Gribov ambiguity and degenerate systems

The relation between Gribov ambiguity and degeneracies in the symplectic structure of physical systems is analyzed. It is shown that, in finite-dimensional systems, the presence of Gribov ambiguities in regular constrained systems (those where the constraints are functionally independent) always leads to a degenerate symplectic structure upon Dirac reduction. The implications for the Gribov-Zwanziger approach to QCD are discussed.

Figure 1

Qualitative flow of the orbits near the degeneracy surface (thick line), which can act as a sink or as a source.

Figure 2

The gauge condition $G_i(u)\approx 0$ (thick line) intersects the gauge orbits (thin lines) more than once, provided there exist points where the orbits run tangent to the gauge condition.

Figure 3

The orbits generated by gauge transformations in the FLPR model are helicoids of the form $(x,y,z)=\mathbf{(}\rho \cos [\alpha ϵ(t)],\rho \sin [\alpha ϵ(t)],ϵ(t)\mathbf{)}$.

Figure 4

The surface defined by the $\lambda$-gauge condition $G=z-\lambda x=0$ is a plane (here plotted for $\lambda =1$). The Gribov ambiguity in the FLPR model is reflected by the fact that this plane intersects some gauge orbits more than once.

Figure 5

In the case of the FLPR model, the Gribov horizon (vertical plane), $y=-{(\alpha \lambda )}^{-1}$, and the constraint surface (inclined plane), $G=0$, are plotted for $\lambda =1$ and $\alpha =1/3$. The GH divides the constraint surface into two dynamically disconnected regions.

Figure 6

Orbits for the Christ-Lee model are given by circles centered at the origin. The GH ($y=-x/\mu$) and the surface $G=0$ ($y=\mu x$) are plotted for $\mu =2$. The GH restricted to the constraint surface corresponds to the point $x=y=0$.