The spectral geometry of de Sitter space in asymptotic safety

Abstract

Within the functional renormalization group approach to Background Independent quantum gravity, we explore the scale dependent effective geometry of the de Sitter solution dS4. The investigation employs a novel approach whose essential ingredient is a modified spectral flow of the metric dependent d’Alembertian, or of similar hyperbolic kinetic operators. The corresponding one-parameter family of spectra and eigenfunctions encodes information about the nonperturbative backreaction of the dynamically gravitating vacuum fluctuations on the mean field geometry of the quantum spacetime. Used as a diagnostic tool, the power of the spectral flow method resides in its ability to identify the scale dependent subsets of field modes that supply the degrees of freedom which participate in the effective field theory description of the respective scale. A central result is that the ultraviolet of Quantum Einstein Gravity comprises far less effective degrees of freedom than predicted (incorrectly) by background dependent reasoning. The Lorentzian signature of dS4 is taken into account by selecting a class of renormalization group trajectories which are known to apply to both the Euclidean and a Lorentzian version of the approach. Exploring the quantum spacetime’s spatial geometry carried by physical fields, we find that 3-dimensional space disintegrates into a collection of coherent patches which individually can, but in their entirety cannot be described by one of the effective average actions occurring along the renormalization group trajectory. A natural concept of an entropy is introduced in order to quantify this fragmentation effect. Tentatively applied to the real Universe, surprising analogies to properties of the observed cosmic microwave background are uncovered. Furthermore, a set of distinguished field modes is found which, in principle, has the ability to transport information about the asymptotic fixed point regime from the ultraviolet, across almost the entire “scale history”, to cosmological distances in the observed Universe.

30 years in: Quo vadis generalized uncertainty principle?

“…Given the operator xa , we can construct the conjugate operator ka such that [x a , ka ] = i, (8) where again the index a is not being summed over. Throughout this paper, we differentiate between the operator ka , here termed wave number, and the momentum operator pa .…”

“…Given a model as in equation ( 2), the operator ka can be obtained as a function of pa such that this constraint (and with it the presence of a minimal length) can be checked by analyzing the domain of ka (p b ). In turn, we can always define a wave number ka satisfying equation (8).…”

“…In the absence of direct empirical guidance, tentative progress can be made by concentrating on common predictions of the existing QG models. In this vein, arguments from different perspectives, such as string theory [1][2][3][4][5][6], asymptotically safe QG [7,8] and loop QG [9][10][11], as well as gedankenexperiments heuristically combining quantum theory and general relativity [12][13][14][15][16][17][18][19][20][21][22][23][24], indicate that the classical-to-quantum transition of gravity could be marked by the emergence of a minimal length of order of the Planck scale (l P ∼ 10 −35 m). It is worth mentioning that, in theories with additional dimensions, the value of the gravitational constant can increase, which implies a larger Planck length, i. e., a larger expected minimallength scale [25][26][27][28].…”

On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety

“…Furthermore, this flow is valid both in the Euclidean and the Lorentzian setting, see refs. [34][35][36] for a recent discussion of this point.…”

On the Lorentz symmetry in conformally reduced quantum gravity

“…However, in AS gravity we should perform the RG blocking up into the UV direction, since the Reuter fixed point is UV attractive. Recently, one of the most important issues for the RG method was to show that the gravity models and their extensions or generalizations remain AS [16][17][18][19][20][21][22][23][24].…”