Approximate ground state in a model with spontaneous symmetry breakdown

The spontaneous symmetry breakdown is treated by means of a variational approach. Use is made of coherent states of Glauber and of pairing states of BCS-type as the translationally invariant vacuum states for the discussion of the real scalar field ⁴ withm ₀ ² 0. The first type of trial states reproduces the usual approach to spontaneous symmetry breakdown (-) in the tree approximation (which is possible only form ₀ ² <0), while the second type of trial states offers the possibility of spontaneous symmetry breakdown even form ₀ ² =0.     

Infinite hierarchy of exponents in a two-component random resistor network

We have studied the voltage distribution for a two-component random mixture of conductances _a and _b. A scaling theory is developed for the moments of the distribution, which predicts, for small values ofh=_a/_b, an infinite number of crossover exponents, one for each moment, for Euclidean dimensiond >2, and only one crossover exponent ford=2. Monte Carlo results on the square lattice confirm this prediction.     

Periodic and flat irreducible representations ofSU(3) q

We construct all the periodic irreducible representations ofU(SU(3)) _q forq am-root of unity. Their dimensions arek(2m) ² fork=1,...,m (onlyk=1,...,m/2 for evenm). Their interest is that they could be a tool to generalize the chiral Potts model. By truncation of these representations, we construct flat representations ofU(SU(3)) _q , in which all the multiplicities of the weights are set to 1.     

Probabilistic bootstrap percolation

In bootstrap percolation, sites are occupied with probabilityp, but those with less thanm occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at leastm occupied first neighbors or the whole lattice is empty) is achieved. Formm ₁ the transition is first order, while form<m ₁ it is second order, withm-dependent exponents. In probabilistic bootstrap percolation, sites have probabilityr or (1–r) of beingm- orm-sites, respectively (m-sites are those which need at leastm occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. Form=2 andm=3, the transition changes from second to first order atr ₁=1/2, and the exponent is different forr<1/2,r=1/2, andr>1/2. The same qualitative behavior is found form=1 andm=3. On the other hand, form=1 andm=2 the transition is always second order, with the same exponents ofm=1, for any value ofr>0. We found, form=z–1 andm=z, wherez is the coordination number of the lattice, thatp _c=1 for a value ofr which depends onz, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents and form=2 andm=1 are equal, for dimensions below 6.On leave from Universidade Federal de Santa Catarina, Depto. de Fisica, 88049, Florianópolis, SC, Brazil     

Heavy-light mesons and quark constantsf B,f D in the vacuum correlator method

Quark constantsf _M and masses ofB, D, andD _S mesons are calculated using the vacuum correlator method, suggested recently for long-distance nonperturbative QCD. The dynamical input is standard current quark masses, string tension and _s( _QCD). The electronic width of , and calculated with the same parameters agrees with experiment. Hyperfine splitting ofD, D _s mesons is also reproduced. The resulting values off _B,f _D,f _Ds are 0.17, 0.21 and 0.25 GeV, respectively;f _D andf _Ds are within recent experimental bounds. It is shown thatf _M does obey the asymptoticM ^–1/2 law forM>5 GeV.     

Separability in distant Jauch-type hybrid macrostates of a quantum and a classical system

It is assumed that for a quantum system (Q) plus a classical one (C) that are in a distant state the actually measurable Hermitian operators are of the formA _kK b _k Q _k (A is any Hermitian operator forQ, and the decomposition _k Q _k =1 of the identity is, after von Neumann, characteristic forC). This leads to Jauch-type macrostates (classes of microstates or statistical operators) forQ+C. On the other hand, it is shown that in theQ+Q case the essence of quantum correlations are the conditional states (or statistical operators) of subsystem I and the reduced state _II. Along these lines, the correlation entities (as a complete set of invariants) for the macrostates of theQ+C system are derived, and it is shown that one can make an isomorphic transition from the-convex set of the latter to that of the hybrid macrostates ( _k ,p _k ) Here _k is the conditional state ofQ under the condition thatQ _k occurs onC, andp _k is a classical discrete probability distribution onK, taking the place of _II as the macrostate ofC. This study indirectly throws new light on the nonseparability in theQ+Q case by contrasting it with a well-understood separability in theC+C andQ+C cases.     

The 1/N F expansion of the γ and β functions in Q.E.D.

The Callan-Symanzik- and-functions are calculated analytically for Q.E.D. in the limit of a large number of leptons (N _F) up to terms of order 1/N _F inclusive. We give closed analytic expressions for the coefficients of these terms in their series expansion in powers ofK N _F/. We have been able to sum these series and to obtain some striking results.     

