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In Matt Strassler's recent post (here) he makes the statement that scale invariant (I assume he means conformally invariant, more generally) theories have no particles in them. What's the reason for this? What are the technical and heuristic explanations, apart from what he mentions in his post?

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    $\begingroup$ +1 for the question. I wonder if it is because, in a particle interacting theory (with necessary massless particles ), we need to have (in $3+1$ dimensions) propagators$\langle \psi(x) \psi(y)\rangle \sim \frac{1}{(x-y)^2}$, but in conformal theory, we may have an other dimension, that is $\langle \psi(x) \psi(y)\rangle \sim \frac{1}{(x-y)^\Delta}$ $\endgroup$
    – Trimok
    Sep 25 '13 at 19:17
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    $\begingroup$ Echoing Trimok, as there are no scales in a CFT, there can be no massive particles in CFT. The existence of massless quasi-particles in CFT is more subtle, see e.g. arxiv.org/abs/cond-mat/9706166. $\endgroup$
    – Qmechanic
    Sep 25 '13 at 22:43
  • $\begingroup$ See also this answer by @Vibert. $\endgroup$
    – Trimok
    Sep 27 '13 at 9:26
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    $\begingroup$ Re massless particles, he says "but in a scale-invariant quantum field theory with at least one force, any such ripples die away and turn into several ripples" presumably that's the get-out clause for free Maxwell theory to be conformally invariant and have particles? $\endgroup$
    – twistor59
    Sep 29 '13 at 16:27
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    $\begingroup$ I think he was just talking about interacting theories, which excludes the trivial free cases. $\endgroup$
    – user26866
    Sep 29 '13 at 20:02
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The precise statement should be that massless fields in conformal field theories in 3+1 dimensions are necessarily free. This result was first proved by Buchholz and Fredenhagen.

There are two modern proofs of this fact, one by Steven Weinberg (please see arXiv: hep-th/1210.3864v1) and the other by Yoh Tanimoto in the framework of algebraic quantum field theory. (Weinberg's proof is a generalization of an unpublished argument by Witten for the spin 0 fields). Weinberg proves that the massless fields neceaarily satisfy free field equations.

The freeness of the massless fields should not be confused with the freeness of the whole theory, as conformal field theories contain massive fields also, where there is no known freeness restriction on the massive fields.

The conformal group in 3+1 dimensions is $SU(2,2)/\mathbb{Z}_4$. Its positive energy representations have been classified by Mack in: All unitary ray representations of the conformal group SU(2,2) with positive energy.

These representations are parameterized by two $SL_2$ quantum numbers $(j_1, j_2)$ and when restricted to the Poincaré subgroup, they reducibly or irreducibly decompose into representations of mass $m$ and spin $s$.The massless representations are the two families $(j, 0)$ or $(0, j)$ which reduce to massless Poincaré multiplets.

It should be emphasized that the existence of a mass parameter does not contradict conformal invariance, because the dilation parameter can shrink the energy to zero and these states are also gapless.

As mentioned above, there is no known restriction on the freeness of the massive states, and there is a construction by Odzijewicz of a massive conformal particle and an attempt to describe its interaction with an external field. The mass of such a particle is not a constant of motion and can change as a consequence of the interaction.

Odzijewicz works in the single particle level and uses the Orbit Method to describe the dynamics of the massive conformal particles. Weinberg and the others work on the field level for the same irreducible representations of the conformal group. It would be interesting to see a unified treatment of the two approaches.

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One nice way of thinking about this is that the correlators in CFTs don't have simple poles but $\sim 1/p^{2\Delta}$ type power-law singularities, so if we Fourier transform this we don't get an exponentially localized wave-packet like we do with the simple poles in the familiar two-point functions in 3+1d QFT. Therefore the elementary excitations of a field with this two-point function, like in CFTs, are not what we usually like to think of as particles.

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  • $\begingroup$ What do you mean regarding exponentially localized wave packets? For a massless scalar, the propagator just turns into ~1/(x-y)^# for some # depending on dimensions $\endgroup$
    – user26866
    Jul 17 '20 at 15:04

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