Correlation inequalities for two-component hypercubicϕ 4 models

A collection of new and already known correlation inequalities is found for a family of two-component hypercubic ⁴ models, using techniques of duplicated variables, rotated correlation inequalities, and random walk representation. Among the interesting new inequalities are: rotated very special Dunlop-Newman inequality _1,x ² ; _1,z ² + _2g ² 0, rotated Griffiths I inequality _{1,x 1,y} ; _1z ² 0, and anti-Lebowitz inequalityu ₄ ¹¹¹¹ >-0.     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.     

Melnikov's criterion for nondifferentiable weak-noise potentials

The stationary probability density of Fokker-Planck models with weak noise is asymptotically of the form exp[–1 /(q)]. If is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption, has the properties of a Liapounov function, and can be used, e.g., as a thermodynamic potential in nonequilibrium steady states. We consider systems having several attractors and show, by applying Melnikov's method to the associated Hamiltonian, that in general is not differentiable. A small perturbation of a model with differentiable leads to a nondifferentiable . The method is illustrated on a model used in the treatment of the unstable mode in a laser.     

Regular quantum field theories

Mathematically regular and more precise versions of quantum field theories are discussed. A new class of representations called minimal wave-packet representations is introduced. Several possibilities of constructing nonconventional, bounded interaction operators (nonpolynomial, nonlinear, explicitly or implicitly nonlocal) corresponding to the traditional ⁴ or ³ interactions are reviewed. The problem of macrocausality is discussed. A procedure of renormalization of regular theories is indicated.     

Strong coupling in the quantum field theory with nonlocal nonpolynomial interaction

In the relativistic quantum field theory the representation for theS-matrix elements is obtained for any coupling constantsg in the case of a one component scalar field (x) with nonlocal nonpolynomial interaction _I ()=gU() when the causal function is bounded in the Euclidean region 0D _c (x _E ² D _c (0)< and the function |U(u)|1 for realu. It is proved that the two point Green function is bounded in the physical region of momenta variablep ².     

A generalized consistency condition for massless fields

Using a generalized consistency condition (gcc), we construct couplings between massless scalar fields and the spin 2 gravitational field. Specifically, we consider all possible third-order interaction terms for scalar fields and a and use the gcc to single out one. We also find three generalized current identities associated with the massless gauge fields.     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

Existence of three phases for aP(ϕ)2 model of quantum field

In the two-dimensional model of the quantum field theory with lagrangean density :()²–(–)^{21/2 4}–⁶: there exist (at least) three different phases for small and some ().     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

The energy-momentum spectrum in theP(ϕ)2 quantum field theory

TheP()₂ interaction with the periodic boundary conditions is considered. It is shown that the energy-momentum spectrum lies in the forward light cone. As a consequence, this result implies that theP()₂ theory in the infinite volume with the periodic boundary conditions is Lorentz invariant.     

Eigenfunctions of the Koch snowflake domain

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions _n and their gradients {_n } for a class of planar domains with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when is the Koch snowflake domain, one of our main results states that {₁()} tends to or 0 as approaches certain types of boundary points (where ₁ > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on ). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve . Then given Ob (resp., Ac), we show that {₁()} (resp., 0) as tends to in within a cone based at . Moreover, we show that blowup of {₁} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the Koch drum.Research partially supported by the National Science Foundation under Grant DMS-9207098.     

Wetting transition on grain boundaries in Al contacting with a Sn-rich melt

The contact angle at the intersection of a grain boundary in Al bicrystals with the solid Al/liquid Al–Sn interphase boundary has been measured for two symmetric tilt <011> {001} grain boundaries with tilt angles of 32° and 38.5°. The temperature dependencies (T) present the evidence of the grain boundary wetting phase transition at T_w. The observed hysteresis is consistent with the assumption that the wetting transition is of first order. The determined discontinuity in the temperature derivative of the grain boundary energy is–5.6 J/m²K (T w1=617°C) for the boundary with a low energy (=38.5°) and –17 J/m²K (T w2=604°C) for the grain boundary with a high energy (=32°).     

