# Non-Lagrangian Models of Current Algebra

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Referee this paper: Phys. Rev. 179 (1969), 1499 by Kenneth G. Wilson

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author: K.G. Wilson

Original Abstract: An alternative is proposed to specific Lagrangian models of current algebra. In this alternative there are no explicit canonical fields, and operator products at the same point [say,$j_\mu(x)j_\mu(x)$] have no meaning. Instead, it is assumed that scale invariance is a broken symmetry of strong interactions, as proposed by Kastrup and Mack. Also, a generalization of equal-time commutators is assumed: Operator products at short distances have expansions involving local fields multiplying singular functions. It is assumed that the dominant fields are the $SU(3)×SU(3)$ currents and the $SU(3)×SU(3)$ multiplet containing the pion field. It is assumed that the pion field scales like a field of dimension $\Delta$, where $\Delta$ is unspecified within the range $1≤\Delta<4$; the value of $\Delta$ is a consequence of renormalization. These hypotheses imply several qualitative predictions: The second Weinberg sum rule does not hold for the difference of the $K^*$ and axial-$K^*$ propagators, even for exact $SU(2)×SU(2)$; electromagnetic corrections require one subtraction proportional to the $I=1$, $I_z=0$$\sigma$ field; $η→3π$ and $π_0→2γ$ are allowed by current algebra. Octet dominance of nonleptonic weak processes can be understood, and a new form of superconvergence relation is deduced as a consequence. A generalization of the Bjorken limit is proposed.

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paper authored to hep-th
retagged Apr 27, 2015

## 2 Reviews

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This paper is heavily cited because it introduced the operator-product expansion (OPE) to the study of quantum fields. This was done a few years before quantum field theory was reestablished - with the acceptance of the standard model - as the fundamental theory of Nature.

Although almost 50 years old, I review the paper because it is still very worthwhile to read this paper today.

The paper gives an introduction to the OPE that is almost fully complementary to the treatment in modern books of the subject, such as Chapter 20 of Volume 2 of The Quantum Theory of Fields by S. Weinberg or Chapter 18 of An Introduction to Quantum Field Theory by M.E. Peskin and D.V. Schroeder.

The latter are unnecessarily abstract, while Wilson's approach is most concrete and gives clear intuition about the meaning and use of the OPE in terms of fields, currents, and strong interactions. This is done without mixing it with the canonical formalism or the path integral, which are essential for renormalized perturbation theory but are clouded by technical complexities and by unsolved mathematical riddles regarding their very definition. It may well be that when we'll have one day a rigorous nonperturbative relativistic quantum field theory in 4 dimensions that Wilson's part will be the part that survived with least alterations.

Wilson starts off with observable currents $j_\mu(x)$ and other local quantum fields $\phi(x)$, which are operator-vector-valued distributions on space-time. It is well-known that the product of two such fields at the same point $x$ is meaningless. Traditionally, these currents were discussed in terms of equal-time
commutation relations (''current algebra''), but these relations may suffer from divergences and in any case do not give enough information about the singular behavior of the distributions.

Thus Wilson proposed to replace the equal-time commutation relations by properties of the product $A(x)B(y)$ of two local fields in the limit where the arguments $x$ and $y$ get arbitrarily close. In the case of free fields and their Wick products, one can use Wick's theorem to evaluate these products, and finds that they can be written as finite linear combinations of local fields (Wick products and their derivatives) in $x$, with distributional coefficients depending on $y-x$ only. One can recover equal-time commutators by evaluating $[A(x),B(y)]$ and taking the weak limit $y\to x$. Thus the formulas for the product contain all information about the commutators, and in fact much more.

For an interacting theory, one can show to all orders of perturbation theory that the product $A(x)B(y)$ can still be written as linear combinations of local renormalized fields in $x$ (''generalized Wick products''), though the number of terms grows with the order of perturbation theory. Thus Wilson proposed that in general, and independent of perturbation theory, for any two local fields $A$ and $B$, there is an expansion of $A(x)B(y)$ as an infinite series in local fields in $x$, with distributional coefficients depending on $y-x$ only. This is the OPE of these fields.