On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation for sufficiently small >0, in which sense we call them semiclassical bound states. We assume that 1p< forn=1,2 and 1p<1+4/(n–2) forn3, and thatV is in the class(V) _a in the sense of Kato for somea. For any finite collection {x ₁,...,x _N} of nondegenerate critical points ofV, we construct a solution of the forme ^–iEt/v(x) forE<a, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx ₁,...,x _N respectively. The concentration gets stronger as 0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller >0. Indeed, for each such collection of critical points we construct 2 ^N–1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.     

Dynamical mass generation

We investigate the possibility of calculating the fermion and the gauge-boson masses within an electroweakSU(2)_L ×U(1)_Y gauge-invariant model without the Higgs fields. Instead of the whole Higgs sector we introduce one Abelian vector boson C with massM renormalizably interacting with leptons and quarks of chiralities L and R with a strengthh. An interplay of all interactions which contribute to the fermion masses results in the fermion mass formulam _f= =M exp[87²/(3y(f _L)y(f _R)h ²)], wherey(f _L,R) are the C hypercharges, and y(f_L). y(f _R) < 0. The intermediate-boson massesm _w andm _z are expressed in terms of the fermion masses via sum rules.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Second order large deviation estimates for ferromagnetic systems in the phase coexistence region

We consider thed-dimensional Ising model with ferromagnetic nearest neighbor interaction at inverse temperature . Let be the magnetization inside ad-dimensional hyper cube , ₊ be the+Gibbs state andm*() be the spontaneous magnetization. For such thatm*()>0 we find a sufficient condition (easily verified to hold for large ) for ₊({M [a,b]}) to decay exponentially with ||^(d–1)/d when –m*<b<m*, –1a<b. Ford=2 this sufficient condition is the exponential decay of a connectivity function. We also prove a partial converse to this result, obtain a sharper result for the magnetization ond–1 dimensional cross sections of the model and prove a similar result ford=2, –m*<a<b<m*, and large, when free boundary conditions are chosen outside .Work partly supported by the U.S. Army Research Office     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

Structural relaxation effects on the low-temperature properties of vitreous silica

Measurements of the low-temperature specific heatC and thermal conductivity of vitreous silica after heat treatment at temperaturesT _a between 900°C and 1,400°C are reported. A decrease ofC and an increase of are observed over the whole temperature range studied (C0.06K<T<6K; 0.5 K<T<20 K). Below 1 K the changes inC and (10%) are attributed to a dependence of the density of tunneling states on the fictive temperature. Measurements of the thermal conductivity show that these changes are reversible, thus strongly supporting the evidence for a connection between the tunneling states and the quasi-equilibrium state which is frozen in when an undercooled liquid drops out of thermal equilibrium. Our results are compared to predictions of the free-volume theory of the glass transition. At higher temperaturesC decreases by roughly the same amount as below 1 K while increases by up to 30%. The dependence ofC and onT _a cannot be explained unambigously in terms of a phonon-fraction crossover in the vibrational density of states. Instead, a recently proposed model of coupled SiO₄ rotations is favored.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Jauch-Piron system of imprimitivities for phonons. I. Localizability in discrete space

This paper is devoted to a discussion of the notion of localizability for phonons, i.e., quasiparticles arising from the harmonic vibrations of a system ofn atoms bound to one another by elastic forces. The natural tools for the analysis of localizability are the projection operatorsÊ() acting on the Hilbert space of one-phonon states, where is an arbitrary subset of the set that consists ofn vectors specifying the equilibrium positions ofn atoms. The expectation value ofÊ() is the probability that the phonon belongs to the atoms whose equilibrium positions are characterized by the elements of . For a strongly localizable phonon all of the projection operatorsÊ() commute with one another, whereas in the case of a weakly localizable phonon the operatorsÊ(₁) andÊ(₂) do not commute when ₁ and ₂ overlap. With the aid of the Jauch-Piron quantum theory of localization in space, the present paper describes the method of obtainingÊ() and also shows that if in the system ofn atoms there exist normal modes of zero frequency, then the phonon is only weakly localizable. Given the explicit expression forÊ(), one can define the number-of-phonons operator as well as the quasiparticle analogue (given in a companion paper) of the Wigner distribution function.     

