Classical Yang-Mills Equations Research Articles

Classical gluon radiation in ultrarelativistic nucleus-nucleus collisions

The classical Yang-Mills equations are solved perturbatively in covariant gauge for a collision of two ultrarelativistic nuclei. The nuclei are taken as ensembles of classical color charges on eikonal trajectories. The classical gluon field is computed in coordinate space up to cubic order in the coupling constant $g$. We construct the Feynman diagrams corresponding to this field and show the equivalence of the classical and diagrammatic approaches. An argument is given which demonstrates that at higher orders in $g$ the classical description of the process breaks down. As an application, we calculate the energy, number, and multiplicity distributions of produced soft gluons and reproduce earlier results by Gunion and Bertsch and by Kovner, McLerran, and Weigert.

Scattering of plane waves in self-dual Yang - Mills theory

We consider the classical self-dual Yang-Mills equation in 3+1-dimensional Minkowski space. We have found an exact solution, which describes scattering of $n$ plane waves. In order to write the solution in a compact form, it is convenient to introduce a scattering operator $\hat{T}$. It acts in the direct product of three linear spaces: 1) universal enveloping of $su(N)$ Lie algebra, 2) $n$-dimensional vector space and 3) space of functions defined on the unit interval.

Exact solutions of the Yang-Mills equations with source in the form of two point color charges

We find a family of exact retarded solutions of the classical Yang—Mills equations with a source that consists of two point color charges that move along arbitrary timelike worldlines in Minkowski space. The solutions are stable with respect to small field perturbations and are therefore suitable for playing the role of a nonperturbative gluon vacuum. It is shown that a description of confinement in the framework of classical Yang—Mills theory is impossible. Confinement is ensured by quantum fluctuations over the constructed gluon vacuum.

HAMILTONIAN DYNAMICS OF YANG-MILLS FIELDS ON A LATTICE

We review recent results from studies of the dynamics of classical Yang-Mills fields on a lattice. We discuss the numerical techniques employed in solving the classical lattice Yang-Mills equations in real time, and present results exhibiting the universal chaotic behavior of non-abelian gauge theories. The complete spectrum of Lyapunov exponents is determined for the gauge group SU(2). We survey results obtained for the SU(3) gauge theory and other nonlinear field theories. We also discuss the relevance of these results to the problem of thermalization in gauge theories.

SU( N) instantons and exotic Skyrmions

The classical Yang-Mills equations are solved for arbitrary semi-simple gauge groups in the Schwinger-Fock gauge. A new class of SU( N) instantons is presented which are not embeddings of SU( N − 1) instantons but have non-trivial SU( N) color structure and carry winding number n = 1 6 [N(N 2 − 1)] . Explicit configurations are given for SU(3) and SU(4) gauge groups. By means of the Atiyah-Manton procedure Skyrmion fields are constructed from the SU( N) instantons. These Skyrmions represent exotic baryon states.

Classical Yang-Mills field generated by two colored point charges

An exact retarded solution to the classical Yang-Mills equations with the source composed of two arbitrarily moving colored point charges is given. The solution is stable against small field disturbances hence for the role of the nonperturbative gluon vacuum. It is shown that the confinement description turns out to be impossible with the scope of the classical Yang-Mills theory. Confinement is accomplished due to quantum fluctuations about the constructed vacuum.

Classical Yang-Mills fields with conformal invariance: from exact solutions to qualitative analysis

The authors consider classical Yang-Mills equations on Minkowski space, with gauge group SO(3), and look for solutions invariant under a four-dimensional subgroup of the conformal group. In that case, the equations do admit non-Abelian solutions, but these cannot be obtained analytically. Instead, they analyse the equations qualitatively (in a restricted case), in effect treating them as a dynamical system, which turns out to be very simple and completely integrable. The behaviour of the solutions is obtained by singular perturbation methods and the result is confirmed by numerical integration of the equations.

Nonlinear effects in the flux-tube model for ultrarelativistic nucleus-nucleus collisions.

We solve the classical Yang-Mills equations numerically for the evolution of the color fields in the flux-tube model for ultrarelativistic collisions. We show that nonlinear effects are small and decrease as the nuclei recede. This result can be understood in the limit of large background fields from an analytic study of the linearized equations.

Yang-Mills equations and solvable groups.

It is shown that a large class of classical Yang-Mills equations do not possess a Lagrangian (or Hamiltonian) and hence cannot be quantized by conventional methods. These theories may be embedded in a larger Yang-Mills theory which does possess a Hamiltonian structure. However, the symplectic structure induced on the embedded theory is degenerate and hence the quantization of the ambient theory does not induce a quantization of the embedded theory.

