Quantum symmetry, the cosmological constant and Planck-scale phenomenology

Abstract

We present a simple algebraic argument for the conclusion that the low energy limit of a quantum theory of gravity must be a theory invariant, not under the Poincaré group, but under a deformation of it parameterized by a dimensional parameter proportional to the Planck mass. Such deformations, called κ-Poincaré algebras, imply modified energy-momentum relations of a type that may be observable in near future experiments. Our argument applies in both 2 + 1 and 3 + 1 dimensions and assumes only 1) that the low energy limit of a quantum theory of gravity must involve also a limit in which the cosmological constant is taken very small with respect to the Planck scale and 2) that in 3 + 1 dimensions the physical energy and momenta of physical elementary particles is related to symmetries of the full quantum gravity theory by appropriate renormalization depending on Λl 2 P lanck . The argument makes use of the fact that the cosmological constant results in the symmetry algebra of quantum gravity being quantum deformed, as a consequence when the limit Λl 2 P lanck → 0 is taken one finds a deformed Poincaré invariance. We are also able to isolate what information must be provided by the quantum theory in order to determine which presentation of the κ-Poincaré algebra is relevant for the physical symmetry generators and, hence, the exact form of the modified energy-momentum relations. These arguments imply that Lorentz invariance is modified as in proposals for doubly special relativity, rather than broken, in theories of quantum gravity, so long as those theories behave smoothly in the limit the cosmological constant is taken to be small.

On κ-deformation and triangular quasibialgebra structure

“…If κ is finite in nature, it is certainly very large. Moreover it is usually supposed [43][44][45] -though cf. [37] -that the role of quantum field theory with κ-deformed Poincaré symmetry, if any, will be that of an effective description in a regime intermediate between standard QFT and full quantum gravity.…”

Severe constraints on the loop-quantum-gravity energy-momentum dispersion relation from the black-hole area-entropy law

“…The possibility of Planck-scale modifications of the dispersion relation has been considered extensively in the recent quantum-gravity literature [9][10][11] and in particular in Loop Quantum Gravity [12][13][14][15].…”

Symmetries of quantum spacetime in three dimensions

“…This shows that when restricting to S 3 , the diffeomorphism constraints can be expressed in terms of smearing functions that do not depend on the phase variables, and in particular they split into two sets of constraints corresponding to the two copies of constraints (10). Therefore, in order to study symmetries of (Euclidean) deSitter space-time it is enough to consider the algebra (11), in which the smearing functions are proportional to δ's.…”

“…It is a straightforward but tedious calculation to show that if in (67), (68) and (69) we change the generators as (see [4] and [10])…”

“…Having obtained the deformed symmetry algebra for the case of the Euclidean de Sitter quantum gravity in 3D, we now want to make the contraction Λ → 0, so as to obtain a symmetry that replaces, according to what was said above, the standard Poincaré symmetry in the case of quantum space-time. It turns out that taking the contraction limit is highly nontrivial and can be performed if the deformation parameters q 1 , q 2 of the two deformed su q (2) algebras satisfy q 1 = q −1 2 [9] (the divergencies are not visible on the level of the algebra, but can be seen when one wants to find the expression of coproducts in the contraction limit; for this reason they were missed in [10].) The so constructed symmetry algebra of flat quantum space-time in 3D turns out to be the κ-Poincaré algebra.…”