Gauge-Invariant Regularization of Quantum Field Theory on the Light Front

We briefly describe problems of the Hamiltonian approach for quantizing gauge fields on the light front for space–time bounded by the inequality |x ^–|L with periodic boundary conditions in the variable x ^– imposed on all fields (the DLCQ method). With these restrictions, we consider the gauge-invariant ultraviolet regularization by passing to a lattice in transverse coordinates. We remove the remaining ultraviolet divergences in the longitudinal momentum p _– by imposing a gauge-invariant finite-mode regularization. It turns out that the canonical formalism on the light front with such a regularization imposed does not contain the usual most complicated second-class constraints between zero and nonzero modes of fields. The described scheme can be used both to regularize the standard gauge theory and to provide a gauge-invariant formulation of effective low-energy models on the light front. Because the manifest Lorentz invariance is broken in our formalism, the vacuum state is poorly defined. We discuss this problem, in particular, in relation to the problem of passing to the continuous space limit.     

Properties of Defect Modes in One-Dimensional Optically Induced Photonic Lattices

In this article, localized defect modes in one-dimensional optically induced photonic lattices are studied comprehensively. First, the origin of these defect modes is investigated analytically in the weak-defect limit by perturbation methods. It is shown that in an attractive defect where the lattice light intensity at the defect site is higher than that of nearby sites, a defect mode bifurcates from the left edge of every Bloch band; while in a repulsive defect, a defect mode bifurcates from the right edge of every Bloch band. When the defect is not weak, defect modes are examined by numerical methods. It is shown that in a repulsive defect, the strongest confinement of defect modes arises when the lattice light intensity at the defect site is nonzero rather than zero. In addition, as the potential strength increases, defect modes disappear from lower bandgaps and appear in higher bandgaps. In an attractive defect, however, defect modes persist in every bandgap as the potential strength increases. Using a piecewise-constant potential model, defect modes are calculated analytically for a general defect. The analytical results qualitatively explain the main features in numerical results.     

Model of quark–antiquark interaction in quantum chromodynamics on the light front

We formulate a model of quark–antiquark interaction related to the limit transition to the light-front Hamiltonian in quantum chromodynamics. As ultraviolet regularization, we use a lattice in the space of transverse coordinates, and we additionally introduce a longitudinal light-front coordinate cutoff and also corresponding periodic boundary conditions. We regard the zero mode with respect to this coordinate as an independent dynamical variable. The state space of the model is limited to a quark and an antiquark that interact only via the zero mode of the gluon field on the light front. In this framework, we obtain a discrete mass spectrum of bound states. This spectrum is determined by an equation that with respect to the longitudinal coordinate turns out to be analogous to the’ t Hooft equation in two-dimensional quantum chromodynamics. The equation also contains a quark–antiquark potential that ensures confinement in the transverse space.     

When do several linear operators share an invariant cone?

We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented.     

A criterion for the integral equivalence of two generalized convex integer polyhedra

We introduce the notion of integral equivalence and formulate a criterion for the equivalence of two polyhedra having certain special properties. The category of polyhedra under consideration includes Klein polyhedra, which are the convex hulls of nonzero points of the lattice ?³ that belong to some 3-dimensional simplicial cone with vertex at the origin, and therefore the criterion enables one to improve some results related to Klein polyhedra. In particular, we suggest a simplified formulation of a geometric analog of Lagrange’s theorem on continued fractions in the three-dimensional case.     

Noncrossing Partitions, Catalan Words, and the Semicircle Law

As is well known, the joint limit distribution of independent Wigner matrices is free with the marginals being semicircular. This freeness is intimately tied to noncrossing pair partitions or, equivalently what are known as Catalan words, each of which contributes one to the limit moments. We investigate the following questions. Consider a sequence of patterned matrices: (i) When do only Catalan words contribute (one), so that we get the semicircle limit? (ii) When does each Catalan word contribute one (with possible nonzero contribution from non-Catalan words)? (iii) For what matrix models do Catalan words not necessarily contribute one each and non-semicircle limits arise, even when non-Catalan words have zero contribution? In particular we show that in a general sense, the semicircle law serves as a lower bound for possible limits. Further, there is a large class of non-Wigner matrices whose limit is the semicircle. This may be viewed as robustness of the semicircle law. Similarly, there is a large class of block matrices whose limit is not semicircular.     

