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.     

Electroweak interactions at an infinite sublayer quark level. IV. Composite model of electron,neutrino, and gauge bosons

We assume that the electron (e ^–), neutrino (v _e), and gauge bosons (W ^±,Z ⁰) are composed of only two kinds of particles, an ultimate particleu at an infinite sublayer quark level and a chargeless fermiont, such thate ^–=(u ^cp u ^cp l),V _e =(u u ^cp l,W ⁺=(u u ),W ^–=(u ^cp u ^cp andZ ⁰=(u u ^cp . It is then shown thatCP is violated in weak interactions associated with these electron, neutrino, and gauge bosons.     

Electroweak interactions at an infinite sublayer quark level

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 doublet (u ,u ^c ) ^T it is shown that there exists only one gauge fieldW _µ ³ , from which the electromagnetic fieldA _µ=W _µ ³ cos _w and the neutral vector boson fieldZ _µ ⁰ =W _µ ³ sin _w are derived.     

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     

A note on the Lie algebras g(m∞)

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

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

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.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in ⁿ , with odd n3. We study the asymptotics of the distribution _t of the random solution at time t as t . It is assumed that the initial measure ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of _t to a Gaussian measure as t , which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.     

Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW ₁ ^B ₊ andw ₁ ^B ₊ of theW-infinity algebrasW _{1 +} andw _{1 +} are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW ₁ ^B ₊ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW ₁ ^B ₊ symmetries are shown to be contracted intow ₁ ^B ₊ symmetries of the dispersionless BKP hierarchy.     

On a regularity problem occurring in connection with Anosov diffeomorphisms

Let be aC -manifold and _s and _u be two Hölder foliations, transverse, and with uniformlyC leaves. If a functionf is uniformlyC along the leaves of the two foliations, then it isC on . The proof is elementary.     

Many boson realizations of universal nonlinearW ∞-algebras,modified KP hierarchies,and graded lie algebras

An infinite number of free field realizations of the universal nonlinear ^(N) ( ₁₊ ^(N) ) algebras, which are identical to the KP Hamiltonian structures, are obtained in terms ofp plusq scalars of different signatures withp –q =N. They are generalizations of the Miura transformation, and naturally give rise to the modified KP hierarchies via corresponding realizations of the latter. Their characteristic Liealgebraic origin is shown to be the graded SL(p, q).     

Absence of asymptotic-time symmetry breaking in zero-bias spin-boson model with partial relaxation

The symmetric spin-boson model without external field is treated for any type of coupling to the boson bath and any initial bath density matrix. With initially fully aligned spin (_z (0)= =1), the proof is given that a partial relaxation (_z (+) t1<) implies that there is no asymptotic-time (up-and-down) symmetry breaking (i.e. that _z (+)=0). For the problem of a particle (interacting with free bosons) in a symmetric double well without spatial symmetry breaking before the infinite time limit, this means that att + the particle distribution becomes symmetric (irrespective of the full initial asymmetry) unless the particle fully remains (att + ) in Ihe starting well.     

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.     

On infinite direct products of Lorentz transformations

An infinite direct product _i =1 U _i (a) of continuous unitary representations of SL(2,C) in Hilbert spaces _i is continuous only on certain incomplete direct product subspaces of _i =1 _i . If no representations of the complementary series occur, then each of these subspaces contains a product vector almost all factors of which are SL(2, C)-invariant.     

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.     

D-Dimensional ideal gas in parastatistics: Thermodynamic properties

We consider a parastatistics ideal gas with energy spectrum ¦k¦ (>0) or even more generally in ad-dimensional box with volumeV (periodic boundary conditions), the numberN of the gas particles being well determined (real particles) or not (quasiparticles). We calculate the main thermodynamic quantities (chemical potential, internal energy, specific heatC, equation of state, latent heat, average numbers of particles) for arbitraryd, ,T (temperature), andp (maximal number of particles per state allowed in the parastatistics). The main asymptotic regimes are worked out explicitly. In particular, the Bose-Einstein condensation for fixed densityN/V appears as a nonuniform convergence in thep limit, in complete analogy with the standard critical phenomena that appear in interacting systems in theN limit. The system behaves essentially like a Fermi-Dirac one forall finite values ofp, and reveals a Bose-Einstein behavioronly in thep limit. For instance, at low temperaturesC T ifp< andC T ^d/ ifp. Finally, the Sommerfeld integral and its expansion are generalized to an arbitrary, finitep.     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant satisfies 12, then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx=± and the initial data. The proof is given by an elementaryL ²-energy method.     

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu _t +f(u) _x =u _xx with the initial datau ₀ which tend to the constant statesu _± asx±. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f(u _±). Moreover, the rate of asymptotics in time is investigated. For the casef(u₊)(u_–), if the integral of the initial disturbance over (–,x) is small and decays at the algebraic rate as |x|, then the solution approaches the traveling wave at the corresponding rate ast. This rate seems to be almost optimal compared with the rate in the casef=u ²/2 for which an explicit form of the solution exists. The rate is also obtained in the casef(u _± =s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.