On the Nature of the “Low-Temperature Anomaly” in Metal–Semiconductor Schottky-Barrier Contacts

It is shown that the low-temperature anomaly of the I-V characteristics of homogeneous metal–semiconductor Schottky-barrier contacts (an increase in the ideality factor n and a decrease in the barrier height _bm, measured using the saturation current, with decreasing temperature, as well as the fact that their product _bn n_bm coincides very closely with the actual barrier height _b(V)) is a consequence of two factors: (1) the nonlinear dependence of the actual barrier height on the bias, resulting in an increase in the ideality factor with increasing bias voltage (current) and (2) measurement of the parameters n and _bm for the same current for all temperatures. A new expression for the flat-band barrier height _bf is derived.     

Generalization of the relativistic string model in the geometrical approach

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation – = Hsh. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.     

The proper formula for relative entropy and its asymptotics in quantum probability

Umegaki's relative entropyS(,)=TrD (logD –logD ) (of states and with density operatorsD andD , respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality TrA logAB TrA(logA+logB) is obtained for positive definite matricesA andB.     

Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc _k (t)s(t)^– (k/s(t)), wherec _k (t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, and(x) is a scaling function. For the rate constantK(i, j) in Smoluchowski's equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a ^– K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j0 asj. We show that gelation occurs if>1, and does not occur if1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation for(x). We present a detailed analysis of the behavior of(x) at large and small values ofx. For all models, we find exponential large-x behavior: (x)A x ^– e ^–x asx and, for different kernelsK(i, j), algebraic or exponential small-x behavior: (x)Bx ^– or (x)=exp(–Cx ^–|| + ...) asx0.     

Nonperturbative construction of nonrenormalizable models of quantum field theory in four-dimensional space-time

The report presents the construction of non-cutoff Euclidean Green's functions for nonrenormalizable interactions _I()= d(ge): exp : in four-dimensional space-time. It is shown that all axioms for the generating functional of E.G.F. are satisfied except perhaps theSO(4) invariance. It is shown that the singularities of E.G.F. for coinciding points are not worse than those of the free theory.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.Supported in part by NSF Grant No INT 73-20002 A 01 (formerly GF-41958).The author would like to thank Professors T.Balaban, J.Fröhlich and A.Uhlmann for interesting discussions and valuable suggestions. He would also like to express his sincere thanks to Professor J.Niederle for the very kind hospitality extended to him during his stay at the Symposium on Mathematical Methods in the Theory of Elementary Particles in Liblice Castle.     

Diffraction of a plane wave in air by a cylindrical shell with two local masses

The diffraction of a plane wave by a cylindrical shell with two inertial masses attached symmetrically (at = ±₁) to its surface is investigated. The frequency response characteristics of the scattered field are calculated for a shell equipped with diametrically opposite masses (₁ = /2) and driven only by symmetric (about the plane =0, =) modes of vibration. It is established that the level of the scattered field in the interval of frequencies corresponding to even symmetric mode orders increases upon transition from the model with a single local mass (at =) to the model with two masses (at =±/2) (i.e., with a split inhomogeneity). At the frequencies of odd modes the scattered field coincides with the field of the homogeneous shell. The directivity patterns of the scattered field of a shell with two masses (at =±/2) are determined.Architecture and Construction Institute, Nizhegorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 3, pp. 289–299, March, 1994.     

Derivations and automorphisms of operator algebras

The theorem that each derivation of aC*-algebra extends to an inner derivation of the weak-operator closure ( )^– of in each faithful representation of is proved in sketch and used to study the automorphism group of in its norm topology. It is proved that the connected component of the identity in this group contains the open ball of radius 2 with centerl and that each automorphism in extends to an inner automorphism of ( )^–.Research conducted with the partial support of the NSF and ONR.     

The relativistic polaron without cutoffs

The (non-Lorentz covariant) system consisting of a relativistic scalar Boson field interacting with a single spinless particle (relativistic polaron) with kinetic energy function (m ²+|p|²)^1/2 is studied ind space demensions, whered3. The interaction Hamiltonian is taken to be (x)* (x) (x)dx where has a momentum cutoff. The physical one polaron Hilbert space _ph for this model, corresponding to no cutoff on , is constructed. The total renormalized HamiltonianH without cutoff is constructed as a semibounded self-adjoint operator on _pH . The time zero physical Boson field is also constructed. First order estimates are established for the local (in momentum space) number operators in terms ofH.This research was partially supported by N.S.F. grants GP 28109 and GP 28443 and U.S.A.F. grant AF-AFOSR 743-67.