One-, Two-, and Three-Dimensional Ising Model in the Static Fluctuation Approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator _f ^z withquadratic fluctuations is replaced with its mean value [( _f ^z )² ( _f ^z )²]. This means that the true quantum mechanical spectrum of the operator _f ^z is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.     

Infrared magnetooptic — A tool for the determination of the effective mass

The analysis of the magnetooptical effects of heavily doped materials at the plasma edge yields the concentration dependence of the effective cyclotron mass. Therefore these experiments support the general diskussion about the nonparabolicity of a band, the position of the Fermi level at high degenerated semiconductors and the determination of the dispersion function. Experimental results of the magnetooptical determinedm _c ^* (N) function are compared with coresponding band structure calculations. A matrix calculation model, which describes the symmetrical magnetooptical transmission effects as well as the asymmetrical magnetooptical reflection effects of arbitrary successions of coherent films and incoherent substrates consitently, is used to determine free carrier density profilesN (z) of inhomogeneously doped semiconductors non-destructively. This application of the matrix formalism requires the knowledge of them ^* (N)-function. The influence of the effective cyclotron mass on thedifferential magnetooptical interference structures caused by buried density profiles is discussed.     

Approximate vacuums in (:φ2m :g(x))2 field theories and perturbation series

A class of perturbation problems is considered, in which the Rayleigh-Schrödinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:^2m :g(x))₂, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex ⁴ anharmonic oscillator are given.Supported in part by the National Research Council of Canada.     

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS ² classical spins asL] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has diagonal form).Supported by USNSF Grant MCS-78-01885     

Quantum qualitative dynamics

We describe our work on qualitative methods for visualizing the quantum eigenstates of systems with nonlinear classical dynamics. For two-degree-of-freedom systems, our approach is based on the use of generalized coherent states, and allows systems with nonoscillator kinematics to be investigated. The general approach is illustrated with two examples involving vibration-rotation interaction in polyatomic molecules. We apply the coherent states of the Lie groupH ₄SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, andSU(2)SU(2) coherent states to study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction. In both systems, quantum states are visualized on the rotational surface of section and compared with the corresponding classical phase space structure. Striking classical-quantum correspondence is observed. We then describe recent results on the quantum states of (N 3)-dimensional systems of coupled nonlinear oscillators, which reveal a quantum delocalization that is reminiscent of classical Arnold diffusion.     

Photoluminescence of AlxGa1−xAs/GaAs quantum well heterostructures grown by molecular beam epitaxy

Low-temperature photoluminescence measurements on nominally undoped Al_xGa_1–xAs/GaAs quantum well heterostructures (QWHs) grown by molecular beam epitaxy (MBE) exemplified the exclusivelyintrinsic free-exciton nature of the luminescence under moderate excitation conditions. Neither any spectroscopic evidence for alloy clustering in the Al_xGa_1–xAs barriers nor any extrinsic luminescence due to recombination with residual acceptors has been detected in single and double QWHs when grown at 670 °C under optimized MBE growth conditions. Carrier confinement in Al_xGa_1–xAs/GaAs QWHs starts at a well width ofL _z30 nm when x0.25. The minor average well thickness fluctuation ofL _z=4×10^–2nm as determined from the excitonic halfwidth allowed the realization of well widths as low asL _z=1 nm and thus a shift of the free-exciton line as high as 2.01 eV which is close to the conduction band edge of the employed Al_0.43Ga_0.57As confinement layer. The measurements further revealed a strongly enhanced luminescence efficiency of the quantum wells as compared to bulk material which is caused by the modified exciton transition probabilities due to carrier localization.     

KL,S → l+l− decays in the quark model of superconductivity type

The probabilities of K_L,S l⁺l^– decays are calculated by considering only the dominant two-lepton channel K_L,S ^{* *} l⁺l^–. The transitions K⁰ X, where X=⁰,, or (700) dominating in the K_L,S ^{* *}decay amplitudes, are described by the effective weak Lagrangian. The matrix elements of four-quark operators are calculated within the framework of the quark model of superconductivity type. It is shown that the leading contribution to matrix elements of K⁰ X transitions comes from the penguin operator matrix elements.     

“Special” states in quantum measurement apparatus: Structural requirements for the recovery of standard probabilities

In a recently proposed quantum measurement theory the definiteness of quantum measurements is achieved by means of special states. The recovery of the usual quantum probabilities is related to the relative abundance of particular classes of special states. In the present article we consider two-state discrimination, and model the apparatus modes that could provide the special states. We find that there are structural features which, if generally present in apparatus, will provide universal recovery of standard probabilities. These structural features relate to the distribution of certain Hamiltonian matrix elements or interaction times. In particular, those quantities must be asymptotically (x ) distributed according to the Cauchy law, C_a(x)=a/(x ² +a ² ).     

