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

Abstract

We show that the Lorentzian Snyder models, together with their Galilei and Carroll 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 such curved 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 Galilei and Carroll 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 $c\to \infty$. 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.

v2 matches version accepted by Class. Quant. Grav

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• Carrollian hydrodynamics from symmetries
• Carrollian manifolds and null infinity: A view from Cartan geometry
• The group structure of dynamical transformations between quantum reference frames
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Heisenberg doubles for Snyder type models
• Double Quantization
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• Carroll-Schrödinger Equation
• Spinorial Snyder and Yang Models From Superalgebras And Noncommutative Quantum Superspaces
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincaré
• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs