QED with vector and axial vector types of interaction and dynamical generation of particle masses

We consider a model of electrodynamics with two types of interaction, the vector \((e\bar \psi (\gamma ^\mu A_\mu )\psi )\) and axial vector \((e_A \bar \psi (\gamma ^\mu \gamma ^5 B_\mu )\psi )\) interactions, i.e., with two types of vector gauge fields, which corresponds to the local nature of the complete massless-fermion symmetry group U(1) ? U _A (1). We present a phenomenological model with spontaneous symmetry breaking through which the fermion and the axial vector field B_μ acquire masses. Based on an approximate solution of the Dyson equation for the fermion mass operator, we demonstrate the phenomenon of dynamical chiral symmetry breaking when the field B_μ has mass. We show the possibility of eliminating the axial anomalies in the model under consideration when introducing other types of fermions (quarks) within the standard-model fermion generations. We consider the polarization operator for the field B_μ and the procedure for removing divergences when calculating it. We demonstrate the emergence of a mass pole in the propagator of the particles that correspond to the field B_03BC when chiral symmetry is broken and consider the problems of regularizing closed fermion loops with axial vector vertices in connection with chiral symmetry breaking.     

Localization of classical waves I: Acoustic waves

We consider classical acoustic waves in a medium described by a position dependent mass density (x). We assume that (x) is a reandom perturbation of a periodic function ₀(x) and that the periodic acoustic operator has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators onL ²( ^d ). We prove that, in the random medium described by (x), the random operatorA exhibits Anderson localization inside the gap in the spectrum ofA ₀. This is shown even in situations when the gap is totally filled by the spectrum of the random opertor; we can prescribe random environments that ensure localization in almost the whole gap.This author was supported by the U.S. Air Force Grant F49620-94-1-0172.This author was supported in part by the NSF Grants DMS-9208029 and DMS-9500720.     

Landau Hamiltonians with Unbounded Random Potentials

We prove the almost sure existence of a pure point spectrum for the two-dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant is assumed to be absolutely continuous. The corresponding densityg has support equal to , and satisfies , for some > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum is , provided the magnetic field B0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum is pure point at all energies except in intervals of width about each lower Landau level , for m < n. We also prove that for any B0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.     

Remarks on spectra of modular operators of von Neumann algebras

It is shown that if is an invariant state of an asymptotically abelianC* algebra , then the spectrum of modular operator for is contained in the spectrum of any other modular operator for the von Neumann algebra .It is also shown that a modular operator can not have an isolated spectrum with a finite multiplicity at 1 unless the associated Hilbert space is of finite dimension. It is further shown that if a modular operator has an isolated spectrum with a finite multiplicity atx 1, then the von Neumann algebra is a direct sum of ₁ and ₂ where ₁ is represented on a finite dimensional Hilbert space and the modular operator for ₂ does not have its spectrum atx.Applications to Connes invariant are given.On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.     

Limiting Absorption Principle at Critical Values for the Dirac Operator

We prove estimates for the resolvent H ₀ – z)^-1 of the Dirac operator , valid, even for z close to the critical points ±m. In particular, it is shown that the operator -smooth. As a by-product, the absence of the singular spectrum as well as the existence and unitarity of the wave operators are obtained for a class of perturbations .     

Mass Generation by Weyl Symmetry Breaking

A massless electroweak theory for leptons is formulated in a Weyl space, W₄, yielding a Weyl invariant dynamics of a scalar field , chiral Dirac fermion fields _L and _R, and the gauge fields , A, Z, W, and W , allowing for conformal rescalings of the metric g and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, , representing the D(1) or dilatation gauge fields. The local group structure of this Weyl electroweak (WEW) theory is given by —or its universal coverging group for the fermions—with denoting the electroweak gauge group SU(2)_W × U(1)_Y. In order to investigate the appearance of nonzero masses in the theory the Weyl symmetry is explicitly broken by a term in the Lagrangean constructed with the curvature scalar R of the W₄ and a mass term for the scalar field. Thereby also the Z and W gauge fields as well as the charged fermion field (electron) acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D ² = 0, where is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory by breaking the D(1) gauge symmetry, is compared to the so-called spontaneous symmetry breaking in the standard electroweak theory, which is, actually, the choice of a particular (nonlinear ) gauge obtained by adopting an origin, , in the coset space representing , with being invariant under the electromagnetic, gauge group U(1)_e.m.. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl symmetry breaking, yielding a pseudo-Riemannian space, V₄, from a W₄ and a scalar field with a constant modulus . The quantity affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. The consequences of the broken WEW theory are worked out and the determination of the parameters of the theory is discussed.     

On equivalent-neighbour,random site models of disordered systems

Models of random systems whose Hamiltonian reads , where and _i ,=1,...,n are independent, identically distributed random variables are discussed.J _ij are assumed to be symmetric, with respect toJ ₀, random variables and also symmetric functions of components of . A question of dependence of a phase diagram on a probability distribution of is addressed. A class of distributions and interactionsJ _ij , which give rise to phase diagrams called typical is selected. Then a problem of obtaining typical phase diagrams, containing a certain region with an infinite number of pure phases, is studied.     

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

Bose-Einstein Condensation and Spontaneous Symmetry Breaking

After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long-range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein condensation is presented. The proof is based on a rigorous validation of Bogoliubov's c-number substitution for the k = 0 mode operator α₀.     

