Quantum Mechanics on a Curved Snyder Space

Abstract

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.

A Paradox of Unity

“…One may recall for example that the behavior of particles is described via Hermite functions [75], i.e., via infinitely smooth functions in S, test functions of tempered distributions S (rough functions). This "smoothness-discreteness" duality in nature is moreover a particular case of Born's principle of reciprocity [21][22][23][24][25][26] because smoothness is the reciprocal (Fourier transform) of discreteness, and, vice versa, discreteness is the reciprocal (Fourier transform) of smoothness [18][19][20]. It is moreover consistent to Scaruffi's idea of a (smooth) ocean (Relativity Theory) and its (rough) ripples (Quantum Theory), [82].…”

“…Simultaneously, one may see mother Earth as blue dot from "outside" and being aware of the homogeneously distributed, still "warm" cosmic background radiation ( Figure 5, left). There is indeed a striking parallel to all these things which is expressed in Max Born's principle of reciprocity [21][22][23][24][25][26]. Unity is scalar (middle) in zero-dimansional spaces.…”

On Boundedness of Entropy of Photon Gas in Noncommutative Spacetime

“…Deformed density of states is not rare in this kind of theories. We find similar deformed density of states in DSR theories, Snyder noncommutative space, and polymerized phase space also [30][31][32]. Interested readers should go through [29,33,34] for the detailed manipulations of algebra associated with this model.…”

Snyder-de Sitter Meets the Grosse-Wulkenhaar Model

“…A relevant property of the SdS model is that it can be rewritten by means of a noncanonical transformation into the usual Snyder space -indeed, the operators X i and P i defined by [23,25]…”

“…In this context a deformation of Snyder spaces in the spirit of triple special relativity [21], the Snyder-de Sitter space (SdS), has also been pursued [22,23,24,25]. Nevertheless, the similarities between SdS and the Grosse-Wulkenhaar model have apparently passed unnoticed -to revert this situation is the goal of this work, which is organized as follows: in Section 2 we review the basics of SdS model.…”