Living in Curved Momentum Space

Abstract

In this paper we review some aspects of relativistic particles' mechanics in the case of a non-trivial geometry of momentum space. We start with showing how the curved momentum space arises in the theory of gravity in 2+1 dimensions coupled to particles, when (topological) degrees of freedom of gravity are solved for. We argue that there might exist a similar topological phase of quantum gravity in 3+1 dimensions. Then we characterize the main properties of the theory of interacting particles with curved momentum space and the symmetries of the action. We discuss the spacetime picture and the emergence of the principle of relative locality, according to which locality of events is not absolute but becomes observer dependent, in the controllable, relativistic way. We conclude with the detailed review of the most studied kappa-Poincare framework, which corresponds to the de Sitter momentum space.

22 pages. In version 2, discussion of Lorentz symmetry on pp. 11 and 18 corrected

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• Generally covariant formulation of Relative Locality in curved spacetime
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• Curved momentum spaces from quantum groups with cosmological constant
• Planck-scale-deformed relativistic symmetries and diffeomorphisms in momentum space
• Deformed Carroll particle from 2+1 gravity
• Snyder Momentum Space in Relative Locality
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• On Poisson Electrodynamics With Charged Fields
• Relative Locality in Curved Space-time
• Noncommutative spaces and covariant formulation of statistical mechanics
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• Noncommutative momentum and torsional regularization
• Early universe thermostatistics in curved momentum spaces
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• Fate of $\kappa$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes
• Orientability of loop processes in relative locality
• Emergence of curved momentum-spacetime and its effect on the cyclotron motion in the antiferromagnetic quantum critical metal
• A symplectic geometric origin of universal quartic modified dispersion relations