Coherent states of theq-canonical commutation relations

For theq-deformed canonical commutation relationsa(f)a (g)=(1-q)f,g 1+qa (g)a(f) forf, g in some Hilbert space we consider representations generated from a vector satisfyinga(f)=<f, >, where . We show that such a representation exists if and only if 1. Moreover, for <1 these representations are unitarily equivalent to the Fock representation (obtained for =0). On the other hand representations obtained for different unit vectors are disjoint. We show that the universal C^*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a naturalq-analogue of the Cuntz algebra (obtained forq=0). We discuss the conjecture that, ford<, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting casesq=±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.Supported in part by the NSF(USA), and NATO Available by anonymous FTPfrom nostrom.physik.Uni-Osnabrueck.DE     

On the geometric origin of the equation ϕ,11 — ϕ,22 = eϕ — e-2ϕ

It is shown that the equation ,11 — ,22 = e — e^-2 determines the intrinsic geometry of the two-dimensional affine sphere in the three-dimensional unimodular affine space like the sine-Gordon equation describes the metric on the surface of a constant negative curvature in the three-dimensional Euclidean space. The linear equations that determine the moving frame on the affine sphere are the Lax operators to the equation ,11 — ,22 = e — e^-2.     

Multicomponent field theories and classical rotators

It is shown that aD-component Euclidean quantum field, =(¹,...,^D), with ||⁴+|²| interaction, can be obtained as a limit of (ferromagnetic) classical rotator models; this extends a result of Simon and Griffiths from the caseD=1. For these Euclidean field models, it is then shown that a Lee-Yang theorem applies forD=2 or 3 and that Griffiths' second inequality is valid forD=2; a complete proof is included of a Lee-Yang theorem for plane rotator and classical Heisenberg models. As an application of Griffiths' second inequality forD=2, an interesting relation between the parallel and transverse two-point correlations is obtained.Research supported in part by the National Science Foundation under grant NSF MPS 74-04870.     

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.     

Shell sources of stationary axisymmetric gravitational fields

A number of solutions for material shell sources of stationary axisymmetric gravitational fields are presented. Explicit solutions are found for shells lying on equipotential hypersurfaces (g _tt = const) and generating static monopole fields in prolate and oblate spheroidal coordinates (Zipoy-Voorhees fields). Numerical solutions are found for shells lying on hypersurfaces of constantg /g and generating Kerr- and Tomimatsu-Sato ( = 2) fields. The shells have minimum areas allowed by the energy conditions of Hawking and Ellis.     

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.     

A class of nonlinear relativistic partial differential equations in elementary particle theory

In mathematical approaches to elementary particle theory, the equation [² - ²/t²]=m² ;+g ³ has been of interest [1,2]; it describes a quartically self-coupled neutral scalar meson field. This paper applies the decomposition method [3-6] to obtain accurate non-perturbative timedevelopment of the field for this equation, or variations involving other nonlinear interactions, without the use of cutoff functions or truncations.     

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.     

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 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.     

Lorentz-symmetric solutions of nonlinear equations

Asymptotic properties of a real scalar self-interacting classical field depending on one variablez = t ²–x ² are studied. The field(z) approaches a minimum of the potentialU() for z + and a maximum forz– ifU((0)) is larger than two minima and smaller than two maxima ofU neighbouring to(0).Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

Internal Lifschitz singularities of disordered finite-difference Schrödinger operators

The integrated density of states has C-like singularities, ln|k(E)–k(E _c )|=–|E–E _c |^–v/2 _c (E), with _c >0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potentiald has a sufficiently large gap. The behaviour of _c nearE _c is determined by the local continuity properties ofd near the relevant edge: _c (E)=O(1) ifd has an atom and =O(ln|E–E _c |) if is (absolutely) continuous and power bounded.     

The effect of substrate work function on work function reduction of Re/W alloy-coated impregnated cathodes

The work function reduction at a metal surface caused by adsorbed barium and oxygen is already known with the resulting expression for the overall work function being =_S–_R. Results from the Re/W alloy-coated cathodes exhibit that _R increases with substrate work function with a slope of 1.09 by linear regression. A qualitative explanation of the above results comes from considering the arrangement of barium and oxygen on the surfaces. A charge transfer model is proposed for two-dimensional barium and oxygen lying on a flat non-interacting substrate. Calculation shows that the dipole strength increases with the charge transfer below the intermediate states.     

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].     

On the large order behavior of Φ 4 4

We continue the rigorous study of the large order behavior of the perturbation series for the ⁴ model in 4 dimensions started in [1]. In this paper we prove a result announced in [1]. We show that the exact radius of convergence of the Borel transform of the renormalized perturbation series for ₄ ⁴ is greater than or equal to the expected value given by the position of the first renormalon [2]. This result holds for any vector (²)² model withN components, and makes use of the Lipatov bound of [1]. This result is based on a partial resummation of counterterms similar to the one of [3], but in a phase-space analysis of the renormalized series.     

ФИЗИЧЕСКИЕ ОСНОВЫ ИЗ МЕРЕНИЯ НАПРАВЛЕНИЯ ВОЗДУШНОГО ПОТОКА МЕ ТОДОМ НАКАЛЕННОЙ ПРО ВОЛОКИ ПРИ ВЫСОКНХ ПОДЗВУКО ВЫХ СКОРОСТЯХ

d - l - t _d R _max - J , ( ) - - , . . - ( ) - - R R _max —R. ,      

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.     

Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions

Let ₁(x) and ₂(y) be two local fields in a conformal quantum field theory (CQFT) in two dimensional spacetime. It is then shown that the vector-valued distribution ₁(x)₂(y)|0 is a boundary value of a vectorvalued holomorphic function which is defined on a large conformally invariant domain. By group theoretical arguments alone it is proved that ₁(x)₂(y)|0 can be expanded into conformal partial waves. These have all the properties of a global version of Wilson's operator product expansions when applied to the vacuum state |0. Finally, the corresponding calculations are carried out more explicitly in the Thirring model. Here, a complete set of local conformally covariant fields is found, which is closed under vacuum expansion of any two it its elements (a vacuum expansion is an operator product expansion applied to the vacuum).Work supported by Deutsche Forschungsgemeinschaft