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.     

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.     

Space-dependent dirac operators and effective quantum field theory for fermions

For the operator, wherem(x) can change sign, we develop a cluster expansion for computing the determinant and Green's functions. We use a local chiral transformation to relate the space-dependent case to the ordinary Dirac operator.Supported in part by National Science Foundation grant PHY/DMS 88-16214Supported in part by National Science Foundation grants DMS 90-08827 and DMS 88-580873Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

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     

A vanishing theorem for supersymmetric quantum field theory and finite size effects in multiphase cluster expansions

We apply cluster expansion methods to to theN=2 Wess-Zumino models in finite volume, in two space-time dimensions. We show that in the region of convergence of the cluster expansion, a vanishing theorem holds for the supercharge of the theory; that is, the dimension of the kernel of the Hamiltonian is equal to the index of the supercharge.Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 90-07206Supported in part by National Science Foundation Mathematical Sciences Postodoctoral Research Fellowship DmS 88-07291     

The phase structure of the two-dimensionalN=2 Wess-Zumino model

We construct a convergent cluster expansion for the two-dimensionalN=2 Wess-Zumino model, in a region of parameter space where there are multiple phase. As a result of this expansion, we are able to construct the infinite volume field theory and demonstrate exponential decay of correlations. We are also able to investigate the different phases of the model, develop the phase diagram, and show that the free energy of each phase vanishes.Supported in part by National Science Foundation grants DMS 90-08827, PHY/DMS 88-16214 and DMS 88-58073Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Axioms for Euclidean Green's functions

We establish necessary and sufficient conditions for Euclidean Green's functions to define a unique Wightman field theory.Supported by the National Science Foundation under grant GP 31239X.Supported in part by the Air Force Office of Scientific Research, contract AF 44620-70-C-0030.     

The smooth cohomology ofN=2 supersymmetric Landau-Ginzburg field theories

We compute the smooth cohomology (both unrestricted and compactly supported) of the supercharge of an ultraviolet cutoffN=2 supersymmetric Landau-Ginzburg field theory.Supported in part by the National Science Foundation under grant DMS-9206936.Supported in part by the Department of Energy under grant DE-FG02-88ER25065.     

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     

The spin statistics connection for dyons

Goldhaber's spin-statistics connection for electric-pole-magnetic-pole composite dyons is deduced in a gauge-invariant way by connecting the spatial interchange operator with a rotation.Supported in part by the National Science Foundation.Supported in part by the National Science Foundation.     

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     

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     

Circle theorem for hard-core binary lattice gases

We study the distribution of zeros in a symmetric, two-component Widom-Rowlinson lattice system (any number of dimensions). We show that for sufficiently large mean activity the system partition function cannot vanish if the magnitude of the ratio of the two (complex) activities is different from one.Supported in part by National Science Foundation Grants CHE76-11253 (to LKR) and Phy 78-15920 (to JLL), and by the Petroleum Research Foundation.     

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     

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.     

Symanzik's improved actions from the viewpoint of the renormalization group

We investigate Symanzik's improvement program in a four-dimensional Euclidean scalar field theory with smooth momentum space cutoff. We use Wilson's renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff. This leads to a sequence of solutions to the renormalization group equation. We define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale. The improved actions are close approximations to the continuum effective action. We prove their existence to every order of improvement and to every order of renormalized perturbation theory.Supported in part by a German National Scholarship Foundation fellowship, and by the National Science Foundation under Grant DMS/PHY 86-45122     

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.     

Chiral two-component spinors and the factorization of Kramers's equation

Kramers's equation specialized to the Coulomb field is factored using a rotationally invariant, angular momentum based, algebra of three anticommuting operators. Comparing the explicit chiral two-component solutions for the factored equation to the two-component solutions defined by the Foldy-Wouthuysen series for the Dirac-Coulomb Hamiltonian, it is concluded that this series cannot converge.Supported in part by the National Science Foundation.On leave from Physics Department, Duke University, Durham, North Carolina 27706.Supported in part by the Fund for Basic Research administered by the Israeli Academy of Sciences and Humanities Basic Research Foundation.     

Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation

We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szegő and Baxter. We also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro-geometric solutions.Supported in part by the US National Science Foundation under Grants No. DMS-0200219 and DMS-0405526.The research of the second and third author was supported in part by the Research Council of Norway     

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