Asymptotic behaviour of spin-momentum distribution observables for fermion fields with cut-off self-coupling

In a previous paper asymptotic creation and annhilation operatorsa _± ^# have been constructed by the Kato-Mugibayashi method from the creation and annihilation operatorsa ^# for spin 1/2 fields with an interaction Hamiltonian density which is an evendegree polynomial in the field with ultra-violet cut-off and its derivatives. For any eigenvector of the total HamiltonianH=H ₀+H _I partial isometries _± have been defined so thata _± ^# equal _± a ^# *_± on the ranges _± of _±. Since the existence of a groundstate ofH has been proved, the existence of at least one pair _± follows. The purpose of this paper is to show that for any _± orthogonal to the distribution of spins and momenta of the interacting Schrödinger states exp[–itH]_± approaches fort the distributions of spins and momenta of the free state exp[–itH ₀] if a wave-amplitude renormalization is carried out in _±. This is achieved by studying the expectation values of the operators in themaximally abelian W*-algebra generated by operators of the form a*a, in terms of whichany information about spins and momenta can be expressed.Supported in part by the National Research Council of Canada.     

A first passage time problem for random walk occupancy

Two recent studies of diffusion and flow properties of polymers in a melt have suggested the problem of finding the average time form Brownian particles to leave a sphere for the first time, given that exited particles can also reenter the sphere. We prove that the asymptotic density (asm) for the time to first emptiness of the sphere for zero-mean Brownian motion is a delta function, characterized by the exit timea(m/lnm)^2/D ,a being a constant andD being the dimension. The presence of a field leaves the delta-function form for the density, but changes the time dependence toa lnm, with only the constanta depending on the dimension. Simulations of the process suggest that the value ofm needed for the validity of the asymptotic result is orders of magnitude greater than 1000.     

Fermion masses

In this paper, we show that 2m _e m /(m _e ² +m ² = (g _V/g _A) _e ² . From this expression, the Weinberg mixing parameter is shown to be 0·2254 or 0·2746. Assuming that the electron and muon neutrino masses are degenerate, we find thatm _v = (g _V/g _A) _e ² . (m _e m )/M _WL, where M_WL is the mass of the standard W^± boson. The neutrino mass turns out to be 6·5 eV. The -neutrino mass is found to be about 81 MeV. The masses of c, t, s and b quarks are found to be respectively 1·7 GeV, 21·2 GeV, 0·57 GeV and 2·18 GeV by assumingm _d=m _u= 0·3 GeV. All these masses agree with other estimates except the b quark which has about half of its expected value.     

Detection on NMR via the anisotropy of X-rays (X-NMR-ON)

Nuclear magnetic resonance on oriented nuclei has been detected for the first time via the destruction of the anisotropy of characteristic L_x-rays. The new method can be applied to isomeric states which decay only via highly converted transitions, for which the standard NMR-ON technique — detection of NMR via the anisotropy of -rays — is not applicable. The X-NMR-ON technique has been used to measure the magnetic hyperfine splitting of^193mpt (I=1322⁺; E=149.8 keV; T_1/2= 4.3 d) to be ¦ g _NB_HF/h¦=111.3 (3) MHz. with the known hyperfine field of –1280(27) kG the magnetic moment of^193mpt is deduced to be ¦¦=0.7417(14) _N. This magnetic moment differs strongly from the known magnetic moments of the 13/2⁺ isomeric states in Hg and Pb and^195mPt.     

On the integrated density of states for crystals with randomly distributed impurities

In the present paper, we discuss spectral properties of a periodic Schrödinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentrationp of impurities, for 0<p<1, we focus our attention on the limiting behavior of the integrated density of states _p of the random Schrödinger operator, inside a spectral gap of the periodic operator, asp0. Denoting byU ₀ the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points ofU ₀, in the sense that _p (U ) is of orderp, for any fixed -neighborhoodU ofU ₀, while _p (K)C·p ², for any compact subsetK of the gap which does not intersectU .Research partially supported by Deutsche Forschungsgemeinschaft     

A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT :R ² R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT be such that 0<J1 or 1J<. LetW _u ⁿ be the unstable manifold of a hyperbolicn-cycle andW _s ^m the stable manifold of a hyperbolicm-cycle. Suppose that for< _c ,W _u ⁿ andW _s ^m have no common points, and that for> _c ,W _u ⁿ andW s/m have a transversal heteroclinic point. Then at= _c ,W _u ⁿ andW _s ^m are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.     

Contributions toΛ c + →πΣ,πΛ andp\bar K^0 from excited states

We consider contributions to _c ⁺ , and from excited states. The calculations are performed within the MIT-bag model and a heavy quark bag model. Because the mass of _c ⁺ is rather big compared to the strange baryons, excited baryon states with mass close to that of _c ⁺ in some cases give significant pole contributions to the decay amplitudes of _c ⁺ .