Approaches to Quantum Gravity

520 F. Girellithe Legendre transform, which is in general hard to invert, if not by perturbationsin M P :dx μds =ẋ μ = λ{x μ , E 2 − m 2 − p 2 dp μ− F(p,μ,M P )},ds = 0.The Lagrangian L then obtained will be in general a non-bilinear functionF(ẋ) of ẋ μ ,∫S = F(ẋ)ds.The key feature is that it is still time reparametrization invariant so that upon rescalingof the vector ẋ → aẋ, wehaveF(aẋ) =|a|F(ẋ). 12 This means that F can beidentified with a norm (pseudo-norm if the kernel of F is not trivial). The particlelives then in a space the metric of which is given byg μν (ẋ) = 1 ∂F 22 ∂ẋ μ ∂ẋ . νThis is a Finsler metric [30] and is the natural generalization of Riemannian metrics:the latter arises from a norm which is a bilinear form on the tangent space,whereas a Finsler metric arises from general normsF riem (x, ẋ) = g μν (x)ẋ μ ẋ ν F fins (x, ẋ) = g μν (x, ẋ)ẋ μ ẋ ν .All the geometrical objects (curvature, Killing vectors) arising in Riemanniangeometry have been generalized by mathematicians to the Finsler case, thoughoften with some ambiguities. In particular the notion of tetrad becomes here clearlyvector dependent, as proposed in the previous section. What is left now is to explorethis new concept of geometry, and to try to understand how these mathematicalstructures can provide a better understanding of the semiclassical spacetimes, butalso to possible new experimental tests.A key feature of this approach is to keep the usual notion of tangent bundle, as avector bundle. Another possible interpretation of the MDR is to say that momentumspace is curved, so that we lose the vector bundle structure for the tangent bundle.This is the standard interpretation of DSR.26.3.2.2 Extended phase spaceThe choice of symplectic structure and therefore the choice of physical configurationcoordinates was pretty arbitrary in the previous section. It is natural to ask ifone can have some canonical way to derive the full (non-trivial) phase space. Forthis it would be convenient to construct a linear momentum in some space, definethe canonical conjugated configuration coordinates and inverse the map to recover12 The homogeneity might be true only for a > 0 in which case the MDR is not invariant under time inversion.