Physical renormalization schemes and asymptotic safety in quantum gravity
aa r X i v : . [ h e p - t h ] J a n Physical renormalization schemes and asymptotic safety in quantum gravity
Kevin Falls
Institut f¨ur Theoretische Physik, University of Heidelberg, Philosophenweg 12, 62910 Heidelberg, Germany (Dated: August 11, 2017)The methods of the renormalization group and the ε -expansion are applied to quantum gravityrevealing the existence of an asymptotically safe fixed point in spacetime dimensions higher thantwo. To facilitate this, physical renormalization schemes are exploited where the renormalizationgroup flow equations take a form which is independent of the parameterisation of the physicaldegrees of freedom (i.e. the gauge fixing condition and the choice of field variables). Instead the flowequation depends on the anomalous dimensions of reference observables. In the presence of spacetimeboundaries 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 ε -expansion neartwo dimensions we consider Einstein gravity coupled to matter. Scheme independence is genericallyobscured by the loop-expansion due to breaking of two-dimensional Weyl invariance. In schemeswhich preserve two-dimensional Weyl invariance we avoid the loop expansion and find a uniqueultra-violet (UV) fixed point. At this fixed point the anomalous dimensions are large and one mustresum all loop orders to obtain the critical exponents. Performing the resummation a set of universalscaling dimensions are found. These scaling dimensions show that only a finite number of matterinteractions are relevant. This is a strong indication that quantum gravity is renormalisable. I. INTRODUCTION
It remains open problem to identify the theory of quantum gravity which Nature has chosen. Due tothe dimensionality of Newton’s constant G we know that a perturbative quantisation of general relativitydoes not lead to a predictive theory [1, 2]. However, this does not rule out the possibility that quantumversion of general relativity may be defined as a local quantum field theory. From the view point of therenormalization group (RG) [3–10], perturbation theory is just the expansion around the non-interactinglow energy fixed point G = 0; which is simply not the right starting point to formulate the fundamentaltheory. What we actually require is an interacting UV fixed point G = G ∗ = 0 where gravity can be definedas an asymptotically safe theory. At this point all reaction rates, and other dimensionless observables,remain finite as the UV cut-off is removed [11]. One may then evade the problem of perturbative non-renormalisability provided there are only a finite number of relevant interactions, i.e. interactions whichget stronger as energies decrease. Quantum field theories possessing such a fixed point have been shown toexist for interactions other than gravity [12, 13], with the recent example of gauge theories in four spacetimedimensions being of particular interest [14]. There is also an increasing amount of evidence in favour ofthis scenario for quantum gravity in four dimensions, coming from functional renormalization group studies[15–47], and lattice regularisations of quantum gravity [48–54]. Additionally, evidence for fixed points inhigher than four dimensions has also been found [55, 56] using the functional renormalization group. Forreviews on asymptotic safety see [57–63].A method to study asymptotic safety from within perturbation theory is provided by the ε -expansionaround two dimensions [64–80]. In this case one sets the spacetime dimension to D = 2 + ε where ε is asmall parameter. The one-loop beta function for Newton’s constant then takes the form β G = εG − b G , (1.1)where one expects b to be scheme independent since G is dimensionless in two dimensions. A non-trivialfixed point G ∗ = ε/b + O ( ε ), which can be made arbitrarily small, then presents itself. Going to higherloops the coefficient b should be replaced by the beta function β κ ( D →
2) for the inverse Newton’s constant κ = 1 /G obtained in the limit D → ε -expansion of the solution to β G ( G ∗ ) = 0 to find a fixed point in integer dimensions D > At this fixed point one would like to know the scaling dimensions θ of interactions to ascertain whether See e.g. [83] for the application of this idea to gauge theories in
D > renormalisability can be achieved; this being the case if only a finite number of the exponents θ have apositive real part.In exactly two dimensions such critical exponents can be obtained exactly, obeying the Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling relation [84–86], and the beta function is just given by the conformal anomaly.It was then shown by Kawai, Kitazawa and Ninomiya [68] that the KPZ scaling relation can be reproducedby starting with gravity in D = 2 + ε dimensions and taking a particular limit. However this limit does notcorrespond to a fixed point in higher dimensions.Now in the continuum approach to quantum gravity one typically has to choose how to parameterise thephysical degrees of freedom. If one calculates observables, i.e. diffeomorphism invariant quantities, thereshould be no dependence on this choice [87–89]. However, in explicit calculations, beta functions appearto depend on the parameterisation via the choice of gauge fixing condition and the choice of field variables[17, 19, 78, 90–93]. This problem leads to apparently scheme dependent value for b (see e.g. [91]) andthus calls into question the physical significance of the fixed point. To make matters worse one also findsa different beta function when the renormalization of boundary terms is considered [64, 65, 94], leading toan apparently inconsistent theory [95]. Further to this, going to two-loops appears to produce non-localdivergencies spoiling the renormalisability of the theory [67].Our hypothesis is that these problems arise from using renormalization schemes based on local correlationfunctions which are not themselves observables. Thus, to alleviate this issue one should use a physicalrenormalization scheme, where we renormalise physical observables directly, as was original proposed byWeinberg [11]. The purpose of this paper is to construct such schemes and then use them to resolve theproblem of scheme dependence. What we shall see is that generically in D > G is dimensionful in dimensions D > b , obtained in the two-dimensional limit.We observe that this problem has its roots in the observation made in [66], namely that the loop expansionclose to two dimensions is actually an expansion in G/ε . This has the consequence that b cannot be uniquelydetermined within a generic scheme and the scaling exponents θ have order one quantum corrections. Thekey insight is to observe that the G/ε expansion is a consequence of schemes breaking two-dimensional Weylinvariance. Using a physical scheme, based on observables that are Weyl invariant in the limit D →
2, avoidsthe expansion in
G/ε and allows for the identification of the fixed point. To calculate the scaling exponentsof dimensionful interactions one must then additionally resum the
G/ε expansion. After this resummationis performed one has the non-perturbative beta functions which do not suffer from scheme dependence.We now outline the rest of the paper. We begin by reviewing the formal definition of the functional measurefor quantum gravity in section II. Several important features are highlighted. In particular we stress thatthe measure takes diffeomorphism and reparameterisation invariant form which is unique up to an overallnormalisation. Furthermore the normalisation is fixed by requiring the absence of non-universal divergencies ∼ δ (0) in the continuum limit [96] (see section II C). The two-dimensional limit of the measure is discussedin section II D and we note that it can be taken in a non-singular fashion provided Newton’s couplingalso goes to zero as the limit is taken. In section III we discuss the origin of gauge and parameterisationdependancies when correlation functions are considered. We note that these dependencies can be removedby a field renormalization and that certain choices of gauge and/or parameterisation can be understoodas giving an implicit renormalization condition for observables. Following from this observation we definephysical renormalization schemes in section IV, giving the explicit example of schemes based on the volumeof spacetime and the volumes of its boundaries. We apply this scheme at one-loop and in general dimension D >
G/ε is avoided and the one-loop beta functions isexact. In section VII E we point out why G IR2D = − G ∗ is the IR fixed point of two-dimensional quantumgravity which is obtained from higher dimensions where ε is the IR regulator. We then use the method ofKawai, Kitazawa and Ninomiya [68] to resum the expansion in G/ε using dimensional regularisation andthe method of steepest descent. We can then show that the exact beta functions are scheme independentin the two dimensional limit. The explicit form of the non-perturbative scaling dimensions at the UV fixedpoint in
D >
Notation and conventions : The notation and conventions used in this paper are as follows.Greek letters from middle of the alphabet µ, ν... = 0 , ..., D − D is thedimension of spacetime which we take to be D >
