2013/01/31 by Arraut, Ivan
#FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)
paper · doi:10.48550/arxiv.1301.7609
I introduce the spatial curvature effects inside the formalism of Relative Locality as a non-commutative structure of the momentum space in agreement with the very well known concepts of Quantum Groups. This gives a natural red-shift effect in agreement with an extended version of the Generalized Uncertainty Principle (GUP) and in agreement with the standard notions of curvature inside General Relativity. I then introduce the concept of Relative Co-locality as a reinterpretation of the usual notions of spacetime curvature. From this redefinition, I obtain the momentum space curvature effects as a non-commutativity in position space. This introduce a natural blue-shift effect in agreement with the extended version of GUP. Both effects, Relative locality and Co-locality are dual each other inside the formalism of quantum groups SUq(n) symmetric Heisenberg algebras and their q-Bargmann Fock representations. When Relative locality and Co-locality are introduced, the q-deformation parameter takes the form q≈ 1+√\frac\vert p\vert \vert x\vertrΛmpl with the spatial curvature effects in Relative Locality appearing like ΔX≈ \frac\vert x\vertmplΔP and the momentum curvature effects in Relative Co-locality appearing like ΔP≈ (\vert p\vert)/(rΛ)ΔX, where rΛ=(1)/(√Λ) is the scale defined by the Cosmological Constant Λ, mpl is the Planck mass and ΔX/ΔP is a scale of position/momentum or time/energy associated with the event, p and x are the momentum and position of the observer relative to the event.