A large class of QFTs called 2 dimensional conformal field theories, can be defined without an action.

The general method for constructing an any dimensional CFT is as follows, you can calculate the action of the symmetry generators of the fields using Ward identities and their commutation relations given by the Virasoro algebra which allows you to construct the whole hilbert space from the knowledge of how your set of "primary" fields transform under conformal transformations which is encoded in the conformal dimension. Why you can do this in CFTs but not in any QFTs is the state-operator correspondence. Each state of the Hilbert space is in 1-1 correspondence with the set of field operators. There is a subset of these called minimal models which are defined through a finite set of primary fields. One can even give them a Lagrangian formulation through the Couloumb gas formalism which consists of coupling it with the Ricci scalar.

What distinguished a d-dimensional CFT to a CFT in 2 dimensions is that the conformal group of a 2D CFT is infinite dimensional. Any locally holomorphic map corresponds to a local conformal transformation. This allows you to use Laurent expansions and other tools of complex analysis.

For "What is a QFT"? This page from nLab maybe useful. It somehow "axiomatizes the assignment of algebras of observables to patches of parameter space".

For further info. on CFT refer to di Francesco et Al's Conformal field theory.

This post imported from StackExchange Physics at 2014-08-29 16:43 (UCT), posted by SE-user ramanujan_dirac