We construct a family of relativistic-invariant generating functionals of the form $$F(f^*,g)=\exp\left\{\gamma\int\frac{d\mathbf k}{\omega(\mathbf k)}f^*(\mathbf k)g(\mathbf k)\right\}$$ for the non-Fock representations of the CCR. We analyze the first order in the coupling constant of the model theory. In this theory, the asymptotic in field coincides with the field $\varphi(x)$ corresponding to such a functional. We prove that in the first order, the in and out fields are unitarily equivalent and the scattering matrix consequently exists. Moreover, the kinematics of the “non-Fock quantum field theory” is much richer than the standard kinematics: in this case, the $S$-matrix does not coincide with the chronologically ordered exponent of the interaction Lagrangian.