Constructing quantum fields in a Fock space using a new picture of quantum mechanics

For any conventional nonrelativistic quantum theory of a finite number of degrees of freedom, we construct a picture which we call ’’the scattering picture,’’ combining the ’’nice’’ properties of both the interaction and the Heisenberg pictures, and show that, in the absence of bound states, the theory could be formulated in terms of a free Hamiltonian and an effective potential. We generalize the equations thus derived to the relativistic case and show that, given a Poincaré invariant self-adjoint operator D densely defined on a Fock space, there exists an interacting field which is asymptotically free and has as the scattering matrix the nontrivial operator S=eiD, provided that D annihilates the vacuum and the one-particle states. Crossing relations could easily be imposed on D, but, apart from a few comments, the problem of analyticity of S is left open.