In his article “Brainwashed by Feynman?” (Physics Today, February 2000, page 11), Philip W. Anderson says that a generation of “field theoretically trained young theorists” now performs essentially irrelevant studies of the interacting fermions in condensed matter: “The obvious assumption is that if one is able, by dint of very hard work, to sum up all the Feynman diagrams, one must arrive at the right answer. The problem is that no one has ever been able, over four decades, even to arrive at the right interaction that way.” Anderson invites these theorists to imitate their particle physics colleagues, who “have long abandoned straightforward diagrams, in favor of a much more varied toolkit of concepts and techniques.”
This view, from a leader of theoretical condensed matter physics, calls for a debate. As a field theoretically trained (no longer young) theorist, I attack the problem of interacting fermions by resumming “all the diagrams.” I agree with Anderson that the phenomenology of the interactions in “borderline materials” is still poorly understood, and that the description of many phenomena in condensed matter may require tools other than straightforward perturbation theory. However, I would not dismiss perturbation theory, for instance, by believing that it is limited to the analysis of essentially boring weakly coupled theories.
The study of interacting fermions in condensed matter by resumming diagrams has at least two scientific godfathers: Richard Feynman and Kenneth Wilson. In the late 1980s, a group of mathematical physicists, Giuseppe Benfatto, Joel Feldman, Giovanni Gallavotti, and Eugene Trubowitz, also made important conceptual progress: They realized that Wilson’s renormalization group should be adapted in a nontrivial way to condensed matter. Indeed the long-range behavior of a system of interacting fermions is governed by an extended singularity, the Fermi surface. As a result, the corresponding scaling analysis and the underlying dynamical flow of effective interactions is much richer than in the ordinary Wilsonian case. No simple analog exists of the “block spin” and rescaling concepts. More important, there is an infinite set of relevant operators, or of coupling constants. Feynman diagrams are essential to analyze the corresponding flows.
Organizing perturbation theory based on the renormalization group around the Fermi surface is therefore not only conceptual progress, it is probably (even numerically 1 ) the best tool to understand which among all these couplings diverges first and dominates the long-range physics. I also do not think that the analogy with the confinement problem for hadrons is relevant; we know that, because of the extended nature of the Fermi surface in more than one dimension, nonperturbative phenomena in condensed matter physics (particularly the formation of bound states such as Cooper pairs) can actually be controlled by analytic methods because of their similarity to large-component vector models. 2 Despite many efforts, this is not yet the case for hadrons, because matrix models are involved in their formation.
But is it the use of Feynman graphs or the pretension to “sum them all” that Anderson considers irrelevant?
If the latter is the case, here is a brief defense of traditional, perhaps old-fashioned, 3 mathematical physics, which consists in proving mathematical theorems about idealized systems inspired by physics. I view it as an indispensable complement to theoretical physics in the long run.
When Lars Onsager proved that the two-dimensional Ising model has a phase transition, or when John Imbrie solved a controversy about the nature of the ground state of the random-field Ising model in three dimensions, 4 they certainly did not believe any real material to be an exact Ising model. Nevertheless, each of their results acquires a particular value because it is mathematically rigorous. Even more than diamonds, mathematical theorems are forever; they are precious strongholds among all the uncertainties of an ever-changing scientific landscape.
The question of whether perturbation theory can be summed up or not goes back at least to Henri Poincaré. Even a negative result, such as his famous observation that the Lindstedt perturbation series of classical mechanics diverges, is a scientific landmark, a starting point for advances such as the convergence theorems of the Kolmogorov-Arnold-Moser type and the investigation of chaos. Therefore, a mathematical physicist educated in the Poincaré spirit, when learning that first-order perturbation theory makes sense in a particular physical context, worries whether adding “all” the diagrams does not change the picture.
To return to condensed matter, Manfred Salmhofer recently formalized a precise mathematical criterion to distinguish Fermi liquids above the Bardeen-Cooper-Schrieffer temperature from, for example, Luttinger liquids. 5 Summing up “all the diagrams,” Margherita Disertori and I proved that an isotropic jellium with a small, short-range interaction in two dimensions is indeed a Fermi liquid in this sense. 6 Certainly, at least for simple models, interacting Fermi liquid theory is mathematically consistent in two dimensions; it is also, one would hope, a step toward rigorously settling the phase diagram of more complicated and realistic models, such as the Hubbard model near half-filling, and towards the rigorous analysis of the two-dimensional Anderson model of free or weakly interacting electrons in a random potential.
