Supersymmetry and classical solutions

We obtain classical solutions to the field equations of the massless supersymmetric Wess-Zumino model and to the field equations of the interacting SU(2) gauge supermultiplet. This is done by applying finite supersymmetry transformations to the known solutions of the scalar field equation with ?⁴ interaction and the Yang-Mills field equations. The relevance of supersymmetry to the solution of classical field equations involving anticommuting fermion fields is discussed.     

New exact solutions of the Yang-Mills field equations

In this paper new exact solutions of the Yang-Mills SU(2) gauge field equations are obtained using the Carmeli-Charach-Kaye null-tetrad formalism. The solutions are classified and briefly discussed.     

Approximate radiative solutions of the Yang-Mills field equations

In this paper we look for the asymptotic radiative solutions of the Yang-Mills field equations. Considering the potential of the Yang-Mills field as a connection in a principal fibre bundle gives us a fully covariant formalism similar to the formalism of the General Relativity. Then we apply directly the results obtained by Mme Choquet-Bruhat for the gravitational field by means of the W.K.B. method. After deriving the equations for the asymptotic waves and interpreting the zero-order conditions as the initial conditions, we consider some known trivial solutions of the Yang-Mills field equations as the background field and construct the asymptotic waves explicitly. All the solutions considered turn out to be of the electromagnetic type, with some extra restrictions of the algebraic type.     

规范场与主纤维丛上的联络

本文较详细地给出物理上的规范场与数学上的主纤维丛上的联络论之间的对应关系,从而以联络论的观点统一地处理杨振宁所说的规范场的“微分方法”与“积分”方法,并指出存在有更广泛的规范场的可能性。此外,讨论了引力场如何作为规范场及比较了一些已知的引力场方程。     

Some global solutions of the Yang-Mills equations in Minkowski space

A class of global solutions of the Yang-Mills equations whose Cauchy data depend on a pair of arbitrary functions is constructed. The asymptotic propagation of the energy in space-time is studied. The same results are valid if the Yang-Mills field is coupled to a scalar field.Research supported in part by NSF Grants MCS 77-01340 and MCS 79-01965     

Stable Yang-Mills fields for an over-critical point source

The classical Yang-Mills energy functional in the presence of an external particle is explored as a function of the coupling strength. In the over-critical coupling region besides the Coulomb field other new saddle points as well as new minima are found. Near the critical point approximate expressions for the new static and stable solutions of the Yang-Mills equations are derived.     

Self-duality,helicity and coherent states in non-Abelian gauge theories

Solutions of the non-linear classical Yang-Mills field equations obtained by iteration from self-dual solutions of the linearized equations are shown to be themselves self-dual. In Minkowski space, such solutions are necessarily complex. We give a quantum theory interpretation of these solutions which relates them to the matrix element of the field operators between the vacuum state and a coherent state of spin-one quanta of definite helicity.     

Gravitational instantons

The path-integral approach to quantum field theory assigns special importance to finite action Euclidean solutions of classical field equations. In Yang-Mills gauge theories, the instanton solutions of classical field equations with self-dual field strength have given rise to a new, nonperturbative treatment of the quantum field theory and its vacuum state. Since gravitation is also a species of gauge theory, one might think that similar phenomena would occur in gravity. The authors recently sought and found a new self-dual solution to Euclidean gravity which plays a role parallel to that of the Yang-Mills instanton. Gravitational instantons now promise to yield new insights into the nature of quantum gravity.This essay received the second award from the Gravity Research Foundation for the year 1979-Ed.     

First integrals for time-independent Yang-Mills equations

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.     

The class of exact cosmological models with a scalar-field nonlinearity induced by the Yang-Mills field

A new class of exact solutions of the system of Einstein-Yang-Mills equations and the equations of a nonlinear scalar field interacting with the Yang-Mills SO(3) field is obtained on the basis of the Friedman spatially homogeneous cosmological models.     

Nonlinear wave modulations in plasmas

A review of the generic features as well as the exact analytical solutions of coupled scalar field equations governing nonlinear wave modulations in plasmas is presented. Coupled sets of equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-KDV system are considered. For stationary solutions, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system which are valid in different regions of the parameter space are obtained. The generic system is shown to generalize the Hénon-Heiles equations in the field of nonlinear dynamics to include a case when the kinetic energy in the corresponding Hamiltonian is not positive definite. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex KDV equation and the complexified classical dynamical equations is also pointed out.     

A fundamental quadratic variational principle underlying general relativity

The fundamental result of Lanczos is used in a new type of quadratic variational principle whose field equations are the Einstein field equations together with the Yang-Mills type equations for the Riemann curvature. Additionally, a spin-2 theory of gravity for the special case of the Einstein vacuum is discussed.     

Exact solutions of the equations of motions of quarks in a non-Abelian field of a flat color wave

Exact solutions of the equations of motion of quarks in a non-Abelian field of a flat color wave of a spectral configuration are obtained. The gauge field of the wave takes values from the SU(2) group and is an exact solution of the Yang-Mills equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 76–78, December, 1986.     

Low energy static Yang-Mills configurations

Non-singular static solutions for the SU(2) Yang-Mills field equations in the presence of external sources are presented. These solutions possess energies less than those of the magnetic dipole solutions, and can in fact have vanishingly small energies.     

Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space

In this paper I will first derive, based on energy estimations and geometric invariance, the asymptotic behavior of solutions of linear spin field equations in Minkowski space. It generalizes the result in [3] where it was proved for the spin-1 and spin-2 cases. The techniques are then applied to Yang-Mills equations, the result improves the previous one in [1] by allowing the initial data to have charge, dipole and quadrupole moments. The Lie derivative operator for spinors and some properties will be also discussed; they can be used to simplify some algebraic calculations of [4].This research is partially supported by a grant from NSF under DMS-8610730     

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL _n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space.A summary of the results of the second part of this work appears in the Introduction.Most of this work was done while Darby Fellow of Mathematics at Lincoln College, Oxford     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.