If I have understood the subject correctly, Wilson correctly explained the second order phase transitions and was able to compute the critical exponents of several of them using the Renormalization Group.

However, since critical phenomena are described by Conformal Field Theories and the Standard Model of Particle Physics is not I wonder why is Wilson's work so important in our current understanding of the Standard Model.


2 Answers 2


Some history: In the pre-Wilsonian interpretation of quantum field theory, renormalisability of theories was considered an essential requirement. That is, you should be able to remove all ultraviolet divergences of the theory (that occur in perturbation theory) by absorbing them in a finite number of parameters in the lagrangian. If you could not do that, then the theory was called "non-renormalisable" and considered pathological. The renormalisability criteria placed strong constraints of the theory, determining which finite number of interactions were allowed. The standard model is renormalisable.

One effect of the renormalization process was that coupling constants became scale (energy) dependent, and such running coupling constants were understood to be physical. eg asymptotic freedom is the result of the QCD coupling constant becoming weak at large energies.

Wilson's goal and approach in condensed matter was different. He was interested in studying the behaviour of complicated theories near the second order phase transition where the correlation length becomes very large, that is when the theory is essentially scale invariant. That means that you could study the system via an effective theory near the phase transition point without worrying about the underlying microscopic degrees of freedom. In this approach you write down a simple field theory with the desired symmetries and you use a cut-off $\Lambda$ in all calculations since your theory is only an approximation at low energies $E \ll \Lambda$ (long distance scales).

So in the Wilsonian approach there are no ultraviolet divergences. But it also meant that you had to include many more interactions in your lagrangian. The coupling constants also became $\Lambda$ dependent and you had a running of the couplings here too. In practice, in the limit $E/\Lambda \to 0$, only a finite number of couplings are important: the ones that are "marginal" and "relevant". The ones that are "irrelevant" are the ones that a particle physicist would have labelled as "non-renormalisable".

So it was initially a different philosophy and approach in high-energy physics and condensed matter, but gradually high-energy physicists understood that the Wilsonian perspective of effective field theories is applicable to their studies of quantum field theory.

In the modern perspective, the standard model is believed to be a low-energy approximation of some as yet unknown theory. Although the current model is renormalisable, it is believed that "non-renormaliable = irrelevant" interactions would become important at higher energies, and they would represent new processes. So you can write down what the next possible interaction could be (eg to give neutrinos small masses) and make some predictions.

Such an effective field theory approach is also useful when you want to make low energy predictions from your high energy theories.

So, in summary, the Wilsonian perspective of quantum field theories and the renormalisation group is important because it gives them a very physical and intuitive meaning.

Now, some comments about "conformal field theories". A bigger symmetry than just scale invariance is "conformal invariance". In two dimensional space conformal invariance results in an infinite dimensional symmetry group which places strong constraints on a theory and allows many exact results to be obtained in condensed matter systems. It is also important in string theory as the world sheet is two dimensional. Conformal symmetry is less powerful in higher dimensions as the symmetry group is finite and the symmetry is typically broken by quantum effects through the generation of mass scales.

  • $\begingroup$ I get what you mean but there's an issues that boders me: Wilson's Renormalization Group was developped to deal with a special class of Field Theories, those that describe 2nd order Ph. Ts. Most of these theories are conformal invariant at their critical points. Was he not addressing a very restricted set of (Q)FTs? Can you extrapolate anything to a set of much less restricted QFTs (those whose symmetry group is just Poincare's group)? Moreover, in Therm. Ph. Trs. you need to adjust just two relevant couplings to be at the critical surface (line). Is this not, again, too restrictive? $\endgroup$ Commented Apr 24, 2018 at 7:33
  • $\begingroup$ You are confusing Wilson's general techniques and philosophy -- using a cutoff, keeping many interaction terms, and the RG with the specific applications he originally used it for. As I explained above the general techniques and philosophy have nothing to do with critical point, or the conformal group or Poincare group etc. They are applicable to all QFT, and give a physically pleasing interpretation to renormalisation etc, and that is why that outlook is now adopted widely. In condensed matter, near a 2nd order PT, scale invariance simply means you can use an effective field to ...(tbc) $\endgroup$
    – rparwani
    Commented Apr 24, 2018 at 13:44
  • $\begingroup$ (contd) .. to describe systems that are actually not field theories away from the second order critical point (eg discrete magnetic spin systems). Such field theory ideas for condensed matter were initiated by Landau and Ginsbarg and Wilson pushed them further and showed how to calculate critical coefficients more accurately. To summarise: Wilson was using very general QFT techniques to study specific systems in condensed matter, not the other way around. Its just that his interpretation and perspective was not widely known to high energy physicists, but now it is. $\endgroup$
    – rparwani
    Commented Apr 24, 2018 at 13:50
  • $\begingroup$ If I understood you correctly, Wilson's method would consist in progressively integrating thin momentum shells starting from the natural cutoff scale to see what the infrarred part of the QFT would look like. This would cause new interaction terms to show up in the theory but that would not matter because their coupling constants tend to zero as you get near the IR end. Finally, you assume that the first few terms that first showed up could be important in an UV extension of the theory precisely because they were created by integrating momentum shells at the original UV end. Is that it? $\endgroup$ Commented Apr 24, 2018 at 16:47
  • $\begingroup$ That is the detailed implementation in principle, but it is rarely implemented like that in QFT because its tedious or impractical. In practice I believe people define the low energy effective theory by just writing down all possible terms allowed by symmetries, put a cutoff, and calculate in some perturbative scheme, eg epsilon expansion, coupling expansion or large N etc., . Then you look for fixed points etc of the beta function if you are doing condensed matter, or calculate other stuff that interests you more in high energy physics. That is the general Wilson RG philosophy I believe. $\endgroup$
    – rparwani
    Commented Apr 24, 2018 at 17:27

I'm not sure if this will directly answer your question, but perhaps it will be helpful. I will simply quote a few sections from the beginning of Conformal Field Theory by Phillipe Francesco, picking only those which relate to high energy particle physics (CFT's are very important in understanding quantum critical points in condensed matter systems).

