Isolated maximal surfaces in spacetime

Maximal surfaces and their implications for the ambient spacetime are studied. Our methods exploit the interplay between contact of the volume functional and energy conditions. Essentially, we find that in closed universes, maximal surfaces are unique; they maximize volume; and they yield future and past singularities.Supported in part by the National Science Foundation under grant No. PHY 70-02077A03 and by the Humboldt FoundationSupported in part by the Sonderforschungsbereich (Theoretische Mathematik) of the University of Bonn     

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.     

Foundational aspects of geometro-stochastic quantization of Dirac fields in curved spacetime

The application of the recently developed method of geometro-stochastic quantization to spin-1/2 fields in curved spacetime does not give rise to any of the conceptual and technical difficulties encountered by more conventional methods of quantization. Field propagation is based on the concept of stochastic parallel transport governed by quantum connections that incorporate Poincaré gauge invariance and obey the equivalence principle. Consequently, under free-fall conditions, such geometro-stochastic Dirac field propagation does not give rise to Bogolubov transformations resulting in spontaneous fermion creation.1. Supported in part by the NSERC Research Grant No. A5206.2. Supported by an NSERC postgraduate fellowship.     

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.     

An elliptic complex associated to the Yang-Mills constraint equations

This paper presents an elliptic complex of linear spaces and operators which resolves the linearized constraint equations for the initial data for a Yang-Mills (gauge) field. The complex exists when the background gauge field satisfies a certain condition corresponding to self-duality in the Riemannian case and is defined on a spacetime which has a spacelike Cauchy surface.Partially Supported by NSF Grant MCS 78-01460     

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     

On the determination of Cauchy surfaces from intrinsic properties

We consider the problem of determining from intrinsic properties whether or not a given spacelike surface is a Cauchy surface. We present three results relevant to this question. First, we derive necessary and sufficient conditions for a compact surface to be a Cauchy surface in a spacetime which admits one. Second, we show that for a non-compact surface it is impossible to determine whether or not it is a Cauchy surface without imposing some restriction on the entire spacetime. Third, we derive conditions for an asymptotically flat surface to be a Cauchy surface by imposing the global condition that it be imbedded in a weakly asymptotically simple and empty spacetime.This research was supported in part by the National Science Foundation grants PHY 70-022077 and PHY 76-20029 as well as the National Aeronautics and Space Administration grant NGR 21-002-010National Science Foundation Pre-doctoral Fellow     

Quantum statistical entropy corresponding to cosmic horizon in five-dimensional spacetime

The generalized uncertainty relation is introduced to calculate the quantum statistical entropy corresponding to cosmic horizon. By using the new equation of state density motivated by the generalized uncertainty relation, we discuss entropies of Bose field and Fermi field on the background of five-dimensional spacetime. In our calculation, we need not introduce cutoff. There is no divergent logarithmic term in the original brick-wall method. And it is obtained that the quantum statistical entropy corresponding to cosmic horizon is proportional to the area of the horizon. Further it is shown that the entropy corresponding to cosmic horizon is the entropy of quantum state on the surface of horizon. The black hole’s entropy is the intrinsic property of the black hole. The entropy is a quantum effect. In our calculation, by using the quantum statistical method, we obtain the partition function of Bose field and Fermi field on the background of five-dimensional spacetime. We provide a way to study the quantum statistical entropy corresponding to cosmic horizon in the higher-dimensional spacetime. Supported by the National Natural Science Foundation of China (Grant No. 10374075) and the Natural Science Foundation of Shanxi Province, China (Grant No. 2006011012)     

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     

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.     

Anomalous quantization of the free Dirac field in three-dimensional spacetime

The possibility for Bose quantization of the free spinor field in three-dimensional spacetime is demonstrated, compared with Fermi quantization, and discussed.This work is supported in part by the Bulgarian National Foundation for Scientific Research (Ph-20-1991).     

