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 $R$.
Highlights
While the quest for exotic quantum theories of gravity captivates many physicists, a much more mundane question remains unanswered: what is the qualitative mechanism for the mismatch between the UV degrees of freedom (d.o.f.) of the standard model and the IR states we observe in the lab
New physics is unlikely needed; from lattice simulations, we know that nonAbelian gauge theory by itself is capable of creating gluonic flux tubes which confine quark-antiquark pairs into mesons at low energy [1]
We have discovered that the noninstantaneous terms vanish because of the invariance under time-dependent gauge transformations gðtÞ
Summary
While the quest for exotic quantum theories of gravity captivates many physicists, a much more mundane question remains unanswered: what is the qualitative mechanism for the mismatch between the UV degrees of freedom (d.o.f.) of the standard model (quarks and gluons) and the IR states we observe in the lab (baryons and mesons). The presence of a long range colorColoumb potential at high temperature challenges this view, and suggests that one might expect a strongly interacting fluid, despite the approximate StefanBoltzmann like behavior witnessed by Karsch et al on the lattice [22] This isn’t contridictory with the renormalization group; recall that in Coulomb gauge, the physical quantity g2DA0A0 is a renormalization-group invariant [23]. The partition function becomes vanishingly small in the Euclidean-time direction, yielding a dimensionally reduced theory, in addition to any instantaneous physics inherited from the higher dimensional theory This heuristic picture is illustrated in [24] and a rigorous treatment of Gribov-Zwanger theory in Coulomb gauge at finite temperature can be found in [27].
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