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].     

Scaling function for energy-density correlations: Dispersion theory andε-expansion forT>T c

A dispersion representation for the static energy-density correlation function ² (q) ²(–q) _c =C(q,T)=A+Bt ^–h(z ²), wherez=q , t=(T—T)_c/T _c and is the correlation length, is discussed.h(z ²) is calculated to order ² in the zero-field critical region (T>T _c) for the standard isotropicn-component ⁴Ginzburg-Landau-Wilson model. Utilizing a procedure similar to that introduced by Bray for the two-point correlation function, the-expansion results are used in conjunction with an approximant for the spectral functionF(z/2) Imh(—z ²) based on the asymptotically exact short-distance expansion resulth ^–1(z ²)z ^/v[D ₀+D ₁ z ^{–(1 —)/v} +D ₂ z ^–1/v ] to predict quantitatively the full momentum dependence ofC(q,T) forT>T _c. In contrast to the two-point correlation function,C(q,T) is found to be a monotonic function as the critical temperature is approached at fixedq (forT>T _c).     

Consequences of the complex character of the internal symmetry in supersymmetric theories

The consequences of the invariance of the superpotential under the complexificationG ^c of the internal symmetry group on the determination of the possible patterns of symmetry and supersymmetry breaking are established in a globally supersymmetric theory. In particular, in the case of global internal symmetry we show that a vacuum associaated to a pointz, whereG _z ^c G _z ^c is always degenerate with a vacuum associated to a pointz, whereG _z ^c =G _z ^c ; all the other degeneracies of the minimum of the potential on an orbit ofG ^c are also determined and shown to be completely removed when the internal symmetry is gauged. The zeroes of theD-term of a supersymmetric gauge theory are characterized as the points of the closed orbits ofG ^c which are at minimum distance from the origin; at these pointsG _z ^c =G _z ^c . It is rigorously proved that the minimum of the potential is zero if the gradient of the superpotential vanishes somewhere. It is also shown that theD-term necessarily vanishes at the minimum of the potential if the direction of spontaneous supersymmetry breaking is invariant byG.Partially supported by the Swiss National Science Foundation and INFN, Sezione di PadovaOn leave of absence from the Department of Physics of the University of Padova, Italy     

Equations for the Magnetization Modes in a System Having a Quadruple Nucleus with the Spin S = 1

Within the framework of the method of a density matrix, differential equations are obtained that describe the evolution of the modes of magnetization <Î₊>, <Î_{+ z} >, and <Î_{+ z} ²> for a spin system containing a nucleus with a 1/2 spin connected scalarly with the quadruple nucleus of the spin S = 1 in the presence of crosscorrelation between the dipole ISinteractions, anisotropy of the chemical shift of the I and S nuclei, and quadruple interaction of the S nucleus. The equations are used for analysis of the evolution of the intensities of each line in the triplet of the Raman spectrum of the 1/2 spin. Expressions for the times of crossrelaxation and the times of transverse relaxation of each of the lines of the triplet are obtained.     

Quantitative universality for a class of nonlinear transformations

A large class of recursion relationsx _{n + 1} = f(x_n) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum . With sufficiently small),z > 1, the universal details depend only uponz. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio ( = 2.5029078750957... forz = 2). This structure is determined by a universal functiong ^*(x), where the 2ⁿth iterate off,f ⁽ⁿ⁾, converges locally to ^–n g ^*( ⁿ x) for largen. For the class off's considered, there exists a _n such that a 2ⁿ-point stable limit cycle including exists; – _n R~ ^–n ( = 4.669201609103... forz = 2). The numbers and have been computationally determined for a range ofz through their definitions, for a variety off's for eachz. We present a recursive mechanism that explains these results by determiningg ^* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.Research performed under the auspices of the U.S. Energy Research and Development Administration.     

Effect of disorder on relaxation in the one-dimensional Glauber model

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking valuesJ ₀ andJ ₁ with probabilitiesp and 1–p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large timest, the magnetizationM(t) varies as [exp(–₀ t](t), where ₀ is a function of the stronger bond strengthJ ₀ only, and (t) decreases slower than an exponential. For very long times, we find that ln (t) varies as –t ^1/3. For low enough temperatures, there is an intermediate time regime when ln (t) varies as –t ^1/2. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming thatM(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum valueJ ₀, we find that ln (t) varies as –t ^1/3(lnt)^2/3 for large times.     

