Electroweak interactions at an infinite sublayer quark level. II

It is shown that the standard model of the electroweak interactions holds at an infinite sublayer quark level, insofar as we consider the weak isospin doublet (u _{L,u L} ^cp , whereu is an infinite number of quarks at an infinite sublayer level.     

A note on the Lie algebras g(m∞)

The algebras g(m) are interpreted as realisations of the infinite rank affine Lie algebras g.     

Random walks on bond percolation clusters: ac hopping conductivity below the threshold

The frequency dependence of the ac hopping conductivity in two and three dimensional lattices with random interruptions is calculated by Monte Carlo simulation of random walks on bond percolation clusters. At low frequencies the real and imaginary parts of the ac conductivity vanish linearly and quadratically with the frequency, respectively. The critical behaviour of the imaginary part of the ac conductivity below the percolation threshold is shown to depend on the long time limit of the mean square displacement of random walksR ² , while the real part depends on the time constant of the system as well.R ² is found to diverge with an exponentu=2- according to the conjecture of Stauffer.     

Spatial fluctuations in reaction-limited aggregation

A general method is used for describing reaction-diffusion systems, namely van Kampen's method of compounding moments, to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areK _ij=ij (i, j=1, 2,). For thediffusion constants D _j of clusters of sizej the following class of models is considered:D _j=D if 1Js andD _j=0 ifj>s. The casess= ands< are studied separately. For the models= the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs< the density fluctuations remain finite att _c, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isd _c=2.     

Influence of boundary conditions on square bond percolation nearpc

The influence of boundary conditions on square bond percolation for system sizes ranging from 10×10 to 240×240 is studied for the quantitiesP , , the effective percolation threshold and the finite-size scaling relations forP and . The Monte Carlo simulations suggest that free edges approximate the infinite system as well as the more complicated periodic boundary conditions.     

A Chern-Simons-Like string withW-symmetry

The constraint algebra of a generalized Chern-Simons string is investigated. It is shown that an infinite number of secondary constraints are present in the model. The connection withW -gravity is discussed.Supported by Bulgarian Foundation for Fundamental Researches under contract -20-91.Supported by Bulgarian Foundation for Fundamental Researches under contract -11-91.     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

Limits on stability of positive molecular ions

It is shown that a positive diatomic molecule consisting of N electrons and two nuclei with charges Z ₁ and Z ₂ is unstable with respect to breakup into two atomic subsystems if the nuclear charges are sufficiently large. Bounds on the critical charge are obtained in the limit as N .Research supported in part by NSF grants DMS-8709805 and DMS-8808112.     

Ground-state structure in a highly disordered spin-glass model

We propose a new Ising spin-glass model on Z ^d of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite-volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2 ^N , whereN=N(d) is the number of distinct global components in the invasion forest. We prove thatN(d)= if the invasion connectivity function is square summable. We argue that the critical dimension separatingN=1 andN= isd _c=8. WhenN(d)=, we consider free or periodic boundary conditions on cubes of side lengthL and show that frustration leads to chaoticL dependence withall pairs of ground states occurring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.     

Two examples illustrating the differences between classical and quantum mechanics

Two examples are presented: The first shows that a potentialV(x) can be in the limit circle case at even if the classical travel time to is infinite. The second shows thatV(x) can be in the limit point case at even though the classical travel time to infinity is finite. The first example illustrates the reflection of quantum waves at sharp steps. The second example illustrates the tunnel effect.This research was partially supported by the National Science Foundation under Grant GP-34260.For help with the diagrams, the authors thank Bob Johnson.     

The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞)

We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL _q () by the so-called quasi-classical limit of GL _q (). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.     

Quasifinite highest weight modules over the superW 1+∞algebra

We study quasifinite highest weight modules over the supersymmetric extension of theW ₁₊ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the (B, C)-system, and the embedding into (|) are also presented.Address after April 1, 1994: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, JapanAddress after April 1, 1994: Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan     

Electroweak interactions at an infinite sublayer quark level. III.CP violation

In a previous paper we proposed the infinite sublayer quark model, in which there exists an infinite number of quarksu and antiquarksu ^c at an infinite sublayer level. By applying the standard model of the electroweak interactions to the weak isospin doublets (u _L ,u _L ^cp ) ^T and (u _L ,u _L ^ct ) ^T , it is shown that there exists only one phase factor, which causesCP violation.     

Solute reaction mediated precipitate patterns in cross gradient free systems

Recent studies on the spontaneous formation of undulatory patterns of precipitate concentration, such as Liesegang and related phenomena, have shown that these effects can be explained on the basis of a competitive particle growth mechanism as underlies Ostwald ripening. However these effects require an overall gradient or boundary perturbation to induce regular patterning on a scale greater than the interparticle spacing to override the single crystal-so called greedy giant instability. Here we show that by allowing for precipitate particle mediated reactions involving the solute species (from which the precipitate is constructed) a fastest growing mode of wave vectork _d , 0<k _d <, exists and thus pattern formation can occur completely autonomously at a well defined wave length without the need of cross gradients. These processes may be much more widely realized in physical and biological self organization processes than the familiar reaction diffusion (Turing) model due to the rather nonspecific nature of the present phenomenon.Research supported in part by grants from the National Science Foundation and the Petroleum Research FundA.P. Sloan Fellow, 1980–82NATO Fellow, 1982     

On the chiral hubbard model and the chiral Kondo lattice model

The integrability of the one-dimensional chiral Hubbard model is discussed in the limit of strong interaction,U=. The system is shown to be integrable in the sense of the existence of an infinite number of constants of motion. The system is related to a chiral Kondo lattice model at strong interactionJ=+.     

Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

LetH _l be the Hamiltonian in aP()₂ theory with sharp space cutoff in the interval (–l/2,l/2). LetE _l =inf(H _l ), (l)=–E _l /l, and let _l be the vacuum forH _l . discuss properties of (l) and _l . In particular, asl, there are finite constants <0 and such that (l), ((l)–)l, and hence (l)=+/l+o(l ^–1). Moreover exp(–c ₁ l) _{l 1}exp(–c ₂ l) forc ₁,c ₂ positive constants, where _{l 1} is theL ¹(Q, d₀) norm of ₁ with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofH _l in the infinite volume limit.Research sponsored by AFOSR under Contract No. F44620-71-C-0108.On leave from Istituto di Fisica Teorica, Universitá di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.A. Sloan Foundation Fellow.     

Differential calculus on quantum spheres

Three-dimensional differential calculus on quantum spheres S _infc ^sup2 ,]–1, 1[{0}, c[0, ], is introduced and investigated. Spectra of generalized Laplacians are found. These operators are expressed by generalized directional derivatives. Classical limits of these objects are obtained and a simple approach to quantum mechanics on a quantum sphere is presented.     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

The logarithm of the derivative operator and higher spin algebras ofW ∞ type

We use the notion of the logarithm of the derivative operator to describeW type algebras as central extensions of the algebra of differential operators. We also provide closed formulae for the truncations ofW ₁₊ to higher spin algebras withsM, for allM2. The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76Sf00098 and in part by the National Science Foundation under grants PHY-85-15857 and PHY-87-17155Address after July 1, 1992: Dept. of Mathematics, Yale University, New Haven, CTO6520, USA