Scattering experiments failed to detect a characteristic length scale when probing the proton deeply with inelastically scattered electrons. This supported the idea that the proton is a composite object made of point-like constituents, the quarks. . .

In other words, in this deep-inelastic range, the internal dynamics of the proton does not provide its own length scale $\ell$ that could justify a separate dependence of the structure functions on the dimensionless variables $\ell^2\nu$ and $\ell^2 q^2$. In the context of quantum chromodynamics (QCD, the modem theory of strong interactions), this reflects the asymptotic freedom of the theory, namely, the quasi-free character of the quarks when probed at very small length scales.

Of course, the quark-gluon system underlying the scaling phenomena of deep inelastic scattering is thoroughly quantum-mechanical, just like systems undergoing quantum-critical phenomena. However, scale invariance manifests itself at short distances in QCD, whereas it emerges at long distances in quantum systems like the Heisenberg spin chain.

In regards to String Theory:

The first-quantized formulation of string theory involves fields (representing the physical shape of the string) that reside on the world-sheet. From the point of view of field theory, this constitutes a two-dimensional system, endowed with reparametrization invariance on the world-sheet, meaning that the precise coordinate system used on the world-sheet has no physical consequence. . . This reparametrization invariance is tantamount to conformal invariance. Conformal invariance of the world-sheet theory is essential for preventing the appearance of ghosts (states leading to negative probabilities in quantum mechanics). The various string models that have been elaborated basically differ in the specific content of this conformally invariant two-dimensional field theory (including boundary conditions). A classification of conformally invariant theories in two dimensions gives a perspective on the variety of consistent first-quantized string theories that can be constructed . . .

string scattering amplitudes were expressed in terms of correlation functions of a conformal field theory defined on the plane (tree amplitudes), on the torus (one-loop amplitudes), or on some higher-genus Riemann surface.

And further regarding the operator product expansion (OPE) and conformal bootstrap:

The modem study of conformal invariance in two dimensions was initiated by Belavin, Polyakov, and Zamolodchikov, in their fundamental 1984 paper. These authors combined the representation theory of the Virasoro algebra . . with the idea of an algebra of local operators and showed how to construct completely solvable conformal theories: the so-called minimal models. An intense activity at the border of mathematical physics and statistical mechanics followed this initial envoi and the minimal models were identified with various two-dimensional statistical systems at their critical point. More solvable models were found by including additional symmetries or extensions of conformal symmetry in the construction of conformal theories.

A striking feature of the work of Belavin, Polyakov, and Zamolodchikov . . . regarding conformal theories is the minor role played (if at all) by the Lagrangian or Hamiltonian formalism. Rather, the dynamical principle invoked in these studies is the associativity of the operator algebra, also known as the bootstrap hypothesis. . . The key ingredient of this approach is the assumption that the product of local quantum operators can always be expressed as a linear combination of well-defined local operators . . . This is the operator product expansion, initially put forward by Wilson . . . The dynamical principle of the bootstrap approach is the associativity of this algebra. In practice, a successful application of the bootstrap approach is hopeless, unless the number of local fields is finite. This is precisely the case in minimal conformal field theories. . .

Following the pioneering work of Belavin, Polyakov, and Zamolodchikov, conformal field theory has rapidly developed along many directions. The work of Zamolodchikov has strongly influenced many of these developments: conformal field theories with Lie algebra symmetry (with Knizhnik), theories with higher- spin fields—the W-algebras—or with fractional statistics—parafermions (with Fateev), vicinity of the critical point, etc. These developments, and their offspring, still constitute active fields of research today and make conformal field theory one of the most active areas of research in mathematical physics.

  • $\begingroup$ I'm sorry but I don't get how a scale invariant theory can be useful at explaining interactions among massive particles, such as quarks and leptons. However there must be some kind of connection, since Wilson is consistently referenced in SM books, such as Peskins and Schröder. $\endgroup$ Commented Apr 22, 2018 at 10:47
  • $\begingroup$ A Wilsonian QFT lies on a path connecting two CFTs... $\endgroup$ Commented Apr 22, 2018 at 13:42
  • $\begingroup$ Would you be so kind as to elaborate on that idea a little more? As far as I know, a CFT is related to a renormalization fixed point and QFTs are not. Non conformal QFTs do not have a fixed point where to flow and, therefore, are not scale invariant. $\endgroup$ Commented Apr 22, 2018 at 16:25
  • $\begingroup$ I'm only a humble student but in my understanding a massless theory has no lengthscale and is therefore conformally invariant. At very high energy scattering particles are approximately massless and therefore can be described by CFT. $\endgroup$
    – Kai
    Commented Apr 22, 2018 at 19:09
  • $\begingroup$ I'm not a physicist and don't know much about the subject. What I understand by CFT is a (Q)FT whose Lagrangian-Hamiltonian coupling constants doesn't depend on the energy scale. If this happens, I think that the Poincaré symmetry group can then be extended to the Conformal symmetry group. To the best of my knowledge, the SM is not scale invariant (the QED coupling constant becomes infinite at high energy scales, for example). $\endgroup$ Commented Apr 23, 2018 at 7:18

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