The 1/N F expansion of the γ and β functions in Q.E.D.

The Callan-Symanzik- and-functions are calculated analytically for Q.E.D. in the limit of a large number of leptons (N _F) up to terms of order 1/N _F inclusive. We give closed analytic expressions for the coefficients of these terms in their series expansion in powers ofK N _F/. We have been able to sum these series and to obtain some striking results.     

Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

LetH _l be the Hamiltonian in aP()₂ theory with sharp space cutoff in the interval (–l/2,l/2). LetE _l =inf(H _l ), (l)=–E _l /l, and let _l be the vacuum forH _l . discuss properties of (l) and _l . In particular, asl, there are finite constants <0 and such that (l), ((l)–)l, and hence (l)=+/l+o(l ^–1). Moreover exp(–c ₁ l) _{l 1}exp(–c ₂ l) forc ₁,c ₂ positive constants, where _{l 1} is theL ¹(Q, d₀) norm of ₁ with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofH _l in the infinite volume limit.Research sponsored by AFOSR under Contract No. F44620-71-C-0108.On leave from Istituto di Fisica Teorica, Universitá di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.A. Sloan Foundation Fellow.     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

The free energy of the spin-boson model

Forn spins 1/2 coupled linearly to a boson field in a volumeV _n, the existence of the specific free energy is proved in the limitn ,V _n withn/V _n=const. The interaction is essentially of the mean field type, in as much as it is proportional to 1/V _n; the coupling constants are allowed to be spin dependent. A variational expression is obtained for the limiting specific free energy, and a critical temperature is identified above which the system behaves as if there were no coupling at all.     

Phase uniqueness and correlation length in diluted-field Ising models

The diluted-field Ising model, a random nonnegative field ferromagnetic model, is shown to have a unique Gibbs measure with probability I when the field mean is positive. Our methods involve comparisons with ordinary uniform field Ising models. They yield as a corollary a way of obtaining spontaneous magnetization through the application of a vanishing random magnetic field. The correlation lengths of this model defined as (lim _n-(1/n) log ₀; _n)^-1, wheren is the site on the first coordinate axis at distancen from the origin and ₀; _n is the origin ton two-point truncated correlation function, is non-random. We derive an upper bound for it in terms of the correlation length of an ordinary nonrandom model with uniform field related to the field distribution of the diluted model.     

On infinite direct products of Lorentz transformations

An infinite direct product _i =1 U _i (a) of continuous unitary representations of SL(2,C) in Hilbert spaces _i is continuous only on certain incomplete direct product subspaces of _i =1 _i . If no representations of the complementary series occur, then each of these subspaces contains a product vector almost all factors of which are SL(2, C)-invariant.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.     

Measure and dimension of solenoidal attractors of one dimensional dynamical systems

Letf:MM be aC -map of the interval or the circle with non-flat critical points. A closed invariant subsetAM is called a solenoidal attractor off if it has the following structure: , where{I _k ⁽ⁿ⁾ is the cycle of intervals of periodp _n. We prove that the Lebesgue measure ofA is equal to zero and if sup(p _n+1/p_n)< then the Hausdorff dimension ofA is strictly less than 1.     

D-Dimensional ideal gas in parastatistics: Thermodynamic properties

We consider a parastatistics ideal gas with energy spectrum ¦k¦ (>0) or even more generally in ad-dimensional box with volumeV (periodic boundary conditions), the numberN of the gas particles being well determined (real particles) or not (quasiparticles). We calculate the main thermodynamic quantities (chemical potential, internal energy, specific heatC, equation of state, latent heat, average numbers of particles) for arbitraryd, ,T (temperature), andp (maximal number of particles per state allowed in the parastatistics). The main asymptotic regimes are worked out explicitly. In particular, the Bose-Einstein condensation for fixed densityN/V appears as a nonuniform convergence in thep limit, in complete analogy with the standard critical phenomena that appear in interacting systems in theN limit. The system behaves essentially like a Fermi-Dirac one forall finite values ofp, and reveals a Bose-Einstein behavioronly in thep limit. For instance, at low temperaturesC T ifp< andC T ^d/ ifp. Finally, the Sommerfeld integral and its expansion are generalized to an arbitrary, finitep.     

On solvable models in classical lattice systems

We consider one dimensional classical lattice systems and an increasing sequence _n (n=1,2, ...) of subsets of the state space; _n takes into account correlations betweenn successive lattice points.If the interaction range of the potential is finite, we prove that the equilibrium states defined by the variational principle are elements of { _n } _n<. Finally we give a new proof of the fact that all faithful states of _n are DLR-states for some potential.Bevoegdverklaard navorser NFWO     

Properties of theL p -solutions of linear impulsive equations in a banach space

Conditions for the existence ofL _p-solutions (1p) of linear impulsive equations in a Banach space are found.     

Influence of boundary conditions on square bond percolation nearpc

The influence of boundary conditions on square bond percolation for system sizes ranging from 10×10 to 240×240 is studied for the quantitiesP , , the effective percolation threshold and the finite-size scaling relations forP and . The Monte Carlo simulations suggest that free edges approximate the infinite system as well as the more complicated periodic boundary conditions.     

Electroweak interactions at an infinite sublayer quark level. II

It is shown that the standard model of the electroweak interactions holds at an infinite sublayer quark level, insofar as we consider the weak isospin doublet (u _{L,u L} ^cp , whereu is an infinite number of quarks at an infinite sublayer level.     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

The size effect correction of d. c. resistance in metal wires

A simple method of determining the bulk value of the d.c. resistance r, of the residual resistance ratiok , and of the mean free path of electrons in isotropic metal wires is presented. For thick wires the linear correction can be applied under the condition that the experimentally found mean free path _e is smaller than the wire diameterd. In both thick and thin wires another type of correction can be used, based on the difference, between the precise and linear correction as a function of /d, carried out by a graphical method.The author acknowledges useful discussions with P.Svoboda, to whom he is indebted for stimulating remarks. Also the kind advice of L.Smrka and P.Steda was welcome.     

Form Factors and Action of on ∞-Cycles

Let be a sequence of skew-symmetric polynomials in X₁,,X_l satisfying deg_XjP_n,ln–1, whose coefficients are symmetric Laurent polynomials in z₁,,z_n. We call p an -cycle if holds for all n,l. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables _a=–logX_a are the integration variables and _j=logz_j are the rapidity variables. To each -cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the -cycles. In this paper, we define an action of on the space of -cycles. There are two sectors of -cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) -cycles which are polynomials in z₁,,z_n is equal to the level (–1) irreducible character of with lowest weight –₀ (resp. –₁). We also suggest a possible tensor product structure of the full space of -cycles.     

A one-parameter family of cylindrically symmetric perfect fluid cosmologies

Non-stationary cylindrically symmetric one-parameter solutions to Einstein's equations are given for a perfect fluid. There is a time singularity (t=0) at which the pressurep and density are equal to + throughout the radial coordinate range 0 r < , but the solutions are well behaved fort > 0,p and decreasing steadily to zero asr increases through the range 0r<, or as t increases through the range 0<t<. The motion is irrotational with shear, expansion and acceleration. The family of solutions, of Petrov type I, are generally spatially inhomogeneous, of class B(ii), having two spacelike Killing vectors which are mutually orthogonal and hypersurface orthogonal, associated with an orthogonally transitive groupG ₂. The particular members for which there are equations of statep=/3 andp= are specially considered.     

On the role of hardy spaces in form factor bounds

Functions of the space H ^p (1p) of the unit disc and upper half-plane are characterized by their distributional boundary values and asymptotic behaviour.