Hamilton geometry: Phase space geometry from modified dispersion relations

Abstract

We describe the Hamilton geometry of the phase space of particles whose motion is characterised by general dispersion relations. In this framework spacetime and momentum space are naturally curved and intertwined, allowing for a simultaneous description of both spacetime curvature and non-trivial momentum space geometry. We consider as explicit examples two models for Planck-scale modified dispersion relations, inspired from the $q$-de Sitter and $\kappa$-Poincaré quantum groups. In the first case we find the expressions for the momentum and position dependent curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. In contrast, for a dispersion relation that is induced by a spacetime metric, as in General Relativity, the Hamilton geometry yields a flat momentum space and the usual curved spacetime geometry with only position dependent geometric objects.

32 pages, section on quantisation of the theory added, comments on additin of momenta on curved momentum spaces extended

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• Planck-scale modified dispersion relations and Finsler geometry
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• Realization of DSR-relativistic symmetries in Finsler geometries
• Speed of particles and a relativity of locality in $\kappa$-Minkowski quantum spacetime
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• Deformed Lorentz symmetry and relative locality in a curved/expanding spacetime
• Modified Dispersion Relations from the Renormalization Group of Gravity
• Planck-scale phenomenology with anti-de Sitter momentum space
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• Radar orthogonality and radar length in Finsler and metric spacetime geometry
• Linking Covariant and Canonical General Relativity via Local Observers

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