This paper is a classic, and all of it is highly original, and it's deductions are mostly accurate up to section VII in their context. But here, I would like to focus only on the inaccuracies, as this is the thing that helps with historical material.
The detailed assumptions about the strong interactions in this paper aren't at all right, since Wilson was writing well before QCD. But there are also a few accuracy problems in the context of pure 1960s physics. These are outweighed by the great originality of the paper's main method. The paper's virtue is not that the physical hypotheses are right, nor that the results of Wilson's model are particularly enlightening in the special case of the strong interactions, but that the OPE produces a formalism to analyse arbitrary strongly coupled field theory fixed points, a language for talking about the algebra of operators in any quantum field theory without paradoxes. The results he gives are examples of the power of the OPE to produce results, it's just that the results themselves are not physically right inside QCD.
The strong interactions, as understood in the 1960s, consisted of families of hadrons of mass <5GeV, interacting by scattering at energies well below the scale where the quark and gluon substructure becomes manifest. It was known that not all these particles could be quanta of elementary quantum fields. The Regge trajectory idea was developed by Regge and Mandelstam, and shown to fit the experimental data starting with Chew and Frautschi. The empirical Regge law said that the mesons form families of spin and mass so that m2 is proportional to J, with a universal proportionality constant. The assumption that this behavior continued made predictions for the scaling laws for the near-beam scattering at higher energy, and Regge theory fits worked to predict the small-angle scattering at higher energies and fit the total cross sections (Regge theory still works as well as ever for this, although people largely stopped talking about it). The trajectories meant that there were going to be infinitely many strongly interacting particles of ever higher mass and spin, extrapolating the m2 vs. J line of the known ones.
The Regge behavior was emphasized by Chew and collaborators, who worked on S-matrix theory. These folks believed that field theory was hopeless for the strong interactions, and that it is best to formulate a new kind of theory from scratch, without using the concept of local fields at all, only using experimental data to infer relativistic amplitudes, and dispersion relations and unitarity. When supplemented with the hypotheses of linear Regge trajectories in a narrow-resonance approximation, and Dolen-Horn-Schmidt duality, this idea leads to string theory.
Separate from the Regge trajectory story, which linked meson resonances of increasing mass and spin, the mesons and baryons of one particular spin and parity clustered into groups according to their isospin, hypercharge, and strangeness, each of them making the simplest SU(3) multiplets which were all broken to lowest order in a universal mass-breaking pattern, which was parametrized by introducing a diagonal SU(3) matrix diag($m_s$,$m_d$,$m_u$) as a noninvariant spurion to contribute to the mass matrix for the hadrons. This SU(3) breaking term was identified as the quark-mass matrix, which was appreciated long before the quarks themselves were understood. The interpretation of the mass-relations in SU(3) flavor multiplets was that the difference in quark-masses (whatever quarks were exactly) was responsible for splitting the hadrons in an SU(3) multiplet apart in mass, and the fundamental underlying strong interaction theory, ignoring quark masses, was SU(3) symmetric (see, e.g., "The Eightfold Way").
The SU(3) relations meant that the strong interactions were pictured as split in two--- one part was the universal SU(3) symmetric super-strong interactions, describing high energies where the quark masses were negligible--- this part today we would identify as the theory of QCD at zero-quark-mass. The other part of the theory was what were called the medium-strong interactions, which produced or included the quark masses as perturbation over the SU(3) symmetric super-strong interactions. Now we also know that the quark masses are the whole story regarding the medium strong interactions, at least in the fundamental Lagrangian.
