On the density of the Ward ansätze in the space of anti-self-dual Yang Mills solutions

A general patching matrixP for the twistor construction of antiself-dual Yang-Mills fields is approximated by a terminating Laurent series. The approximate patching matrixP(^m) is triangularized (so that it becomes one of the Ward ansätze) and the associated Riemann-Hilbert problem is solved, thereby generating an anti-self-dual solution of the Yang-Mills equations. The approximate patching matrices and the associated (exact) anti-self-dual Yang-Mills solutions are then shown to converge onP and its corresponding solution so that the Ward ansätze forms a dense subset in the solution space in the Weierstrass sense.     

On spherically symmetric fields in gauge theories

Gauge fields admitting spherical symmetry are listed. Spherically symmetric solutions of Yang-Mills equations and spherically symmetric magnetic monopoles are studied. A simple exact solution of the Yang-Mills and Einstein equations is found.     

Cosmological models with Yang-Mills fields

Cosmological models with an SU(2) Yang-Mills field are studied. For a specific model with a minimally coupled Yang-Mills Lagrangian, which includes an arbitrary function of the second-order term and a fourth-order term, a corresponding reconstruction program is proposed. It is shown that the model with minimal coupling has no de Sitter solutions, for any nontrivial function of the second-order term. To get de Sitter solutions, a gravitational model with nonminimally coupled Yang-Mills fields is then investigated. It is shown that the model with non-minimal coupling has in fact a de Sitter solution, even in absence of the cosmological constant term.     

A relation between the Einstein and the Yang-Mills field equations

We show that any solution of the vacuum Einstein equations is a double self dual solution of the O(4) Yang-Mills equations.     

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

A Yang-Mills formulation of Einstein gravity with spin-affine connection as the dynamical variable of gravitational field is suggested based on the Stephenson-Kilmister-Yang (SKY) equation. A physically interesting property of the present formalism is that the Einstein field equation appears as a first-integral solution to the Yang-Mills type gravitational gauge field equation. The gravitational current density, the law of conservation and the gravitational gauge field strength in vierbein formulation are discussed. The present scheme could provide us with new insight into a possible way to include both Yang-Mills field and gravitational gauge field into one framework of generalized vierbein fields.     

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.     

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。     

General solution of the spherically symmetric Yang-Mills equations

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

The interaction of classical Yang-Mills charges

Beginning with Khriplovich's formulation of the two point charge Yang-Mills problem, the gauge freedom is used to reduce by one the number of quantities entering the field equations. While a Coulomb solution is shown to exist for any orientation of the color charges, any solution which has a magnetic field present will have lower energy. When the field equations are solved for the simple charge configurations of parallel plates and cylindrical capacitors, one finds screening due to the vector potential. This screening is shown to be complete for any isolated charged plate or line charge. This result may be an indication that isolated charge cannot occur physically for the classical Yang-Mills theory.     

A QCD Model Using Generalized Yang-Mills Theory

Generalized Yang-Mills theory has a covariant derivative, which contains both vector and scalar gauge bosons. Based on this theory, we construct a strong interaction model by using the group U（4）. By using this U（4） generalized Yang-Mills model, we also obtain a gauge potential solution, which can be used to explain the asymptotic behavior and color confinement.     

On the Yang-Mills mass gap problem

Theorem 4.1 of the author’s paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, RJMP 21 (2), 169–188 (2014) claims the relative ellipticity of cutoff Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces. This implies that the spectra of cutoff self-adjoint Yang-Mills energy-mass operators in a nonperturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge group) are nonnegative sequences of the eigenvalues converging to +∞. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have self-similar positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem. The present note shows that the fundamental spectral value of a cutoff quantum Yang-Mills energy-mass operator is the simple zero eigenvalue with the vacuum eigenvector. The direct proof (without von Neumann algebras) is based on the domination over the number operator (with simple fundamental eigenvalue) and the standard spectral variational principle.     

Painleve Analysis and the Yang-Mills Equation in Two Specific Cases

Two important configurations of Yang-Mills fields are discussed applying to them the Painleve test of Ablowitz. The case with a constant electric E-field does not admit an exact solution while the cylindrically symmetric self-dual system admits an inverse scattering approach for instanton solution.     

Vortex solutions with a smooth core

A Yang-Mills field is used to smooth over the core of a vortex solution.     

Geometrodynamics in Multidimensional Unified Theories

The unified theory of gravitation and a Yang-Mills field is formulated as a dynamical theory of (r+3) geometries presumed to be principal bundles endowed with a Riemannian metric. Beyond the usual constraint equations the second fundamental form should satisfy a third constraint equation. It is shown that they have a wormhole-type solution describing a pair of Yang-Mills charges.     

On the Yang-Mills effective action with instanton gauge field

The calculation of the determinant for the second order covariant derivative operator is analyzed in the space R⁴ and for Yang-Mills instanton field configurations. The problems inherent to the ultraviolet ζ-function renormalization method and infrared divergencies of this operator are reviewed. A method for estimating the asymptotic coefficients of its determinant has been discussed and the failure of the general conformal invariance induced by any regularization technique has also been considered. A particular solution valid for a Yang-Mills multi instanton configuration, valuable in order to get the non conformally invariant piece of the general solution, is the main result obtained here. We give it as an integral equation, namely in a semi-explicit form.     

On multimeron solutions of the classical Yang-Mills equation

The double-meron solution and the solution indicated by Protogenov [1] of the SU(2) Yang-Mills equation in four-dimensional euclidean space are generalised. It is noted that the double-meron solution has an analogous solution in Minkowski space which has a finite energy integral.     

The Continual Approach in the Yang-Mills Theory in the External Non-Abelian Field

The theory of Yang-Mills field in interaction with matter fields is considered in the presence of external gauge field. A closed expression for the generating functional of the Green functions is obtained, and a detailed analysis of the Green functions of the scalar, spinor, ghost and Yang-Mills fields is performed. The path-integral solution for all these Green functions is obtained, which includes the functional averaging over the classical trajectories in the space of commuting and anticommuting variables, the latter being anociated with the particle spin and isospin. For illustration an arbitrary Abelian-like external field is considered, as well as non-Abelian-like constant external field.     

Stability of instanton-induced compactification in 8 dimensions

We consider the 8-dimensional Einstein-SU(2) Yang-Mills system when the Yang-Mills system assumes a 1-instanton configuration on an internal S₄ with the vacuum solution possessing the geometry of M₄ × S₄ and the invariance group P₄(Poincaré) × SO(5). We demonstrate the classical stability of the solution by computing the spectrum of the physical states and showing the absence of ghosts and tachyons amongst them. We also discuss the possibility of obtaining SO(5) multiplets of massless fermions.     

A non-singular spherically symmetric separable copy of the vacuum in SU(2) Landau gauge

We give a new non-singular spherically symmetric and separable copy of the vacuum in Landau gauge for SU(2) Yang-Mills theories. The solution can be considered as a generalization of Gribov's solution in Coulomb gauge and thus it demonstrates the similarity between vacua in Coulomb and Landau gauges.     

Stability in the Yang-Mills system with an external source

We investigate stability of the Coulomb solution in the Yang-Mills system under finite perturbations below the critical coupling strength. The role of recently discovered low-energy solutions is discussed.