Asymptotically safe canonical quantum gravity: Gaussian dust matter

Abstract: In a recent series of publications we have started to investigate possible points of contact between the canonical (CQG) and the asymptotically safe (ASQG) approach to quantum gravity, despite the fact that the CQG approach is exclusively for Lorentzian signature gravity while the ASQG approach is mostly for Euclidean signature gravity. Expectedly, the simplest route is via the generating functional of time ordered N-point functions which requires a Lorentzian version of the Wetterich equation and heat kernel methods employed in ASQG. In the present contribution we consider gravity coupled to Gaussian dust matter. This is a generally covariant Lorentzian signature system, which can be considered as a field theoretical implementation of the idealisation of a congruence of collision free test observers in free fall, filling the universe. The field theory version correctly accounts for geometry -- matter backreaction and thus in principle serves as a dark matter model. Moreover, the intuitive geometric interpretation selects a preferred reference frame that allows to disentangle gauge degrees of freedom from observables. The CQG treatment of this theory has already been considered in the past. For this particular matter content it is possible to formulate the quantum field theory of observables as a non-linear $\sigma$ model described by a highly non-linear conservative Hamiltonian. This allows to apply techniques from Euclidean field theory to derive the generating functional of Schwinger N-point functions which can be treated with the standard Euclidean version of the heat kernel methods employed in ASQG. The corresponding Euclidean action is closely related to Euclidean signature gravity but not identical to it despite the fact that the underlying Hamiltonian is for Lorentzian signature gravity.

An Expert Analysis of "Asymptotically Safe Canonical Quantum Gravity: Gaussian Dust Matter"

The paper by R. Ferrero and T. Thiemann titled "Asymptotically Safe Canonical Quantum Gravity: Gaussian Dust Matter" tackles the integration of two principal approaches to quantum gravity: canonical quantum gravity (CQG) and asymptotically safe quantum gravity (ASQG). This work focuses on the unprecedented synthesis of these frameworks, primarily by considering the gravitational interaction with Gaussian dust matter within a Lorentzian signature, which functions as a crucial element in addressing the problem of time in quantum gravity.

The primary goal of this research involves constructing a unified perspective that bridges CQG — predominantly developed for Lorentzian gravity — with ASQG, which typically operates within Euclidean regimes. The authors examine the potential for coherence between these approaches using the Gaussian dust model, which acts as a field-theoretic representation of collision-free observers in free fall. This model is significant as it naturally disentangles gauge degrees from observables due to its geometric interpretation and the preferred reference frame it provides.

Model and Methodology

Ferrero and Thiemann's exploration begins with established Lorentzian gravitational theories intertwined with Gaussian dust matter, portrayed canonically. The novelty arises from employing canonical transformations which eliminate the challenging measure Jacobians typically present in phase space path integrations. Consequently, the generating functional of time ordered N-point functions can be transformed to yield configurations invoking Euclidean signature for spacetime metrics.

The methodology leverages both the CQG’s canonical quantization strategies and ASQG’s renormalization techniques. In particular, the authors utilize adapted heat kernel methods to define the flow of couplings in the quantum gravity regime via the Wetterich Equation—a tool pivotal to the ASQG framework for pinpointing fixed points indicative of asymptotic safety.

Results and Assertions

The paper furnishes a detailed examination of the flow of Newton’s constant and the cosmological constant under typical truncations, discerning the flow properties and locating the UV fixed point, which serves as an analogue to the Reuter fixed point, well-recognized in ASQG. Intriguingly, the authors underscore how the critical exponents associated with these

fixed points show that relevant directions exist in their flow analysis in three-dimensional spacetime with a density weight transformation corresponding to the choice $r = \frac{D - 4}{4D}$.

Implications and Future Directions

Ferrero and Thiemann's analysis posits meaningful implications, both theoretically and practically, for the understanding of quantum gravity and its quantization consistency. The proposal of canonical transformations deploying density-weighted metrics systems has the potential to refine the renormalization pathways, eliminating complexities inherent to measure factors on non-linear sigma models in gravity.

Looking ahead, an expansion of theory space may unveil further intricacies and possibly confirm the subtle effects of the canonical transformation on the renormalization group properties in broader truncations. Additionally, the opportunity to compare and contrast this approach with Lorentzian ASQG ventures, especially those involving foliated structures, may provide pivotal insights that refine current methodologies and lay groundwork for either concurrent or alternative interests in quantum gravity.

In summary, this paper serves as a noteworthy step into the intersection of CQG and ASQG, addressing fundamental concerns of time and observables within the quantum field, and offering a preliminary yet insightful glimpse into the practical translation of operator Hamiltonian evolution within Euclidean frameworks—a field that opens new vistas in the quest for a consistent and full-fledged quantum gravity theory.