But further, it was known, from the Nambu Jona-Lasinio model and the sigma model, that the true (flavor) symmetry group of the strong-stong interactions must be an $SU(2)\times SU(2)$ symmetry and that it was fundamentally breaking down to Isospin, since the pions provided three Goldstone bosons for a broken SU(2). This was generalized to broken flavor $SU(3)\times SU(3)$. The formalism for analyzing the symmetries was provided by Gell-Mann, who proposed that the currents for these symmetries were still useful to consider as local fields, even though the underlying theory might not be a field theory. He suggested to use the local current commutators to make predictions, because the form of these local commutators would be determined by the symmetry of the theory. Then you could interpret the current commutation relations, which were determined by group theory, by considering the unbroken symmetry generators to be integrals of the unbroken currents (these are the "conserved" currents), while the Goldstone boson fields were identified with the broken symmetry currents (these are the "partially conserved" currents).
It was a very important idea, and the primitive precursor to the OPE. The OPE extends the idea of a current algebra to cover all local fields in a local field multiplication algebra, rather than just currents (although it works for currents too). Current algebra today is most often just identified with some particular universal coefficients of singular parts of the current-current OPE.
The quark masses set a length-scale, and the symmetry breaking set a scale also. At the time, it was not clear whether, at high energies, the strong-strong interactions would have a separate independent scale inside, or whether they would be scale invariant. Wilson's fundamental starting point in this paper is the idea, which he attributes to Kastrup and Mack, that the strong-strong interactions are scale free, they are a scale invariant theory with the only scales coming from spontaneous symmetry breaking, which also produces the quark mass matrix effects. This is false in QCD. The strong interactions are not scale invariant except at extremely high energies where they are free, and they go to this limit slowly, by logarithms, not by power-laws, as in the case of scale invariance broken by masses or condensates.
But Wilson further makes the then radical assumption that this theory will not be a pure S-matrix bootstrap-type string theory, but a scale invariant quantum field theory, with local operators at every spacetime point and multiplication laws for these fields. What makes this paper radical is that the field theory is not assumed to be a traditional weakly coupled field theory, rather a completely different kind of field theory which is always strongly coupled, a theory at a scale invariant RG limit. He renounces a traditional perturbative Lagrangian description and introduces the OPE for the fields, and makes an assumption on the scale dimension for the fields. He calls the scale invariant super-strong interaction the "Hadronic skeleton theory". It is defined as the hypothetical ultraviolet fixed point of the strong interactions.
This assumption is not correct for the strong interactions. In QCD, the theory at zero quark mass has its own intrinsic scale, $\Lambda_{QCD}$, which is determined by the renormalization group running of the QCD coupling, and is separate and independent of the quark masses, which just happen to have the same order of magnitude in nature by an unexplained coincidence (The quark masses come from the Higgs scale while the QCD scale is independent, but they match up to a few orders of magnitude).
The "hadronic skeleton theory" in modern QCD would just be the ultraviolet fixed point, which is the free theory of quarks and gluons. This is ultimately a perturbative theory. In other theories, like Banks-Zaks theories, where you introduce enough flavors of quarks and colors to make a weakly coupled scale invariant limit, you can have scale-invariant infrared limits, and then you can imagine a theory which is scale invariant forever, with only a little bit of breaking. But these are not QCD.
The theory Wilson is proposing and examining here is an entirely different possibility from the then mainstream bootstrap approach, and Wilson devotes the paper to showing that the operator products subsume and extend the current algebra approach, extending it to the case where the fields have arbitrary scale dimensions. This was a type of field theory which had not been considered in the 1950s, but it was clear that there was at least one example, because the 2 dimensional Thirring model provided one. Wilson would later give the canonical example, the Wilson Fisher fixed point in 3d, with the scaling dimensions the Ising model anomalous dimensions. Wilson had the courage to imagine and propose that such a thing is happening in 4 dimensions and in the strong interactions.
