Kevin Falls

A bstract: We study gauge dependence in proper-time renormalisation group flows for asymptotically safe quantum gravity. Working in an essential scheme, we use field redefinitions to separate redundant off-shell contributions from the on-shell running of physical couplings. We consider two approximations: in the first, we work on general backgrounds and project onto all terms with up to four derivatives, neglecting boundary terms; in the second, we work on a maximally symmetric background and retain all orders in the Ricci scalar. Although the flow equation depends on the gauge parameters, this dependence can be cancelled order by order once the redundant terms are absorbed by field redefinitions. In the four-derivative approximation, the dependence on the retained gauge parameter drops out of the beta function for Newton’s constant to third order in the coupling. On the sphere, the flow becomes independent of both parameters of the general gauge to all curvature orders, and all orders in the coupling. Crucially, this cancellation depends on the choice of regulator. We verify the mechanism explicitly at one loop, where the gauge-dependent fluctuation contributions are cancelled by the ghost sector on shell. The resulting essential flow displays a gauge-independent non-Gaussian fixed point, supporting the interpretation that universal information in quantum gravity is encoded in the on-shell essential sector.

A bstract: For the same quantum field theory distinct effective actions can be obtained by coupling sources to different choices of field variables. This is the same as considering effective actions for theories related by a change of variables and thus differ only by the values of so called inessential couplings. The effective actions will appear quite different since they generate correlation functions of different operators. Here we show that the effective actions are related by an implicit change of variables of the mean field, i.e. the argument of the effective action. Conversely, one can go the other way: by making a change of variables of the mean variable we obtain a new effective action which generates correlation functions of an implicitly defined composite field. The existence of the implicit transformations in both cases rests on the existence of solutions to initial value problems where the “time” parameter is an inessential coupling. Non-perturbatively the solutions may only exist for some non-zero amount of this time. However, at each order in perturbation theory one obtains linear equations which implies that unique solutions exist. We then show that scattering amplitudes are independent of inessential couplings without use of perturbation theory. For gauge theories we expect our correspondence to extend to effective actions which differ by a choice of gauge.

The detection of gravitational waves has intensified the need for efficient, high-precision modeling of the two-body problem in General Relativity. Current analytical methods, primarily the Post-Minkowskian and Post-Newtonian expansions, are inherently perturbative, while numerical relativity remains computationally expensive. In this Letter, we introduce a middle path: an exact renormalization group (RG) equation for classical gravitational systems. We demonstrate that our equation correctly reproduces the first three orders of the Post-Minkowskian expansion. Crucially, it provides a framework for non-perturbative approximations. As a first application, we show that our method efficiently recovers the 1PN two-body action, bypassing the need for complex three-graviton vertex calculations by leveraging the intrinsic nonlinearity of the RG flow. This establishes the exact RG as a powerful new tool for tackling strong-field dynamics in gravity.

We investigate the renormalisation of Einstein gravity using a novel subtraction scheme in dimensional regularisation. The one-loop beta function for Newton’s constant receives contributions from poles in even dimensions and can be mapped to the beta function obtained using a proper-time cutoff. Field redefinitions are used to remove off-shell contributions to the renormalisation group equations. To check the consistency of our approximations we use a general parametrisation of the metric fluctuation. Within truncations of the derivative expansion and the expansion in Newton’s constant, we show that the parametrisation dependence can be removed order by order. Going to all orders in the scalar curvature an all-order beta function for Newton’s constant is obtained that is independent of the parameterisation. The beta function vanishes at the Reuter fixed point and the critical exponent is in good agreement with non-perturbative calculations. Finally, we compare the critical exponent to the counterpart computed via Causal Dynamical Triangulations (CDT).

Much like the action, diffeomorphism invariance can be used to fix the form of the path integral measure in quantum gravity. Moreover, since there is a redundancy between what constitutes “the action” and what constitutes “the measure” one can always pick a minimal form of the latter. However, the authors of recent papers [1, 2] have advocated a form of the path integral measure for quantum gravity, proposed long ago by Fradkin and Vilkovisky, that is not invariant. This is easily seen since it depends explicitly on the g ⁰⁰ component of the inverse metric without being contracted to form a scalar. An equally non-invariant measure was proposed in [3]. As noted by their proponents, when these measures are used, certain divergences that typically appear are absent. However, the divergences that remain with the proposed measures are, unsurprisingly, neither diffeomorphism-invariant nor is the regulated effective action. We demonstrate this explicitly by computing the free scalar field contribution to the divergent part of the gravitational effective action using different measures and a proper-time cutoff. We support our findings with a thorough discussion of the path integral measure. In particular, we see how the contributions from the measure, obtained in a canonical setting, could be reinterpreted in a relational way compatible with diffeomorphism invariance.