2. Lowercase letters from the start of the latin alphabetare DeWitt indices a, b, c = { A, x } , { B, x } .... for the fields that parameterise the geometry, and the matterfields when they are present, with the uppercase letters denoting the components (e.g. a symmetric pairof spacetime indices A = ( µν ) which may be covariant or contravariant) and x denoting the spacetimecoordinates e.g. φ A ( x ) = g µν ( x ). Greek letters from the start of the alphabet α, β etc. are used for DeWittindices for the diffeomorphisms e.g. ξ α = ǫ µ ( x ). When we go to a parameterisation where gauge variantand gauge invariant fields are identified a = { ¯ a, α } where ¯ a runs over the gauge invariant components and α the gauge variant components. From the middle of the latin alphabet m, n, o = { M, x } , { N, x } .... areused for super-fields including Fadeev-Popov ghosts e.g. ϕ N ( x ) = { g µν ( x ) , η µ ( x ) , ¯ η ν ( x ) } . When we discussboundaries i, j, k, l will denote tangential indices and n normal coordinates (no confusion should occur withthe DeWitt notation). The covariant derivative with respect to the boundary metric γ ij is denoted with by | i.e γ ij | k = 0 and ∇ µ denotes a covariant derivative with respect to the bulk metric ∇ ρ g µν = 0.The Einstein sum rule is used throughout and is extended to imply an integral for DeWitt indices e.g. J a φ a ≡ Z d D x J A ( x ) φ A ( x ) , (1.2)and similarly for other indices. We also use a · to denote “matrix” multiplication( C · M ) a b ≡ C ac M cb ≡ Z d D y C AC ( x, y ) M CB ( y, z ) , ( C · φ ) a ≡ Z d D yC AB ( x, y ) φ B ( y ) (1.3)The notation det M ab denotes the determinant of the matrix M with components M ab and similarly forthe super-determinant sdet M ab (and similarly for other index sets.). We use commas and superscripts todenote functional derivatives e.g. F (1) a [ φ ] ≡ F ,a [ φ ] ≡ δδφ a F [ φ ] ≡ δδφ A ( x ) F [ φ ] . (1.4)We work in units where the reduced Planck’s constant ¯ h and the speed of light c are one. When weconsider renormalization group equations we will typically work in units of the cutoff scale Λ and thus itshould be understood that all fields and couplings are made dimensionless by the corresponding power ofΛ such that Λ does not appear explicitly in the equations. When we work in units of the cutoff we indicatedependence on Λ via the dimensionless RG time t = log(Λ / Λ ) (1.5)and derivatives with respect to the cut off will be denoted by ∂ t . When we work in dimensionful units Λwill appear explicitly in expressions and we use Λ ∂ Λ to denote a derivative. II. THE FUNCTIONAL INTEGRAL IN QUANTUM GRAVITY
In this paper we will consider euclidean quantum gravity with spacetime dimension
D > D → S Λ takes the Einstein-Hilbert form, S Λ ≈ S EH = − πG Z d D x √ g ( R − λ ) + ... (2.1)within a semi-classical regime where the cutoff scale Λ is sub-Planckian Λ ≪ M Pl ≡ G − D − . Here G denotes Newton’s constant and ¯ λ is the cosmological constant, which is related to the vacuum energy λ by λ ≡ ¯ λ/ (8 πG ). If the spacetime manifold involves boundaries the action should be supplemented by therequired boundary terms denoted by the ellipsis.Similarly to the action, the functional measure d M ( φ ) in the sub-Planckian regime should be determinedby the canonical quantisation of Einstein’s theory. We therefore have the functional integral Z = Z d M ( φ ) e − S EH [ φ ] , (2.2)where φ denotes the fields which are being integrated over. Here we have not yet introduced the gaugefixing and thus the measure still includes a formal factor of V − where V diff is the gauge volume which mustfactor out from the integral over φ . In this section we will not concern ourselves with the regularisationof (2.2) or the gauge fixing procedure. Instead the purpose of this section is to find an appropriate formalexpression for the measure before regularisation and gauge fixing. A. Geometry of geometries
The fields φ , on which both the action and measure depend, parameterise the (gauge variant) degrees offreedom. Here we assume that they are related to the metric g µν by an invertible relation g µν = g µν ( φ ) , φ A = φ A ( g µν ) , (2.3)the choice of which cannot affect the physics. Some typical choices for φ A are φ A = g µν , φ A = g µν , φ A = √ gg µν , (2.4)which are independent of any background field. With the introduction of a background metric ¯ g µν twopopular choices for the fields φ A are the linear and exponential parameterisations respectively: g µν = ¯ g µν + φ µν , g µν = ¯ g µρ ( e φ ) ρ ν , (2.5)where in the latter case we have the matrix exponential of a field tensor field φ . While the choice of φ is unphysical, the geometries which are being integrated over affect the functional integral at least atthe non-perturbative level. This observation motivates the use of the exponential parameterisation [97],since the positive definiteness of g µν is ensured even for large values of the field, whereas for the linearparameterisation this is not the case. However, at the perturbative level we do not expect this to be anissue; as we argue below in section II E.In a more general case we can consider variations of the metric δ n g µν = T µνA ...A n ( φ, ∂ µ φ, ... ) δφ A ...δφ A n , (2.6)where the coefficients T can depend on the dynamical fields and its derivatives as well as on the backgroundgeometry if present. Furthermore, in the most general case T are differential operators acting on thevariations δφ A . Here let us first assume that neither the transformation (2.3) nor the measure d M ( φ )involve spacetime derivatives.Since without any sources present the fields φ are just integration variables, Z is invariant under a changein the choice of field variables provided we take into account the Jacobian in the measure and re-write S [ φ ]in terms of the new variables. A useful point of view [98] is to consider the fields φ A ( x ) ≡ φ a as coordinateson the ‘space of geometries’ Φ, to which we associate a metric C ab [ φ ] with the line element δl = C ab [ φ ] δφ a δφ b , (2.7)which is invariant under a change of coordinates. Thus if we wish to use a different set of field variables φ ′ a which are related to the original variables φ a = φ a [ φ ′ ] the metric in the coordinate system corresponding to φ ′ a is given by C ′ ab [ φ ′ ] = C cd [ φ [ φ ′ ]] δφ c δφ ′ a δφ d δφ ′ b (2.8)We can write a covariant measure on Φ as d M [ φ ] = Y a dφ a (2 π ) / V − [ φ ] p | det C ab [ φ ] | , (2.9)where p | det C ab [ φ ] | provides the volume element and V diff [ φ ] is the volume of gauge orbit correspondingto diffeomorphisms which we take to be a scalar on Φ such that V ′ diff [ φ ′ ] = V diff [ φ [ φ ′ ]]. The Jacobianencountered by a change of variables is automatically taken into account by transforming C ab [ φ ] since Y a dφ a (2 π ) / V − [ φ ] p | det C ab [ φ ] | = Y a dφ ′ a (2 π ) / det δφ a δφ ′ b V − [ φ [ φ ′ ]] p | det C ab [ φ [ φ ′ ]] | (2.10)= Y a dφ ′ a (2 π ) / V ′− [ φ ′ ] q | det C ′ ab [ φ ′ ] | (2.11)Thus by specifying the form of C ab for a one set of field variables φ we can then determine the form of themeasure in any other set of variables φ ′ by determining the the components of C ′ ab . B. Determining the measure
In principle the metric C ab (or equivalently the measure) of any field can be determined by canonicalquantisation [96] or by invoking Becchi-Rouet-Stora-Tyutin (BRST) invariance [99]. In fact up to an overallnormalisation C ab can be determined by demanding that (2.7) is diffeomorphism invariant [100] whichcoincides with the BRST invariant form after gauge fixing. On the other hand, Fradkin and Vilkovisky[96] argue that C ab should be such that the strongest divergencies, which otherwise renormalise the vacuumenergy, are removed. They then claim [101] that this is can be achieved by a non-covariant factor of g entering C ab . However, Toms [100] argues that it is in fact the phase space metric that is non-covariant,leading to a covariant metric on Φ after integrating out the canonical momentum.Following Toms’ argument the ‘correct measure’ is that of Fradkin and Vilkovisky but without the factorsof g , which are replaced by mass scales µ and µ ǫ . This coincides with the BRST invariant measure ofFujikawa [99] fixing the measure up to an overall normalisation parameterised by µ and µ ǫ . We will employthis form below and then, following Fradkin and Vilkovisky, use the freedom to normalise the measure toremove the strongest divergencies.To take some simpler examples [100] we can consider the quantisation of a single scalar field s (with acanonical action) in curved space where the line element is given by δl = µ s Z d D x √ gδs ( x ) δs ( x ) , (2.12)which depends on the metric over spacetime via the volume element √ g and is invariant under a spacetimediffeomorphism where g µν transforms as a tensor and δs ( x ) as a scalar. For a vector field v µ we have δl = µ v Z d D x √ gg µν δv µ δv ν (2.13)involving again the metric tensor. In both cases the form is unique under the assumption that the metric C is ultra-local. Here the mass scales µ s and µ v are needed to ensure that the measure is dimensionless.We have written (2.12) and (2.13) in terms of the scalar field s and the vector v µ . However we can now usea different set of field variables while keeping δl invariant. For example instead of using a scalar s ( x ) wecould instead use a density ˜ s = g w/ s of weight w in which case the (2.12) is then given by [100] δl = µ s Z d D x √ g − w δ ˜ s ( x ) δ ˜ s ( x ) (2.14)the choice w = 1 / C ab becomes independent of g µν .In the case of a vector we can also choose to use densities ˜ v µ = g w/ v µ or use a one form v µ instead of acontravariant vector.Returning to the gravitation degrees of freedom themselves, if we choose φ A ( x ) = g µν ( x ) the metric onΦ is written in the DeWitt form: C ab δφ a δφ b = µ πG Z d D x √ g ( g µρ g νσ + g µσ g νρ + ag µν g ρσ ) δg µν δg ρσ . (2.15)where a is the DeWitt parameter. Here we will take a = − γ ij , then coincides with the DeWitt metric G ijkl = √ γ ( γ ik γ jl + γ il γ jk − γ ij γ kl ) appearingin the Hamiltonian: H = 116 πG Z Σ d D − y (cid:16) π ij G − ijkl π kl − √ γ R Σ (cid:17) (2.17)where π ij are the canonical momenta and R Σ is Ricci scalar on Σ. Equally, by quantising the theory in acovariant gauge the measure is determined by the part of the action involving two time derivatives [96]. Inparticular the metric (2.15) with (2.16) can be found via g C ab = µ δ S EH [ φ ] δ∂ φ a δ∂ φ b (2.18)in Feynman-’t Hooft gauge where the hessian of S EH is a minimal differential operator. Finally, Vilkovisky[102] also arrives at the same form via arguments based on the connection on Φ used to define a covariantfunctional derivative. We therefore take (2.15) with the DeWitt parameter (2.16) as defining the measure.For the diffeomorphisms we also need a measure in order to define the gauge volume V diff . Here againwe can choose any parameterisation we like for diffeomorphisms ξ α since they are only coordinates on thegauge orbit. To be concrete we consider an infinitesimal diffeomorphism g µν → g µν + ∇ µ ǫ ν + ∇ ν ǫ µ , (2.19)then the metric on the space of diffeomorphisms, in these coordinates, is δξ α δξ β G αβ = µ ǫ πG Z d D x √ gg µν ǫ µ ǫ ν , (2.20)giving the invariant measure for the gauge volume V diff = Z Y α dξ α (2 π ) / p det G αβ . (2.21)One can then choose a different parameterisation of the gauge orbit transforming G αβ appropriately.Some comments are in order. Here we have assumed that the line element (2.7) and transformations(2.3) do not involve derivatives which then determines the measure by diffeomorphism invariance up toan overall normalisation. However, with the reparameterisation invariant measure we can now make moregeneral, even non-local, field transformations. The important point is that the measure should be ultra-local in a parameterisation φ which leads to a local action second order in derivatives. Here we considerthe unregulated functional integral which is only a formal expression. Once we regulate the theory we willintroduce a cutoff scale Λ where we will regain the unregulated form of the measure (and/or action) onlyin the UV limit Λ → ∞ . C. Normalisation and local divergencies
Let’s return to the choice of measure and the renormalization of the vacuum energy. It is useful to quoteFradkin and Vilkovisky [96]:“It is essential for the present discussion that whichever definite, but unique, way of calculating the localmeasure and the local term in the functional integral is chosen, one will always obtain as a result thecancellation of divergent terms ∝ δ (0) by the local measure.”As we have defined it, the measure depends on the scales µ and µ ǫ and thus it is these that we must fixsuch that the strongest divergencies are removed. Ultimately they will be identified with the cut-off scale µ ∝ µ ǫ ∝ Λ when the continuum limit is taken. Let us define the relation between the two scales as µ ǫ = ζ ǫ µ (2.22)where we treat ζ ǫ as a parameter with the ratio between µ and Λ fixed. Then if we start with ζ ǫ = 1 theeffect of shifting the ration µ ǫ /µ → ζ ǫ will be to change the normalisation of the functional integral Z → e − D R d D xδ (0) log ζ ǫ Z d M ( φ ) e − S [ φ ] (2.23)in the continuum limit. This suggests that when the continuum limit is taken we should adjust ζ ǫ so that Z is finite and non-zero e.g. Z = 1. If this is not done then there is a factor involving R d D xδ (0) whichclearly has no geometrical interpretation, and can be understood as a breaking of general covariance. Toelaborate on this point, imagine we want to give meaning to the quantity Q x ζ ǫ we could do so by writingit as the determinant of some operator Y x ζ ǫ = det ˆ o (2.24)Written out in components we could then say this operator acts on a scalar like Z d D x ′ ˆ o ( x, x ′ ) s ( x ′ ) = ζ ǫ s ( x ) (2.25)which leads to (2.23). Now if we consider a diffeomorphism it is evident that ˆ o ( x, x ′ ) must transform asa scalar at x and a scalar density at x ′ , but a priori it knows nothing of the dynamical fields. So onemust introduce some auxiliary background structure or make some arbitrary choice for the field dependenceˆ o = ˆ o [ φ ].When the theory is regularised such divergencies will appear only when the limit Λ → ∞ is taken and theform of these divergencies will depend on ζ ǫ which now appears as a parameter of the regularisation scheme.In particular in the regulated theory which preserves diffeomorphism invariance the R d D xδ (0) appear asdivergence proportional to the dimensionful spacetime volume ∼ Z d D x √ g Λ D , (2.26)which appears to renormalise the dimensionful vacuum energy by a term proportional to Λ D . On the otherhand this must follow from some implicit choice of how the operator ˆ o depends on the dynamical fields in itsregulated form. Similarly if spacetime boundaries Σ are present we will get terms ∼ R Σ √ γd D − y Λ D − whichrenormalise a boundary volume term. One should then fix ζ ǫ (or more generally the overall normalisationof the measure) in order to remove such divergencies as the continuum limit is taken. This will be possiblesince such terms are always non-universal.We note that in the (causal) dynamical triangulation approaches to gravity such a parameter generallyneeds to be tuned to uncover phase transitions in four dimensions, either by including a discrete version of(2.23) in the euclidean version [54], or by introducing an anisotropy in the regularisation scheme for causaldynamical triangulations [51] (which was actually originally advocated by Fradkin and Vilkovisky [101]).The main point however is not that we must tune a non-universal parameter to obtain a continuum limit,rather we need to tune the parameter if the continuum action is to be of the Einstein-Hilbert form. D. The two-dimensional limit
Here we have assumed that the dimensionality of spacetime is greater than two. A key question iswhether two-dimensional quantum gravity can be recovered in a particular limit. In two dimensions theEinstein-Hilbert action with a vanishing cosmological constant ¯ λ = 0 is a topological invariant and theclassical theory also enjoys Weyl invariance in addition to diffeomorphism invariance. The Weyl invariancecan also be seen in the functional measure since (2.15) is degenerate in the limit D →
2. In particular if wedecompose the metric as g µν ( x ) = e σ ( x ) ˆ g µν ( x ) where ˆ g µν is a uni-modular metric with a fixed determinantand σ parameterises the conformal modes then (2.15) reads C ab δφ a δφ b = µ πG Z d D x √ g ((ˆ g µρ ˆ g νσ + ˆ g µσ ˆ g νρ ) δ ˆ g µν δ ˆ g ρσ − D ( D − δσδσ ) (2.27)which reveals that C ab has vanishing eigenvalues in two dimensions. Thus to take the limit D → G → G/ ( D −
2) fixed this limit can betaken since the total measure R d M ( φ ) is proportional to factors of G/ ( D − G ∝ ( D − E. Integration limits
Now we return to the question of integration limits; the important point is the following. Imagine wehave a standard integral of the form Z ab dφ √ G − e − G S ( φ ) , (2.28)where we can think of G as the small parameter in which the integral will be expanded in. To performsuch an integral in perturbation theory one first expands the field φ about a saddle point b < ¯ φ < a andcanonically normalises the fluctuations φ → ¯ φ + √ Gδφ . (2.29)After this the integral is of the form Z ( a − ¯ φ ) √ /G ( b − ¯ φ ) √ /G dδφ e − G S ( ¯ φ + √ Gδφ ) , (2.30)and we can proceed with the expansion order by order in G . This appears to depend on the limits a and b .On the other hand if G << Z ( a − ¯ φ ) √ /G ( b − ¯ φ ) √ /G dδφ e − G S ( φ ) ≈ Z ∞−∞ dδφ e − G S ( ¯ φ + √ Gδφ ) , (2.31)where the corrections are exponentially suppressed (i.e. by factors e − const . G ) and hence do not contributeto the asymptotic expansion in G . Evidently the same conclusion is reached at the level of the functionalintegral since it is just a multiple integral of the same form. Hence the perturbative expansion does notdepend on the integration limits for the fields φ ( x ). III. GAUGE AND PARAMETERISATION DEPENDENT BETA FUNCTIONSA. Legendre effective action
With the measure in place Z is manifestly gauge and field parameterisation invariant. The problems ofgauge and parameterisation dependence arise when we instead consider correlation functions which do notshare this property. The first step to obtain correlation functions is to add a gauge fixing action to S alongwith the corresponding Faddeev-Popov determinant which can be expressed in terms of ghost fields. Thisstep ensures that Z is unchanged and V diff can be factored out. To make this step implicitly let us simplyinclude the ghosts in the set of fields ϕ n e.g. ϕ n ≡ ϕ N ( x ) = { g µν , η µ , ¯ η ν } and denote the metric on thisenlarged field space by C nm . We then can put the functional integral in the Faddeev-Popov form: Z = Z Y n dϕ n (2 π ) / p | sdet C nm ( ϕ ) | e −S [ ϕ ] , (3.1)where S [ ϕ ] = S [ φ ] + S gf [ φ ] + S gh [ η, ¯ η, φ ] now depends on the gauge fixing condition and both S and themeasure are invariant under BRST transformations. A typical choice for the gauge fixing action is S gf [ φ ] = 132 πGα Z √ ¯ gF µ ( φ ) F µ ( φ ) , F µ ( φ ) = ¯ ∇ ν φ νµ −
12 ¯ ∇ µ φ νν , (3.2)where the barred quantities depend on the background metric ¯ g µν . Since Z is unchanged it is still indepen-dent of the choice of parameterisation and gauge. The dependence on these unphysical choices enters in thenext step in which we couple a source J n to the fields ϕ n to obtain e −W [ J ] = Z Y n dϕ n (2 π ) / p | sdet C nm ( ϕ ) | e −S [ ϕ ]+ J n ϕ n . (3.3)From here one defines the Legendre effective action which is related to W [ J ] by a Legendre transformationΓ[ ¯ ϕ ] = W [ J ] + ¯ ϕ n J n , (3.4)being a functional of the classical fields ¯ ϕ = h ϕ i J where the subscript denotes that the expectation valueis source dependent. The functional Γ[ ¯ ϕ ] is the generating functional for one-particle irreducible correla-tion functions. However with J = 0 the Legendre effective action is neither gauge nor parameterisationindependent. By differentiating the effective action we haveΓ (1) n [ ¯ ϕ ] = J n , (3.5)and consequently it is only when Γ[ ¯ ϕ ] is evaluated on a solution to the equation of motion that the sourceis zero. The off shell action Γ[ ¯ ϕ ] will therefore depend on both the gauge and the field parameterisation. B. Origin of gauge and parameterisation dependence
As mentioned in the introduction the renormalization of Newton’s constant at one-loop suffers fromunphysical dependencies on the gauge and parameterisation and furthermore acquires a different form whenobtained from bulk or boundary terms in the action. To trace the origin of these issues we now look athow beta functions are typically derived from Γ[ ¯ ϕ ] at the one-loop level. An important point here will beto rebut the claim of [103] that: the dependence of beta functions the on parameterisation is physicallyacceptable and due to the fact that the Jacobian in the path integral measure is not taken into account.Here we will automatically keep track of the Jacobian by transforming the field space metric C nm observingthat the dependence on the parameterisation is due to the source, rather than the measure.Employing the background field method [104] the one-loop effective action for gravity can be cast in agauge invariant (but not independent) formΓ[ g µν ] = S [ g µν ] + 12 STr log (cid:16) C − · S (2) (cid:17) (3.6)where Γ[ g µν ] is a gauge invariant functional of the metric g µν . In (3.6) we have combined the contributionfrom the measure and the Gaussian integrals. This expression is only formal since it still needs to be regulatedto remove divergencies. Beta functions can then be found by either demanding that Γ[ g µν ] is independentof the UV cutoff Λ or by introducing an IR cutoff k on which Γ will depend but S is independent of. Notethat these are just two different ways of formulating the renormalization group and typically Γ k is relatedto S Λ by a Legendre transformation [7, 105] at the exact level.Let’s now look into the structure of the operator C − · S (2) involved in the super-trace. This two pointfunction is made of the product of the inverse field space metric C − and the hessian S (2) nm ≡ S ,nm . First0let us assume for simplicity that the hessian in a convenient parameterisation is a second order minimaldifferential operator such that it takes the form: S (2) nm = c no ( −∇ δ om − E o m ) ≡ c no ∆ o m (3.7)where c nm is structure that appears in front of the Laplacian. The operator E depends on the curvature andthe cosmological constant. To understand the dependence of the hessian on the parameterisation we canconsider a different set of fields ˜ ϕ and find the corresponding hessian. Here we assume that the Jacobian isultra-local, i.e ϕ ( ˜ ϕ ) does not involve derivatives and as such the transformed hessian is again second orderin derivatives. One then observers that the hessians are related by:˜ S (2) nm = δϕ o δ ˜ ϕ n S (2) op δϕ p δ ˜ ϕ m + δϕ o δ ˜ ϕ n δ ˜ ϕ m S (1) o (3.8)where the second term vanishes on the equations of motion S (1) o ≡ S ,o = 0. On the other hand ˜ S (2) nm alsotakes the form (3.7) by replacing c → ˜ c and E → ˜ E . It follows that the coefficients ˜ c nm and c nm are relatedby ˜ c nm = δϕ r δ ˜ ϕ n c rs δϕ s δ ˜ ϕ m (3.9)and thus they transform as components of metric on field space, just like C nm .Now the way in which divergencies of (3.6) are typically regulated is to suppress the modes of ∆ definedin (3.7). However this regulates only the super-trace12 STr log (∆) (3.10)where we take units µ = 1 with ζ ǫ = 1. As a result there is an unregulated UV divergence ∼ STr log( C − · c ) = δ (0) Z d D x str log( C − · c ) (3.11)where we have performed the spacetime integral of the super-trace leaving the super-trace str over theindices A . Usually this divergence is simply neglected, which is justified only ifsdet C nm = sdet c nm , (3.12)otherwise we will be left with the divergence (3.11). However, for the BRST invariant functional measurei.e. that based on (2.15) and (2.20) one finds that C nm = c nm . For example the hessians for the metric andghosts are given by: δ S δg µν ( x ) δg ρσ ( y ) = C µν,ρσ ( −∇ + ... ) δ ( x − y ) , δ S δη µ ( x ) δ ¯ η ( y ) = G µν ( −∇ + ... ) δ ( x − y ) (3.13)with the tensor structures those of (2.15) and (2.20). Thus either one adopts the BRST invariant measurewhich leads to (3.12) or one has additional UV divergencies unregulated by cutoffs for the modes of theLaplace operator −∇ . This is inline with Fradkin and Vilkovisky’s [96] observation that the correct measureleads to the cancelation of (3.11).The main point here is that regulators of the Laplacian do not lead to beta functions dependent on themeasure (notwithstanding the field independent normalisation). To give an example we can add an IRregulator to the one loop expression (3.6) obtaining:Γ k [ g µν ] = S [ g µν ] + 12 STr log (cid:0) C − · c · (∆ + R k ( −∇ )) (cid:1) (3.14)where R k ( −∇ ) is a momentum dependent mass which depends on the IR cutoff scale k such IR modes p < k are suppressed. Then taking a k derivative we get the flow equation [6, 7] for the effective averageaction at one-loop k∂ k Γ k = 12 STr [ k∂ k R k · (∆ + R k ) − ] , (3.15)1which is studied in [103]. Since both C and c fall out of this equation the beta functions will not be dependon them. This shows that the dependence of the beta function on the parameterisation is not due to thefunctional measure; there is no dependence of k∂ k Γ k on C . The dependence on the parameterisation insteadarises due to the second term in the RHS of (3.8) which vanishes on shell.It is also the off shell corrections to the hessian that introduces the gauge dependence since it is only onshell that the hessian S (2) is guaranteed to be gauge invariant. To see this we observe that the action isinvariant under φ a → φ a + L aα ξ α (3.16)for infinitesimal ξ α and hence S ,a L aα ξ α = 0 . (3.17)Taking a further derivative of the above equation we have S (2) ab L aα ξ α + S (1) a L aα,b ξ α = 0 (3.18)which shows that generically only when the equations of motion apply will the quadratic action be gaugeinvariant. This lack of gauge invariance then leads to the dependence of Γ on the gauge fixing condition(see e.g. [27, 106]). We can then conclude that the results of [103] still hold even after taking into accountthe Jacobian and that it is the off shell nature of the calculations that is responsible for both gauge andparameterisation dependence. C. Gauge and parameterisation dependent beta functions
Let’s now discuss how the gauge and parameterisation dependence affects the beta functions. First wecan consider the form of the at the UV divergencies which remain relevant in the limit D →
