Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Bäcklund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.     

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     

Solutions of the Maxwell and Yang-Mills equations associated with hopf fibrings

It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang-Mills equations correspond to natural connections defined on the principal bundlesU(2)/U(1)=S ₃ S ₂ andSp(2)/Sp(1)=S ₇ S ₄, respectively. This observation leads to a general methods of constructing new, topologically nontrivial solutions of the Maxwell and Yang-Mills equations. Among them is an electromagnetic instanton defined over the two-dimensional complex projective space endowed with the Fubini-Study metric.On leave from the Institute of Theoretical Physics, Warsaw University, Hoza 69, Warsaw, Poland.     

Null solutions of the Yang-Mills equations

We investigate the correspondence between null solutions of the Yang-Mills equations and shearfree geodesic null congruences. We give an example of a non-Abelian null solution with twisting rays.Alexander von Humboldt Fellow. Address after 31 May 1986: Institute of Theoretical Physics, University of Warsaw, Hoa 69, 00-681 Warsaw, Poland.     

Yang-Mills equations on the two-dimensional sphere

We construct a 1:1 correspondence between the equivalence classes of Yang-Mills fields overS ² and the conjugacy classes of closed geodesics of the structure group. Furthermore, we give an explicit isolation theorem for any Yang-Mills field overS ².     

Pseudoparticle solutions of the Yang-Mills equations

We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.     

Monopole solution of Yang-Mills equations

A monopole solution of the classical Yang-Mills field equations is presented. The new solution includes both “electric” and “magnetic” parts.     

Yang-Mills equations are formally linearizable

It is proved, in the framework of a recent theory of non-linear group representations elaborated by the authors, that the non-linear Yang-Mills equations are formally linearizable: there exists a formal non-linear operator which intertwins the non-linear Yang-Mills equations with their free (linear) part.     

Lorentz-invariant solutions of the classical Yang-Mills equations

A gauge representation of the noncompact group SL(2,C) is defined. All the corresponding invariant (singular) solutions of the classical Yang-Mills equations in Minkowski space are found. It is shown that they are related to a family of real SU(2) x SU(2)-invariant Euclidean solutions containing the self-dual (one-instanton) configurations.     

Geometry of the Einstein and Yang-Mills equations

It is shown that the Einstein and Yang-Mills equations arise from the conditions for the space-time to be a submanifold of a pseudo-Euclidean space with dimension greater than 5. Some possible applications to cosmology, spin-2 fields, and geometrodynamics are discussed.     

Sandwich data leading to elliptic equations

It is shown that a large class of conformal transformations of the metric tensor on a three-dimensional space-like hypersurface leads to sandwich problems that result in elliptic differential equations for the unknown variables.Research supported by the National Science Foundation under Grant GP19378.Helpful discussions with A. Komar are gratefully acknowledged.     

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations are shown to be conditions of compatibility of two linear equations. All the N-instanton fields are constructed explicitly.     

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 O(3,1) Yang-Mills equations and the Einstein equations

The vacuum Einstein equations (with cosmological constant) written in a slightly unconventional manner, can be decomposed into three parts: the first two parts are the ordinary self dual Yang-Mills equations and the anti-self dual Yang-Mills equations for anO(3,1) gauge group, on an unspecified background space-time, the third part are equations that solder or relate these two Y-M fields and connections to the curvature and connection of that unknown space-time. It is the purpose of this note to take this point of view seriously and concentrate on the first two parts in their own right. We apply to them generalizations of solution construction techniques which have arisen from the study of self dual Yang-Mills equations on Minkowski space. At the end we discuss how to solder or bootstrap these results to the determination of the space-time itself.     

Self-dual Yang-Mills fields and Nahm's equations

For the gauge fields with values in arbitrary semisimple Lie algebra we introduce the ansatzes which reduce the self-duality equations in the Euclidean spaceR ^4,0 to the well-known Nahm equations.     

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.     

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.     

General solution of the spherically symmetric Yang-Mills equations

A general solution of the spherically symmetric Yang-Mills equations is presented.