This is a working paper summarizing results of an ongoing research project whose aim is to uniquely characterize the uncertainty mea sure for the Dempster-Shafer Theory. A set of intuitive axiomatic requirements is pre sented, some of their implications are shown, and the proof is given of the minimality of re cently proposed measure AU among all mea sures satisfying the proposed requirements.

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Abstract:A quantum entropy space is suggested as the fundamental arena describing the quantum effects. In the quantum regime the entropy is expressed as the superposition of many different Boltzmann entropies that span the space of the entropies before any measure. When a measure is performed the quantum entropy collapses to one component. A suggestive reading of the relational interpretation of quantum mechanics and of Bohm's quantum potential in terms of the quantum entropy are provided. The space associated with the quantum entropy determines a distortion in the classical space of position, which appears as a Weyl-like gauge potential connected with Fisher information. This Weyl-like gauge potential produces a deformation of the moments which changes the classical action in such a way that Bohm's quantum potential emerges as consequence of the non classical definition of entropy, in a non-Euclidean information space under the constraint of a minimum condition of Fisher information (Fisher Bohm-entropy). Finally, the possible quantum relativistic extensions of the theory and the connections with the problem of quantum gravity are investigated. The non classical thermodynamic approach to quantum phenomena changes the geometry of the particle phase space. In the light of the representation of gravity in ordinary phase space by torsion in the flat space (Teleparallel gravity), the change of geometry in the phase space introduces quantum phenomena in a natural way. This gives a new force to F. Shojai's and A. Shojai's theory where the geometry of space-time is highly coupled with a quantum potential whose origin is not the Schrödinger equation but the non classical entropy of a system of many particles that OPEN ACCESSEntropy 2013, 15 3603 together change the geometry of the phase space of the positions (entanglement). In this way the non classical thermodynamic changes the classical geodetic as a consequence of the quantum phenomena and quantum and gravity are unified. Quantum affects geometry of multidimensional phase space and gravity changes in any point the torsion in the ordinary four-dimensional Lorenz space-time metric.

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