The averaged Lagrangian and high-frequency gravitational waves

The averaged Lagrangian technique of Whitham is applied to the second variation Lagrangian for the perturbations of a general-relativistic spacetime. This gives a variational principle for (sums of) approximately periodic gravitational waves which in turn leads to the rederivation of some results of Isaacson. Examples of the use of the method are discussed.Supported by National Science Foundation grant GP-31358.     

On potential and field fluctuations in classical charged systems

Using electrostatic identities the potential and microfield in a plasma, important for determining line shapes, are expressed as limits of local quantities. These are shown to be well defined for typical configurations of macroscopic, i.e., infinite systems (under some mild clustering assumptions). Their covariance contains a slowly decaying part (¦x¦^–1, for the potential) whose coefficient is universal whenever the Stillinger-Lovett second moment condition holds. We show further that the contributions from distant regions (which are equal to suitable averages over local regions) have a Gaussian distribution.Supported in part by AFOSR Grant No. 82-0016.Supported in part by the Swiss National Foundation for Scientific Research.     

Graded manifold theory as the geometry of supersymmetry

Building upon Kostant's graded manifold theory, we present a new way of introducing spinors into the spacetime manifold, by expanding the algebra of functions on spacetime to a graded algebra. The elements of differential geometry are generalized to accomodate the expanded algebra of functions and in this enriched geometry we find the elements of supersymmetry and of supergravity theory. The geometrical role of the supergravity fields is discussed and a derivation of their transformation rules is given.Research supported in part by the National Science Foundation under Grant Nos. PHY77-22864 and PHY77-05299     

The Casimir effect of Reissner-NordstrSm black hole

The stress tensor of a massless scalar field satisfying a mixed boundary condition in a （1 ＋ 1）-dimensional Reissner- Nordstrom black hole background is calculated by using Wald＇s axiom. We find that Dirichlet stress tensor and Neumann stress tensor can be deduced by changing the coefficients of the stress tensor calculated under a mixed boundary condition. The stress tensors satisfying Dirichlet and Neumann boundary conditions are discussed. In addition, we also find that the stress tensor in conformal flat spacetime background differs from that in flat spacetime only by a constant.     

The stochastic quantum mechanics approach to the unification of relativity and quantum theory

The stochastic phase-space solution of the particle localizability problem in relativistic quantum mechanics is reviewed. It leads to relativistically covariant probability measures that give rise to covariant and conserved probability currents. The resulting particle propagators are used in the formulation of stochastic geometries underlying a concept of quantum spacetime that is operationally based on stochastically extended quantum test particles. The epistemological implications of the intrinsic stochasticity of such quantum spacetime frameworks for microcausality, the EPR paradox, etc., are discussed.Supported in part by NSERC Grant A5208.     

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     

Magnetic field dependence of impurity-induced muon depolarization in noble metals

We have measured the magnetic field dependence of the muon depolarization rate up to 5 kOe in AuGd (350 ppm), AgGd (340 ppm) and AgEr (300 ppm). A simple model which includes both dipolar and nearest-neighbor contact interactions between the muon and the magnetic impurity does not fit the data. An axial crystal-field interaction, arising from the electric field gradient induced by the muon at the site of the impurity, is found to dominate the Hamiltonian, and may have a large effect on the field dependence.Supported in part by the U.S. Department of Energy and the National Science Foundation.Work performed at Sandia National Laboratories supported by the U.S. Department of Energy under contract DE-AC04-76DP000789.     

Non-Abelian isometries of multidimensional universes

A field theory on a(d + n)-dimensional manifold in the presence of ann-dimensional isometry group spanningn-dimensional orbit spaces may be reduced to a field theory on ad-dimensional manifold. The field content of such reduced theories is completely worked out when the isometries may be non-Abelian and the resultant space may have torsion. The equations of motion of the dimensionally reduced theory are obtained directly from the higher-dimensional theory. The reduced theory is given in terms of the metric tensor, a set of scalar fields, and a set of antisymmetric tensor fields.Supported in part by the Department of Energy under Contract DE-AS-2-76ER02978 and in part by the National Science Foundation under Grant NSF Phy 83 134 10.