On gauge-invariant Green function in 2+1 dimensional QED

Both the gauge-invariant fermion Green function and gauge-dependent conventional Green function in $2+1$ dimensional QED are studied in the large $N$ limit. In temporal gauge, the infra-red divergence of gauge-dependent Green function is found to be regulariable, the anomalous dimension is found to be $\eta= \frac{64}{3 \pi^{2} N}$. This anomalous dimension was argued to be the same as that of gauge-invariant Green function. However, in Coulomb gauge, the infra-red divergence of the gauge-dependent Green function is found to be un-regulariable, anomalous dimension is even not defined, but the infra-red divergence is shown to be cancelled in any gauge-invariant physical quantities. The gauge-invariant Green function is also studied directly in Lorentz covariant gauge and the anomalous dimension is found to be the same as that calculated in temporal gauge.