Asymptotic Safety and Canonical Quantum Gravity

TL;DR

The paper argues that Asymptotic Safety and Canonical Quantum Gravity are complementary rather than opposed, each addressing non-perturbative, background-independent aspects of quantum gravity while aiming to identify physically meaningful observables.It develops a framework that connects the CQG path integral for relational observables with the AS FRG program, using the Effective Average Action and a background-independent flow to relate UV completions to IR physics.Two concrete realizations—one with four scalar clocks and one with Gaussian dust as material references—demonstrate how deparametrization shapes the reduced dynamics, the nature of fixed points, and the calculational trade-offs between Lorentzian and Euclidean methods.The work highlights how a relational, gauge-invariant observable sector can be evolved non-perturbatively across scales, offering a route toward predictive quantum gravity and potential connections to cosmology, while noting open issues about frame dependence and the precise mapping between canonical and covariant descriptions.

Abstract

In the context of gravity the Lagrangian and Hamiltonian formalisms have been developed largely independently, emphasizing renormalization and quantization, respectively. The formalisms use a different methodology to distinguish between gauge and physical degrees of freedom. In this review we analyze the connection between the Asymptotically Safe and Canonical Quantum Gravity approaches. Based on the Hamiltonian formulation, the Canonical Quantum Gravity approach inherently provides a natural framework for defining observables. This serves as the foundation for constructing the generating functional of the $n$-point correlation functions of physical degrees of freedom. By means of background-independent, non-perturbative renormalization methods well-established in the Lagrangian framework and typically employed in Asymptotic Safety, the resulting generating functional can be handled. In particular, we employ the Functional Renormalization Group to regularize the path integral and to compute the flow connecting the bare theory in the ultraviolet with the effective infrared theory. An important advantage of this approach is that it establishes an explicit, systematic relation between the quantization procedure and the systematics of quantum field theory-based renormalization group methods. More importantly, this synthesis not only bridges canonical and covariant approaches but also paves the way for a consistent and predictive quantum theory of gravity grounded in physically meaningful, gauge-invariant observables.

Figures (4)

• Figure 1: LQG, Spin Foams, and GFT offer complementary perspectives on CQG. They are connected through a coherent mathematical structure. LQG is formulated in the canonical framework, where quantum states of geometry are described by spin networks, graphs labeled by SU(2) representations that encode quantized areas and volumes. These spin networks form the kinematical basis for spin foam models, which provide a covariant, path-integral approach by assigning transition amplitudes to histories of spin networks, represented as 2-complexes (foams). Spin foams thus aim to implement the dynamics of LQG in a spacetime setting. GFT, in turn, offers a field-theoretic generalization of this picture: it is a quantum field theory on group manifolds whose Fock space recreates spin network states as many-body excitations of geometric quanta. Perturbative expansions of GFT yield Feynman diagrams that correspond to spin foam amplitudes, allowing the theory to generate sums over discrete spacetime geometries. Through coarse-graining, GFT also provides a pathway toward recovering the continuum limit, unifying the discrete structures of LQG and spin foams within a second-quantized framework.
• Figure 2: Flow diagram in the $\lambda$-$g$ plane, dimensionless $\Lambda(k)$ and $G_{N}(k)$. The IR fixed point at $\lambda_*, g_* =(0,0)$ and the UV fixed point $\lambda_*, g_* =(0.19, 0.71)$ are depicted with a black dot. All the trajectories stem from the UV fixed point (the arrows' direction points towards decreasing RG scale $k$. The thick black line represent the trajectory which connects the UV to the IR fixed point. Furthermore, the flow cannot be trusted in regions beyond the red dashed line, since it becomes singular.
• Figure 3: Case (a), four scalar fields Ferrero:2024rvi. Projection of the real ($\lambda_r,\; g_r$) and the imaginary part ($\lambda_i,\; g_i$) of the flow near the UV fixed point (black dot) at $(\lambda_*,g_*)= (0.45962 + 0.0496779\; i,1.0127 + 0.420496\; i)$. The arrows point towards decreasing $k$.
• Figure 4: Case (b), Gaussian dust Ferrero:2025idz. The trajectories stem from the UV fixed point and flow towards decreasing $k$. The thick line is the trajectory ending at $k = 0$. The dashed red line represents the zone after which the flow becomes singular.