Green functions based on Tsallis nonextensive statistical mechanics: normalized q-expectation value formulation

In this paper, the Green function theory of quantum many-particle systems recently presented is reworked within the framework of nonextensive statistical mechanics with a new normalized q-expectation values. This reformulation introduces a renormalization of temperature of the earlier theory and a self-consistency condition. The importance of these two features is nontrivial and to emphasize this, we explicitly contrast the maximum entropy density matrices derived for an exactly solvable model based on the two types of the constraints. The linear response theory is also presented, along with its two-particle Green function version. In order to emphasize the importance of the new formalism, we collect here the results where both the formalisms have been used to examine the same set of problems. This reveals clearly that the new formalism is the method of choice because the numerical results are much more physically meaningful than those found in the old version, even though the general features or the answers retain the same characteristics in certain cases. In the case where thermodynamic entities are to be examined as in the case of the q-dependence of Bose–Einstein condensation, the self-consistent requirement in the new formalism is numerically much more subtle, and thus the earlier results are modified as shown in Fig. 1.