Ja n 20 02 Some remarks on the “ classical ” large N limit

It has been proposed some time ago that the large N−limit can be understood as a “classical limit”, where commutators in some sense approach the corresponding Poisson brackets. We discuss this in the light of some recent numerical results for an SU(N) gauge model, which do not agree with this “classicality” of the large N−limit. The world sheet becomes very crumpled. We speculate that this effect would disappear in supersymmetric models. To Holger Bech Nielsen on his 60th birthday In the 1970ies Holger and I discussed the confinement problem quite a lot. In those days the picture of confinement by means of a string connecting the quarks became more and more accepted. It was known that the Nambu-Goto action could be written as a square root of the Poisson bracket squared. One of the subjects we discussed was whether somehow the Poison bracket was related to the commutator like in quantum mechanics, in some limit. However, we did not gain any insight in this. After the work by Eguchi and Kawai it became of course clear that the relevant limit was the large N limit, where the action could be expressed in terms of commutators. Later on Hoppe [1] constructed a very suggestive relation between Poisson brackets and commutators. This was then used in ingenious ways by a number of authors [2, 3, 4], in connection with the Eguchi-Kawai formulation of the large N theory. To give a brief review of this approach, let us consider SU(N) with the gauge field Aμ and the generators lk,k = (k1, k2), (Aμ) j i = ∑ k akμ (lk) j i , (1) where akμ are expansion coefficients to be integrated in functional integrals. This expansion can be compared to the one for a string variable Xμ(σ, τ), Xμ(σ, τ) = ∑