  We consider manifestly crossing symmetric dispersion relations for Mellin
amplitudes of scalar four point correlators in conformal field theories (CFTs).
This allows us to set up the non-perturbative Polyakov bootstrap for CFTs in
Mellin space on a firm foundation, thereby fixing the contact term ambiguities
in the crossing symmetric blocks. Our new approach employs certain "locality"
constraints replacing the requirement of crossing symmetry in the usual
fixed-$t$ dispersion relation. Using these constraints we show that the sum
rules based on the two channel dispersion relations and the present dispersion
relations are identical. Our framework allows us to connect with the
conceptually rich picture of the Polyakov blocks being Witten diagrams in
anti-de Sitter (AdS) space. We also give two sided bounds for Wilson
coefficients for effective field theories in AdS space.