A new theory on scattering of electrons due to spiral dislocations

The topologically singular lattice structure around a spiral dislocation scatters electrons in a manner much different from the deformation-potential-scattering. Thet-matrix is represented ast(k, k)=t ₀ sin (k _z/2) exp (±i _kk/2)/sin ( _kk/2) with the scattering angle _kk, the componentk _z of the wave vector parallel to the dislocation, andt ₀(Fermi energy). This kind of scattering alone does not cause a perpendicular resistivity, but in a dirty limit a specific dislocation resistivity does exist, whose relaxation time per unit length of dislocation is of the order of/(Fermi energy).Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-KölnOn leave of absence from Department of Materials Science, Hiroshima University, Higashisenda-machi, Hiroshima, Japan     

Mössbauer effect in superconducting tin-europium-molybdenum-sulfides

Mössbauer spectra of¹¹⁹Sn in Sn_1–z Eu _z Mo₆S₈ (z=0, 1/3, 2/3) have been measured in the temperature range 4.9 KT293 K. The spectra consist of a quadrupole doublett with unequal intensities at all temperatures. The Debye-Waller factor, the isomeric shift and the asymmetry of the intensity of the quadrupole components show irregularities between 50 K and 110 K which are interpreted as arising from a structural phase transition. The temperature dependence of the Debye-Waller factor can be approximated by a simple phonon spectrum. No quantitative explanation can be given for the large values of the asymmetry. Evidence is presented that this behavior is connected with the properties of the librational modes of the Mo₆S₈-units in the crystal.     

Critique of Home and Sengupta's derivation of a Bell inequality

Home and Sengupta claim to derive a Bell-type inequality without assuming locality. I show that their derivation implicitly assumes determinism and a weak locality condition. Their derivation also relies upon further, highly implausible conditions that severely limit the physical implications of their argument.1. If by dispersion-free H&S mean that measurement results depend deterministically on the system's state, then they agree with my argument in this section.2. The first term in inequality (1) implicitly assumes that measurement results of two incompatible (noncommuting) observables, L_z and L_a, are simultaneously well-defined. This assumption is not problematic given an underlying determinism, because L_z ^t can be defined as the result wewould obtain upon measuring L_z. Similarly for L_a ^t. Since we can define those two values counterfactually, it is irrelevant for H&S's derivation that we cannot measure them simultaneously.3. Formally, in going from (1) to (2), H&S make the followingperfect correlations assumption: If QM ascribes zero probability to a joint measurement result, then that joint measurement result does not occur. Specifically, H&S assume that, since QM assigns (for state ) zero probability to the joint measurement result S_z ^tL_z ^t, it therefore is not the case that S_z ^tL_z ^t. Standard Bell derivations (cf. Redhead [2]) do not assume that the hidden-variable theory incorporates QM's perfect correlations. But an alternate, Stapp-Eberhard style derivation of an H&S inequality, using four spatial directions instead of three, also avoids this perfect correlations assumption.For deterministic theories in which all observables possess values and measurement always faithfully reveals those values (independent of which other measurements occur simultaneously), Kochen and Specker's functionality condition (FUNC) follows from the perfect correlations assumption. Since an H&S derivation need not assume perfect correlations, Home and Sengupta rule out a slightly broader class of possessed-values theories than Kochen and Specker do.4. This criticism of local noncontextuality also applies under certain conditions to Kochen and Specker's FUNC condition. FUNC, applied to the class of theories discussed in the second paragraph of footnote 3, implies LNC. For those theories, therefore, my critique of LNC also constitutes an attack on the physical plausibility of FUNC.     

From quantum to elliptic algebras

It is shown that the elliptic algebra at the critical level c = –2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p ^m = q ^c+2 for , they commute when in addition p = q ^2k for k integer non-zero, and they belong to the center of when k is odd. The Poisson structures obtained for t(z) in these classical limits contain the q-deformed Virasoro algebra, characterizing the structures at p q ^2k as new algebras.     

Survival probability in one dimension for theA+B→B reaction with hard-core repulsion

We study the effect of hard-core repulsion (known as the bus effect) betweenB particles on the reaction-diffusion systemA+BB in the continuous-time random walk model in one dimension with theA particles stationary. We show rigorously that the survival probability of theA particles is asymptotically bounded asC ₁lim _t{[–logS(t)]/t ^0.5}C ₂, whereC ₁ andC ₂ are constants. We also do simulations to confirm our results.     

