Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry

In this new paper (arXiv:1912.12878) we show that the Lorentzian Snyder models, together with their non-relativistic (c→∞) and ultra-relativistic (c→0) limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of these spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-) de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton–Hooke) spaces. This exchange between the non-relativistic and ultra-relativistic limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.

Coreductive Lie bialgebras and dual homogeneous spaces

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M=G/H equipped with an additional Poisson structure π which is compatible with a Poisson-Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra 𝔤=Lie(G), in this new paper (arXiv:1909.01000) we exploit the idea of Lie bialgebra duality in order to introduce the notion of dual homogeneous space of a given homogeneous space M=G/H with respect to the Lie bialgebra δ. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to the dual space thus showing that an even richer duality framework arises. In order to analyse the physical implications of this new duality, the case of M being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding dual reductive and symmetric spaces are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that the dual space is reductive is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally we show that, despite the dual spaces are not endowed in general with an invariant metric, their geometry can be described by making use of K-structures.

The κ-(A)dS noncommutative spacetime

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed in this new paper (arXiv:1905.12358) by quantizing its semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. Under minimal physical assumptions, it is explicitly proven that this is the only possible generalization to the case of non-vanishing cosmological constant of the well-known κ-Minkowski spacetime. The κ-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the κ-Minkowski spacetime. When expressed in ambient coordinates, the quantum κ-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.

Life of cosmological perturbations in MDR models, and the prospect of travelling primordial gravitational waves

In this new paper (arXiv:1905.08484) we follow the life of a generic primordial perturbation mode (scalar or tensor) subject to modified dispersion relations (MDR), as its proper wavelength is stretched by expansion. A necessary condition ensuring that travelling waves can be converted into standing waves is that the mode starts its life deep inside the horizon and in the trans-Planckian regime, then leaves the horizon as the speed of light corresponding to its growing wavelength drops, to eventually become cis-Planckian whilst still outside the horizon, and finally re-enter the horizon at late times. We find that scalar modes in the observable range satisfy this condition, thus ensuring the viability of MDR models in this respect. For tensor modes we find a regime in which this does not occur, but in practice it can only be realised for wavelengths in the range probed by future gravity wave experiments if the quantum gravity scale experienced by gravity waves goes down to the PeV range. In this case travelling -rather than standing- primordial gravity waves could be the tell-tale signature of MDR scenarios.