2. They takethe form such that the RG equation in terms of dimensionful quantities form is given by:Λ ∂ Λ S Λ = Z d D x √ g (cid:20) B Λ D + Λ D − (cid:18) B R + ¯ B (cid:18) R − DD − λ (cid:19)(cid:19)(cid:21) (3.19)where Λ is the UV cut-off scale upon which the couplings depend. The coefficient of the trace of the Einsteinequations ¯ B depends on the gauge and parameterisation whereas the coefficients B and B are indepen-dent of these choices. More generally, while the one-loop divergencies will be gauge and parameterisationdependent off-shell, going on-shell unphysical dependencies will cancel. For example such cancellations havebeen shown explicitly for scalar-tensor theories performed in the Jordon and Einstein frames where thedivergencies in the two frames differ off-shell but agree once the equations of motion are exploited [92, 93].To obtain the beta functions one goes to dimensionless variables in units of Λ such that the metric andcouplings are now dimensionless (here we take coordinates to be dimensionless [ x µ ] = 0 from the start andthus g µν has dimensions of length squared). To take into account for the modified dimensions we then let g µν scale as − ∂ t g µν = − g µν , (3.20)which is consistent with g µν depending on t = log(Λ / Λ ) as g µν ( t ) = e t g µν (0) . (3.21)After the transforming to dimensionless quantities we replace the lhs of (3.19) with ddt S [ g µν ] = ∂ t S [ g µν ] + 2 Z d D xg µν ( x ) δδg µν ( x ) S [ g µν ] (3.22)where ∂ t S [ g µν ] ≡ ∂ t | g µν S [ g µν ] is the partial derivative and thus acts only on the couplings and not on themetric and the second term implements the dilatation step of the RG transformation. In turn in rhs of2(3.19) we simply set Λ = 1 since we are now working in units of the cutoff scale. Then the beta functionfor the vacuum energy, obtained by keeping track of the terms in (3.19), is given by β λ = − Dλ + B − DD − B πGλ , (3.23)which depends on the gauge and parameterisation via the last term. The beta function for Newtons constantdepends on whether we use the bulk or boundary term to obtain the running. In the bulk case we have β G = ( D − G + 16 π ( B + ¯ B ) G . (3.24)Let’s also note that if we were to obtain the beta function from the boundary action we would obtain abeta function of the form β G = ( D − G + 8 πA G (3.25)where for the relative factor between the bulk action and the Gibbons-Hawking-York (GHY) boundary termto be preserved we require that A = 2( B + ¯ B ). In four dimensions it was shown [95] that A = 2 B whenemploying diffeomorphism invariant boundary conditions. Thus it appears that the relative factor betweenbulk and boundary is not preserved also due to the term proportional to the equations of motion. D. Field renormalization and ‘preferred’ parameterisations
From the analysis of this section we can conclude that it is the presence of a source term which leads togauge and parameterisation dependent beta functions at one-loop. Furthermore this may also be respon-sible for the bulk and boundary terms being renormalised differently. However if we now allow for moregeneral dependence of the field on the cut-off scale we can generate other terms in the renormalization of S proportional to the equations of motion. For example we can allow for an anomalous dimension of themetric by replacing (3.20) with − ∂ t g µν = ( − η g ) g µν , (3.26)which is equivalent to replacing (3.21) by g µν ( t ) = e (2 − η g ) t g µν (0) , (3.27)which leads to an RG equation of the form ∂ t S Λ + (2 − η g ) Z d D xg µν ( x ) δδg µν ( x ) S [ g µν ] = Z d D x √ g (cid:20) B + (cid:18) B R + ¯ B (cid:18) R − DD − λ (cid:19)(cid:19)(cid:21) (3.28)Thus by choosing the anomalous dimension η g to be non vanishing we can effectively modify the coefficient¯ B , as well as all other coefficients which multiply terms proportional to the equations of motion. One canthen use this freedom to satisfy a renormalization condition leading gauge and parameterisation indepen-dence beta functions [66, 107, 108]. Investigating asymptotic safety near two dimensions, using dimensionalregularisation, two such renormalization schemes have been proposed [64, 66].The first proposal [64] considered the theory where the cosmological constant was set to zero but theboundary terms were retained. There it was argued that the renormalization of Newton’s constant shouldbe determined by divergencies proportional to Z d D x √ gR + 2 Z Σ d D − y √ γK (3.29)which includes the boundary term, rather than the coefficient of R d D x √ gR alone, which vanishes on shell.In this case the beta function for Newton’s coupling would be identified with (3.25). Unfortunately theboundary conditions used in these calculations were not diffeomorphism invariant and therefore the physicalsignificance of the result β G = ( D − G − G + O ( G ) , (3.30)3obtained this way [64, 65, 94] is questionable. Furthermore redefining the metric will also affect the boundaryterms so it is not clear that this method is fully consistent. Nonetheless the general philosophy behind thisproposal, which highlights the importance of the boundary terms, should play a role in alleviating the issuessurrounding bulk and boundary terms.In [66] a different renormalization condition was used involving the cosmological constant and othermatter couplings where the boundary terms were absent. In this case one can use a redefinition of themetric to remove the terms proportional to the equations of motion in (3.19) such that a coupling, e.g.the cosmological constant, is not renormalised. For the case of the cosmological constant one enforces indimensionless form ∂ t λ = − Dλ , (3.31)where λ is dimensionless in units of Λ. When using dimensional regularisation only the logarithmic termsare retained (and hence B = 0) this results in the beta function β G = ( D − G − G . (3.32)It is clear that the requirement (3.31) is not unique and one could choose a different condition. Indeedrequiring that different couplings g , other than the cosmological constant, are not renormalised will lead toa different beta function which depends on this choice [66].More recently several works [91, 103, 106, 109–111] investigating the the gauge and parameterisation ofbeta function for Newton’s constant have noted that the dependencies can be minimised by certain choices.In particular one can make use of partial gauge fixings and/or parameterisations such that all additionaldependencies are either removed or otherwise satisfy a principle of minimum sensitivity [112]. To understandwhy these choices have this effect follows from observing that the beta functions for G and λ can be obtainedassuming the trace-free Einstein equations hold. As a result the beta functions depend on the gauge andparameterisation due only to the source for the conformal factor J σ ( x ) ∝ R − DD − λ . (3.33)Here is the field σ ( x ) parameterises conformal fluctuations of the metric such that g µν = f ( σ )ˆ g µν where thedeterminant of ˆ g µν is fixed and f ( σ ) is a function. The dependence on the source can then be removed eitherby gauge fixing the conformal factor [91, 110] or picking a parameterisation [106] where the trace of theEinstein equations does not enter S (2) . In the latter case this can be achieved by choosing a parameterisationwhere the volume element is linear in the field σ ( x ) √ g ( x ) − √ ¯ g ( x ) = σ ( x ) (3.34)with √ ¯ g ( x ) denoting the background volume element. The effect of these choices is that no terms involvingthe equation of motion appear and hence ¯ B = 0 . Furthermore there is no dependence on the cosmologicalconstant which leads to a real scaling critical exponents for the vacuum energy given simply by its canonicaldimension θ λ = D , (3.35)obtained in this case by simply differentiating the beta function ∂β λ ∂λ = − θ λ . After removing the non-universal divergencies ∼ Λ D these gauges then automatically satisfy the renormalization condition that thevacuum energy λ is not renormalised (3.31) in pure gravity. They therefore lead generically to the betafunction (3.32). In the case of gauge fixing the conformal factor this can only be done to remove the non-constant modes. As a result thetrace of the equation of motion will enter beta functions via the contribution of constant mode ∂ µ σ = 0. This does not affect¯ B in dimensions D > ∼ ¯ B ( R − λ ) in four dimensions. σ ( x ) we obtain the volume ofspacetime (cid:28)Z d D x √ g ( x ) (cid:29) = Z d D x √ ¯ g ( x ) + Z d D x h σ ( x ) i . (3.36)One can then understand the classical scaling exponent (3.35) as expressing the trivial scaling of the space-time volume − ∂ t (cid:28)Z d D x √ g ( x ) (cid:29) = − D (cid:28)Z d D x √ g ( x ) (cid:29) , (3.37)and thus for these parameterisation there is an implicit renormalization condition that fixes the scalingof an observable. However (3.37) only applies if η g = 0 and thus allowing for a non-vanishing anomalousdimension of the metric then leads to a nontrivial scaling dimension for the volume. IV. PHYSICAL RENORMALIZATION SCHEMES
Following from the discussion in the last section we now wish to define physical renormalization schemes where, instead of any explicit dependence on the parameterisation of the physical degrees of freedom, therenormalization group equations are written in terms of the scaling dimensions of observables. This can beachieved by giving renormalization conditions which relates the renormalization of the fields to the scalingof a set of reference observables . As a result one can maintain both reparameterisation and diffeomorphisminvariance (provided of course that they are not broken by regularisation scheme).To achieve our aim we work with a regulated functional integral in the absence of sources Z = Z d M Λ [ φ ] e − S Λ [ φ ] , (4.1)where the measure and the action depend on the UV cut off scale Λ as indicated by the subscript. Thisdependence should be such that Z itself is independent of the scale Λ, while modes p ≫ Λ are suppressedin the functional integral. The RG flow of S Λ will then generally encode the coarse graining of degreesof freedom, renormalization of the fields and a dilatation [114]. Provided we do not break reparameteri-sation invariance we can then avoid dependence of the choice of parameterisation. Instead, by utilising aphysical renormalization scheme, beta functions will depend on the anomalous dimensions of the referenceobservables.By an observable here we mean a function of the fields which is invariant under the symmetries ofthe theory. For scalar field theories in flat spacetime the local fields s ( x ) are observables and as suchthere renormalization group equations can depend on the anomalous dimension of the fields η s themselveswithout breaking any symmetry of the theory. If we would consider an O ( N ) symmetric theory then s n isnot an observable and instead the anomalous dimension must refer to ̺ = P n s n s n which is an observableand hence renormalization group equations can depend on η ̺ without breaking the O ( N ) symmetry. Sinceobservables are necessarily non-local in quantum gravity we cannot simply identify reference observable witha local function of the fields. Instead here we shall consider observables which are formed by integratingover spacetime or its boundary Σ O = Z d D x √ gO ( x ) , O = Z Σ d D − y √ γO Σ ( y ) (4.2) Similar ideas have been explored in the context of lattice quantisation of quantum gravity [113]. O ( x ) or O Σ ( y ) which are local functions of the fields. Additionally we can formobservables by taking functions of observables of the form (4.2). In physical renormalization schemes thebeta functions will depend on the anomalous dimensions which should be self consistently determined atfixed points since there they correspond to universal critical exponents. A. Volumes as the reference observables
