Today on arxiv I have found a very beatiful paper by Reinhardt and Schleifenbaum (see here). This paper is an important event as the authors present a full account of Yang-Mills theory in D=1+1. As we know, Axel Maas produced a lattice computation of this theory (see here) and found a perfect agreement with truncated Dyson-Schwinger equations. These results disagree completely with those obtained on lattice for D=3+1. From ‘t Hooft’s work we also know that Yang-Mills theory in D=1+1 is completely trivial having no dynamics. This means that the agreement between Maas’ lattice computations and truncated Dyson-Schiwnger equations implies that the truncation eliminates any dynamics from Yang-Mills theory and this explains the disagreement between truncated Dyson-Scwinger equations and lattice Yang-Mills in D=3+1.

In their paper Reinhardt and Schleifenbaum confirm all this but they do a smarter thing. They consider a non trivial Yang-Mills theory in D=1+1 taking a compact manifold . In this case they introduce a length and this means that the “thermodynamic limit” should recover the trivial limit of Yang-Mills theory in D=1+1. Of course, due to this deep link between the theory on the compact manifold and the one on the real line, again this case is not representative for Yang-Mills in D=3+1 but, anyhow, can give some hints on how truncated Dyson-Schwinger equations recover these results. However, it should be emphasized that Gribov copies in D=1+1 have a prominent role and this is not generally true in D=3+1. This can yield the false impression to have caught something of the disagreement between functional methods and lattice computations. Of course, this is plainly false. In order to give an idea of what is going on they get a gluon propagator going like and this goes to zero in the thermodynamic limit as no dynamics is expected in this case. In D=3+1 there is nothing like this. On a compact manifold for this case, the limit is absolutely not trivial. Finally, they get an infrared enhanced ghost propagator and the authors claim that the reason why this is not seen on lattice computations for the D=3+1 case is due to Gribov copies. This conclusion cannot be accepted as the trivial limit of this theory is the D=1+1 case on the real line that has an enhanced ghost propagator too and this must not necessarily be true for D=3+1 where, as said, Gribov copies play no role. This latter fact is the reason of the failure of functional methods and also the reason why dynamics is removed by this approach. Indeed, to account for Gribov copies in D=3+1 one is forced to remove dynamics. This works for D=1+1 where no dynamics exists but fails otherwise.

A note on the running coupling should have been done by the authors. They did not do that but if the gluon propagator goes like , whatever else the ghost propagator does, the thermodynamic limit grants that the coupling goes to zero. No dynamics no interaction.

Another interesting result given by the authors is the spectrum for the theory on the compact manifold. They get the spectrum of a rigid free rotator going like . This is very nice indeed.

Finally, the conclusion by the authors that functional methods turn out to have got a strong support by their computations cannot be sustained. They just give an understanding, a deep one indeed, of the reason why these methods blatantly fail for the D=3+1 case. This is the role of computations in D=1+1 as already seen with Maas’ work.