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     

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     

A unified approach to phase diagrams in field theory and statistical mechanics

We construct the phase diagram of any system which admits a low-temperature polymer or cluster expansion. Such an expansion turns the system into a hard-core interacting contour model with small, but not necessarily positive, activities. The method uses some of Zahradnik's ideas [Z1], but applies equally well to systems with complex interactions. We give two applications. First, to low-temperatureP()₂ models with complex couplings; and second, to a computation of asymptotics of partition functions in periodic volumes. If the index of a supersymmetric field theory is known, the second application would help determine the number of phases in infinite volume.Alfred P. Solan Research Fellow. Supported in part by the National Science Foundation under Grants PHY87-064220, DMS 88-58073, and PHY/DMS 86-45122     

The two-dimensional,N=2 Wess-Zumino model on a cylinder

We construct a family of supersymmetric, two-dimensional quantum field models. We establish the existence of the HamiltonianH and the superchargeQ as self-adjoint operators. We establish the ultraviolet finiteness of the model, independent of perturbation theory. We develop functional integral representations of the heat kernel which are useful for proving estimates in these models. In a companion paper [1] we establish an index theorem forQ, an infinite dimensional Dirac operator on loop space. This paper and, another related one [2], provide the technical justification for our claim thatQ is Fredholm, and for our computation of its index by a homotopy onto quantum mechanics.Supported in part by the National Science Foundation under Grant DMS/PHY 86-45122Hertz Foundation Graduate Fellow     

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     

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     

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.     

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.     

General Zakharov-Shabat equations,multi-time Hamiltonian formalism,and constants of motion

We construct a Hamiltonian formalism for general Zakharov-Shabat equations (zero curvature equations with rational dependence on a parameter) as well as their constants of motion, and prove that the latter are in involution. The field-theoretical (multi-time) Hamiltonian formalism is used.Supported by National Science Foundation, Research Grant DMS 8703407.     

Ward identities for non-commutative geometry

We interpret the cocycle condition in quantum field theory as a set of integrated Ward identities for non-commutative geometry.Dedicated to Res Jost and Arthur WightmanSupported in part by the National Science Foundation under Grants DMS/PHY 88-16214 and INT 87-22044     

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     

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 block spin construction of ondelettes. Part I: Lemarié functions

Using block spin assignments, we construct anL ²-orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions. These functions have exponential localization, and for even values of a construction parameterM one can make them classC ^M–1 with vanishing moments up to orderM inclusive. Such a basis has an important application to phase cell cluster expansions in quantum field theory.Supported in part by the National Science Foundation under Grant No. DMS 8603795 and by the Mathematical Sciences InstituteOn leave from the Mathematics Department, Texas A & M University, College Station, Texas 77843 USA     

On the 1/n expansion

The 1/n expansion is considered for then-component non-linear -model (classical Heisenberg model) on a lattice of arbitrary dimensions. We show that the expansion for correlation functions and free energy is asymptotic, for all temperatures above the spherical model critical temperature. Furthermore, the existence of a mass gap is established for these temperatures andn sufficiently large.Supported in part by the National Science Foundation under Grant PHY 79-16812     

Dynamical approximation entropies and topological entropy in operator algebras

Dynamical entropy invariants, based on a general approximation approach are introduced for C^*-and W^*-algebra automorphisms. This includes a noncommutative extension of topological entropy.Supported in part by Grant No. DMS 9204174 from the National Science Foundation.     

Rarefactions and large time behavior for parabolic equations and monotone schemes

We consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes. For an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate.Supported by NSF Postdoctoral Fellowship Grant No. DMS 85-11476 This research was supported in part by the Institute for Math and its Applications with funds provided by the National Science Foundation     

QuantumK-theory

We construct a cocycle on an infinite dimensional generalization of ap-summable Fredholm module. Our framework is related to Connes' cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds. Thep-summability condition is characteristic of dimensionO(p). We replace this assumption by the requirement that there exists an underlying heat kernel which is trace class. Then we use the heat kernel to regularize states in dimension-independent fashion. Our cocycle may be interpreted as an infinite dimensional Chern character.Supported in part by the National Foundation under Grant DMS/PHY 86-45122     

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.     

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     

Supersymmetry and Fredholm modules over quantized spaces

The purpose of this paper is to apply the framework of non-commutative differential geometry to quantum deformations of a class of Kähler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute an explicit formula for the Chern character on generators of the Toeplitz ^*-algebra.Supported in part by the National Science Foundation under grant DMS-9206936Supported in part by the Department of Energy under grant DE-FG02-88ER25065Supported in part by the Consiglio Nazionale delle Ricerche (CNR)