Let us now give one specific example of such a scheme which we will exploit in the following two sections.Here we consider the case where we have a compact spacetime manifold with disconnected boundaries Σ m .Then we have classical observables consisting of functions O ( V , V , V , ... ) , (4.3)of the spacetime volume V ≡ R d D x √ g and the volumes of the boundaries V m ≡ R Σ m d D − y √ γ . Here theobservables (4.3) will be the reference observables which form the basis of the scheme. To this end weconsider the renormalization condition − Λ ∂ Λ hO ( V , V , V , ... ) i = 0 , (4.4)such that the expectation values of the observables (4.3) are renormalization group invariants in the absenceof any renormalization or dilation of the fields. This condition can then be understood as a restriction ofthe RG flow of the Wilsonian effective action which takes the form S = λ (Λ) V + X m ρ m (Λ) V m + X n g n O n , (4.5)with the coupling constants λ and ρ m corresponding to the different volumes respectively and O n denotingthe set of all other terms in the action with coupling constants g n . In particular the renormalizationcondition (4.4) can be expressed as the requirement that the RG flow of S is independent of the couplings λ and ρ m , ∂∂λ Λ ∂ Λ S = 0 = ∂∂ρ m Λ ∂ Λ S . (4.6)This follows since then the RG flow of the couplings λ and ρ m decouples from the flow of all other couplings g n such that the solution to a flow of the type (4.6) involves λ = λ (Λ ) + Z log(Λ / Λ )0 dt Y ( t ; g n (Λ )) , ρ m = ρ m (Λ ) + Z log(Λ / Λ )0 dt y m ( t ; g n (Λ )) (4.7)where Y ( t ; g n (Λ ) = Λ ∂ Λ λ and y m ( t ; g n (Λ )) = Λ ∂ Λ ρ m are determined from the flow of the essentialcouplings and Λ is an arbitrary reference renormalization scale where the boundary conditions for the floware set. We then observe that the these couplings are linear in λ (Λ ) and ρ m (Λ ) whereas the couplings g n will be independent of λ (Λ ) and ρ m (Λ ). Next note that the functional integral can be viewed as afunction of the renormalised couplings Z = Z ( λ (Λ ) , ρ m (Λ ) , g n (Λ )) (4.8)which generates the expectation values of observables (4.3) by taking derivatives with respect to λ (Λ ) and ρ m (Λ ). For example we obtain the expectation value of the volume via − ∂∂λ (Λ ) log Z ( λ (Λ ) , ρ m (Λ ) , g n (Λ )) = hVi . (4.9) From now on we drop the subscript Λ on the action Wilsonian S . ∂ Λ Z ( λ (Λ ) , ρ m (Λ ) , g n (Λ )) = 0 . (4.10)However taking derivatives with respect to the couplings g n (Λ ) will not generate the corresponding observ-able. Thus while the scaling properties of the observables (4.3) will be trivial the scaling of observables O n will receive quantum corrections.So far we have assumed that the fields do not receive any anomalous scaling and we have not takenthe step of rescaling the fields by the cutoff to implement the dilatation step of the RG transformation.To regain generality we have to allow for φ a to transform under an RG transformation. Without anyrenormalization of the field the transformation is just a dilatation as in (3.20). In this case the scaling of the hO ( V , V , V , ... ) i would just give the canonical mass dimension of the observables fixing the scaling of theobservables upon which are renormalization scheme is based. This then limits our search for fixed pointsunnecessarily [115]. To undo this restriction we can allow for a more general ‘scaling’ of the field whichinvolves quantum corrections to (3.20) taking the form − ∂ t φ a = d a [ φ ] , (4.11)where d a [ φ ] is some field redefinition φ a → φ a − d a [ φ ] δ ΛΛ , (4.12)which can be quite general in principle. Here we will assume for the most part that the transformation(4.11) is a dilatation plus some anomalous scaling given by − ∂ t g µν = ( − η g ) g µν (4.13)where the anomalous dimension η g = η g ( G ) should vanish at the gaussian fixed point η g (0) = 0 for D > − ∂ t (cid:28)Z d D x √ g (cid:29) = d V (cid:28)Z d D x √ g (cid:29) , d V ≡ − D + η V = − D + 12 Dη g . (4.14)and similarly for the boundary volumes we have − ∂ t (cid:28)Z d d y √ γ (cid:29) = d V (cid:28)Z d D − y √ γ (cid:29) , d V ≡ − D + 1 + η V = − D + 1 + 12 ( D − η g . (4.15)If we do not restrict the form of d a [ φ ] a general expression for the scaling of the observables (4.3) willthen be given by − ∂ t hO ( V , V , V , ... ) i = (cid:28) d a δδφ a O ( V , V , V , ... ) (cid:29) . (4.16)Let us note that this expression for the scaling of the observables has no dependence on the gauge or theparameterisation of the fields. This follows since the averages are being taken without any source term inthe functional integral and since d a transforms as a vector on Φ. The flow equation should then be of thegeneral form ∂ t S = d a δδφ a S + F{ S } , with ∂∂λ F{ S } = 0 = ∂∂ρ m F{ S } , (4.17)where F{ S } is the part of the flow equation which represents the coarse graining step of the RG transfor-mation, which depends on the action as indicted by the brackets. The first term on the rhs of (4.17) allowsfor general field redefinitions (4.11) which involves a dilatation plus quantum corrections which are of order G . Thus while the flow equations will now depend on d a [ φ ] its relation to observables is known. One thenexpects that in order to find fixed points where ∂ t S ∗ [ φ ] = 0 we should self consistently determine d a ∗ [ φ ]leading to a discrete set of physical fixed points as is the case for scalar field theories [116].7 B. General physical schemes
In the next two sections we will employ the renormalization scheme based volumes in
D > O m with coupling J m then we can imposethat − ∂ t hO m i = (cid:28) d a [ φ ] δδφ a O m (cid:29) (4.18)which leads to a flow equation of the form ∂ t S = d a δδφ a S + F{ S } , with ∂∂J m F{ S } = 0 , (4.19)Close to two dimensions we will exploit a general set of schemes based on observables of different dimen-sionality. As we shall see this becomes essential to uncover the unique fixed point. Furthermore it is verynatural to consider all observables which appear as terms in the action as reference observables. This wayone can spot when scheme dependence is broken by an approximation.Let us finally note that at the exact level any scheme which is not of the form (4.19) but has the form ∂ t S = ˜ d a δδφ a S + ˜ F{ S } (4.20)can still be brought into the form (4.19). This will be the case since generically ˜ F{ S } and F{ S } will differby a term proportional to the equation of motion˜ F{ S } = F{ S } + ∆ d a δδφ a S (4.21)and thus ˜ d a → d a − ∆ d a restores scheme independence at the exact level. This applies equally to the caseswhere ˜ F{ S } is some other physical scheme (i.e independent of some couplings ˜ J m ) or to generic ‘unphysicalschemes’. Thus at the exact level scheme independence should be preserved [117] but, when approximationsare made, it may not be possible to see this if information in ∆ d a [ φ ] has been neglected. V. ONE-LOOP CALCULATION ON A CLOSED MANIFOLD
We now consider the case where there are no boundaries present to determine the one-loop running of thevacuum energy and Newton’s constant using our renormalization scheme based on the spacetime volume.The functional integral takes the form: Z = V − , Λ Z Y a dφ a (2 π ) / q | det C Λ ab ( φ ) | exp (cid:26) − λ Z d D x √ g + 116 πG Z d D x √ gR + ... (cid:27) , (5.1)where the ellipsis denotes terms which enter as loop-corrections not present in the initial action. Theregularisation will be implemented by a modification of the measure V − C ab ( φ ) → V − , Λ C Λ ab ( φ ). Theregulated measure is required both to suppress modes p ≫ Λ and to ensure the renormalization condition(4.6). Here we do not include the gauge fixing and ghosts in (5.1) and will instead factor out the gaugevolume via a change of variables in the functional integral as we detail in the appendices A and B. Ageneralisation of (5.1) in the the presence of spacetime boundaries will be given in section VI.
