William B. Campbell

Maxwell form of the linear theory of gravitation

A treatment of the linear theory is given that closely parallels the usual E and B formulation of electromagnetic theory. In Part I, an introductory exposition of gravitational radiation is presented. The tidal gravitational field strength E is defined, and it is shown that E is the measurable quantity which characterizes a gravitational wave. The important point is made that the problem of detecting gravitational waves is really the same problem as detecting tidal accelerations due to a time‐varying Newtonian gravitational field. In Part II, the magnetic‐type gravitational field strength B is defined and Maxwell‐like field equations are used to determine the polarization states of a gravitational wave. The detection of gravitational waves by Weber‐type cylinders is discussed.

Tensor and Spinor Spherical Harmonics and the Spin‐s Harmonics sYlm(θ, φ)

A set of Cartesian tensor spherical harmonics is constructed from the spin weighted harmonics of Newman and Penrose, sYlm(θ, φ). It is shown that these tensor harmonics are eigenfunctions of total angular momentum, z component of total angular momentum, total spin and radial component of spin. In particular, − s may be thought of as a helicity for outgoing radiation. Tensor operators are introduced which lower and raise this helicity. They are shown to correspond to the operators ð and ð introduced by Newman and Penrose. Because the sYlm(θ, φ) can be defined for half‐integer values of l, m, and s, a set of spinor spherical harmonics is also constructed which has properties paralleling those of the tensor harmonics.

Non-abelian plane-fronted waves

Abstract Spherical plane-fronted wave solutions for non-abelian gauge theories are presented. Colemans “non-abelian plane-waves” are a special-case of these solutions. These solutions obey a principle of linear superposition and are “force free”.

Unique specification of yang-mills solutions

Abstract Screened time-independent cylindrically-symmetric solutions of Yang-Mills equations are given which show that the source does not uniquely determine the field. However, these particular solutions suggest a natural way of uniquely specifying solutions in terms of a physical realization of a symmetry group.

Path integrals with arbitrary generators and the eigenfunction problem

Abstract We generalize the path integral formalism of quantum mechanics to include the use of arbitrary infinitesimal generators, thus providing explicit expressions for solutions of a wide class of differential equations. In particular, we develop a method of calculating the eigenfunctions of a large class of operators.

First integrals for time-independent Yang-Mills equations

Abstract It is shown that a natural gauge exists for considering time-independent solutions to SU(2) Yang-Mills field equations. Integral theorems are obtained which allow one to draw general conclusions about the energy, momentum and angular momentum in the field. One conclusion is that for any source function there could exist many solutions with an energy lower than the Coulomb energy.

Gravitational radiation: Sources

A useful expression for the tidal gravitational field due to a radiating source is found. In electromagnetic theory the electric field E in the radiation zone is determined by two time derivatives of the component of the dipole moment which lies in the plane perpendicular to the line of sight. This result generalizes in a natural fashion for the tidal gravitational field E. The gravitational field in the radiation zone for several physical systems of interest is determined. A critical discussion of the applicability of the linear theory of gravitation and of the long‐wavelength or quadrupole approximation is given.

Yang-Mills fields due to an infinite charge cylinder

The problem of determining time-independent solutions of the classical Yang-Mills equations for infinitely long charge cylinders is studied. A useful expression for the total energy in the field in terms of just the sources is derived. Numerical solutions have been found in the special cases of a small charge cylinder with a magnetic field B that either lies along the axis of symmetry or encircles the axis. It is as if these two solutions were due to currents encircling the axis or paralleling it, respectively. The condition that the solution behave well at infinity implies an exponential fall off for the fields in the azimuthal B field case and a fall off more rapid than 1/R in the axial B field case, so that in both cases the existence of a B field requires the charge on the axis to be shielded. Consequently, these solutions do not behave at infinity at all like the Maxwell solution for a charge cylinder, and they have a lower energy per unit length. They show that in Yang-Mills theories the source does not determine a unique field. A classical interpretation of this is that the field remembers how the charges were transported during the construction of the cylinder. It also suggests that a quantum mechanical version of this problem would exhibit a “spontaneous symmetry breaking” to a less symmetric, lower energy vacuum. These solutions exhibit a twofold degeneracy, as the magnetic field may be either left- or right-handed in the azimuthal B field case, or point along the +z or −z axis in the axial B field case.

Fermion bound states in the negative energy continuum: A QED for Z≳170 nuclei

Certain negative energy continuum solutions for the Dirac Equation with strong external gauge fields have stress tensors which are zero everywhere on a closed surface. These solutions correspond to bound states which have moved into the negative continuum. Such solutions can be truncated at this stress‐free surface without having to introduce new external forces. The truncated wave functions are normalized and interpreted as bound state wave functions for fermions confined to the interior of the stress‐free surface.In this paper these wave functions are used to construct a viable Furry picture QED for electrons moving near a nucleus with Z≳170. For Z’s this large the K‐shell levels have moved into the negative continuum. The QED I propose allows one to make systematic calculations such as the spontaneous pair production near a bare Z≳170 nucleus.

Glue condensation around a point source

Abstract Exact point source solutions in a classical Yang-Mills theory are considered. It is shown that only above the Mandula threshold can solutions exist with energy lower than Coulomb. Scaling is used to show the energy can be arbitrarily lower.