In historical reminiscences, Wilson describes his thinking about the OPE by considering the momentum-indexed variables field theory divided into concentric sectors of momentum $\lambda a^k <|k|<\lambda a^{k-1}$ for some $|a|<1$. Then he considered integrating out the momenta in consecutive shells, from outside in, producing a discrete version of the renormalization group flow. He realized that operators which were non-coincident before integrating out would have to become coinciding after integration. The history is discussed in reminiscences of Michael Peskin which may be found in this video and the associated historical arxiv paper: http://www.physics.cornell.edu/events-2/ken-wilson-symposium/ken-wilson-symposium-videos/michael-peskin-slac-ken-wilson-solving-the-strong-interactions/. This idea is a momentum space version of block-spinning, which was developed, I believe earlier and independently, by Kadanoff and Migdal in statistical physics. But Wilson was studying statistical physics at the time, and perhaps he was influenced by Kadanoff style block spinning. I don't know.
This idea of operator product expansions is as basic as Taylor series in calculus, or Ito calculus for Brownian walks. The main point is that the coefficients are only diverging locally, there are only finitely many singular operators in the product of any two operators, and the scaling behavior of the coefficients are determined by scaling laws for the fields, and you can do calculus on the operator products expansion and the local operators without fear of contradiction or paradox. Wilson introduces these main points correctly, and these main points are used in all subsequent work on the OPE.
The OPE is also important sociologically in rehabilitating field theory, as the confusions with quantum field theory were caused by people proving "theorems", like the Sutherland-Veltman theorem, and then others then showing by explicit calculation that the theorem is false. The failures of theorems intoduced a whole zoo of diseases--- Schwinger terms, anomalies, and so on. All of these are subsumed into the analysis of the singular terms in the OPE, and when you have a systematic calculus for these, you aren't going to be surprised anymore by false theorems. In principle, you can compute the OPE coefficients, define the composite operators appearing in the expansion, and check if your differential-algebraic identities still hold with the singular coefficients of the OPE. This makes it instrumental in demonstrating that quantum field theory actually was a well defined thing.
But because Wilson's physical strong interaction picture is off, the physical deductions regarding the specific family of models are incorrect, the material in section VII (applications), and the specific predictions regarding strong interaction behavior are mostly obsolete and can be disregarded. The analysis of the SU(2)xSU(2) symmetry breaking is recapitulating current algebra, in a context where Wilson doesn't know the dimensions of the pion field anymore, as he is imagining it is something other than a normal weakly interacting field theory. In real life QCD, the pion field can be taken to be the divergence of the quark axial current $\partial_\mu \psi^i \gamma^5 \gamma^\mu \psi^i$ , and the dimension at short distances is exactly four by free-field dimensional analysis. The paper assumes that the rho will become massless in a hadronic skeleton theory, as all scales collapse. This is not what happens in zero quark mass QCD, where the rho mass doesn't vanish, as the chiral condensate doesn't go away. Neither does the Baryon mass vanish, as it is determined by $\Lambda_{QCD}$
In section VIIA, Wilson discusses some sum rules of Weinberg. I didn't review this yet.
In section VIIB, To be reviewed.
In section VII C,D: The paper notes the existence of an axial anomaly, and attempts to apply the OPE to calculate it. It formulates the calculation in OPE language, but doesn't carry it out. It is certainly possible to do in a specific field theory model with a broken axial current, but Wilson finds it sufficient to pinpoint the exact place where the Veltman-Sutherland theorem fails in differentiating the product of the current and photon field without differentiating the singular coefficients to find the extra finite residue. This calculation I think is important, because the OPE makes all the identities of quantum field theory error-free and systematic.
In section VII E, Wilson proposes ideas for weak interactions, which were known to be coupled to the same axial currents which are spontaneously broken in the strong interaction. This was what allowed the PCAC relation between neutron decay (a weak process) and the pion-nucleon coupling constant (strong process). The pion is a goldstone boson of the same current which is involved in the weak interaction. Wilson proposes that the different scaling laws for the different fields in the skeleton theory are responsible for the enhancement or decay of certain hadronic weak processes over others. Since the skeleton theory is not interacting, this is not what is going on.
Q&A (4869)
