arXiv subjects

Kevin Falls

Publications and source records attributed to Kevin Falls.

At least 19 recordsLinked to original sources

arXiv2026

Towards gauge independence in asymptotically safe quantum gravity

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.

hep-th↗

arXiv2025

Equivalence of effective actions

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.

hep-th↗

arXiv2025

Gauge invariant effective actions for dressed fields

A new fundamental form of the path integral for theories with local symmetry is introduced. It is utilised to construct effective actions that generate correlation functions of dressed fields in Yang-Mills theories and quantum gravity. The construction entails a novel BRST symmetric gauge fixing which imposes that the on-shell correlation functions are those of gauge invariant fields. We demonstrate that the effective actions are gauge and diffeomorphism invariant respectively, with appropriate transformations for the Faddeev-Popov ghosts and Nakanishi-Lautrup fields. As a consistency check, the on-shell one-loop effective actions are shown to take the expected gauge independent form. The effective action which satisfies Zinn-Justin's master equation is also gauge invariant with the anti-fields transforming accordingly. While our choice of gauge will in general be non-linear and non-local, we argue that these gauges are in fact stable under renormalisation if one allows for general renormalised field variables.

hep-th↗

arXiv2025

Path integral measures and diffeomorphism invariance

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 the recent papers arXiv:2412.14108, arXiv:2412.10194 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 arXiv:2009.00728. 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.

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arXiv2024

Asymptotic Safety within on-shell perturbation theory

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).

hep-th↗

arXiv2023

Robustness of the derivative expansion in Asymptotic Safety

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.

hep-th↗

arXiv2021

Relational observables in Asymptotically safe gravity

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.

hep-th↗

arXiv2021

Essential Quantum Einstein Gravity

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.

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arXiv2021

Essential renormalisation group

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 $\partial^2$ 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.

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arXiv2020

Background independent exact renormalisation

A geometric formulation of Wilson's exact renormalisation group is presented based on a gauge invariant ultraviolet regularisation scheme without the introduction of a background field. This allows for a manifestly background independent approach to quantum gravity and gauge theories in the continuum. The regularisation is a geometric variant of Slavnov's scheme consisting of a modified action, which suppresses high momentum modes, supplemented by Pauli-Villars determinants in the path integral measure. An exact renormalisation group flow equation for the Wilsonian effective action is derived by requiring that the path integral is invariant under a change in the cutoff scale while preserving quasi-locality. The renormalisation group flow is defined directly on the space of gauge invariant actions without the need to fix the gauge. We show that the one-loop beta function in Yang-Mills and the one-loop divergencies of General Relativity can be calculated without fixing the gauge. As a first non-perturbative application we find the form of the Yang-Mills beta function within a simple truncation of the Wilsonian effective action.

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arXiv2020

Towards the determination of the dimension of the critical surface in asymptotically safe gravity

We compute the beta functions of Higher Derivative Gravity within the Functional Renormalization Group approach, going beyond previously studied approximations. We find that the presence of a nontrivial Newtonian coupling induces, in addition to the free fixed point of the one-loop approximation, also two nontrivial fixed points, of which one has the right signs to be free from tachyons. Our results are consistent with earlier suggestions that the dimension of the critical surface for pure gravity is three.

hep-th↗

arXiv2019

Curvature dependence of quantum gravity

We investigate the phase diagram of quantum gravity with a vertex expansion about constantly-curved backgrounds. The graviton two- and three-point function are evaluated with a spectral sum on a sphere. We obtain, for the first time, curvature-dependent UV fixed point functions of the dynamical fluctuation couplings $g^*(R)$, $μ^*(R)$, and $λ_3^*(R)$, and the background $f(R)$-potential. Based on these fixed point functions we compute solutions to the quantum and the background equation of motion with and without Standard Model matter. We have checked that the solutions are robust against changes of the truncation.

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arXiv2019

Frame (In)equivalence in Quantum Field Theory and Cosmology

We revisit the question of frame equivalence in Quantum Field Theory in the presence of gravity, a situation of relevance for theories aiming to describe the early Universe dynamics and Inflation in particular. We show that in those cases, the path integral measure must be carefully defined and that the requirement of diffeomorphism invariance forces it to depend non-trivially on the fields. As a consequence, the measure will transform also non-trivially between different frames and it will induce a new finite contribution to the Quantum Effective Action that we name frame discriminant. This new contribution must be taken into account in order to asses the dynamics and physical consequences of a given theory. We apply our result to scalar-tensor theories described in the Einstein and Jordan frame, where we find that the frame discriminant can be thought as inducing a scale-invariant regularization scheme in the Jordan frame.

