Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

Abstract

In this paper, we review two approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called $q$-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces.

arXiv comment22 pages. Matches published version. Invited contribution for Physics. Special Issue "New Advances in Quantum Geometry"

References in corpus (49)

• Lorentz invariance with an invariant energy scale
• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• Modern tests of Lorentz invariance
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Quantum Spacetime Phenomenology
• Gravity's Rainbow
• Tests of Lorentz invariance: a 2013 update
• The principle of relative locality
• Planck-scale modified dispersion relations and Finsler geometry
• Lorentz-violation-induced arrival delays of cosmological particles
• Thermodynamics of modified black holes from gravity's rainbow
• In-vacuo-dispersion features for GRB neutrinos and photons
• Finsler spacetime geometry in Physics
• Hamilton geometry: Phase space geometry from modified dispersion relations
• Realization of DSR-relativistic symmetries in Finsler geometries
• Cosmological particle production in emergent rainbow spacetimes
• Geometry of physical dispersion relations
• Constraints on Lorentz invariance violation from HAWC observations of gamma rays above 100 TeV
• Light cones in Finsler spacetime
• Causal Dynamical Triangulations and the Quest for Quantum Gravity
• Spacetime-noncommutativity regime of Loop Quantum Gravity
• Investigation on Finsler geometry as a generalization to curved spacetime of Planck-scale-deformed relativity in the de Sitter case
• Curved spacetimes with local $\kappa$-Poincaré dispersion relation
• Rainbow metric from quantum gravity
• Deformed relativity symmetries and the local structure of spacetime
• Planck-scale-modified dispersion relations in homogeneous and isotropic spacetimes
• Redshift and lateshift from homogeneous and isotropic modified dispersion relations
• Landau Levels in the Presence of a Cosmic String in Rainbow Gravity
• Reaching the Planck scale with muon lifetime measurements
• Mathematical foundations for field theories on Finsler spacetimes
• Linking loop quantum gravity quantization ambiguities with phenomenology
• Foundations of Finsler spacetimes from the Observers' Viewpoint
• Palatini Actions and Quantum Gravity Phenomenology
• The Soccer-Ball Problem
• Natural Cutoffs via Compact Symplectic Manifolds
• Experimental constraints on the second clock effect
• Extended disformal approach in the scenario of Rainbow Gravity
• Kinematics of particles with quantum de Sitter symmetries
• Could quantum gravity slow down neutrinos?
• Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime
• Phenomenology of curvature-induced quantum-gravity effects
• Gravity's rainbow: a bridge between LQC and DSR
• Rainbows without unicorns: Metric structures in theories with Modified Dispersion Relations
• Constraining the loop quantum gravity parameter space from phenomenology
• Electrostatic self-interaction of charged particles in the space-time of a cosmic string in the context of gravity's rainbow
• Two-body decays in deformed relativity
• The geometry of null-like disformal transformations
• The extended phase space thermodynamics of Planck-scale-corrected Reissner-Nordström-anti-de Sitter black hole
• Revisiting Legendre transformations in Finsler geometry

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• Quantum-spacetime effects on nonrelativistic Schrödinger evolution
• Modified particle lifetimes as a signature of deformed relativity
• Revisiting Legendre transformations in Finsler geometry
• Muon accelerators -- Muon lifetime measurements as window to Planck scale physics