The goal of this review is to present an introduction to loop quantum gravity—a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.

Background independent quantum gravity: A Status report

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. The evidence from symmetry truncations and from the truncated flow of the effective average action is presented in detail. A dimensional reduction phenomenon for the residual interactions in the extreme ultraviolet links both results. For practical reasons the background effective action is used as the central object in the quantum theory. In terms of it criteria for a continuum limit are formulated and the notion of a background geometry self-consistently determined by the quantum dynamics is presented. Self-contained appendices provide prerequisites on the background effective action, the effective average action, and their respective renormalization flows.

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The Asymptotic Safety Scenario in Quantum Gravity

Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation

We discuss the lower Higgs boson mass bounds which come from the absolute stability of the Standard Model (SM) vacuum and from the Higgs inflation, as well as the prediction of the Higgs boson mass coming from the asymptotic safety of the SM. We account for the three-loop renormalization group evolution of the couplings of the SM and for a part of the two-loop corrections that involve the QCD coupling α
 s
 to the initial conditions for their running. This is one step beyond the current state-of-the-art procedure (“one-loop matching-two-loop running”). This results in a reduction of the theoretical uncertainties in the Higgs boson mass bounds and predictions, associated with the SM physics, to 1–2 GeV. We find that with the account of existing experimental uncertainties in the mass of the top quark and α
 s
 (taken at the 2σ level) the bound reads M
 H
  ≥ M
 min (equality corresponds to the asymptotic-safety prediction), where $ {{M}_{{\min }}}=\left( {129\pm 6} \right) $
 GeV. We argue that the discovery of the SM Higgs boson in this range would be in agreement with the hypothesis of the absence of new energy scales between the Fermi and Planck scales, whereas the coincidence of M
 H
 with M
 min would suggest that the electroweak scale is determined by Planck physics. In order to clarify the relation between the Fermi and Planck scales a construction of an electron-positron or muon collider with a center-of-mass energy ~ (200 + 200 GeV) (Higgs and t-quark factory) would be needed.

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Higgs Boson Mass and New Physics

We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in quantum Einstein gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the causal dynamical triangulation approach.

Quantum Einstein gravity

Nonperturbative treatments of the UV limit of pure gravity suggest that it admits a stable fixed point with a positive Newton constant and a cosmological constant. We prove that this result is stable under the addition of a scalar field with a generic potential and nonminimal coupling to the scalar curvature. There is a fixed point where the mass and all nonminimal scalar interactions vanish, while the gravitational couplings have values which are almost identical to those in the pure gravity case. We discuss the linearized flow around this fixed point and find that the critical surface is four dimensional. In the presence of other, arbitrary, massless minimally coupled matter fields, the existence of the fixed point, the sign of the cosmological constant, and the dimension of the critical surface depend on the type and number of fields. In particular, for some matter content, there exist polynomial asymptotically free scalar potentials, suggesting a gravitational solution to the well-known problem of triviality.

Asymptotic safety of gravity coupled to matter

Recent studies of the ultraviolet behavior of pure gravity suggest that it admits a non-Gaussian attractive fixed point, and therefore that the theory is asymptotically safe. We consider the effect on this fixed point of massless minimally coupled matter fields. The existence of a UV attractive fixed point puts bounds on the type and number of such fields.

http://active.pd.infn.it/g4/seminars/2014/files/percacci.pdf

Constraints on matter from asymptotic safety

We clarify a point concerning the ultraviolet behaviour of the quantum field theory of gravity, under the assumption of the existence of an ultraviolet fixed point. We explain why Newton's constant should scale like the inverse of the square of the cutoff, even though it is technically inessential. As a consequence of this behaviour, the existence of an UV fixed point would seem to imply that gravity has a built-in UV cutoff when described in Planck units, but not necessarily in other units.

On the ultraviolet behaviour of Newton's constant

We clarify a point concerning the ultraviolet behaviour of the Quantum Field Theory of gravity, under the assumption of the existence of an ultraviolet Fixed Point. We explain why Newton's constant should to scale like the inverse of the square of the cutoff, even though it is technically inessential. As a consequence of this behaviour, the existence of an UV Fixed Point would seem to imply that gravity has a built-in UV cutoff when described in Planck units, but not necessarily in other units.

Daniele Perini

Papers

Asymptotic safety of gravity coupled to matter

Constraints on matter from asymptotic safety

On the ultraviolet behaviour of Newton's constant

On the Ultraviolet Behaviour of Newton's constant

Should We Expect a Fixed Point for Newton's Constant?