Hamilton geometry: Phase space geometry from modified dispersion relations

Abstract

Quantum gravity phenomenology suggests an effective modification of the general relativistic dispersion relation of freely falling point particles caused by an underlying theory of quantum gravity. Here we analyse the consequences of modifications of the general relativistic dispersion on the geometry of spacetime in the language of Hamilton geometry. The dispersion relation is interpreted as the Hamiltonian which determines the motion of point particles. It is a function on the cotangent bundle of spacetime, i.e. on phase space, and determines the geometry of phase space completely, in a similar way as the metric determines the geometry of spacetime in general relativity. After a review of the general Hamilton geometry of phase space we discuss two examples. The phase space geometry of the metric Hamiltonian $H_g(x,p)=g^{ab}(x)p_ap_b$ and the phase space geometry of the first order q-de Sitter dispersion relation of the form $H_{qDS}(x,p)=g^{ab}(x)p_ap_b + \ell G^{abc}(x)p_ap_bp_c$ which is suggested from quantum gravity phenomenology. We will see that for the metric Hamiltonian $H_g$ the geometry of phase space is equivalent to the standard metric spacetime geometry from general relativity. For the q-de Sitter Hamiltonian $H_{qDS}$ the Hamilton equations of motion for point particles do not become autoparallels but contain a force term, the momentum space part of phase space is curved and the curvature of spacetime becomes momentum dependent.