Quark confinement in the two-dimensional lattice Higgs-Villain model

We prove quark confinement in the two-dimensional lattice Higgs-Villain model in the weak coupling region by using a Kirkwood-Salsburg equations for unbounded spins.Research partially supported by the National Science Foundation under Grant PHY-77-18762On leave from Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada. Research partially supported by National Research Council under Grant A4015On leave from Istituto di Fisica Teorica, Napoli, Italy     

Exponential clustering for long-range integer-spin systems

By using Kirkwood-Salsburg equations for classical spin systems with unbounded integer values we prove exponential decay (resp. power law decay) for exponential (resp. power law) decaying potentials. We use these results to prove the mass gap in the two-dimensional Higgs-Villain model in the weak coupling region.Research partially supported by the National Science Foundation under Grant PHY-77-18762On leave from Department of Mathematics, University of British Columbia, Vancouver, B. C., Canada. Research partially supported by National Research Council under Grant A4015On leave from Istituto di Fisica Teorica, Napoli, Italy     

Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems IV

We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method ofinfrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.Research partially supported by US National Science Foundation under Grant MPS-75-11864Research partially supported by Canadian National Research Council under Grant A4015Research partially supported by US National Science Foundation under Grant MCS-75-21684-A01     

Propagators and renormalization transformations for lattice gauge theories. I

Lattice gauge theories may be looked at as perturbations of the theory of a vector field with a Gaussian action. We study this theory here and in following papers obtaining crucial results for understanding the renormalization group method in more complicated non-Abelian gauge field theories.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions

We discuss applications of the abstract scheme of part I of this work, in particular of infrared bounds and chessboard estimates, to proving the existence of phase transitions in lattice systems. Included are antiferromagnets in an external field, hard-core exclusion models, classical and quantum Coulomb lattice gases, and six-vertex models.Research partially supported by Canadian National Research Council Grant A4015 and U.S. National Science Foundation Grants PHY-77-18762, MCS-75-21684-A02, and MCS-78-01885.     

Propagators for lattice gauge theories in a background field

We prove regularity and decay properties for propagators connected with the renormalization group method in lattice gauge theories. These propagators depend on an external gauge field configuration, called a background field.Research supported in part by the National Science Foundation under Grant PHY-82-0369     

Propagators and renormalization transformations for lattice gauge theories. II

We continue the studies of the Paper I and extend the results of this paper to operators defined by restrictions on different scales, or by renormalization transformations of different orders.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Spaces of regular gauge field configurations on a lattice and gauge fixing conditions

We consider spaces of lattice gauge field configurations satisfying gauge invariant regularity conditions, and intersections of these spaces with a surface given by gauge fixing conditions. We prove that if these conditions are chosen properly then configurations belonging to the intersection are small and regular.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Gross-Neveu model through convergent perturbation expansions

We construct a continuum limit for the effective low energy Lagrangians of the Gross-Neveu model in two euclidean dimensions by showing that they are related to each other through convergent perturbation expansions. This provides a rigorous control of the ultraviolet problem in a renormalizable quantum field theory.Supported in part by the National Science Foundation under Grant MCS-81-20833Supported in part by the National Science Foundation under Grant PHY-82-03669     

Probing new physics from top-charm associated productions at Linear Colliders

Using the scalar prototype and gauge theory as the simplest possible examples, diagrammatic methods are developed for the recently proposed phasespace form of continuum regularization. A number of one-loop and all-order applications are given, including general diagrammatic discussions of the no-growth theorem and the uniqueness of the phase-space stochastic calculus. The approach also generates an alternate derivation of the equivalence of the large-β phase-space regularization to the more conventional coordinate-space regularization. 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 US Department of Energy under contract DE-AC03-76SF00098 and in part by the National Science Foundation under Research Grant No. PHY-85-15857. The work of Z. Bern was also supported by the Danish Research Council     

Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures

We prove that the self-avoiding walk has Ornstein-Zernike decay and some related properties for all noncritical temperatures at which the model is defined.National Science Foundation Postdoctoral Research Fellows. Work supported in part by the National Science Foundation under Grants No. PHY-82-03669 and MCS-81-20833     

