The Coleman-Mandula theorem states that under certain seemingly-mild assumptions on the properties of the S-matrix (roughly: one particle states are left invariant and the amplitudes are analytic in external momenta) the largest possible Lie algebra of symmetries of a (non-trivial) S-matrix is given by Poincaré times an internal symmetry.

On the other hand, there are (interacting) field theories whose Lagrangians are symmetric under the conformal extension of the Poincaré group, and in some rare case this property is retained even at the quantum level.

Why (interacting) conformal invariant QFTs do not contradict the theorem? Is it possible to define an S matrix in these theories? I have read somewhere that they do not admit a particle interpretation, what does it mean exactly?


2 Answers 2


You sort of answered the question yourself. In CFT there is no Notion of "particles" - individual lumps of energy which exists independently of each other when sufficiently far away from each other. Other ways to say the same thing - The Hilbert space of the theory does not organize naturally into a Fock space, or there is no cluster decomposition. All of this follows from lack of scale in the theory. As a result LSZ reduction which depends on contraction of appropriate asymptotic states does not work (if you try to force it you find IR divergences which cannot be resumed). So, as you say the S matrix does not exist, which is the loophole to the Coleman Mandula theorem.

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    $\begingroup$ good points, i'm sure everyone know this by now, but it is good to notice that all IR divergences are effectively removed if the space-time is compact, so these theories should be still interesting in such spaces $\endgroup$
    – lurscher
    Apr 8, 2011 at 3:52
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    $\begingroup$ Sure, if you put any IR cutoff in the form of finite space, the theory is no longer conformal and there are IR finite observables. Still, these are not the S-matrix because there is no longer "infinity". Other type of IR cutoff, like finite masses, will makes the S-matrix well-defined but then the theory is no longer conformal, and correspondingly the Coleman-Mandula theorem does hold. $\endgroup$
    – user566
    Apr 8, 2011 at 3:59
  • $\begingroup$ sorry, what i said made no sense at all. I think i've never get a real sense of how much are we constrained to accept super Lie algebras as the only way to relax the conditions of this highly constraining theorem. Is it possible to get around by doing mild perturbations to the Poincare algebra? (like for instance, commutators for space-time generators have small-order corrections in other inner space generators and viceversa), what other alternatives are there? are they completely exhausted? $\endgroup$
    – lurscher
    Apr 8, 2011 at 15:58
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    $\begingroup$ @Lurscher: CM theorem is part of an intense period when people tried to extend the idea of grand unification to include spacetime symmetries. They all failed, which motivated CM to give a general result. There are other loopholes except for CFT, some already noted in the CM paper itself (for example two-dimensional QFT). Any no-go theorem might have some further loopholes that violate the assumptions in some subtle way, but at this stage the burden of proof is on whichever person trying to revive this old idea of mixing internal and spacetime symmetries (I hear there might be one or two..). $\endgroup$
    – user566
    Apr 8, 2011 at 16:18
  • $\begingroup$ Thank you for the answer. After posting the question I realized that in the hypothesis there is also the requirement of having a mass gap in the theory and CFTs are clearly excluded. If I understand well what you are saying, also the LSZ construction relies on mass gap, otherwise one would get IR divergences. If this is the case how is it possible to define the scattering in theories with massless particles? What about gauge theories? $\endgroup$ Apr 12, 2011 at 20:08

Haag-Lopuszkanski-Sohnius in their 1975 paper where they discuss about supersymmetry explained how to extend the Coleman-Mandula theorem in the case the theory has a spectrum that do not contain any massive excitations. Their result is that the Poincaré group can be extended to the corresponding conformal group. This is also carefully reviewed in Weinberg III, ch. 24, app B.


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