Schwinger-Dyson Equations in Coulomb Gauge Consistent with Numerical Simulation

TL;DR

This study demonstrates that in Coulomb gauge, non-instantaneous terms in the Schwinger-Dyson equations vanish due to a remnant gauge symmetry, leading to a purely instantaneous gluon propagator consistent with lattice simulations and supporting confinement via a linearly rising potential.

Contribution

The paper proves that all non-instantaneous terms in the Coulomb gauge Schwinger-Dyson equations are zero, simplifying the analysis and confirming the instantaneous nature of the gluon propagator.

Findings

Non-instantaneous terms in the SDE vanish due to gauge symmetry.

The gluon propagator is purely instantaneous, proportional to a delta function in time.

Results support confinement through a linearly rising Coulomb potential.

Abstract

In the present work we undertake a study of the Schwinger-Dyson equation (SDE) in the Euclidean formulation of local quantum gauge field theory, with Coulomb gauge condition $\partial_i A_i = 0$. We continue a previous study which kept only instantaneous terms in the SDE that are proportional to $\delta(t)$ in order to calculate the instantaneous part of the time component of the gluon propagator $D_{A_0 A_0}(t, R)$. We compare the results of that study with a numerical simulation of lattice gauge theory and find that the infrared critical exponents and related quantities agree to within 1\% to 3\%. This raises the question, "Why is the agreement so good, despite the systematic neglect of non-instantaneous terms?" We discovered the happy circumstance that all the non-instantaneous terms are in fact zero. They are forbidden by the symmetry of the local action in Coulomb gauge under time-dependent gauge transformations $g(t)$. This remnant gauge symmetry is not fixed by the Coulomb gauge condition. The numerical result of the present calculation is the same as in the previous study; the novelty is that we now demonstrate that all the non-instantaneous terms in the SDE vanish. We derive some elementary properties of propagators which are a consequence of the remnant gauge symmetry. In particular the time component of the gluon propagator is found to be purely instantaneous $D_{A_0 A_0}(t, R) = \delta(t) V(R)$, where $V(R)$ is the color-Coulomb potential. Our results support the simple physical scenario in which confinement is the result of a linearly rising color-Coulomb potential, $V(R) \sim \sigma R$ at large $R$.