High scale perturbative gauge coupling in R-parity conserving SUSY <Emphasis Type="Italic">SO</Emphasis>(10) with longer proton lifetime

It is well known that in single step breaking of R-parity conserving SUSY SO(10) that needs the Higgs representations , the GUT gauge coupling violates the perturbative constraint at mass scales a few times larger than the GUT scale. Therefore, if the SO(10) gauge coupling is to remain perturbative up to the Planck scale ( GeV), the scale M_U of the GUT symmetry breaking is to be bounded from below. The bound depends upon specific Higgs representations used for SO(10) symmetry breaking but, as we find, cannot be lower than $1.5 \times 10$¹⁷ GeV. In order to obtain such a high unification scale we propose a two-step SO(10) breaking through SU(2)_L $\times$ SU(2)_R $\times$ U(1)_B-L $\times$SU(3)_C ( ) intermediate gauge symmetry. We estimate the potential threshold and gravitational corrections to the gauge coupling running and show that they can make the picture of perturbative gauge coupling running consistent at least up to the Planck scale. We also show that when by the Higgs representations , gravitational corrections alone with negligible threshold effects may guarantee such perturbative gauge coupling. The lifetime of the proton is found to increase by nearly 6 orders over the present experimental limit for . For the proton decay mediated by a dim = 5 operator a wide range of lifetimes is possible, extending from the current experimental limit up to values 2-3 orders longer. Received: 1 July 2005, Revised: 21 August 2005, Published online: 11 October 2005     

$$\mathfrak{s}\mathfrak{u} $$ q(2)-Invariant Schrödinger Equation of the Three-Dimensional Harmonic Oscillator

We propose a q-deformation of the -invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for and generic , and study the effects of the q-value range and of the arbitrariness in the Casimir operator choice. We then show that the quadrupole operator in l=0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.     

Point Interactions: $$\mathcal{P}\mathcal{T}$$ -Hermiticity and Reality of the Spectrum

General point interactions for the second derivative operator in one dimension are studied. In particular, -self-adjoint point interactions with the support at the origin and at points ±l are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.     

Renormalization of binary trees derived from one-dimensional unimodal maps

For one-dimensional unimodal mapsh (x):I I, whereI=[x ₀,x ₁] when =_max, a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval, we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period-doubling fixed point and the scaling constant. The period-doubling fixed point depends on the details of the maph (x), whereas the scaling constant equals the derivative . The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequenceQ and the scaling constant ofQ is found to be approximately 1.     

Application of the Landau theory on the2d structural phase transition of CH4 on (0001) graphite

A few degrees before melting CH₄ absorbed on (0001) Graphite undergoes a structural phase transition from a commensurate to an incommensurate2d solid. In the commensurate phase the molecules show a hexagonal structure. Since it is known from specific heat measurements that the corresponding phase transition is of continuous type, Landau's theory of second order phase transition should be applicable. Application of this theory allows predicting that the structure of the CH₄ molecules in the incommensurate phase should show a rectangular distortion, driven by a one or two component symmetry breaking order parameter. The rectangular distortion is in agreement with other more general theoretical investigations.     

Exact monopole solutions in gauge theories for an arbitrary semisimple compact group

We construct an exact n-parametric monopole and dyon solutions for an arbitrary compact gauge group G of rank n by using the symmetry between cylindrically symmetric instanton equations in Euclidean space R ₄ and monopole equations in Minkowski space R _3,1 (with Higgs scalar field in adjoint representation). The solutions are spherically symmetric with respect to the total momentum operator represents the minimal embedding of SU(2) in G. Explicit expressions for the monopole magnetic charge and mass matrices are obtained. The remarkable aspect of our results is the existence of discrete series of the monopole solutions, which are labelled by n quantum numbers and degenerated in the latter ones at a fixed monopole mass matrix.     

Integral Presentations for the Universal R-Matrix

We present an integral formula for the universal R-matrix of quantum affine algebra U _q with Drinfeld comultiplication. We show that the properties of the universal R-matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For U _q we describe precisely the cycles and present a new simple expression for the universal R-matrix as a result of calculation of corresponding integrals.     

Automorphisms that commute with a modular automorphism

For M a factor of type III₁ we can find for every automorphism group _s that commutes with a modular automorphism group _t and another modular automorphism group , an automorphism group that commutes with is connected with _s by an inner cocycle.     

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator ;f0,q _i ()0, has Lifschitz singularities at the edges of the gaps inSp(H ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.     

Towards a model-independent analysis of rareB decays

Motivated by the experimental accessibility of rareB decays in the ongoing and planned experiments, we propose to undertake a model-independent analysis of the inclusive decay rates and distributions in the processesBX _s andBX _s ^+– (B=B ^± orB _d ⁰ ). We show how measurements of the decay rates and distributions in these processes would allow us to extract the magnitude and sign of the dominant Wilson coefficients of the magnetic moment operator and the fourfermion operators and . Non-standard-model effects could thus manifest themselves at low energy in rareB decays through the Wilson coefficient having values distinctly different from their standard-model counterparts. We illustrate this possibility using the examples of the two-doublet Higgs models and the minimal supersymmetric models. The dilepton invariant mass spectrum and the forward-backward asymmetry of ⁺ in the centre-of-mass system of the dilepton pair in the decayBX _s ^+– are also worked out for the standard model and some representative solutions for the other two models.