The Hartree-Fock theory for Coulomb systems

For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.Research partially supported by U.S. National Science Foundation Grant MCS-75-21684Research partially supported by U.S. National Science Foundation under Grants MPS-75-11864 and MPS-75-20638. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, NJ08540, USA     

Local Ward identities and the decay of correlations in ferromagnets

Using local Ward identities we prove a number of correlation inequalities forN-component, isotropically coupled, pair interacting ferromagnets; some for allN2 and some forN=2, 3, 4. These are used to prove a mass gap above the mean field temperature, for allN2. ForN=2, 3, 4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges asm0.Research partially supported by US National Science Foundation under Grant PHY-7825390 A01.Research partially supported by US National Science Foundation under Grant MCS-78-01885.     

Existence of conserved currents in the perturbative sine-Gordon and massive thirring models

The existence of infinitely many conserved currents in the quantized sine-Gordon and massive Thirring models is proved in renormalized perturbation theory.Research supported in part by National Science Foundation Grant No. PHY 74-22218 A03Research supported in part by National Science Foundation Grant No. MP 574-05783 A01     

Dark matter and inflation

I demonstrate that dark matter consisting of any type or types of stable weakly interacting elementary particle is incompatible with the minimal predictions of inflation, based on present observation of galaxy clustering, and assuming galaxies are good tracers of mass in the universe. If we wish to resolve this problem by particle physics alone, we seem to be driven to the possibility that the initial dark matter was unstable.Research supported in part by the National Science Foundation under Grant No. PHY-82-15249 and also by the NSERC (Canada).This essay received the first award from the Gravity Research Foundation for the year 1984-Ed.     

Hydrodynamical limit for a Hamiltonian system with weak noise

Starting from a general hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prolve that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed timet provided that the Euler equation has a smooth solution with a given initial data up to timet. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit.Research partially supported by U.S. National Science Foundation grants DMS 89001682, DMS 920-1222 and a grant from ARO, DAAL03-92-G-0317Research partially supported by U.S. National Science Foundation grants DMS-9101196, DMS-9100383, and PHY-9019433-A01, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship     

Bound on the ionization energy of large atoms

We present a simple argument which gives a bound on the ionization energy of large atoms that implies the bound on the excess charge of Fefferman and Seco [2].Supported by a Sloan Dissertation Fellowship. Address from September 1989: Department of Mathematics, Caltech, Pasadena, CA 91125, USASupported in part by NSERC Grant N. A7901Supported by a Danish Research Academy Fellowship and U.S. National Science Foundation Grant PHY-85-15288-A03. Address from September 1989: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA     

Reality in the differential calculus on q-Euclidean spaces

The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.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 grant PHY-85-15857.     

The Thomas-Fermi-von Weizsäcker theory of atoms and molecules

We place the Thomas-Fermi-von Weizsäcker model of atoms on a firm mathematical footing. We prove existence and uniqueness of solutions of the Thomas-Fermi-von Weizsäcker equation as well as the fact that they minimize the Thomas-Fermi-von Weizsäcker energy functional. Moreover, we prove the existence of binding for two very dissimilar atoms in the frame of this model.on leave from Universidad de Chile, Santiago, ChileResearch supported by U. S. National Science Foundation under Grants MCS78-20455 (R. B.), PHY-7825390 A 01 (H. B. and E. L.), and Army Research Grant DAH 29-78-6-0127 (H. B.)     

Can a fourth lepton family exist without its quark counterpart?

Solely within the minimal standard model, we show that it is possible to have a fourth lepton familywithout its quark counterpart provided the Higgs sector is strongly coupled and can accommodate Skyrmion solutions. The triangle and Witten anomalies of the fourth lepton family are cancelled by those of the “weak” Skyrmions if the latter are quantized as fermions. Thus a fourth quark family is not needed to cancel the anomalies of the new leptons. If indeed such a family is discovered in the near future bye ⁺ e ⁻ machines and if no Higgs boson nor new quarks of masses less than 1TeV are found, this intriguing scenario can be a very viable one. Arguments are presented to show that if such a scheme is realized, it also implies that the Higgs fields are composite. work supported by the U.S. National Science Foundation under Grant No. PHY-83-01 186 work supported by the U.S. Dept. of Energy under Grant No. DE-AS05-80ER10713