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arXiv2018

On de Sitter solutions in asymptotically safe $f(R)$ theories

The availability of scaling solutions in renormalisation group improved versions of cosmology are investigated in the high-energy limit. We adopt $f(R)$-type models of quantum gravity which display an interacting ultraviolet fixed point at shortest distances. Expanding the gravitational fixed point action to very high order in the curvature scalar, we detect a convergence-limiting singularity in the complex field plane. Resummation techniques including Padé approximants as well as infinite order approximations of the effective action are used to maximise the domain of validity. We find that the theory displays near de Sitter solutions as well as an anti-de Sitter solution in the UV whereas real de Sitter solutions, for small curvature, appear to be absent. The significance of our results for inflation, and implications for more general models of quantum gravity are discussed.

gr-qc↗

arXiv2018

Physical renormalization schemes and asymptotic safety in quantum gravity

The methods of the renormalization group and the $\varepsilon$-expansion are applied to quantum gravity revealing the existence of an asymptotically safe fixed point in spacetime dimensions higher than two. To facilitate this, physical renormalization schemes are exploited where the renormalization group flow equations take a form which is independent of the parameterisation of the physical degrees of freedom (i.e. the gauge fixing condition and the choice of field variables). Instead the flow equation depends on the anomalous dimensions of reference observables. In the presence of spacetime boundaries we find that the required balance between the Einstein-Hilbert action and Gibbons-Hawking-York boundary term is preserved by the beta functions. Exploiting the $\varepsilon$-expansion near two dimensions we consider Einstein gravity coupled to matter. Scheme independence is generically obscured by the loop-expansion due to breaking of two-dimensional Weyl invariance. In schemes which preserve two-dimensional Weyl invariance we avoid the loop expansion and find a unique ultra-violet (UV) fixed point. At this fixed point the anomalous dimensions are large and one must resum all loop orders to obtain the critical exponents. Performing the resummation a set of universal scaling dimensions are found. These scaling dimensions show that only a finite number of matter interactions are relevant. This is a strong indication that quantum gravity is renormalizable.

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arXiv2017

Conformal anomaly from gauge fields without gauge fixing

We show how the Weyl anomaly generated by gauge fields, can be computed from manifestly gauge invariant and diffeomorphism invariant exact renormalization group equations, without having to fix the gauge at any stage. Regularisation is provided by covariant higher derivatives and by embedding the Maxwell field into a spontaneously broken $U(1|1)$ supergauge theory. We first provide a realisation that leaves behind two versions of the original $U(1)$ gauge field, and then construct a manifestly $U(1|1)$ supergauge invariant flow equation which leaves behind only the original Maxwell field in the spontaneously broken regime.

hep-th↗

arXiv2016

Renormalization Group Equation for $f(R)$ gravity on hyperbolic spaces

We derive the flow equation for the gravitational effective average action in an $f(R)$ truncation on hyperbolic spacetimes using the exponential parametrization of the metric. In contrast to previous works on compact spaces, we are able to evaluate traces exactly using the optimised cutoff. This reveals in particular that all modes can be integrated out for a finite value of the cutoff due to a gap in the spectrum of the Laplacian, leading to the effective action. Studying polynomial solutions, we find poorer convergence than has been found on compact spacetimes even though at small curvature the equations only differ in the treatment of certain modes. In the vicinity of an asymptotically free fixed point, we find the universal beta function for the $R^2$ coupling and compute the corresponding effective action which involves an $R^2 \log R^2$ quantum correction.

hep-th↗

arXiv2016

Further evidence for asymptotic safety of quantum gravity

The asymptotic safety conjecture is examined for quantum gravity in four dimensions. Using the renormalisation group, we find evidence for an interacting UV fixed point for polynomial actions up to the 34th power in the Ricci scalar. The extrapolation to infinite polynomial order is given, and the self-consistency of the fixed point is established using a bootstrap test. All details of our analysis are provided. We also clarify further aspects such as stability, convergence, the role of boundary conditions, and a partial degeneracy of eigenvalues. Within this setting we find strong support for the conjecture.

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