I think the answer is that such a construction is in general *impossible*, for two reasons:

(1) The two-point function (or functions, if the field multiplet is not a singlet) says little by itself about the higher-order correlation functions of the theory. It does fully encode the theory if the latter is free (see (2) below).

(2) A two point function need not come from a quantum field theory given by a Lagrangian on the *same* space-time the fields live on. For instance, a conformally covariant scalar two-point function in Minkowski space-time with non-canonical scaling degree yields a well-defined, free field theory if we set the higher-order truncated correlation functions to zero. This quantum field theory has a dynamics which cannot be given by any Lagrangian in Minkowski space-time.

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro