Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Baryon decays and gamma-spectroscopy of hypernuclei

The applicability of (K ^–,^–) - and (K ^–,^– N)-reactions on^13,14C and^14,15N nuclei to the study of -transitions in primary and daughter -hypernuclei is discussed. The intensity of -deexcitation of ¹³C state |S ¹²C(15·11 MeV; 1⁺1): 1/2⁺ has been shown to be comparable with the intensity of baryon decay. Isospin selection rules are used to distinguish excitation energy ranges of primary hypernuclei, where the identification of the secondary -lines is probable.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Study of the influence of iron and lanthanum oxides on the band structure of Pb(Zr,Ti) O3

The measurements of diffuse reflectance spectra in the range of 400; 2500 nm and of photoacoustic spectra in the range of 350; 800 nm were used for the study of the influence of iron (III) oxide and lathanum oxide additions on the band structure of polycrystalline lead zirconate — lead titanate solid solutions. The molar ratio of zirconium and titanium in the studied solid solutions was 60 40, and the concentrations of the dopants werex FeO_1.5, andy LaO_1.5, werex 0; 0·05 mol, andy 0; 0·10 mol, respectively. The diffuse reflectance spectra were expressed in terms of the Kubelka-Munk function values versus incident electromagnetic radiation energy. These dependences were decomposed into partial absorption bands using a computer. Transitions of charged carriers between energy levels or bands which differ mutually by the energy of 0·9, 1·2–1·3, 1·6, 1·65–1·7, 2·4–2·5, 2·75–2·8, 2·9–3·2, 3·4–3·8 eV, were identified. Their intensities depend on the concentration of Fe and/or on La in the Pb(Zr_0.60 Ti_0.40) solid solutions.The authors are very grateful to Mrs. Iva Havlasová, Vra Andrsová, Ing. Jan Netuka, and Mrs. Liana Karlíková from the first authors' institute, for their technical assistance.     

Decay of the Two-Point Function in One-Dimensional O(N) Spin Models with Long-Range Interactions

Using Griffiths and Lieb–Simon type inequalities, it is shown that the two-point function of ferromagnetic spin models with N components in one dimension decays like the interaction J(n)n ^– provided that 1N4 and T>T _c.     

Expansion of the frequency criterion of absolute stability for controllable systems with one nonlinear nonstationary element

It is demonstrated that for nondegenerate systems having a nonstationary nonlinearity the frequency condition k^–1 + Re W(i) 0 (k < ) ensures exponential absolute stability.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 319–322, March, 1972.     

Radiation effects in lepton decays of neutral vector mesons and μ-e universality

A study has been made of the effects of radiation corrections of order as regards the probability of lepton decay of a moving neutral vector meson V e⁺e^– and V ^+– (V = ⁰, , ); -e universality has been checked by examining the effects of radiation corrections due to exchange of virtual photons between lepton and antilepton, and also to vacuum polarization by e⁺e^– electron and ^+–muon closed loops; in addition, a study has been made of the emission of soft photons as regards the partial width for the electron and muon modes of V-meson decay. Closed expressions (1), (4), and (5) for the corrections imply that the ratio ¦R_v¦ for the probabilities of decay of a V meson to e⁺e^– and ^+–, which are given by (11), agree well with the available evidence on ⁰ e⁺e^{– +–} decays and indicates a possible deviation from -e universality in e⁺e^–+– decays.Read at the meeting on high-energy physics, Nuclear Physics Section Academy of Sciences of the USSR (Moscow, February 11–15, 1974).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 84–89, July 1975.We are indebted to Professor A. A. Sokolov for a discussion.     

Polarization effects in the photoproduction of a fermion pair and a gamma quantum on nuclei

Integration of the cross section of the process+(Ze)(Ze)+₁+e^–+e⁺ over the angles of emission of the electron and positron was carried out, and the angular, energy, and spin dependences of the cross section were obtained. The cross section for photoproduction of a lepton-antilepton pair and a quantum on nuclei in the case where the emitted photon moves in the same direction as the incident quantum can exceed the Bethe-Heitler cross section for ordinary photoproduction of an e^–e⁺ pair. The probabilities of some electron and positron spin states are analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 106–110, June, 1972.In conclusion I express my thanks to A. A. Sokolov and A. I. lViukhtarov for valuable advice.     

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.     

Self-diffusion in simple models: Systems with long-range jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self-diffusion coefficientD _N() in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set ofN neighboring sites. We obtain positive upper and lower bounds onF _N()=N{(1–)–[D_N()/D_N(0)]}/[(1–)]x for [0, 1]. Computer simulations for the square, triangular, and one-dimensional lattices suggest thatF _N becomes effectively independent ofN forN20.     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Generating fields from data onH − ⋃H +andH − ⋃I +

There are series solutions for characteristic boundary value problems for fields on black hole backgrounds that converge when the data are given on =^{– +}, or on =^{– +}, but may not converge when the data are given on ^{– –}, or on ^{+ +}. We specialize to oscillatory data of frequency and calculate approximate reflection and transmission coefficientsR() andT(), using a field generated by data on =^{– +}, and again, using a field generated by data on ^{– –}. The first calculation gives qualitatively good results at all frequencies at each order of approximation, and quantitatively better results at higher orders of approximation; the second calculation, using the series which may not converge, gives bad results except at very high frequencies. Thus for the physically unnatural case of a field that vanishes on ^– and goes toe ^iv on ⁺ we have a series that is convergent, and uniformly so with respect to frequency, while for the natural case of a field that vanishes on ^– and goes toe ^iv on ^– we are limited to high frequencies. It is argued that a frequency-dependent renormalization of a series of the first type provides an approximation scheme that is convergent, and uniformly so with respect to frequency, for the physically important problems of the second type. The difficulties posed by the -dependent renormalization for the study of incident pulses are discussed.     

Pauli-Dirac matrix generators of Clifford Algebras

This article presents a Pauli-Dirac matrix approach to Clifford Algebras. It is shown that the algebra C₂ is generated by two Pauli matrices i₂ and i₃; C₃ is generated by the three Pauli matrices ₁, ₂, ₃; C₄ is generated by four Dirac matrices ₀, ₁, ₂, ₃ and C₅ is generated by five Dirac matrices i₀, i₁, i₂, i₃, i₅. The higher dimensional anticommuting matrices which generate arbitrarily high order Clifford algebras are given in closed form. The results obtained with this Clifford algebra approach are compared with the vector product method which was described in a recent article [Found. Phys. 10, 531–553 (1980) by Poole, Farach and Aharonov] and with the Dirac, Rashevskii and Ramakrishnan methods of matrix generation.Supported by the National Science Foundation under Grant ISP-80-11451.     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

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 equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.     

Parametric relaxation of spectra

We study the change of an quasienergy spectrum upon variation of the weight of a perturbation in the Floquet operatorF=F ₀e^–iV . Employing ideas from level dynamics and random matrix-theory we show that the distribution of nearest-neighbor spacings can display effectively irreversible behavior. Small deviations from equilibrium relax in a certain collision time which scales with the numberN of levels as _collN ^–3/2.