R2phase diagram of quantum Einstein gravity and its spectral dimension

Abstract

Within the gravitational asymptotic safety program, the RG flow of the R 2 truncation in three and four spacetime dimensions is analyzed in detail. In particular, we construct RG trajectories which emanate from the non-Gaussian UV fixed point and possess long classical regimes where the effective average action is well approximated by the classical Einstein-Hilbert action. As an application we study the spectral dimension of the effective QEG spacetimes resulting from these trajectories, establishing that the picture of a multi-fractal spacetime is robust under the extension of the truncated theory space. We demonstrate that regimes of constant spectral dimensions can either be attributed to universal features of RG fixed points or singular loci in the β functions.

A functional renormalization group equation for foliated spacetimes

“…In a sense our novel flow equation contains the continuum analogue of the foliation structure that is supposed to be responsible for the well-defined classical limit in CDT. Following the ideas [57,58,59,56] one could, e.g., use the foliated flow equation to study diffusion processes on spatial slices and examine if these can be matched to the CDT data recently reported in [60]. This may shed new light on the question whether the continuum limit of CDT corresponds to an isotropic or anisotropic gravity theory as suggested in [61,62].…”

“…Addressing this problem systematically presumably requires the inclusion of higher derivative operators like the square of the intrinsic Ricci scalar ( (d) R) 2 in the truncation ansatz. Since the inclusion of such higher-derivative terms in the truncation ansatz is known to be very laborious [55,9,56] and beyond a first investigation of foliated RG flows, we do not pursue this direction further and leave it to future research.…”

On a renormalization group scheme for causal dynamical triangulations

“…As I demonstrate in section 3.3, the rate of decay for σ > σ cl sets the scale of the large scale curvature, which is also the scale dS that characterizes the central accumulation. 5 If one possessed a model of the spectral dimension's behavior for σ < σ cl , of which there are several [15,18,19,21,22,23,29,32,34,37,38], then one could presumably also extract a scale qm from the rate of increase. Once again, as expected, based on numerical measurements alone, these three scales are characterized by dimensionless numbers.…”

Probing the quantum nature of spacetime by diffusion

“…Within analytical approaches to quantum gravity, it has been proposed that quantum effects in spacetime, foremost a dynamical dimensional change, can be captured by a modification of the Laplacian operator appearing in the classical diffusion equation [6][7][8][9][10][11][12][13]18]. We will call this new operator "generalized Laplacian" or, as done in probability theory, spatial generator.…”

“…causal dynamical triangulations (CDT) in four [1,2] and three spacetime dimensions [3,4], Euclidean dynamical triangulations (EDT) [5], loop quantum gravity (LQG) [6], asymptotically safe gravity (or quantum Einstein gravity, QEG) [7][8][9][10], nonlocal gravity [11], and also Hořava-Lifshitz (HL) gravity [12,13]. Remarkably, many of these works assess that d S = 2 at microscopic scales [14][15][16].…”