A no-go theorem concerning the cluster decomposition property of direct-interaction scattering theories

Abstract

An axiomatic framework for relativistic direct‐interaction single channel scattering theories is formulated in terms of two representations U’ and U of the Poincaré group. The infinitesimal generators P (momentum), J (angular momentum), E (energy), N (’’boost’’) of U, and P’, J’, E’ N’ of U’ are assumed to be related by the formulas of Bakamjian, Thomas, and Foldy: P’=P, J’=J, E’= (M’²+RP²)^1/2, and N’= (E’X+XE’)/2+(J−X×P)(M’+E’)⁻¹, where X is the center‐of‐mass position operator of U given by X=T−P× (J−T×P) M⁻¹(M+E)⁻¹ with T= (E⁻¹N+NE⁻¹)/2 and M’ is a positive operator that commutes with P, J, and X. Then, it is proved that, within the above‐mentioned framework, the Moller operators W_±=lim exp(−itE’)exp(itE) for t→±∞ cannot satisfy the cluster decomposition property (also known as separability) except for interactions that vanish if any one of the particles is removed to infinity.