  We study loop amplitudes in anti de-Sitter space via the spectral
representation. We consider loops of spinning fields and in particular gauge
fields, and derive various identities connecting different families of loop
diagrams, at different number of loops, different spins, different masses. Such
identities are useful for the computation of Witten diagrams. Considering the
theory of large-$N_f$ conformal scalar QED defined on AdS space, we derive an
analytic expression for the exact 4-point correlation function at sub-leading
order in $\frac{1}{N_f}$. Additionally, we derive analytic expressions for bulk
2-point functions and boundary 4-point functions for various families of
diagrams, which we denote as ``blob diagrams''. Finally we study 4-point ladder
diagrams with spinning fields, and we derive integral expressions for the
spectral representation of a $k$-loop ladder diagram.
