ANDERSON REPLIES: The question Vincent Rivasseau raises is the subject of a recent paper of mine, 1 which I believe explains the situation satisfactorily. It appears that there is actually no mathematical contradiction between our two results.
Rivasseau and Margherita Disertori explicitly point out in Rivasseau’s reference 6 that their perturbative renormalization group procedure eventually fails due to lack of convergence, and is only valid above a certain explicitly derived temperature. I would agree completely with that statement. My arguments stem from a second rigorous approach to the many-body problem, the method of Kerson Huang, T.-D. Lee, and C. N. Yang, and of Viktor Galitskii, which was applied to the two-dimensional case by Paul Bloom in 1975. 2 Bloom also claimed to have proved rigorously that the system is a simple Fermi liquid, in a different limit, near T = 0 and for arbitrarily strong interactions, but for low density. I show that Bloom’s calculation is in error, but the difference between Bloom’s and my results would only be visible below Rivasseau’s limiting temperature—so I agree that the electron gas would look like a Fermi liquid above that temperature. I think it must be significant that the temperature limits derived from two completely different approaches are the same. But it is worth noting that for real physical interactions the Rivasseau limitation excludes any T appreciably below the Fermi temperature.
Of course, I cannot be accused of too much opposition to the use of the renormalization group in the many-body problem, since I invented it. Rivasseau’s thumbnail history does not mention that I was the first to use the renormalization group on a many-body problem, with Gideon Yuval, in a paper submitted in 1969 on the Kondo Hamiltonian. 3 We predate Kenneth Wilson by some months, though the paper was delayed by an eminent but unperceptive referee. I may also have been the first to suggest using it on the Fermi liquid. 4 Thus, the subsequent work by Rivasseau and others is not “an adaptation of Wilson’s renormalization group” but of mine, though of course I make no claim to its more important use in statistical mechanics.
Some of the above resulted from valuable discussions with Manfred Salmhofer at the Institute for Theoretical Physics in Santa Barbara, California.
