Reinterpreting deformed Heisenberg algebras

Abstract

Minimal and maximal uncertainties of position measurements are widely considered possible hallmarks of low-energy quantum as well as classical gravity. While General Relativity describes interactions in terms of spatial curvature, its quantum analogue may also extend to the realm of curved momentum space as suggested, e.g. in the context of Relative Locality in Deformed Special Relativity. Drawing on earlier work, we show in an entirely Born reciprocal, i.e. position and momentum space covariant, way that the quadratic Generalized Extended Uncertainty principle can alternatively be described in terms of quantum dynamics on a general curved cotangent manifold. In the case of the Extended Uncertainty Principle the curvature tensor in position space is proportional to the noncommutativity of the momenta, while an analogous relation applies to the curvature tensor in momentum space and the noncommutativity of the coordinates for the Generalized Uncertainty Principle. In the process of deriving this map, the covariance of the approach constrains the admissible models to an interesting subclass of noncommutative geometries which has not been studied before. Furthermore, we reverse the approach to derive general anisotropically deformed uncertainty relations from general background geometries. As an example, this formalism is applied to (anti)-de Sitter spacetime.

Sodium Alginate Doped with Magnesium Perchlorate as Solid Biopolymer Electrolytes for Energy Storage Applications

“…Figure shows the direct current (DC) polarization curve of higher ionic conducting biopolymer membrane placed between two stainless steel blocking electrodes subjected to standard potential of 1.5 V at room temperature. [ 28 ] The polarization of current flow is detected as a function of time. As time increases the current gradually decreases and it gets constant after a long polarization period and the initial and final current are noted.…”

Momentum gauge fields from curved momentum space through Kaluza–Klein reduction

“…Recently, it has been shown that modified Heisenberg algebras follow a similar pattern [16][17][18][19]: non-relativistic single-particle theories exhibiting a generalized uncertainty principle as well as a noncommutative geometry can be mapped onto theories yielding non-relativistic single-particle dynamics on curved momentum space. Here, the concept of curved momentum space refers to the theory of generalized Hamilton spaces (see [18,31] for more information).…”

“…This ansatz may as well be applied in reversed order [18] such that a noncommutative space emerges from an underlying momentum space geometry. Along these lines, the kinematics of the non-relativistic single-particle quantum theory at hand are to be described in terms of the canonical positions Xi and momenta Pi , satisfying the algebra…”

“…1. Curved momentum space [16][17][18][19] and 2. Momentum-dependent gauge fields 6 [23] arising from a momentum-space local U (1) symmetry.…”

Towards quantum mechanics on the curved cotangent bundle

“…Recent times have seen a merger of the areas of the GUP and curved momentum space, first through indication [90][91][92][93][94][95][96] and finally explicitly [97]. In particular, in [97][98][99] it was shown by direct construction that GUP-deformed theories can be mapped into quantum mechanics on nontrivial momentum space. Thus, in quantum theory a minimal length can be translated into a momentum-dependent metric.…”