Quantum statistical mechanics for nonextensive systems

The traditional basis of description of many-particle systems in terms of G reen functions is here generalized to the case when the system is nonextens ive, by incorporating the Tsallis form of the density matrix indexed by a n onextensive parameter q. This is accomplished by expressing the many-partic le q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends o n this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observ able quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-pa rticle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions sim ilar to the traditional ones follows because the equations of motion for bo th of these an formally similar. Approximation schemes for one-particle q G reen functions such as Hartree and Hartree-Fock schemes are given as exampl es. This extension enables us to predict possible experimental tests for th e validity of this framework by expressing some observable quantities in te rms of the q averages.