If you have the representation, every operator A_0 defines a field satisfying 2) by mean of the construction given in my previous mail. Thus 2) in itself is an empty requirement.I am very interested in this question rised by Eugene. I would like to reformulate it in a more general form as follows. Basically standard QFT is based on the following two assumptions:
1) A representation of the Poincaré group is defined on the Hilbert space of a relativistic quantum system;
2) All the operators of the Hilbert space are generated by fields, i.e., operator valued distributions defined on Minkowski space-time and transforming covariantly under the above representation.
For example, these two assumptions can be easily recognized in Wightman's axiomatic formulation of QFT. Of course 1 and 2 are different and independent assumptions, and a theory which only satisfies 1 can obviously be developed.
My problem is that I am not completely conviced of the need of assumption 2
The important missing thing is that there must be such a (distribution-valued) field that is nonzero and satisfies causal commutation rules.