# Massive excitations in Conformal Quantum Field Theory

+ 4 like - 0 dislike
1419 views

Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of quantum states into irreducible representations of the Poincare group). Classification of irreducible unitary representations of the Poincare group leads to the notions of mass and spin.

Now, suppose we have a conformal QFT and are doing the same trick with the conformal group. Which irreducible representations do we have?

We still have the massless particles (at least I'm pretty sure we do although I don't immediately see the action of special conformal transformations). However, all representations for a given spin s and any mass m > 0 combine into a single irreducible representation.

What sort of physical object corresponds to this representation? Is it possible to construct a scattering theory for such objects? Is it possible to define unstable objects of this sort?

This post has been migrated from (A51.SE)
Very very naive question: you say there will be (irreducible) representations with a fixed spin $s$ and any mass $m>0$. Since any mass $m$ introduce a length scale $L\sim \frac 1m$, conformal transformations would transform states of different masses into each other. So you would need a theory of uncountable number of particles with any mass $m>0$? If this is correct, doesn't it (naively) seem to be quit hopeless to construct any consistent quantum field theory of this kind? Has such a theory ever been constructed?

This post has been migrated from (A51.SE)
@Heidar, these states would not be particles. This is because the mass spectrum within each such representation is continuous.

This post has been migrated from (A51.SE)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.