How are shadows and projector related?

While computing the conformal partial waves, it seems to me that

$$\int d^dx |O\rangle\langle\tilde O| = \mathcal{N}^{-1}\sum_{n}|P^n O\rangle\langle P^n O|$$ where $$\tilde O$$ is the shadow dual of $$O$$, and the RHS is the projector. Can anyone provide a proof of the above statement ?

• You should probably add some references to put your problem in context, e.g. arxiv.org/abs/1204.3894. For instance it is not clear what P is on the RHS. Is it related to the generator of translations? – M.Jo Apr 23 at 14:54

No, because this statement is wrong. You cannot make sense of the left hand side as an operator on a Hilbert space, since $$|O(x)\rangle$$ is not a normalizable state in radial quantization if $$x$$ is not inside unit sphere. Furthermore, $$|\tilde O(x)\rangle$$ cannot be defined as a state either, for the same reason.
• @PeterKravchuck What is the difference between a conformal block and a conformal partial wave ? Also, if you look at David Simons' TASI review on bootstrap, he uses the same notation for the projector (which is $|\mathcal{O}|$) and which he uses for the shadow representation. Even in the paper you cited, he says in section 2.6 LHS can be used to compute conformal blocks as well. – user44690 Apr 24 at 4:51