Mills Field Equations Research Articles

Non-linear composition and infinite conformal symmetry of topologically non-trivial solutions in (3+1)-dimensional Yang\u2013Mills theory

An infinite-dimensional family of analytic solutions in pure SU(2) Yang–Mills theory at finite density in (3+1) dimensions is constructed. It is labelled by two integeres (p and q) as well as by a two-dimensional free massless scalar field. The gauge field depends on all the 4 coordinates (to keep alive the topological charge) but in such a way to reduce the (3+1)-dimensional Yang–Mills field equations to the field equation of a 2D free massless scalar field. For each p and q, both the on-shell action and the energy-density reduce to the action and Hamiltonian of the corresponding 2D CFT. The topological charge density associated to the non-Abelian Chern–Simons current is non-zero. It is possible to define a non-linear composition within this family as if these configurations were “Lego blocks”. The non-linear effects of Yang–Mills theory manifest themselves since the topological charge density of the composition of two solutions is not the sum of the charge densities of the components. This leads to an upper bound on the amplitudes in order for the topological charge density to be well-defined. This suggests that if the temperature and/or the energy is/are high enough, the topological density of these configurations is not well-defined anymore. Semiclassically, one can show that (depending on whether the topological charge is even or odd) some of the operators appearing in the 2D CFT should be quantized as Fermions (despite the Bosonic nature of the classical field).

Graded Lie Algebroids: A Framework for Geometrization of Matter and Forces Unification

In this paper, we introduce a geometric structure that is capable of describing matter and forces simultaneously. This structure can be established by using the notion of $$Z_{2}$$ -graded Lie algebroid structures and graded semi-Riemannian metrics on them. Using calculus of variations, we derive field equations from the extended Hilbert–Einstein action. The derived equations contain Yang–Mills and Einstein field equations simultaneously. The even part of the graded Lie algebroid describes forces and its odd part is related to matter and particles.

HYM-flation: Yang–Mills cosmology with Horndeski coupling

We propose new mechanism for inflation using classical SU(2) Yang–Mills (YM) homogeneous and isotropic field non-minimally coupled to gravity via Horndeski prescription. This is the unique generally and gauge covariant ghost-free YM theory with the curvature-dependent action leading to second-order gravity and Yang–Mills field equations. We show that its solution space contains de Sitter boundary to which the trajectories are attracted for some finite time, ensuring the robust inflation with a graceful exit. The theory can be generalized to include the Higgs field leading to two-steps inflationary scenario, in which the Planck-scale YM-generated inflation naturally prepares the desired initial conditions for the GUT-scale Higgs inflation.

Covariant extrinsic gravity and the geometric origin of dark energy

In this paper, we construct the covariant or model independent induced Einstein–Yang–Mills field equations on a four-dimensional brane embedded isometrically in an D-dimensional bulk space, assuming the matter fields are confined to the brane. Applying this formalism to cosmology, we derive the generalized Friedmann equations. We derive the density parameter of dark energy in terms of width of the brane, normal curvature radii and the number of extra large dimensions. We show that dark energy could actually be the manifestation of the local extrinsic shape of the brane. It is shown that the predictions of this model are in good agreement with observation if we consider an 11-dimensional bulk space.

MAPPING A MASSLESS SCALAR FIELD THEORY ON A YANG–MILLS THEORY: CLASSICAL CASE

We analyze a recent proposal to map a massless scalar field theory onto a Yang–Mills theory at classical level. It is seen that this mapping exists at a perturbative level when the expansion is a gradient expansion. In this limit the theories share the spectrum, at the leading order, that is the one of a harmonic oscillator. Gradient expansion is exploited maintaining Lorentz covariance by introducing a fifth coordinate and turning the theory to Euclidean space. These expansions give common solutions to scalar and Yang–Mills field equations that are so proved to exist by construction, confirming that the selected components of the Yang–Mills field are indeed an extremum of the corresponding action functional.

No energy to be extracted from the vacuum**A review of a former article in this journal.