Fractional Moment Estimates for Random Unitary Operators

We consider unitary analogs of d-dimensional Anderson models on l²( $$\mathbb(z)$$^d) defined by the product U=D S where S is a deterministic unitary and D is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman–Molchanov to get exponential estimates on fractional moments of the matrix elements of U(U –z)^–1, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for U.     

Fully Entangled Quantum States in C^{N^2 } and Bell Measurement

Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding, and quantum teleportation. We describe how to generate fully entangled states in the Hilbert space C ^N C ^N starting from a unitary matrix and show that they form an orthonormal basis in this space.     

On the distribution of zeros of a Ruelle zeta-function

We study the limit distribution of zeros of a Ruelle -function for the dynamical systemzz ²+c whenc is real andc–2–0 and apply the results to the correlation functions of this dynamical system.Supported by NSF grant DMS-9101798     

Spin alignment and parity violation effects inρ 0 production in neutrino and antineutrino charged-current interactions

Most of the measured events on hydrogen, deuterium and neon targets from BEBC at CERN and the 15-foot Bubble Chamber at Fermilab are combined to measure spin-density matrix elements for ⁰ production in the current-fragmentation region in neutrino and antineutrino charged-current interactions. The spin-alignment parameter is found to be 0.08±0.11 for neutrino and 0.41±0.11 for antineutrino interactions. The parity-odd density matrix element Im(₁₀+_–10) is consistent with zero in both reactions. In the combined neutrino and antineutrino data an indication of parity violation is observed: Im _1–1=0.38±0.16 atx _B >0.2,Q ²>8 GeV² andz>0.6.     

New Coherent States for the BDS-Hamiltonian

In the paper we construct a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, previously introduced by Beckers, Debergh, and Szafraniec, which we have called the BDS-Hamiltonian. This Hamiltonian depends on the new creation operator a ⁺, i.e. the usual creation operator displaced with the real quantity . In order to construct the coherent states, we use a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact we change the inner product. This ansatz assures that the set of eigenstates be orthonormalized and complete. In the new inner product space the BDS-Hamiltonian is self-adjoint. Using these coherent states, we construct the corresponding density operator and we find the P-distribution function of the unnormalized density operator of the BDS-Hamiltonian. Also, we calculate some thermal averages related to the BDS-oscillators system which obey the quantum canonical distribution conditions.     

Stochastic and quantum space-time metrics and the weak-field limit

In this review we present a simple method of introducing stochastic and quantum metrics into gravitational theory at short distances in terms of small fluctuations around a classical background space-time. We consider only residual effects due to the stochastic (or quantum) theory of gravity and use a perturbative stochastization (or quantization) method. By using the general covariance and correspondence principles, we reconstruct the theory of gravitational, mechanical, electromagnetic, and quantum mechanical processes and tensor algebra in the space-time with stochastic and quantum metrics. Some consequences of the theory are also considered, in particular, it indicates that the value of the fundamental lengthl lies in the interval 10^–23l10^–22 cm.     

Discrete Painlevé equations and their appearance in quantum gravity

We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize similarity solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp [–t ₁ z ² –t ₂ z ⁴ –t ₃ z ⁶].     

Computational Logic on Fock Space

Fock space may provide an important mathematical model for quantum computation. For this reason, it may be useful to generalize previous work on computational logic to the Fock space framework. The basic construction of this computational logic is the set D(H) of density operators on a Fock space H. We first define n-sector p _n() and total probabilities p() of elements D(H). We next discuss NOT, AND, and OR operations on D(H). Natural equivalence classes and Scotian elements are described. We also discuss minimal and maximal elements and quantum numbers for the equivalence classes. We finally treat the operation and the stronger equivalence classes associated with this operation.     

The stability and instability of relativistic matter

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namelyp ²/2m is replaced by (p ² c ²+m ² c ⁴)^1/2–mc ². The electrons are allowed to haveq spin states (q=2 in nature). For one electron and one nucleus instability occurs ifz>2/, wherez is the nuclear charge and is the fine structure constant. We prove that stability occurs in the many-body case ifz2/ and <1/(47q). For smallz, a better bound on is also given. In the other direction we show that there is a critical _c (no greater than 128/15) such that if > _c then instability always occurs forall positivez (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A02The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978     

Quantum mechanical version of classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz^*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz^* with |s|²-|r|²=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.     

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, , with real holomorphic ,f _k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion . The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen.     

Observation of meson central production in baryon exchange processes

The central production of ⁰,f ₂ and ₃ ⁰ mesons is observed for the first time in processes which are originated by ⁺ p reactions proceeding via baryon exchange mechanism. The data come from the CERN WA56 experiment designed to separate the baryon exchange reactions in ⁺ p-collisions at 20 GeV/c. We report on the measured integral and differential cross sections and also give the density matrix elements of the meson resonances observed.