Dispersion Formulas in QFTs, CFTs, and Holography
  We study momentum space dispersion formulas in general QFTs and their
applications for CFT correlation functions. We show, using two independent
methods, that QFT dispersion formulas can be written in terms of causal
commutators. The first derivation uses analyticity properties of retarded
correlators in momentum space. The second derivation uses the largest time
equation and the defining properties of the time-ordered product. At four
points we show that the momentum space QFT dispersion formula depends on the
same causal double-commutators as the CFT dispersion formula. At $n$-points,
the QFT dispersion formula depends on a sum of nested advanced commutators. For
CFT four-point functions, we show that the momentum space dispersion formula is
equivalent to the CFT dispersion formula, up to possible semi-local terms. We
also show that the Polyakov-Regge expansions associated to the momentum space
and CFT dispersion formulas are related by a Fourier transform. In the process,
we prove that the momentum space conformal blocks of the causal
double-commutator are equal to cut Witten diagrams. Finally, by combining the
momentum space dispersion formulas with the AdS Cutkosky rules, we find a
complete, bulk unitarity method for AdS/CFT correlators in momentum space.
