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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?

Qmechanic
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Morrissey87
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2 Answers2

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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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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.

Old Fart
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