Origin of external sources for classical Yang-Mills fields.

An effort to derive the classical Yang-Mills equation with an external color four-current density ${j}^{a\ensuremath{\mu}}$(x) from a more fundamental theory is presented. It is shown that in the case of an arbitrary static external color charge, i.e., when ${j}^{a\ensuremath{\mu}}$=${\ensuremath{\delta}}_{0}^{\ensuremath{\mu}}$${\ensuremath{\rho}}^{a}$(x), the classical Yang-Mills equation follows from the closed set of classical Yang-Mills and Dirac equations in the limit m\ensuremath{\rightarrow}\ensuremath{\infty}, where m is the mass of the Dirac particle. In the case of the classical Yang-Mills equation with an external color current ${j}^{\mathrm{ai}}$, i=1,2,3, no such derivation is found.

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.

Classical Yang-Mills fields with non-compact symmetry and an arbitrary compact gauge group

The authors consider classical Yang-Mills equations on Minkowski space. When the gauge group is SU(2), they have shown previously that the only solutions invariant under a non-compact maximal subgroup of the conformal group are Abelian Maxwell fields. They now extend the analysis to the case where the gauge group is an arbitrary semisimple compact group, with the same negative result.

Color screening and topological index in the classical Yang-Mills theory with sources.

We reexamine the relation between the topological charge M and the occurrence of color screening in the classical Yang-Mills equations with sources. We find that the M=0 sector allows partial-screening solutions as well as the total-screening solutions.

Intersection of separatrices of periodical trajectories and non-integrability of the classical Yang-Mills equations

Intersection of separatrices of periodical trajectories and non-integrability of the classical Yang-Mills equations

Classical Yang-Mills fields on Minkowski space with a non-compact invariance group

The authors consider classical Yang-Mills equations on (compactified) Minkowski space, in the fibre bundle formulation, and look for real solutions invariant under a subgroup G of the conformal group C(3,1). When the gauge group is SU(2) and G is a non-compact maximal subgroup of C(3,1), the only non-zero invariant solutions are Abelian Maxwell fields.

Yang-Mills equations with external sources: a connection with scalar theories

It is found that classical Yang-Mills equations with external four-vector current densities are equivalent, under a particular ansatz, to the equation Square Operator phi + mu 2 phi 3eK=0 with mu an arbitrary constant and K an arbitrary differentiable function. Kink, sine-Gordon and Liouville equations for radially (spherically) symmetric ansatz functions are obtained with different parametrisations of K.

On colour screening of classical Yang-Mills fields

By working in a temporal gauge the author shows that solutions of classical Yang-Mills equations with external charge and current densities with vanishing magnetic field may not be useful in explaining classical confinement.

Variational methods in the master field formulation for (3+1)-dimensional quantum chromodynamics

Master fields are formulated for finite-$N$ (3+1)-dimensional QCD. They satisfy classical Yang-Mills equations with an infinite number of internal indices and an infinite number of constraints. Master fields and constraints on them in the large-$N$ limit ($N\ensuremath{\rightarrow}\ensuremath{\infty}$ with fixed ${g}^{2}N$) are derived from the finite-$N$ master fields and constraints using vacuum dominance among color-singlet states. Explicit solutions for the large-$N$ constraints are given and used as trial functions in a Hartree-Fock variational calculation. This Hartree-Fock method can include essential features of gluon condensation effects in the QC${\mathrm{D}}_{3+1}$ vacuum which is expected to cause confinement. Inclusion of quark fields and Hartree-Fock equations for the meson energy spectrum are also discussed.

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

Classical Yang-Mills (Y.M.) equations with static external sources are formulated as a Hamiltonian system with gauge symmetry in A 0 = 0 gauge. Using the concept of a “momentum mapping” (J. Marsden and A. Weinstein, Rep. Math. Phys. 5 (1974), 121) on symplectic manifolds with symmetry, an analogue of centrifugal potential of a mass point in a spherically symmetric potential is derived. This gives rise to an effective potential V eff, whose critical points are rigorously proved to be in one-to-one correspondence with static Y.M. solutions. V eff additionally depends on the prescribed external source ϱ, which is as a constant of motion analogous to angular moment of the mass point. Thus bifurcation of static solutions is caused by bifurcation of critical points of V eff under variation of the external parameter ϱ. Some closing remarks on dynamics and stability on gauge orbit space are added.

Partial color screening by classical Yang-Mills fields

Non-coulombic axially (but not spherically) symmetrical static solutions to the classical Yang-Mills equations in the presence of an external point source are numerically found.