A poisson—lie structure on the diffeomorphism group of a circle

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S ¹) of the vector fields on the circleS ¹ and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr X(S ¹) X(S ¹) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S ¹), and a sort of Poisson-Lie structure on the group of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to quantum diffeomorphism groups.Research supported by the Erwin Schrödinger International Institute for Mathematical Physics.     

Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

On the time evolution automorphisms of the CCR-algebra for quantum mechanics

In ordinary quantum mechanics for finite systems, the time evolution induced by Hamiltonians of the form is studied from the point of view of *-automorphisms of the CCRC*-algebra (see Ref. [1, 2]). It is proved that those Hamiltonians do not induce *-automorphisms of this algebra in the cases: a) and b)V L (,dx) L ¹ (,dx), except when the potential is trivial.     

The spectrum of a quasiperiodic Schrödinger operator

The spectrum (H) of the tight binding Fibonacci Hamiltonian (H _mn= _m,n+1+ _m+1,n + _m,n v(n),v(n)= ((n–1)), 1/ is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any , and (H) is a Cantor set for 4. Combining this with Casdagli's earlier result, one finds that the spectrum is singular continuous for 16.On leave from the Central Research Institute for Physics, Budapest, Hungary     

Averaged electron Green function and CDW pinning in one-dimensional random potential

The averaged retarded electron Green functionG ⁺(,k) in 1d disordered metal is calculated using the Berezinsky diagram technique. Using the Gorkov's theory it is shown, that the substitution of inG ⁺ (,k) by the square of the external frequency atk=0 gives the dependence of Fröhlich conductivity _F(). This dependence describes the impurity pinning of CDW in 1d disordered metals. The good agreement of this dependence with experimental data Zeller et al. about _F() in quasi-1d conductor KCP is found     

Zeros of partition functions via correlation inequalities

We show analyticity of the pressure for some classical ferromagnetic systems in the region ¦Im ¦ < Re of the external field. The proof, via correlation inequalities, is simpler than existing proofs for the Lee and Yang region {Re 0} and applies, without any approximation procedure, to more general continuous spin variables, e.g., distributed as , where _2n is an arbitrary real number and the other parameters are positive. It also applies directly to plane rotators in the region ¦Im ¦ ¦Re ¦ (Euclidean norms), but the proof will be given in a subsequent article, together with new inequalities between truncated correlation functions.     

Scaling characteristics of ac conductivity and dielectric function in fractal regime of the metal-insulator transition

An analysis of the ac conductivity _ac(), and the ac dielectric constant, (), of the metal-insulator percolation systems is presented in the critical regime near the transition threshold. It is argued that the polarization and relaxation of the finite fractal metallic clusters play dominant roles in controlling the dynamic response of the system on both sides of the threshold. The relaxation time constant of a fractal cluster is shown to scale with its size as withd _t = 4 – 2d +d _c + /, whered is tge Euclidean dimension, andd _c , , and are the scaling indices for the charging, the dc conductivity, and the correlation length respectively. The average time dependent response of the system is shown to scale with a new time scale , where is the correlation length and ₀ is a microscopic time constant. It is shown that at frequencies and with /d_t 1, in close agreement with experiments. The effects of the anomalous transport along the infinite cluster and the medium polarizability are also discussed.     

Contribution of Sea Quarks and Gluons to the Proton Spin

Contributions of sea quark and gluon spins to the proton spin in Drell-Jahn processes and direct production of photons in proton-proton and proton-antiproton collisions are studied in the present work. Analytical expressions for two-spin asymmetries and are derived. In both processes, these asymmetries are studied and analyzed as functions of the kinematic variables , x _T, and x _F. Measurements of two-spin asymmetries and make it possible to determine the individual contributions of sea quark and gluon spins to the proton spin.     