A few years ago a hopeful article (Evans 2000 Phys. Scr.61 513–7) appeared in this journal promising that according to its 15 authors' opinion the pending energy crisis could be solved by ‘extracting energy from the vacuum’. However, in the past years the energy price has grown to unthinkable heights: a reason for having a look at the promised ‘energy from the vacuum’ in (Evans 2000). So we shall do so below and shall arrive at a great disappointment: the 15 authors were in error; their vacuum energy stems from a simple flaw of thinking by misinterpreting the well-known Lorenz term of the classical Maxwell gauge theory. Their miraculous conclusion should have made the authors suspicious, that just the Lorenz term should yield a vacuum current. Surely, its a pity that vacuum currents and vacuum energy in (Evans 2000) have their origin merely in a simple flaw of thinking, and all further speculations for a vacuum energy density are in vain. However, better to return to reality. Quote: of [1]: Great announcements…It is shown that if the Loren(t)z condition is discarded, the Maxwell–Heaviside field equations become the Lehnert equations, indicating the presence of charge density and current density in the vacuum. The Lehnert equations are a subset of the O(3) Yang–Mills field equations. Charge and current density in the vacuum are defined straightforwardly in terms of the vector potential and scalar potential, and are conceptually similar to Maxwell's displacement current, which also occurs in the classical vacuum. A demonstration is made of the existence of a time dependent classical vacuum polarization which appears if the Loren(t)z condition is discarded. Vacuum charge and current appear phenomenologically in the Lehnert equations but fundamentally in the O(3) Yang–Mills theory of classical electrodynamics. The latter also allows for the possibility of the existence of vacuum topological magnetic charge density and topological magnetic current density. Both O(3) and Lehnert equations are superior to the Maxwell–Heaviside equations in being able to describe phenomena not amenable to the latter. In theory, devices can be made to extract the energy associated with vacuum charge and current.

Solutions of the massive SU(N) Yang–Mills equations by harmonic maps

We consider classical static solutions of the pure massive SU(N) Yang–Mills field equations in (3+1)-dimensional space–time. By applying harmonic map ansatz, constructed from the harmonic maps S2→CPN−1, we construct some bounded spherically symmetric solutions having finite energies for N=2, 3, and 4 cases.

On the integrability of the generalized Yang–Mills system

We consider a hamiltonian system which, in a special case and under the gauge group $SU(2)$, can be considered as a reduction of the Yang–Mills field equations. We prove explicitly, using the Lax spectral curve technique and the van Moerbeke–M

On global properties of static spherically symmetric EYM fields with compact gauge groups

The set of all possible spherically symmetric magnetic static Einstein–Yang–Mills field equations for an arbitrary compact semisimple gauge group G was classified in two previous papers. Local analytic solutions near the centre and a black-hole horizon as well as those that are analytic and bounded near infinity were shown to exist. Some globally bounded solutions are also known to exist because they can be obtained by embedding solutions for the G = SU(2) case which is well understood. However, more general global solutions for more complicated (so-called irregular) actions for larger gauge groups are very difficult to find numerically since they are very unstable. Here we derive rigorously some asymptotic properties of an arbitrary global solution, namely one that exists locally near a radial value r0 (in Schwarzschild-type coordinates), has positive mass m(r) at r0 and is defined for all r > r0. The set of asymptotic values of the Yang–Mills potential (in a suitable well-defined gauge) is shown to be finite in the so-called regular case, but may form a more complicated real variety for models obtained from irregular rotation group actions.

Nonsymmetric unified theory of gravitation, electromagnetism and Yang–Mills field

The recently studied new variation of Einstein's metric nonsymmetric unified field theory of gravitation and electromagnetism is enlarged to include the Yang–Mills field theory. It is shown that the antisymmetric part of the metric tensor, now a 2 × 2 matrix, can be made to describe both a field obeying Maxwell's equations and a field obeying Yang–Mills's field equations in the flat space linear approximation, thereby making its identification with the sum of a generalized electromagnetic and isotopic field strength tensors a possibly consistent procedure. The theory is shown to be free of unphysical ghost-negative energy radiative modes even when expanded on a curved Riemannian background. The Einstein–Maxwell–Yang–Mills theory is contained in the first approximation of the field equations on a curved general relativity background.

The space of super light rays for D=10, N=1 superconformal structures

We construct the space of super light rays for a D=10, N=1 superconformal structure. A super Ward correspondence is then established between D=10, N=1 supersymmetric Yang–Mills field equations on 10-dimensional spaces of the form M 4× M 6 (where M 4 is a four-dimensional complex space–time and M 6 is six-dimensional complex Minkowski space) and superbundles over the space of super light rays which are trivial on normal embedded quadrics. This result reduces in four dimensions to the equivalence of connections satisfying the D=4, N=4 SSYM field equations and superconnections integrable along super light rays also satisfying an added geometrical constraint coming from dimensional reduction. This extends the work of Witten and of Harnad and Shnider done on flat Minkowski space.