On finite rank operators in CSL algebras II

For finite rank operators in a commutative subspace lattice algebra algℒ we introduce the concept of correlation matrices, basing on which we prove that a finite rank operator in algℒ can be written as a finite sum of rank-one operators in algℒ, if it has only finitely many different correlation matrices. Thus we can recapture the results of J.R. Ringrose, A. Hopenwasser and R.Moore as corollaries of our theorems. Research supported by NSF of China     

Partly zero eigenvectors

This paper is concerned with the forced presence or absence of zero components in an eigenvector. Relative to a fixed matrix Awith eigenvalue λ, we characterize the strictly nonzero part of a partly zero eigenvector associated with λ. We also give a sufficient condition for a fixed matrix to have a partly zero eigenvector, and discuss several examples in which a matrix has one or more partly zero eigenvectors. Our main results, however, are qualitative in nature. We associate a zero/nonzero pattern class of matrices with a digraph, and characterize the set of pattern classes which requires all eigenvectors to be strictly nonzero. A sufficient condition is also given that identifies the components of a partly zero eigenvector which may be nonzero.     

Asymptotic expansions for potential functions of i.i.d. random fields

Summary For sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two. Under standard conditions, formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters. If this convergence fails, the distribution of the sum can be approximated by a mixture of lattice distributions. The result applies to m-dependent random fields generated by independent random variables.     

Schwarzschild-Type Solutions in Dilaton Gravity

In the D-dimensional dilaton gravity model, we obtain and study stationary Schwarzschild-type solutions, i.e., centrally symmetric solutions in a vacuum. They form a two-parameter family of solutions: one of the parameters is an analogue of the gravitational radius, and the second parameter characterizes the intensity of the dilaton field. If the second parameter is zero, which means that the dilaton field is constant, then the expression obtained for the metric becomes the well-known Schwarzschild solution describing a black hole. But if this parameter is nonzero, then the black hole horizon does not appear. We find the parameters of quasielliptic orbits of test particles for the obtained solutions in the weak gravitational field limit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 133–143, October, 2005.     

On the second term in the spectral asymptotics of the maxwell operator in a filled resonator

It is shown that the characteristics of the Maxwell operator in a resonator with a smooth inhomogeneous anisotropic filler have constant multiplicity if and only if the matrices of dielectric permittivity and magnetic permeability are connected by the relation ε≡f μ, where f is a scalar-valued function. When ε≡f μ and the boundary is smooth and ideally conducting, the coefficient of λ² in the asymptotic expansion of the distribution function of the eigenvalues of the Maxwell operator turns out to be zero. When the multiplicity of the characteristics is variable, this coefficient can be either zero or nonzero. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 48–79.     

Quantum inverse scattering method on a spacetime lattice

The formalism of the quantum inverse scattering method is developed on a lattice in spacetime that imitates light cone coordinates. The zero-curvature representation makes it possible to introduce in a natural manner operators of displacement along the coordinate axes. The method is illustrated for the example of the sine—Gordon system.St Petersburg Branch, V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 2, pp. 207–214, August, 1992.     

Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians

On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function. Received: 7 July 1999 / Revised version: 23 February 2000 / Published online: 5 September 2000     

Weakly Gibbsian representations for joint measures of quenched lattice spin models

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov's constructions. Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 RID="*" ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexit?t'     

Algebraic transition matrices in the Conley index theory

We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

     

一个本原矩阵类的指数集的完全刻划

本文证明了至少有一对非零对称元但非对称的ｎ（＞５）阶本原矩阵所成的类的指数集是：（１）迹非零时，无论ｎ（＞５）是奇数还是偶数，都有指数集．（２）迹为零时，（ｉ）若ｎ（＞５）是奇数，则指数集（ｉｉ）若ｎ（＞５）是偶数，则指数集．