Unitarity representations of CFT in arbitrary dimensions

Question

There is a well defined notion of unitarity of representations in Euclidean Conformal field theories that follows from the requiring unitarity in the Lorentzian space. Under this notion, all states that are created by operator insertions have positive inner product, $$\langle \mathcal O|\mathcal O\rangle = \lim_{z\to\infty} z^{2\Delta} \langle\mathcal O(z)\mathcal O(0)\rangle>0$$ where I am thinking of radial quantization, or equivalently, of a field theory on a cylinder. This also goes by the name of reflection positivity. In arbitrary dimensions this requirement imposes constraints on the dimensions of physical operators:

• $\Delta\ge\frac{d-2}2$ for non-spinning bosonic operators
• $\Delta\ge\frac{d-1}2$ for non-spinning fermionic operators
• $\Delta\ge d+\ell-2$ for operators with spin $\ell$

However, in the study of Harmonic analysis on the Euclidean Conformal groups ($SO(d+1,1)$ for a field theory in d-dimensional Euclidean space), one talks about unitary representations of the conformal group with dimensions $\Delta = \frac d2 + i s, \ s\in\mathbb R$ (the principal series) and, in odd dimensions, additionally $\Delta = \frac d2+\mathbb Z_+$ (the discrete series). Clearly these operators are excluded from the class of 'unitary' operators that follow from the Lorentzian unitarity requirement. From what I understand these class of representations provide a basis of $\mathbb L_2$ normalizable functions on the group manifold.

What is the difference between the two different notions of unitarity and how are they related? Moreover, clearly the operators in the first definition are normalizable in that they have a finite positive inner product ($\langle \mathcal O|\mathcal O\rangle >0$). In what sense are they not normalizable on the group manifold (thereby being excluded from Harmonic analysis basis of functions)?

This post imported from StackExchange Physics at 2019-05-05 13:08 (UTC), posted by SE-user nGlacTOwnS

Comments

I have accepted the answer provided at the Physics Stack Exchange at this link: https://physics.stackexchange.com/a/478127/12225