Extension of Haag’s theorem in the case of the lorentz invariant noncommunitative quantum field theory in a space with arbitrary dimension

The generalized Haag theorem was proven in $SO(1,k)$ invariant quantum field theory. Apart from the $(k{+}1)$ variables, an arbitrary number of additional coordinates, including noncommutative ones, can occur in the theory. In $SO(1,k)$ invariant theory new corollaries of the generalized Haag theorem are obtained. It has been proven that the equality of four-point Wightman functions in the two theories leads to the equality of elastic scattering amplitudes and thus to the equality of the total cross sections in these theories. It was also shown that at $k > 3$ the equality of $(k{+}1)$ point Wightman functions in the two theories leads to the equality of the scattering amplitudes of some inelastic processes. In the $SO(1,1)$ invariant theory it was proven that if in one of the theories under consideration the $S$-matrix is equal to unity, then in another theory the $S$-matrix equals unity as well.