Since understanding of the strong interaction was at the time restricted to very high energy (short distance) experiments, Wilson continues to explore the short distance limit where all masses can be neglected and one gets a scale-invariant theory. (The latter theories are often conformal field theories, and indeed the OPE is the main tool in the analysis of conformal field theories.) In this case, the form of the OPE coefficients is much restricted, and the singularities are characterized by one number per field, its scaling dimension. The traditional field dimensions for power counting arguments apply to free fields only, except for currents and the energy-momentum tensor, where conservation laws enforce a scaling dimension of 3 and 4, respectively. All other fields may have anomalous dimensions differing from the nominal power-counting dimension. These anomalous dimensions are due to field  renormalization and already appear in the simplest nontrivial field theories, such as the exactly solvable Thirring model in 1+1 dimension.

Finally, the influence of the masses is restored by perturbation theory around the renormalized scale-invariant theory. Applications are given among others to sum rules, $\eta\to\pi^+\pi^0\pi^-$ decay, and $\pi^0\to\gamma\gamma$ decay, with very little additional theory.

A truly remarkable paper.

[added April 4:] The above paper was preceded by an influential unpublished preprint by Wilson, On products of quantum field operators at short distances (Cornell preprint 1964); footnote 6 in his paper on the OPE for the Thirring model explains why he didn't publish the preprint.

As a result, a number of other papers discussed the OPE before Wilson's 1969 publication appeared in print. In particular, a paper by Brandt and a paper by Zimmermann, both from 1967, discuss the relationship between the OPE, renormalized field equations, and conventional renormalized perturbation theory for Yukawa theory and for $\phi^4$ theory (both in 4 dimensions). Another paper by Brandt from 1969 (see also the sequel from 1970) does the same for QED.

reviewed Mar 31, 2015 by (12,570 points)
edited Apr 6, 2015

Thanks for the nice review, I should definitely read this paper sometime. This reminds me of a confusion I've had for a while: does the fact that operator product at the same spacetime point is ill defined have anything to do with ultraviolet divergences? On the first look I'm tempted to say yes since it involves short distance, but then I can immediately think of conterexamples: (1)The product is ill defined even for free theories.(2)Even in an interacting theory most of the ultraviolet divergences arise from loop diagrams which do not necessarily involve equal spacetime operator product. If you think this short question needs to be answered in a new post I'll do so.

@VladimirKalitvianski, in calculations of say amplitudes you are right, but there's nothing preventing us thinking about local products.

@JiaYiyang As you say, the product is singular even for free fields. This is one (not the only)  reason for the infinities in naive perturbation theory. In 2 dimensions it is the only reason for this, therefore normal ordering is sufficient to renormalize a 2D relativistic field theory. In dimensions 3 and 4, there are other sources of divergences, and renormalization takes the more complex form discussed in all QFT textbooks.

@JiaYiyang: This is all related somehow, but at present I don't understand the connections well enough to write something clear and sensible about it. The currently best understanding is through causal perturbation theory (see the two books by Scharf).

The only classification of UV divergences currently available is according to power counting into ordering divergences, super-renormalizable, renormalizable and non-renormalizable divergences. In the super-renormalizable case no field renormalization is needed, which makes it sometimes tractable by rigorous constructive methods. In 4D, scalar electrodynamics is super-renormalizable, whence it might well be the first interacting relativistic QFT in 4D to be constructed rigorously; but due to not fully understood IR problems, even this so far defied a constructive approach.

Anyway, the OPE looks like the right tool for studying the divergences more systematically.

@JiaYiyang: This paper uses counterterm techniques to obtain OPEs. It is not quite what you asked for, but it shows some of how things are related.

Underlying all singular stuff in QFT and its correct treatment are the microlocal conditions governing products of distributions with singularity on the mass shell (or the light cone in the scaling limit). Its simplest manifestation are the subtractions needed in some dispersion relations; on this level, everything is still governed by clean mathematics. Thus a thorough understanding should start there.

Thanks for the reference!

Underlying all singular stuff in QFT and its correct treatment are the microlocal conditions governing products of distributions with singularity on the mass shell (or the light cone in the scaling limit). Its simplest manifestation are the subtractions needed in some dispersion relations; on this level, everything is still governed by clean mathematics. Thus a thorough understanding should start there.

It is a too narrow understanding. But I have already lost my faith in you. Do not forget one can always transform a wrong expression into the right one if one knows the right one. The pretext is a solid one: the theory must describe the reality, so this mathematical expression must be equal to that one.

+ 4 like - 0 dislike

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.

reviewed Apr 23, 2015 by (7,535 points)
edited Apr 27, 2015

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