Commutator of an arbitrary superoperator with the 'asymptotic' evolution operator of a many-body system

In the frame of the theory of subdynamics, a general picture of the time evolution of a many-body system has been developed. In this paper, the perturbation scheme is avoided so that compact notations and demonstrations are obtained. A general expression for the commutator of an arbitrary superoperator W with the 'asymptotic' evolution superoperator Sigma(t) equals PiU(t) is derived where U(t) is the complete evolution superoperator and where Pi is the usual superprojector of the theory of subdynamics. In the frame of relativistic nonequilibrium statistical mechanics, the application of this commutator to all the generators of the Poincare group is then given and we get back in particular the fundamental property of commutation with Pi of the Lorentz-Liouville superoperator.