We analyse the renormalisation group flow of quantum gravity at sixth order in the derivative expansion within the background field approximation. Non-linear field redefinitions are used to ensure that only essential couplings flow. Working within the universality class of General Relativity, with a vanishing cosmological constant, redundant couplings are fixed to their values at the Gaussian fixed point. This reduces the theory space to two dynamical essential couplings given by Newton's and the Goroff-Sagnotti coupling. Furthermore, it implements the condition that no extra degrees of freedom are present beyond those of General Relativity, in contrast to higher derivative theories and derivative expansions in a conventional renormalisation scheme. We find a unique ultraviolet fixed point with a single relevant direction and analyse the phase diagram of the theory. Our results suggest resilience of the gravitational Reuter fixed point under the inclusion of higher order curvature invariants and show several signs of near-perturbativity. The regulator dependence of our results is investigated in detail and shows that qualitative and quantitative features are robust to a large extent.

We propose a novel scheme for the exact renormalisation group motivated by the desire of reducing the complexity of practical computations. The key idea is to specify renormalisation conditions for all inessential couplings, leaving us with the task of computing only the flow of the essential ones. To achieve this aim, we utilise a renormalisation group equation for the effective average action which incorporates general non-linear field reparameterisations. A prominent feature of the scheme is that, apart from the renormalisation of the mass, the propagator evaluated at any constant value of the field maintains its unrenormalised form. Conceptually, the simplifications can be understood as providing a description based only on quantities that enter expressions for physical observables since the redundant, non-physical content is automatically disregarded. To exemplify the scheme’s utility, we investigate the Wilson-Fisher fixed point in three dimensions at order two in the derivative expansion. In this case, the scheme removes all order \partial^2 ∂ 2 operators apart from the canonical term. Further simplifications occur at higher orders in the derivative expansion. Although we concentrate on a minimal scheme that reduces the complexity of computations, we propose more general schemes where inessential couplings can be tuned to optimise a given approximation. We further discuss the applicability of the scheme to a broad range of physical theories.

We introduce an approach to compute the renormalisation group flow of relational observables in quantum gravity which evolve from their microscopic expressions towards the full quantum expectation value. This is achieved by using the composite operator formalism of the functional renormalisation group. These methods can be applied to a large class of relational observables within a derivative expansion for different physical coordinate systems. As a first application we consider four scalar fields coupled to gravity to represent the physical coordinate frame from which relational observables can be constructed. At leading order of the derivative expansion the observables are the inverse relational metric and the relational scalar curvature. We evaluate their scaling dimensions at the fixed point, both in the standard renormalisation group scheme and in the essential scheme. This represents the first steps to describe running observables within asymptotic safety; this treatment can be generalised to other observables constructed from different tensors and in different physical coordinate systems.

The non-perturbative renormalisation of quantum gravity is investigated allowing for the metric to be reparameterised along the RG flow, such that only the essential couplings constants are renormalised. This allows us to identify a universality class of quantum gravity which is guaranteed to be unitary, since the physical degrees of freedom are those of general relativity without matter and with a vanishing cosmological constant. Considering all diffeomorphism invariant operators with up to four derivatives, only Newton's constant is essential at the Gaussian infrared fixed point associated to the linearised Einstein--Hilbert action. The other inessential couplings can then be fixed to the values they take at the Gaussian fixed point along the RG flow within this universality class. In the ultraviolet, the corresponding beta function for Newton's constant vanishes at the interacting Reuter fixed point. The properties of the Reuter fixed point are stable between the Einstein--Hilbert approximation and the approximation including all diffeomorphism invariant four derivative terms in the flow equation. Our results suggest that Newton's constant is the only relevant essential coupling at the Reuter fixed point. Therefore, we conjecture that quantum Einstein gravity, the ultraviolet completion of Einstein's theory of general relativity in the asymptotic safety scenario, has no free parameters in the absence of matter and in particular predicts a vanishing cosmological constant.