Coulomb Gauge Fixing Research Articles

Fourier accelerated conjugate gradient lattice gauge fixing

We provide details of the first implementation of a non-linear conjugate gradient method for Landau and Coulomb gauge fixing with Fourier acceleration. We find clear improvement over the Fourier accelerated steepest descent method, with the average time taken for the algorithm to converge to a fixed, high accuracy, being reduced by a factor of 2 to 4.

Wong’s equations in Yang-Mills theory

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

Counting photons in static electric and magnetic fields

We describe the electromagnetic field by the massless limit of a massive vector field in the presence of a Coulomb gauge fixing term. The gauge fixing term ensures that, in the massless limit, the longitudinal mode is removed from the spectrum and only the two transverse modes survive. The system, coupled to a classical conserved current, is quantized in the canonical formalism. The classical field configurations due to time-independent electric charges and currents are represented by coherent states of longitudinal and transverse photons, respectively. The occupation number in these states is finite. In particular, the number of longitudinal photons bound by an electric charge q is given by N=q^2/(16\pi\hbar).

Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory

Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)–Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb gauge methods, with an infinite lattice critical point near β=3.2. The theory with both Z2 vortices and monopoles and SO(3)–Z2 monopoles eliminated is simulated in the strong-coupling (β=0) limit on lattices up to 604. Here, as in the high-β phase of the Wilson-action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any β. Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of α∼1.4. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.

TOWARD A PROOF OF LONG RANGE ORDER IN 4D SU(N) LATTICE GAUGE THEORY

An extended version of four-dimensional (4D) SU(2) lattice gauge theory is considered in which different inverse coupling parameters are used, [Formula: see text] for plaquettes which are purely space-like, and βV for those which involve the Euclidean time-like direction. It is shown that when βH = ∞ the partition function becomes, in the Coulomb gauge, exactly that of a set of non-interacting three-dimensional (3D) O(4) classical Heisenberg models. Long range order (LRO) at low temperatures (weak coupling) has been rigorously proven for this model. It is shown that the correlation function demonstrating spontaneous magnetization in the ferromagnetic phase is a continuous function of gH at gH = 0 and therefore, that the spontaneously broken phase enters the (βH, βV) phase plane (no step discontinuity at the edge). Once the phase transition line has entered, it can only exit at another identified edge, which requires the SU(2) gauge theory within also to have a phase transition at finite β. A phase exhibiting spontaneous breaking of the remnant symmetry left after Coulomb gauge fixing, the relevant symmetry here, is non-confining. Easy extension to the SU (N) case implies that the continuum limit of zero-temperature 4D SU (N) lattice gauge theories is not confining, in other words, gluons by themselves do not produce a confinement.

PERTURBATIVE ANALYSIS OF CHERN–SIMONS FIELD THEORY IN THE COULOMB GAUGE

In this paper, we analyse the perturbative aspects of Chern–Simons field theories in the Coulomb gauge. We show that in the perturbative expansion of the Green functions there are neither ultraviolet nor infrared divergences. Moreover, all the radiative corrections are zero at any loop order. Some problems connected with the Coulomb gauge fixing, like the appearance of spurious singularities in the computation of the Feynman diagrams, are discussed and solved. The regularization used here for the spurious singularities can be easily applied also to the Yang–Mills case, which is affected by similar divergences.

Quantization of Chern-Simons field theory in the Coulomb gauge.

We consider here the Chern-Simons field theory with the gauge group SU(N) in the presence of a conformally flat gravitational background that describes a two-dimensional expanding ``universe.'' The propagator and the vertices are explicitly computed at the second order in perturbation theory imposing the Coulomb gauge fixing. It is shown in this gauge that the correlation functions of the gauge fields are independent of the metric.

An investigation into a wavelet accelerated gauge fixing algorithm

We introduce an acceleration algorithm for coulomb gauge fixing, using the compactly supported wavelets introduced by Daubechies. The algorithm is similar to Fourier acceleration. Our provisional numerical results for SU(3) on 8 4 lattices show that the acceleration based on the DAUB6 transform can reduce the number of iterations by a factor up to 3 over the unaccelerated algorithm. The reduction in iterations for Fourier acceleration is approximately a factor of 7.

Structure of the Gauge Transformation Group in the Square Integrable Space and Gribov's Ambiguity in the Coulomb Gauge

It is found that a well-defined gauge transformation group and the well-known discon­ nected structure of it emerge only from the square integrability with some natural assump­ tion of the boundedness on the gradient of the gauge field. The geometrical formulation of the Coulomb gauge fixing degeneracy is restated with more careful treatment to the dual space considered as space of distribution, than in our previous paper. A comment on the suitably restricted space version of the formulation is made. As such an example, '0 (3) symmetric' case is treated. Together with the discon­ nected structure of the gauge transformation group, it is found that there exist a lot of Coulomb gauge fixing degeneracies such that topological number jumps by any integer value between transverse fields.