Approaches to Quantum Gravity

Asymptotic safety 125Dimensional analysis implies that in the FP regime 〈g μν 〉 k = k −2 ( ˜g 0 ) μν , where ˜g 0 ,defined as in (8.13), is a fiducial dimensionless metric that solves the equations ofmotion of Ɣ k0 . For example, in the Einstein–Hilbert truncation, the effective metric〈g μν 〉 k is a solution of the equation R μν = k g μν ,so〈g μν 〉 k = k 0 k〈g μν 〉 k0 ≈(k0k) 2〈g μν 〉 k0 = k −2 ( ˜g 0 ) μν , (8.26)where ≈ means “in the FP regime”. The fractal spacetime is described by thecollection of all these metrics.A phenomenon characterized by an energy scale k will “see” the effective metric〈g μν 〉 k . For a (generally off-shell) free particle with four-momentum p μ it is naturalto use k ∝ p, where p = √ ( ˜g 0 ) μν p μ p ν . Its inverse propagator is then 〈g μν 〉 p p μ p ν .At low energy 〈g μν 〉 k does not depend on k and the propagator has the usual p −2behaviour; in the FP regime, (8.26) implies instead that it is proportional to p −4 .ItsFourier transform has a short-distance logarithmic behaviour which is characteristicof two dimensions, and agrees with the aforementioned numerical results on thespectral dimension in causal dynamical triangulations. This agreement is encouraging,because it suggests that the two approaches are really describing the samephysics. When applied to gravitons in four dimensions (and only in four spacetime“dimensions”) it also agrees with the general prediction, derived at the end ofsection 8.3, that η g = 2 at a nontrivial Gravitational FP.The presence of higher derivative terms in the FP action raises the old issue ofunitarity: as is well-known, the action (8.11) describes, besides a massless graviton,particles with Planck mass and negative residue (ghosts). From a Wilsonianperspective, this is clearly not very significant: to establish the presence of a propagatorpole at the mass m P one should consider the effective action Ɣ k for k ≈ m P ,which may be quite different from the FP action. Something of this sort is knownto happen in the theory of strong interactions: at high energy they are described bya renormalizable and asymptotically free theory (QCD), whose action near the UV(Gaussian) FP describes quarks and gluons. Still, none of these particles appears inthe physical spectrum.As in QCD, matching the UV description to low energy phenomena may turn outto be a highly nontrivial issue. A change of degrees of freedom could be involved.From this point of view one should not assume a priori that the metric appearingin the FP action is “the same” metric that appears in the low energy descriptionof GR. Aside from a field rescaling, as discussed in section 8.2, a more complicatedfunctional field redefinition may be necessary, perhaps involving the matterfields, as exemplified in [44]. Unless at some scale the theory was purely topological,it will always involve a metric and from general covariance arguments itwill almost unavoidably contain an Einstein–Hilbert term. This explains why the