Collision operator in the physical representation, the friedrichs model

Recent work by the Brussels group has shown the importance of parity properties of the dynamical operators for large systems with respect to the inversion L → -L, L being the Liouville-Von Neumann operator. This has led to a causal formulation of dynamics. The transformation Λ(L) leading from the initial representation (Liouville-Von Neumann equation) to the causal (physical) representation is a nonunitary, called starunitary, transformation. We first summarize the general theory and recall the properties of the evolution operator for a dynamical dissipative system in the physical representation. The Friedrichs model is then studied in detail. We restrict ourselves to the discussion of the diagonal elements of the density operator in the physical representation and their evolution. This is possible because these elements obey independent equations. We start with a finite system and keep only dominant terms with respect to the volume L3. We then show that the Friedrichs model belongs to the class of dynamical dissipative systems, for which an 1-197 theorem can be established. We also discuss, for this model, the relation between the causal formulation of dynamics and probability theory. © 1973.SCOPUS: ar.j