemical potential terms in the Bethe Ansatz solution. Explicit examples given are a mixed spin- spin-1 ladder, a mixed spin- spin- ladder and a spin-1 ladder with biquadratic interactions.     

Hyperbolic Conformal Geometry with Clifford Algebra

In this paper, we study hyperbolic conformal geometry following a Clifford algebraic approach. Similar to embedding an affine space into a one-dimensional higher linear space, we embed the hyperboloid model of the hyperbolic n-space in into . The model is convenient for the study of hyperbolic conformal properties. Besides investigating various properties of the model, we also study conformal transformations using their versor representations.     

A Generator for the Weert Superpotential

Weert found a superpotential for the bounded part of the Maxwelltensor associatedto the Lienard–Wiechert field. Here we obtain afourth-rank generator for the superpotential .     

Minimal Model Fusion Rules From 2-Groups

The fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group in the following manner. There is a partition into disjoint subsets and a bijection between and the sectors of the (p,q)-minimal model such that the fusion rules correspond to where .     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

“Principle of Indistinguishability” and Equations of Motion for Particles with Spin

In this work we review the derivation of Dirac and Weinberg equations based on a principle of indistinguishability for the (j,0) and (0,j) irreducible representations (irreps) of the homogeneous Lorentz group (HLG). We generalize this principle and explore its consequences for other irreps containing j1. We rederive Ahluwalia–Kirchbach equation using this principle and conclude that it yields equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j= , there exists no Dirac-like equation for the (j,0)(0,j) representation nor for the ( , ) representation. We rederive Kemmer–Duffin–Petieau (KDP) equation for the (1,0)( , )(0,1) representation by this method and show that the (1, )( ,1) representation satisfies a Dirac-like equation which describes a multiplet of with masses m and m/2, respectively.