Many-boson averages for model systems of the Dicke type

Many-boson averages and quasi-averages for the Dicke-type model many-body systems representing substance coupled to a finite number of boson field modes are discussed. “Substitution rules” are established which imply that when calculating the many-boson averages the boson operators can be replaced by the operators of the substance subsystem, averaging being performed over the linearised approximating hamiltonian. Also some general inequalities for equilibrium Gibbs averages are derived. The results are rigorous.