Classical Electrodynamics without the Lorentz Condition: Extracting Energy from the Vacuum

It is shown that if the Lorentz condition is discarded, the Maxwell–Heaviside field equations become the Lehnert equations, indicating the presence of charge density and current density in the vacuum. The Lehnert equations are a subset of the O(3) Yang–Mills field equations. Charge and current density in the vacuum are defined straightforwardly in terms of the vector potential and scalar potential, and are conceptually similar to Maxwell's displacement current, which also occurs in the classical vacuum. A demonstration is made of the existence of a time dependent classical vacuum polarization which appears if the Lorentz condition is discarded. Vacuum charge and current appear phenomenologically in the Lehnert equations but fundamentally in the O(3) Yang–Mills theory of classical electrodynamics. The latter also allows for the possibility of the existence of vacuum topological magnetic charge density and topological magnetic current density. Both O(3) and Lehnert equations are superior to the Maxwell–Heaviside equations in being able to describe phenomena not amenable to the latter. In theory, devices can be made to extract the energy associated with vacuum charge and current.

Invariant Solutions of the Yang–Mills Field Equations Coupled to a Scalar Doublet

The Yang–Mills system of field equations which includes coupling to an SU(2) scalar matter doublet is developed. It is shown that an SU(2) current for a scalar matter doublet can be developed. The basic structure which fits the Yang–Mills system is somewhat different from the case of the scalar triplet. Using this form for the scalar current, it is possible to write down the Yang–Mills system which couples to the scalar matter doublet. It is shown that several sets of solutions to this system of equations can be obtained.

Yang–Mills field equations for the Coriolis tensor in the n-body problem

The configuration space of a system of n particles has the structure of a principal fiber bundle under the action of the rotation group. In this paper, identities are derived for the curvature of the natural connection on this bundle. These identities are translated into statements about the Coriolis tensor, the local expression for this curvature on the base space of the bundle. The Coriolis tensor is shown to obey field equations of the Yang–Mills type.

Analytic solutions of the SU(3) Yang–Mills field equations with external sources

For external sources specified in the radial gauge frame, it is demonstrated how the SU(3) Yang–Mills field configuration can be constructed from the SU(2) solutions. This is explicitly illustrated for the type-II solutions that exhibit the bifurcation phenomenon.

Nonabelian progressive waves

We present two families of nonabelian wave solutions of the Yang–Mills field equations, some of the previous known solutions occur as special cases of our solutions.

Yang–Mills theory in null-path space

We define a GL(n,C) matrix-valued function G on a six-dimensional space of null paths in Minkowski space. Such paths are defined to begin at an arbitrary spacetime point xa and end at future null infinity. The space of these paths can thus be parametrized by giving the point xa and null direction. We show how knowledge of G can be used to obtain the GL(n,C) Yang–Mills connection at xa. We also derive a single equation for G, involving characteristic data given on null infinity, which is equivalent to the currentless or vacuum Yang–Mills field equations. The self-dual (anti-self-dual) nonabelian fields and the general abelian cases are described as special examples.

Real solutions of the Yang–Mills field equations in Minkowski space

A system of nonlinear partial differential equations is obtained by considering complex self-dual Yang–Mills fields in Minkowski space and then selecting those fields whose real part is again a solution of the Yang–Mills field equations. From solutions of this system families of real solutions of the Yang–Mills field equations in Minkowski space are obtained, which depend on one or more arbitrary solutions of the two-dimensional Laplace equation and one or more arbitrary functions.

Minkowski space Yang–Mills fields from solutions of equations in the three-dimensional Euclidean space

Equations in the three-dimensional Euclidean space are derived by combining the Yang–Mills field equations with the conditions which are imposed on a Yang–Mills field in Bernreuther’s method of constructing Yang–Mills fields in Minkowski space from Yang–Mills fields in Euclidean space. By a proper ansatz for the Yang–Mills fields these equations are reduced to a single differential equation. The differential equation is identical with merons’ equation in Euclidean space if we consider solutions of the latter equation which are functions of the ratio t/ρ, where t is the Euclidean time and ρ is the three-dimensional radius. One such solution is the single meron solution in Euclidean space. Starting from this and applying the method we get the de Alfaro–Fubini–Furlan solution in Minkowski space. Then, a more general ansatz is considered, which leads to a system of three nonlinear differential equations.

Periodic solutions of the Yang—Mills field equations

Exact periodic solutions of the classical SU(2) Yang—Mills equations in Minkowski space-time are constructed. These solutions can be interpreted, respectively, as non-abelian plane waves and spherical waves. The corresponding non-self-dual static solutions are also exhibited, some of which possess vanishing energy-momentum density.