In this paper we revisit a gauge theory of supergravity, where the tetrad, the spin 3=2 field and the spin connection are identified with components of the gauge field. This theory naturally incorporates a cosmological constant term. Furthermore, when adding the topological -term, as in the Yang-Mills theory, an analogue of the Immirzi term for N ¼ 1 supergravity is obtained.

Modified entropies have been extensively considered in the literature [1]. Among the most well known are the Rényi entropy [2] and the Havdra-Charv 'a [3] and Tsallis entropy [4,5]. All these depend on one or several parameters. By means of a modification to Superstatistics [6], one of the authors [7] has proposed generalized entropies that depend only on the probability [7,8]. There are three entropies: S I = k ∑

We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and in this context, such entropies arise when a system is exposed to non-equilibrium conditions, whose general effects can be described by a generalized Boltzmann factor and correspondingly by a generalized probability distribution defining a different statistics. We generalize the usual statistics to their quantum counterparts, and we will focus on the properties of the corresponding generalized quantum ideal Bose gas. The most important consequence of the generalized Bose gas is that the critical temperature predicted for the condensation changes in comparison with the usual quantum Bose gas. Conceptual differences arise when comparing our results with the ones previously reported regarding the q-generalized Bose–Einstein condensation. As the entropies analyzed here only depend on the probability, our results cannot be adjusted by any parameter. Even though these results are close to those of non-extensive statistical mechanics for q ∼ 1 , they differ and cannot be matched for any q.

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