The symmetries of Kerr black holes

The Kerr solution describes, in Einstein's theory, the gravitational field of a rotating black hole. The axial symmetry and stationarity of the solution are shown here to arise in a simple way from properties of the curvature tensor.A portion of this work was completed, in part, during the summer of 1972 at the University of Texas at Dallas, Division of Mathematics and Mathematical Physics, and at the Black Hole session of the Ecole d'été de Physique Théorique in Les Houches; supported, in part, by the National Science Foundation, Grants GP-8868, GP-3463 9X, GP-20023, and GU-1598; the Air Force Office of Scientific Research, Grant 903-67; the National Aeronautics and Space Administration, Grant 44-004-001; the Westinghouse Corparation; the Clark Foundation; and the Rhodes Trust at Oxford.     

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.     

A generalized Reissner-Nordström solution

A new class of solutions to the Einstein-Maxwell equations is presented; it stands in the same relation to the Robinson & Robinson (1969) metrics as the Reissner-Nordström solution to Schwarzschild's.This work was supported in part by the Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, under Contract F33615-68-C-1675; Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 903-67; National Aeronautics and Space Administration under NASA Grant No. NGL 44-004-001; and National Science Foundation under Travel Grant GP-14505.     

Operator product expansions and composite field operators in the general framework of quantum field theory

The short distance behavior of field operator products is analyzed. It is shown that under certain conditions operator product expansions can be derived which give complete information on the short distance behavior and lead to the construction of composite field operators.Suported in parts by funds from the National Science Foundation Grant No. GP-25609.     

On a quadratic first integral for the charged particle orbits in the charged Kerr solution

Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.The research reported herein has been supported in part by the National Science Foundation, grants GP-8868, GP-20023, and GU-1598; the Air Force Office of Scientific Research, grant 903-67; the National Aeronautics and Space Administration, grant 44-004-001; and the Westinghouse Corporation.     

Pair function for the rectangular Ising ferromagnet

A representation of the pair correlation function for the rectangular Ising model in zero magnetic field is derived using a new spinor technique; this enables the scaling limit to be established, as well as several analytical properties of the scaling functions.Supported by the Fonds National Suisse de la Recherche Scientifique, by the National Science Foundation Grant No. PHY76-17191, and by the National Research Council of Canada Grant No. NRC A9344     

Distributional limits of renormalized Feynman integrals with zero-mass denominators

It is shown that the 0 limits of renormalized Feynman integrals exist and define Lorentz invariant tempered distributions in the external momenta. The proof applies to the case where some or all particle masses vanish.Research supported in part by the National Science Foundation Grant MPS-74-21778.Research supported in part by the National Science Foundation Grant GP-43758.     

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.     

Relation between classical and quantum mechanics

Expressions for the Lie derivatives of functions of non-commuting variables are derived and used to reformulate classical mechanics. This is possible only if the phase space variables commute, or if they satisfy Heisenberg's commutation relations.This work was supported in part by the National Science Foundation Grant Number NSF GP-14803, and by the Air Force Office of Scientific Research, contract number AFOSR 68-1524.     

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.     

Geometry in a manifold with projective structure

Parallel transport of line elements, surface elements etc. along geodesics and more general curves in a projectively connected manifold is investigated analytically and in terms of geometrical constructions. Projective curvature is characterized geometrically by a projective analogue of the geodesic deviation equation and by a geometrical construction. The results are interpreted physically as statements about free fall world lines in space-time.This paper is dedicated to our friend John Archibald Wheeler, geometer and physicist, who celebrated his sixtieth birthday on July 9, 1971.This work was supported in part by the National Science Foundation (Grant No. GP-34639X). One of the authors (A.S.) did much of this work while visiting the Université Libre de Bruxelles (summer, 1968), Cambridge University (summer, 1970), and the Nordic Institute for Theoretical Atomic Physics (1970-71); he wishes to thank these institutions and Drs. I. Prigogine, D. Sciama, and C. Møller for their kind hospitality.     

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).     

Shear-free,twisting Einstein-Maxwell metrics in the Newman-Penrose formalism

The problem of finding algebraically special solutions of the vacuum Einstein-Maxwell equations is investigated using the spin coefficient formalism of Newman and Penrose. The general case, in which the degenerate null vectors are not hypersurface orthogonal, is reduced to a problem of solving five coupled differential equations that are no longer dependent on the affine parameter along the degenerate null directions. It is shown that the most general regular, shearfree, nonradiating solution of these equations is the Kerr-Newman metric.Based in part on a doctoral thesis submitted to the University of Pittsburgh (1970) while the author was NASA Predoctoral Trainee. Research also supported in part by the National Science Foundation under Grant GP-19378.     

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.     

The monodromy rings of one loop Feynman integrals

The monodromy rings of Feynman integrals for one loop graphs with an arbitrary number of lines are determined.Research sponsored by the National Science Foundation, Grant No. GP-16147.     

Phase space cell expansion and borel summability for the Euclidean φ 3 4 theory

The stability of the free energy is proved for complex values of the coupling constant by the way of a convergent expansion. As a consequence, one obtains the Borel summability of the perturbation series.On leave of absence from the Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, FranceSupported in part by the National Science Foundation under Grant PHY 75-21212     

Stability for a class of bilocal Hamiltonians

We introduce a method to establish stability of non-local interactions, and we apply this method to certain polynomial non-linear field theory. The non-local potential must satisfy the property ofslow decrease at infinity in Fourier space (SDI).Dedicated to Huzihiro ArakiSupported in part by the National Science Foundation under Grant PHY91-20626     

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     

Dimensionally renormalized Green's functions for theories with massless particles. II

In the framework of dimensional renormalization the existence of Green's functions to all orders of perturbation theory is proved for theories with massless particles, provided all terms in the interaction Lagrangian have infrared degree 4. If the vanishing of masses is enforced by some symmetry and this symmetry is respected by dimensional regularization, Schwinger's action principle holds for these Green's functions as in the massive case.On leave of absence from Max-Planck-Institut für Physik und Astrophysik, D-8000 München, Federal Republic of GermanySupported in part by the National Science Foundation Grant # MPS-74-13208 A01Supported in part by the National Science Foundation Grant # GU 3186     

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