A note on reflection positivity

We establish reflection positivity of covariance operators, using properties of Dirichlet or Neumann boundary data.Supported in part by National Science Foundation Grant No. PHY 78-08066.Supported in part by National Science Foundation Grant No. PHY 77-18762.     

Heat kernel regularization of quantum fields

We discuss some consequences of the existence of a heat kernel regularization (HKR) for quantum fields. We demonstrate that HKR applies in certain examples, using methods which should be useful more generally.Supported in part by the National Science Foundation under Grant PHY/DMS 86-45122Supported in part by a German National Scholarship Foundation fellowship     

An existence theorem for multimeron solutions to classical Yang-Mills field equations

In this paper we prove the existence of solutions to a class of boundary value problems for a singular nonlinear elliptic partial differential equation in a half plane. By a recent paper of J. Glimm and A. Jaffe, this proves the existence of multimeron solutions to the classical SU(2) Yang-Mills field equations in Euclidean space.Supported in part by the National Science Foundation under Grant PHY 77-18762Supported in part by the Icelandic Science FoundationSupported in part by Grant MCS 76-06524     

Supersymmetry and the spectral condition on a cylinder

We study the spectral condition for supersymmetric Wess-Zumino field theories on a cylindrical spacetime. This condition is preserved under certain ultraviolet cutoff procedures and also leads to analyticity of HKR regularized field operators in a complex neighborhood of spacetime.Supported in part by the National Science Foundation under Grant DMS/PHY86-45122.Supported in part by a German National Scholarship.     

Phase diagrams and cluster expansions for low temperatureP(φ)2 models

We give a cluster expansion for the Schwinger functions of the stable phases found in Part I. The Wightman axioms, the mass gap, and asymptoticity of perturbation theory follow.Supported in part by the National Science Foundation under Grant No. PHY 79-16812National Science Foundation predoctoral fellow, 1979–80. Currently Junior Fellow, Harvard University Society of Fellows     

A consistent approximation scheme in QFT

An approximation scheme for the generating functionals for the Schwinger function is introduced using the expansion of these functionals into series of Laguerre polynomials.Supported in part by the National Science Foundation, under Grant INT 73-20002 A01.     

Surface integrals and monopole charges in non-abelian gauge theories

We derive a formula which gives all the magnetic charges (topological invariants) of a monopole in the adjoint representation of a non-abelian gauge theory in terms of surface integrals at infinity.Supported in part by the National Science Foundation under Grant No. PHY 79-16812Junior Fellow, Harvard University Society of Fellows     

Absence of interaction in Lie field theories

It is shown that theories of relativistic Lie fields cannot lead to scattering or reactions, even if an infinite number of Lie fields is present.Supported in part by the National Science Foundation under Grant NSF GP 6036.     

Stable potentials,II

A family of stable one-dimensional potentials is shown not to be decomposable into the sum of a non-negative function and a function of non-negative type. This settles in the negative a question raised by Ruelle.Supported by National Science Foundation Grant GP 13627.Supported by National Science Foundation Grant GP 7469.     

Generalized Vaidya metrics

We investigate criteria under which one may construct the energy tensor of a null radiation field from an algebraically special vacuum metric. The field bears the same relationship to the original metric as does Vaidya's to Schwarzschild's. As an example we generate a class of null radiation fields from a class of vacuum metrics without symmetry discovered by Robinson and Robinson.Supported in part by the National Science Foundation (GP-8868, GP-20033, and GU-1598), Air Force Office of Scientific Research under Grant AF-AFOSR-903-67 and by the National Aeronautics and Space Administration under NASA Grant No. NGL 44-004-001.     

Eigenvalue inequalities for fermions in gauge theories

We show that QCD with a sufficient number of fermions of zero bare mass has physical massless particles. That result also follows from triangle anomalies, so only our method is novel. Our method involves proving special cases of recently conjectured paramagnetic inequalities for fermions. The proofs are simple applications of the Atiyah-Patodi-Singer theorem on spectral flow.Supported in part by the National Science Foundation Grant No. PHY80-19754Supported in part by DOE Grant No. DE-AC02-76ER02220     

On Witten's proof of the positive energy theorem

This paper gives a mathematically rigorous proof of the positive energy theorem using spinors. This completes and simplifies the original argument presented by Edward Witten. We clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant Dirac operator.Supported in part by the National Science Foundation under Grant PHY79-16812Junior Fellow, Harvard University Society of Fellows     

On the renormalization group transformation for scalar hierarchical models

We give a new proof for the existence of a non-Gaussian hierarchical renormalization group fixed point, using what could be called a beta-function for this problem. We also discuss the asymptotic behavior of this fixed point, and the connection between the hierarchical models of Dyson and Gallavotti.Supported in Part by the National Science Foundation under Grant No. DMS-8802590Supported in Part by the Swiss National Science Foundation     

Analysis of Elliptical Thin Ridged Waveguide

Elliptical waveguides and ridged waveguides have found broad applications in many microwave structures. The elliptical waveguide with double infinite thin ridges has been formulated using the mode-matching method. Exact closed-form expressions for eigenvalue problem of all TE and TM modes are presented. Numerical results suggest that the elliptical ridged waveguides have larger bandwidth than that of circular ones. Supported in part by the National Natural Science Foundation of China under Grant 60532010, and in part by the Scientific Research Foundation for the Returned Overseas Chinese Scholars under Grant 04LXJ01.     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

Heisenberg proof of the Balian-Low theorem

We give an alternate proof of the fact that a function generating a basis of coherent states must have an infinitely long tail in either position space or momentum space. Our argument is a very natural one in which the Heisenberg Uncertainty Principle enters directly.Supported in part by the National Science Foundation under Grant No. DMS 8603795.     

Complex free energies and metastable lifetimes

For two simple models we consider the analytic continuation of the free energy across a first-order phase transition. For each system we also study an associated stochastic dynamics and decay rates for passage to the stable equilibrium. We then investigate the relation between the imaginary part of the free energy and the decay rate per unit volume.Supported in part by U.S.-Israel Binational Science Foundation and by National Science Foundation Grant MCS 77-20683.Alfred P. Sloan Research Fellow. On leave from Indiana University, Bloomington, Indiana.     

A time dependent Born-Oppenheimer approximation

We consider the dynamics of a quantum mechanical system which consists of some particles with large masses and some particles with small masses. As we increase the large masses to infinity we obtain the following results: The particles of smaller mass move adiabatically and determine an effective potential in which the heavier particles move semiclassically. Our methods can be applied to diatomic molecules with Coulomb forces.Supported in part by the National Science Foundation under Grant PHY 78-08066     

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     

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