TT deformed scattering happens within matrices

We show the TT deformation of two-dimensional quantum field theories is equivalent to replacing the spacetime dependence of the fields with dependence on the indices of infinitely large matrices. We show how this correspondence explains the CDD phase dressing of the S -matrix and the general formula for the deformation of arbitrary correlation functions. We also describe how the Moyal deformation of self-dual gravity is a TT deformation of the theory described by the Chalmers-Siegel action, where the TT deformation is defined on the two-dimensional plane of interactions.