Conformal Quantum Effects and the Anisotropic Singularities of Scalar-Tensor Theories of Gravity

We show that the inclusion of a term C _abcd C ^abcd in the action can remove the recently described anisotropic singularity occurring on the hypersurface F () = 0 of scalar-tensor theories of gravity of the type

Layering transition in SOS model with external magnetic field

For the SOS model defined by the Hamiltonian , where _x , _x ,{1,2,...},h>0,x ^d ,d2 it is shown that in the low-temperature region an infinite sequence of first-order phase transitions takes place whenh»0 and the temperature is fixed.     

The Relation of Generalized Scalar-Tensor Theory with the 5D Space-Time-Matter Theory

The relationship between cosmological solutionsof five-dimensional Space-Time-Matter (STM) theory anda Generalized Scalar-Tensor (GST) theory is investigatedin which the cosmological term Lambda depends not only on a scalar field but also onits time derivative .Identification of these solutions allows us to solve forthe functional form of the cosmological term, and mayhave relevance for the early Universe.     

Boson fields with bounded interaction densities

We consider interaction densities of the formV((x)), where (x) is a scalar boson field andV() is a bounded real continuous function. We define the cut-off interaction by , where _E(x) is the momentum cut-off field. We prove that the scattering operator S_r(V) corresponding to the cut-off interaction exists, and we study the behavior of the scattering operator as well as the Heisenberg picture fields, as the cut-off is removed.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.At leave from Mathematical Institute, Oslo University.     

A mass zero cluster expansion

A cluster expansion is developed and applied to study the perturbation ()⁴ of the massless lattice field in dimension 3. The method is loosely inspired by the work of Gawedzki and Kupiainen on block spin techniques for the system. The cluster expansion is given in terms of expansion coefficients for the field as a sum of certain special block spin functions. These functions are chosen with a large number of moments zero, so that the interaction couples spatially separated functions with an interaction falling off as a high inverse power of the separation distance. The present techniques, with some technical development, should work for broad classes of other models, including the lattice dipole gas and the model. Models (,,^; )^2s , >1/2, are essentially included in the present work.This work was supported in part by the National Science Foundation under Grant No. PHY 79-05688     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).     

Solutions of the Boltzmann equation for Maxwell interactions and singular angle-dependent cross sections

We consider the spatially homogeneous and isotropic Boltzmann distribution function in the case of nonisotropic, binary cross sections inversely proportional to the relative speed of the colliding particles. Further, we allow the angle dependence of the differential cross section() to be singular in the forward direction ( 0). We assume (), d < which includes the case of a Maxwellian interaction. We explicitly show how to construct the solutions of the Boltzmann equation, study their properties, and obtain for a class of solutions sufficient conditions for their existence at any positive time value. We extend the formalism to the more general case of arbitrary dimensionality. We observe an effect noticed previously by Krook, Wu, and Tjon in other models of the Boltzmann equations-namely, for special initial distributions, we find solutions which exhibit an excess of higher energy particles at later time.     

Isosystolic inequalities and the topological expansion for random surface and matrix models

Using the isosystolic inequalities on Riemann surfaces, we prove that for many random surface or matrix models the radius of convergence of the perturbative series at fixed genus is independent of the genus. This result applies for instance to the dynamically triangulated random surface model in any dimension or to many matrix models with regular propagators in the superrenormalizable domain, for instance ³ in dimensiond<6, in dimensiond<4, and various otherP()₂ models (in particular all those containing an odd power of ). We hope that this result is a first step towards a more rigorous understanding of the genus dependence of surface models or of quantum gravity coupled with matter fields.     

Theory of fiber bundle with local metric of internal space and gravitation with torsion

In this paper we propose a new theory of a fiber bundle provided with a local metric of internal space. The fibers differ from usual fibers, having an enlarged factor. The enlargement may be procured by a differential mapping(x) from structure groupG to the fiberF _x atx M, and(x)R. The torsion presented stems from the local metric of internal space and the local metric stems from a induced mapping _*(x) of(x). From the theory we can get the Brans-Dicke theory with torsion. If we assume the spin density of the gauge field determines the enlarged factor of the fiberF _x, our theory is an extended Cartan theory.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations S_t S _x ² =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(^x, ^t) should be written as a product of the elementary solution x²/(2^t) and a function f=f() where =(x, |t|) owing to the symmetry property which is that S(x, –t)=–S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of (x, |t|)=₀ |t/t₀| (x/x ₀)0 with , results in a soluble ordinary differential equation, of first order in u=ln and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called trivial. The phase plane (f_u, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the trivial fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of von Neumann algebras in standard form

It is shown that for normal states ρ and φ of a W ^*-algebra , where P(.,.) is the transition probability considered by Uhlmann [1], and ζ(ω) is the vector in the natural positive cone of some standard faithful representation of A, associated with the normal state ω. The above inequality is equivalent to: , where d(.,.) is the Bures distance function [5].     