A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)= (t). The equation of motion is (t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d ,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v (t)v(t/²) converges weakly to a diffusion Markov processv(t), and ² x (t) converges weakly to , wherex=lim ² x (0).     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Duality for free bose fields

An elementary proof of Araki's duality theorem for free fields is presented. The theorem says that for a certain class of regionsO in Minkowski space, the commutant of (O), the von Neumann algebra generated by all observables belonging to measurements withinO, is exactly (O), whereO is the spacelike separated complement ofO.Supported by the National Science Foundation under grants GP 24003, GP 30819X and GP 31239X.     

Lattice vibrations in ordered and disordered thiourea

The critical dynamics of the Syozi model for dilute ferromagnetism is considered by the use of master equations. The dynamics is soluble as it is assumed that the time scale of motion on the sublattice on which the impurities move is so much faster than on the other sublattice that fast relaxing variables may be adiabatically eliminated, leaving a new soluble master equation. It is found that the linear and non-linear relaxation of magnetization exponents ^(l) and ^(nl) increase on dilution to ^(l)/(1–) and ^(nl)/(1–) respectively ( is the specific heat exponent for the pure system, which itself changes on dilution to –/(1–)). Thus if the exponents for the pure system obey the scaling law of Rácz and Fisher ^(nl)= ^(l)– ( is the magnetization exponent which changes on dilution to /(1–)) then so do the exponents for the diluted system. Similarly the exponent for spin diffusion changes on dilution to /(1–).     

Additivity of the entropy and definition of the temperature for quantum systems

A closed quantum systemL is considered which is described by a microcanonical ensemble.L consists of two rather weakly interacting subsystemsL ₁,L ₂. In a rigorous way, the additivity of the entropy is proved by deriving an expression for the entropy density ofL in terms of the entropy densities ofL ₁ andL ₂. Rigorous implies that the thermodynamic limit is taken. In the second part, it is shown how a microcanonical state of the composite system — provided this limit exists — gives rise to a canonical state , when restricted toL ₁, providedL ₁ is very small as compared toL ₂; is defined as a limit of Gibbs states. This yields a definition of the equilibrium temperature ^–1.On leave of absence from the Institut für Theoretische Physik, Universität Göttingen.     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Effective Potential for \mathcal{P}\mathcal{T}-Symmetric Quantum Field Theories

Recently, a class of -invariant scalar quantum field theories described by the non-Hermitian Lagrangian = () ² +g ² (i) was studied. It was found that there are two regions of . For <0 the -invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the -invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for -symmetric quantum field theories.     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

Slow flow of a conducting fluid past a non-conducting porous sphere with variable permeability

Slow flow of a conducting fluid past a non-conducting porous sphere of variable permeability in presence of a uniform radial magnetic field is studied. The drag experienced by the sphere is shown graphically and compared to that for a non-conducting fluid.Notation velocity vectors of the porous matrix and the conducting fluid - P, p pressures in the porous material and the free fluid - K permeability at a point of the porous medium - viscosity - v (=) kinematic viscosity - magnetic induction - current density - (r, , ) spherical coordinates - dimensionless constant - conductivity of the liquid The authors remain thankful to the referee for his valuable comments and helpful suggestions for improvement of the quality of the paper.     

On the Fokker-Planck equation with time-dependent drift and diffusion coefficients and its exponential solutions

The static critical behavior of a mixed hydrogen-bonded ferro-antiferroelectric system is investigated with use of a pseudo-spin Ising model in a transverse field, which describes the tunneling of protons. Assuming an infinite-ranged random interbond coupling, then0 replica method is applied to evaluate the averaged free energy, from which the replica-symmetric solution of the Sherrington-Kirkpatrick type is obtained. It is shown that a proton pseudo-spin glass (PG) phase exists below a freezing temperatureT _f , whereT _f and the order parameters for the PG and ferroelectric phase are parametrized by the tunneling frequency . For _c, where _c = and is the random distribution width, no ordering is possible even forT0. Numerical solutions for the PG order parameter and the dielectric susceptibility atT0 are obtained, and the phase diagram for a simple model of a mixed system atT0 is presented.Dedicated to Prof. Dr. H.E. Müser on the occasion of his 60th birthday