Local quantum field theories involving the U(1) current algebra on the circle

The OPE algebra Q=Q(g ² ) generated by a pair of oppositely charged currents (z,±g)(|z|=1) of spin is specified by the leading terms in the small distance expansions of (z ₁,g)(z ₂, -g) and (z ₁,g)(z ₂,g). The current (z,g) splits into a product of a U(1)-Thirring field and a Zamolodchikov-Fattev parafermionic current. The quasilocal(i.e.single-or double-valued) representations of Q are classified. The level k states involve 2(k+1) (ks–k+1) lowest weights (dimensions). The results can be viewed as an extension of the (known) representation theory of the SU(2) current algebra in the bosonic case corresponding to even values of g ² and of the N=2 extended superconformal algebra in the fermionic case corresponding to odd g ².     

The effect of low-temperature electron irradiation on the transition temperature of YBa2Cu3O7−δ superconductors

Measurements of the complex susceptibility =–i of electron-irradiated YBa₂Cu₃O_7– show a strong influence of the electron irradiation dose, ·t on the transition temperatureT _c . For irradiation doses of ·t=2.2·10¹⁹ e^–/cm² we find a damage rate of T _c /(·t)=–1.6·10^–19 K/(e^–/cm²). It is assumed that the decrease ofT _c is mainly a bulk effect due to the production of atomic defects like vacancies and interstitials in the Cu–O–Cu chains and in the basal planes of the unit cells.     

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.     

Contribution ofp-wave (3)+1 subamplitudes to4He binding energy and scattering observables below four-body breakup threshold

The four-body equations of Alt, Grassberger and Sandhas, in the form that the (2)+(2) subsystem amplitudes are treated by the convolution method, are solved for separable two-body interactions between nucleon pairs in the channels³S₁-³D₁ (t ₀₀ alone) and¹S₀. AllL=0 andL=1 (3)+1 subamplitudes are represented in a separable form using the energy dependent pole expansion. The contribution ofp-wave (3)+1 subamplitudes to the⁴He binding energy, threshold scattering observables and low energy cross sections forn ³Hn ³H,p ³Hep ³He,dddd andddp ³H reactions is studied for different two-body potentials.     

Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid

Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vector such that _;v + _v; =2g _v are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.This work was supported in part by the United States Atomic Energy Commission under contract number AT(04-3)-34 Project Agreement No. 125. It was completed when one of the authors (A.H.T.) was on sabbatical leave from the University of California, Berkeley and in residence at the Department of Applied Mathematics and Theoretical Physics, Cambridge University.     

LARGE-SCALE PATHOLOGY OF UNIVERSES WITH VII0×VIII ISOMETRIES

The goal of this paper is to show somewhat unexpected globally pathologic properties in universes described by a class of static planary symmetric exact solutions with G ₆-group of motion. In order to achieve this aim, the Killing vectors, the null geodesics and the Penrose diagrams corresponding to different expressions of g ₄₄=–e ^2f(z), with f(z) solutions of Einstein's equations, have been employed. Finally, woking in a particular gauge, we focus on the behaviour of the radiating electromagnetic modes and derive the observable components of E and B and the expressions of the essential component of the Umov-Poynting vector.     

Adjoint solutions of the Brans-Dicke equations

It is shown that if the Brans-Dicke equations have the solution,g _ij generated by the trace free sourceT _n (T-O) then there exists an adjoint solution ^–1, ²g_ij of these equations generated by the source ^-2 T _u. An example is considered.     

Far-infrared laser stark spectroscopy of CH3OH and13CH3OH

The high resolution laser Stark spectra of methanol and¹³C-substituted methanol have been studied up to Stark fields of about 60 000V/cm with the HCN and DCN lasers. Numerous families of absorption lines have been observed in both parallel and perpendicular polarizations. For methanol, the transitions J _k =7₅ 6₄ A, _t=0; J _k =11₄ 10₃ E _l , _t=0; and J _k =17₃ 16₂ E₂, _t=0 have been identified while the assignments for¹³C-substituted methanol are J _k =14₈ 15₇ A, _t=0; J _k =15₃ 14₂ A⁺, _t=0; J _k =10₇ 9₆ A, _t=0; and J _k =27₉ 27₈ E₁, _t=0. Zero-field frequencies for the assigned transitions are given with improved accuracy over those calculated from available molecular constants, especially for¹³CH₃OH.