The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in ℏ and evanescent. For ℝ4 spacetime, each renormalized physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale Λp > 0, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale L, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds O(1) + 2πL2Λp2. Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with space-times of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton’s constant but non-perturbative in ℏ.
Models of Quantum Gravity Renormalization Group Renormalization Regularization and Renormalons
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G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor.A20 (1974) 69 [INSPIRE].
J.F. Nicoll and T.S. Chang, An Exact One Particle Irreducible Renormalization Group Generator for Critical Phenomena, Phys. Lett.A 62 (1977) 287 [INSPIRE].ADSCrossRefGoogle Scholar
M. Bonini, M. D’Attanasio and G. Marchesini, Perturbative renormalization and infrared finiteness in the Wilson renormalization group: The Massless scalar case, Nucl. Phys.B 409 (1993) 441 [hep-th/9301114] [INSPIRE].ADSCrossRefGoogle Scholar
N. Sakai and I. Senda, Vacuum Energies of String Compactified on Torus, Prog. Theor. Phys.75 (1986) 692 [Erratum ibid.77 (1987) 773] [INSPIRE].
J. Feldbrugge, J.-L. Lehners and N. Turok, No rescue for the no boundary proposal: Pointers to the future of quantum cosmology, Phys. Rev.D 97 (2018) 023509 [arXiv:1708.05104] [INSPIRE].ADSGoogle Scholar