Reappraisal of a model for deformed special relativity

Abstract

We revisit one of the earliest proposals for deformed dispersion relations in the light of recent results on dynamical dimensional reduction and production of cosmological fluctuations. Depending on the specification of the measure of integration and addition rule in momentum space the model may be completed so as to merely deform Lorentz invariance, or so as to introduce a preferred frame. Models which violate Lorentz invariance have a negative UV asymptotic dimension and a very red spectrum of quantum vacuum fluctuations. Instead, models which preserve frame independence can exhibit running to a UV dimension of 2, and a scale-invariant spectrum of fluctuations. The bispectrum of the fluctuations is another point of divergence between the two casings proposed here for the original model.

Dimension and Dimensional Reduction in Quantum Gravity

“…Another rationale comes from the possible role of simple noncommutative geometries in quantum gravity, as in [46], and their implied dispersion relations. For instance, a form of "doubly special relativity"-a deformation of special relativity that preserves frame independence while introducing an invariant energy scale-leads to a spectral dimension that falls to d S = 2 at high energies, but the result depends on a free parameter [112]. A similar reduction of spectral dimension occurs another popular model of noncommutative geometry, κ-Minkowski space, but the limiting value again depends on a nonunique choice, now of a generalized Laplacian [113,114].…”

“…A minimum length may also be incorporated through a "generalized uncertainty principle" [122], which alters the Heisenberg commutation relations. Such a change leads to modified dispersion relations, as in section 3.7, so, not surprisingly, dimensional reduction appears at short distances [49,112]. Maziashvili has also argued that the finite resolution due to a minimum length causes the box-counting dimension d b to decrease at small distances [127].…”

Accelerated horizons and Planck-scale kinematics

“…A first natural guess for a momentum space measure would be just to adopt the usual flat Lebesgue measure 6 dµ(p) = d 4 p. It turns out that such measures is not invariant under the deformed boosts (60)-(62). This feature has been recently noted in [42] elaborating on the original Magueijo-Smolin DSR model. It turns out that, as one could have easily guessed, a covariant measure can be obtained from a transformation of the ordinary flat measure into the following deformed measure via the twist map (51):…”

Photon gas thermodynamics in dS and AdS momentum spaces

“…[17] for more general case). In the case of cosmological coordinatization of dS momentum space the metric is given by (see Ref.…”