The physical properties of linear and action-angle coordinates in classical and quantum mechanics

The quantum harmonic oscillator is described in terms of two basic sets of coordinates: linear coordinates x, p_x and angular coordinates eⁱ, P (action-angle variables). The angular coordinate eⁱ is assumed unitary, the conjugate momentum p is assumed Hermitian, and eⁱ and p are assumed to be a canonical pair. Two transformations are defined connecting the angular coordinates to the linear coordinates. It is found that x, p_x can be physical, i.e., Hermitian and canonical, only under constraints on the p eigenvalue spectrum. The conclusion is that eⁱ can be a unitary operator. A parallel analysis of the classical harmonic oscillator is done with equivalent results.     

On a limiting procedure for obtaining physical states for an infinite Fermi system

We propose a limiting procedure for obtaining physical states for an infinite non-relativistic Fermi system. We take the thermodynamic limit of vector states in the Fock representation of the C.A.R. algebra, representing a condensate state of atoms each of which is formed by 4 fermions. In a simplified example considered in detail, the limit state has a simple decomposition into the product of two B.C.S. states. IfB ⁺ is the operator creating the atom from the vacuum |_0F , it is proved that the states obtained by taking the thermodynamic limit of the vector states corresponding to (B ⁺) ⁿ |_0F and respectively, coincide on the gauge-invariant elements of the algebra for a suitable value ofz.Partially supported by C.N.R.     

Cosmological Dynamics of the Tachyon with an Inverse Power-Law Potential

We investigate tachyon dynamics with an inverse power-law potential V () ^–. We find global attractors of the dynamics leading to a dust behavior for > 2 and to an accellerating universe for 0 < 2. We study linear cosmological perturbations and we show that metric fluctuations are constant on large scales in both cases. In the presence of an additional perfect fluid, the tachyon with this potential behaves as dust or dark energy.     

The Volume Element of Space-Time and Scale Invariance

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S= L ₁ d ⁴ x+ L ₂ d ⁴ x where the volume element d ⁴ x is independent of the metric. For global scale invariance, a dilaton has to be introduced, with non-trivial potentials V()=f ₁ e in L ₁ and U()=f ₂ e ² in L ₂ . This leads to non-trivial mass generation and a potential for which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for Quintessential models, which are scale invariant but formulated with the use of volume element d ⁴ x alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants.     

Branched polymers in a wedge geometry in three dimensions

We investigate the statistical and dimensional properties of uniform star polymers attached by the branching vertex of degreef in a wedge geometry in three dimensions and described by the wedge angles and. We show that the growth constant is equal to ^f , where is the self-avoiding walk limit. Thef and (, ) dependences of the corresponding critical exponent _f (, ) are studied using Monte Carlo techniques. In the casef=1, our results are compared with existing predictions obtained from series expansion and renormalization group methods. We have also estimated the amplitudes for the mean square radius of gyration and the mean square end-to-end branch length. Our results for the ratio of the mean square radius of gyration of anf-star to that of a linear polymer of the same degree of polymerization attached in a similar wedge, and the analogous ratio for the mean square end-to-end branch length, are consistent with these ratios being lattice-independent quantities.     

Nonequilibrium Brownian dynamics simulations of shear thinning in concentrated colloidal suspensions

The effect of interparticle forces on shear thinning in concentrated aqueous and nonaqueous colloidal suspensions was studied using nonequilibrium Brownian dynamics. Hydrodynamic interactions among particles were neglected. Systems of 108 particles were studied at volume fractions of 0.2 and 0.4. For the nonaqueous systems, shear thinning could be correlated with the gradual breakup of small flocs present because of the weak, attractive secondary minimum in the interparticle potential. At the highest shear rate for=0.4, the particles were organized into a hexagonally packed array of strings. For the strongly repulsive aqueous systems, the viscosity appeared to be a discontinuous function of the shear rate. For=0.4, this discontinuity coincided with a transition from a disordered state to a lamellar structure for the suspension.     

Traces,Dispersions of States and Hidden Variables

No Heading The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let be a state on a C^*-algebra A with the projection structure P(A). The dispersion () is defined as () = sup{(p) – (p)² | p P(A)}. It is proved that () 2/9 whenever is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.     

Monte carlo calculations of the conformal charge

The conformal charge is an important quantity which characterizes the nature of the two-dimensional phase transition. We report a first attempt to use a new numerical method to calculate the conformal charge. In this paper, we apply our method to the 2-dimensional, ⁴, continuous-spin Ising model. By varying the parameters in the Hamiltonian, one can change continuously from the known Gaussian limit to the Ising limit. It is well known that the critical points for these two systems are not in the same universality class. We study this behavior for the Gaussian model, the single-well ⁴ model, the border model, and the double-well ⁴ model for a large lattice. Our results, while giving a good general picture, are not so far sufficient to differentiate whether the non-Gaussian cases studied belong to the Ising model universality class or not. Further studies of other lattice sizes should serve to improve greatly our conclusions.