A. Perturbative expansion and regularisation
To compute Z to leading order in G we make the split φ a = ¯ φ a + δφ a , (5.2)8expanding the integrand of (5.1) around the saddle point ¯ φ = ¯ φ ( λ, G ) which depends on the couplings. Itfollows that the saddle point geometry must be an Einstein space where the Ricci curvature R µν ( ¯ φ ) = g µν ( ¯ φ ) 16 πGD − λ , (5.3)depends explicitly on the couplings. Since λ is related to the curvature we can then avoid counter terms inthe RG flow that depend on λ and hence satisfy (4.6) by renormalising curvature dependent terms instead.This allows us to implement (4.6) at each order in perturbation theory if we do not include any anomalousdimension for the metric.To obtain the one-loop quantum corrections we have to compute the Gaussian integral over the gaugeinvariant modes by first extracting the gauge orbit from the integral over the gauge variant fields. This canbe done by fixing the gauge and is most easily achieved by adopting the Feynman-’t Hooft gauge (3.2) where α = 1. However it is possible to factor out the gauge orbit without fixing the gauge [98, 118–120] but insteadusing the freedom to pick coordinates φ a which split the field into physical and gauge degrees of freedom.Gauge independence is then just reflected in the fact that appropriate coordinate systems, correspondingto different gauges, are just related by transformations with a trivial Jacobian. This procedure is outlinedin Appendix A and the resulting determinants, along with the Gaussian integrals, are evaluated explicitlyin Appendix B. The same result can be obtained from the standard Faddeev-Popov gauge fixing procedure[106] apart from a complete treatment of zero modes which we treat here in detail (see Appendix D and[121]). The final result is manifestly gauge independent and is invariant under field reparameterisations: − log Z = S Λ [ ¯ φ ] + 12 Tr log(∆ /µ ) − Tr ′ log(∆ /µ ǫ ) + log Ω( µ ǫ ) . (5.4)where here all quantities evaluated at the saddle point (5.3). The differential operators ∆ and ∆ act onvectors and symmetric tensors respectively and are given by∆ ǫ µ = (cid:18) −∇ − RD (cid:19) ǫ µ , ∆ h µν = −∇ h µν − R µ ρ ν σ h ρσ , . (5.5)The prime indicates that the zero modes should be removed from the vector trace. These correspond toKilling vectors i.e. the subgroup of diffeomorphisms H which are isometries of the saddle point geometry¯ φ A . The invariant volume Ω( µ ) on H (given explicitly by (D1) in Appendix D) then appears in the lastterm of (5.4) to ensure these modes are removed from the functional integral.Since (5.4) is divergent we need to regulate the traces. Our regularisation procedure is implemented atthe level of the measure via a modification of the field space metrics C ab and G αβ which implements aproper-time regularisation. Working in dimensionful units the explicit form of the regulated measure canbe expressed in terms of the metric C Λ ab δφ a δφ b = 132 πG Z d D x √ g (cid:0) g µα g νβ + g µβ g να − g µν g αβ (cid:1) δg µν ∆ e γ (∆ / Λ ) δg αβ , (5.6)while the metric on the space of diffeomorphisms (2.20) is replaced by G αβ ξ α ξ β = 116 πG Z d D x √ gg µν δǫ µ ∆ e γ (∆ / (Λ ζ ǫ ) δǫ ν , (5.7)where: γ ( z ) ≡ Z ∞ dss e − sz , (5.8)is the incomplete gamma function. Here the measures depends on the dynamical fields φ rather than thesaddle point geometry which is necessary for the renormalization condition (4.6) for S [ φ ].This regularisation ensures that Z is UV regulated at one-loop order, in particular it has the effect toreplace (5.4) by the regulated expression − log Z = S Λ [ ¯ φ ] − (cid:18)
12 Tr γ (∆ / Λ ) − Tr ′ γ (∆ / (Λ ζ ǫ )) (cid:19) + log Ω( ζ ǫ Λ e − γ E / ) . (5.9)9where all field dependent quantities are evaluated on the source dependent saddle point and here γ E isEuler’s constant. We then observe that for low momentum modes − γ (∆ / Λ →
0) = log( e γ E ∆ / Λ ) , − γ (∆ / (Λ ζ ǫ )) = log( e γ E ∆ / (Λ ζ ǫ )) , (5.10)which is of the form (5.4) with µ = Λ e − γ E , (5.11)and µ ǫ = ζ ǫ µ . For high momentum modes we have − γ (∆ / Λ → ∞ ) = Λ ∆ e − ∆ / Λ , − γ (∆ / (Λ ζ ǫ ) → ∞ ) = Λ ζ ǫ ∆ e − ∆ / (Λ ζ ǫ ) , (5.12)which vanishes exponentially quickly such that Z is finite. As such the modified measure regulates theone-loop divergencies while introducing the cut-off scale Λ. Sending Λ → ∞ the measure returns to theunregulated form as required. B. One-loop flow equation
We now want to calculate the RG flow of S [ φ ] where we will incorporate a renormalization of the fieldsand a dilation. Let us denote the overall volume element in modified functional measure as M = V − , Λ q | det C Λ ab | , (5.13)then it is straight forward to show that before renormalization of the fields we haveΛ ∂ Λ M = (cid:16) Tr [ e − ∆ / Λ ] − [ e − ∆ / (Λ ζ ǫ ) ] (cid:17) M + O ( G ) (5.14)with the right hand side given by the trace of the heat kernels. Here we made use of the scaling property(D3) of Ω, which implies Λ ∂ Λ log Ω(Λ ζ ǫ e − γ E / ) = 2 N KV , to absorb this contribution into the vector traceby dropping the prime. As a result the flow equation is unaffected by the number of Killing vectors.Now when we go to scaled and renormalised fields we absorb Λ into the fields and use dimensionlesscouplings such that G and λ now denote couplings in units of Λ. Then we have that the measure for thescaled and renormalised fields scales according to ddt M ≡ ∂ t M − d a δδφ a M = (cid:18) Tr [ e − ∆ ] − [ e − ∆ /ζ ǫ ] + δd a δφ a (cid:19) M + O ( G ) (5.15)where the last term accounts for the the Jacobian picked up when transforming to the scaled and renor-malised fields and we again drop terms of order G . Then we note that exact RG equations follows from[117]: ∂ t ( M e − S ) = δδφ a (Ψ a M e − S ) (5.16)for some choice of Ψ a giving different schemes. The invariance of Z follows since the integral of (5.16) iszero. Note that this implicitly sets the boundary of integration for the functional integral since we musthave that Ψ a M e − S vanishes on the boundary. Here set Ψ a = d a to obtain the one-loop flow equation ∂ t S = d a δδφ a S + Tr [ e − ∆ ] − [ e − ∆ /ζ ǫ ] . (5.17)which is of the form (4.17) with F = Tr [ e − ∆ ] − [ e − ∆ /ζ ǫ ] , (5.18)0and (4.16) follows by integrating by parts. Note that in principle any term proportional to the equationof motion can be removed from (5.17) by a specific choice of d a however the repercussion of such a choiceis to induce a non-trivial scaling (4.16) for observables which depend on the volumes. For our choice ofregularisation the flow equation has the form of a proper-time flow but with the additional term that accountsfor the renormalization of the fields. Proper-time flows have been studied previously in the context ofasymptotic safety [122, 123]. Here we stress that these flow equations only regulate the one-loop divergencies.Later we will exploit dimensional regularisation to go beyond one-loop.The point to recognise is that in the one-loop approximation we can choose any regulator which regulatesthe gaussian integral which is performed at the saddle point. This decides that the differential operatorswhich appear in (5.6) and (5.7) are given by (5.5) when evaluated for the saddle point geometry φ = ¯ φ .The additional renormalization condition (4.6) then decides that for general φ they are independent of thevacuum energy. It is then ensured that the dependence of Λ ∂ Λ S on λ comes from the first term in (5.17)and hence d a is related to the anomalous scaling of the volume via (4.16). We could add to the differentialoperators (5.5) terms involving the trace-free Ricci tensor S µν ≡ R µν − D g µν R since for the saddle point S µν ( ¯ φ ) = 0. These won’t modify the renormalization of Newton’s coupling however. C. One-loop beta functions
Expanding the heat kernel for the operators (5.5) in the early-time expansion we obtainTr [ e − ∆ ] − [ e − ∆ ] = D ( D + 1) − Dζ Dǫ (4 π ) D Z d D x √ g + 16 (cid:0) D ( D + 1) − − (2 D + 12) ζ D − ǫ (cid:1) (4 π ) D Z d D x √ gR + ... . (5.19)Then acting the dilatation operator on the action and allowing for an anomalous scaling of the metric (4.13)we have d a δδφ a S = − ( − η g ) 12 (cid:18) Dλ Z d D x √ g − ( D −
2) 116 πG Z d D x √ gR (cid:19) (5.20)where η g is the anomalous dimension of the metric. The flow equation (5.17) then leads to the beta functionsfor the dimensionless couplings β G = ( D − (cid:16) − η V D (cid:17) G − (cid:0) D ( D + 1) − − (2 D + 12) ζ D − ǫ (cid:1) (4 π ) D − G , (5.21) β λ = ( − D + η V ) λ + (cid:18) D ( D + 1) − Dζ Dǫ (cid:19) π ) D , (5.22)which are completely independent on the gauge or parameterisation and instead are written in terms ofthe anomalous scaling dimension of the volume η V given by (4.14). Note that since η V must vanish at theGaussian fixed point it must be order G η V = Gη V , + ... , . (5.23)where η V , is a constant. For ζ ǫ = 1 and η V = 0 the beta functions (5.21) agree with [106]. Here we seethat the beta functions take a more general form in terms of the anomalous dimension and the measureparameter ζ ǫ . Note that in the limit D → ζ ǫ . Ultimately the value of ζ ǫ should be fixed in the continuum limit. If we only consider the one-loopbeta functions its value should be such that the constant term in β λ vanishes which leads to the value ζ crit ǫ = 14 D (1 + D ) D (5.24)which is of order one for all 2 < D < ∞ and is given by ζ ∗ ǫ = √ in the limit D →
2. For this choice of ζ ǫ there exists a fixed point for which λ = 0.1 D. Discussion
The beta functions (5.21) and (5.22) are independent of any gauge fixing parameters and the parameter-isation of the quantum fields by virtue of our approach based on observables. However they do depend onthe anomalous dimension of the reference observable which we have chosen to be the spacetime volume. Thesituation here is similar to that of scalar field theories where renormalization group equations will depend onthe anomalous dimension of the fields. While at fixed points we expect that anomalous dimensions shouldbe scheme independent, provided they do not correspond to the scaling of a redundant operator [124] , awayfrom fixed points the anomalous dimensions are scheme dependent and as such they appear in the one-loopbeta functions as undetermined parameters. In section VII we shall see how the anomalous dimensions canbe determined at the the UV fixed point in D = 2 + ε dimensions.Given the dependence of the one-loop beta functions on η V one may ask whether there is a quantity whichis independent of η V . Since the volume is dimensionful one can in principle only measure its ratio with another dimensionful scale to form a dimensionless number. This suggests that we consider the spacetimevolume measured in Planck units ˜ V = G − DD − V , (5.25)then if we consider its scaling we obtain − ∂ t h ˜ Vi = DD − β G G h ˜ Vi + ( − D + η V ) h ˜ Vi = − DD − (cid:0) D ( D + 1) − − (2 D + 12) ζ D − ǫ (cid:1) (4 π ) D − G h ˜ Vi , (5.26)which is indeed independent of the anomalous dimension of the volume. VI. AMPLITUDES AND THE RENORMALIZATION OF BOUNDARY TERMS
In the preceding section we assumed that the spacetime manifold had no boundary. We now wish toconsider the case where we have a boundary which allows for us to compute amplitudes h φ | φ i = Z [ φ , φ ] , (6.1)where φ and φ denote boundary data which constrains the fields on the two boundaries Σ and Σ .Provided these boundary conditions are diffeomorphism invariant they correspond to different quantumstates and Z [ φ , φ ] constitutes a physical observable i.e. an amplitude in the physical Hilbert space.Subject to these boundary conditions the action must be supplemented with boundary terms [126, 127]such that the action has a meaningful variational principle and amplitudes have the required compositionproperties [ ? ]. This typically leads to a requirement that the bulk and boundary terms be interrelated. A. Action and boundary conditions
Quantum gravity on manifolds with boundaries faces a problem [128] due to the generic lack of diffeomor-phism invariant boundary conditions which lead to a well defined heat kernel for differential operators, suchas ∆ and ∆ . However, such boundary conditions [129, 130] do exist for geometries where the extrinsiccurvature K ij on the boundary Σ takes the form K ij = 1 D − K γ ij , ∂ i K = 0 , (6.2) Recall that a redundant operator corresponds to any coupling that is an eigen-perturbation of a fixed point which can beremoved by a field redefinition. For a discussion on redundant operators in the context of quantum gravity see [125]. i, j etc. denote tangential coordinates, γ ij is the induced metric and K = γ ij K ij . Explicitly theseboundary conditions are given by [95, 129, 130]: h in = 0 = ǫ n (6.3)˙ ǫ i − K ji ǫ j = 0 (6.4)˙ h nn + Kh nn − K ij h ij = 0 (6.5)˙ h ij − K ij h nn = 0 (6.6)where the dot is a normal derivative and n denotes the normal components components of tensors h µν = δφ A and vectors ǫ µ on which ∆ and ∆ act. One can explicitly check that these boundary conditions are gaugeinvariant under the transformation h µν → h µν + ∇ µ ǫ ν + ∇ ν ǫ µ (6.7)provided K ij takes the form (6.2). Some important results concerning the application of these boundaryconditions as well our conventions are given in Appendix E.When we make loop expansion of the amplitude (6.1) the boundary conditions (6.3) are to be imposedon fluctuation fields δφ A = h µν where the saddle point ¯ φ A is a geometry with extrinsic curvature (6.2).We therefore seek an action which has an extremum for such a geometry while giving rise to the linearisedEinstein equations for δφ A . To this end we consider the action S = λ Z d D x √ g − πG (cid:18)Z d D x √ gR + 2 Z Σ d D − y √ γK (cid:19) + ρ Z Σ d D − y √ γ (6.8)where we have included the Gibbons-Hawking-York (GHY) boundary term as well as a boundary termcorresponding to the volume of the boundary (here we include a single boundary for simplicity). Expandingthe action via g µν → g µν + h µν we obtain: S = S [ g µν ] + Z d D x √ g E µν h µν + 12 Z d D − y √ γ (cid:18) ργ ij + 2 116 πG ( K ij − Kγ ij ) (cid:19) h ij + ... where E µν = πG R µν − πG Rg µν + λg µν denotes the Einstein field equations. One then observers thatthe boundary terms vanishes provided the extrinsic curvature evaluated on the saddle point ¯ φ A is given by(6.2) with K = D − D − πGρ , (6.9)whereas the bulk term vanishes for a solution to the Einstein field equations.Computing the action to quadratic order in the fluctuation h µν around this background and applying theboundary conditions (6.3) (along with the identities given in Appendix E) one finds that all boundary termsvanish and we obtain the linearised Einstein field equations for h µν . The hessian is gauge invariant havingthe same form as the one obtained without a boundary, in particular we recover (B11), (B12) and (B13).If instead of (6.8) a different relative coefficient for the GHY term is chosen the hessian involves involvesboundary terms and hence we cannot use such an action to derive the linearised Einstein equations aroundan on shell background. B. Functional integral
It follows that we may generalise our one-loop calculation including the boundary terms with the functionalintegral now given by: Z [ φ , φ ] = V − , Λ Z Y a dφ a (2 π ) / q det C Λ ab ( φ ) exp (cid:26) − λ Z d D x √ g − ρ Z Σ d D y √ γ − ρ Z Σ d D y √ γ + 116 πG (cid:18)Z d D x √ gR + 2 Z Σ d D − y √ γK + 2 Z Σ d D − y √ γK (cid:19) + ... (cid:27) (6.10)3Where we include two separate boundaries to give the interpretation of W = log Z [ φ , φ ] as a an amplitudewith the total boundary being the disjoint union Σ = Σ ∪ Σ . To compute Z at one-loop we proceed asbefore but now the saddle point geometry has extrinsic curvature (6.2) with K Σ , ( ¯ φ ) = D − D − πGρ , , (6.11)dependent on the couplings. Our requirement that the counter terms do not involve the couplings ρ and ρ can be satisfied by renormalising terms which depend on K ( φ ) in a similar manner to how the dependenceon λ is evaded. It follows from (6.11) that the boundary data φ and φ corresponds to defining φ A , = ¯ φ A , + δφ A , (6.12)and requiring that the backgrounds ¯ φ , have extrinsic curvature (6.2) with (6.11) fixing the constantbackground K on each boundary and the boundary conditions of the fluctuations given by (6.3). Sinceby varying ρ and ρ we can set different values K Σ and K Σ we have access to a two parameter familyof amplitudes. Importantly the steps needed to calculate Z to one-loop, detailed in Appendix B, can becarried out with the boundaries present. The non-local operators appearing in the measure are then definedwith the boundary conditions (6.3). C. One-loop RG flow with boundary terms
The flow equation then takes the form (5.17) but where now the heat kernels are subject to the boundaryconditions (6.3) leading to an RG flow for the boundary terms. To satisfy the renormalization condition(4.6) the heat kernel traces depend on K ( φ ) and thus have no off shell dependence on ρ or ρ such that theflow equation is of the form (4.17). We note that strictly the heat kernel traces can only be evaluated when(6.2) applies and therefore we cannot identify terms involving the trace free part of the extrinsic curvature K T ij ≡ K ij − D − Kγ ij . However the first term in (5.17) can be used to produce any term proportional to K T ij and this anyway does not affect the renormalization of Newton’s constant.Utilising the early-time heat kernel expansion on a manifold with a boundary [131, 132] we find the flowof the action S with the boundaries present. Explicitly for F we findTr [ e − ∆ ] − [ e − ∆ ] = D ( D + 1) − Dζ Dǫ (4 π ) D Z d D x √ g + 1(4 π ) D Z Σ d D − y √ γ √ π (cid:0) D − D − − D + 4 ζ D − ǫ (cid:1) (6.13)+ 16 D ( D + 1) − − (2 D + 12) ζ D − ǫ (4 π ) D (cid:18)Z d D x √ gR + 2 Z Σ d D − y √ γK (cid:19) + ... where we see that the required balance between the GHY term and the Einstein-Hilbert action is preserved.This result can be anticipated from the results of [95] where it was shown that the required balance holds forthe on shell Legendre effective action in D = 4. There the result did not lead to a consistent picture sincethe balance needs to hold also off shell to identify the beta function. Here we see the required balance holdsoff shell. This is presumably the case since we have been careful not to break diffeomorphism invariance inderiving the RG equation (5.17). Allowing for a field renormalization (4.13), the balance is also preservedfollowing the fact that both terms have the same canonical dimension. As such the beta function forNewton’s constant is given by (5.21) derived either from the bulk or boundary action.The renormalization of the boundary volumes is given by β ρ m = ( − D + 1 + η V ) ρ m + 1(4 π ) D √ π (cid:0) D − D − − D + 4 ζ D − ǫ (cid:1) . (6.14)Let us note that we cannot put the constant term to zero if we also demand that the constant term for β λ isabsent. However, this is just a short coming of our regularisation scheme. We could add more parametersby normalising different components of the fields differently or by including matter fields (or even auxiliaryfields) and adjusting their normalisation. Once this is done we can also remove the constant term from(6.14) and will have a fixed point for ρ ∗ m = 0.4 VII. THE ε -EXPANSION IN QUANTUM GRAVITY As discussed in the introduction the ε -expansion in quantum gravity appears to give a fixed point forNewton’s constant as an expansion in ε . However as first discussed in [66] the loop-expansion is genericallyan expansion in G/ε rather than in G . As such the ε -expansion is not as one would naively expect.To understand this first recall that in two dimensions the Einstein-Hilbert action with a vanishing cosmo-logical constant is a topological invariant. In consequence the theory in two dimensions is invariant underboth diffeomorphisms and Weyl transformations g µν → Ω( x ) − g µν . Nonetheless when the limit D → s is of the form (see (B13)): S (2) ss = − D ( D − D − πG √ g ( −∇ + ... , (7.1)which appears to vanish in the limit D →
2, leading to a singular propagator. However the functionalmeasure also involves the factor − ( D − D − πG and hence to perform the perturbative expansion one shouldcanonically normalise s by s → s − πGD ( D − D − s (7.2)which removes the singular behaviour from the propagator. In consequence the vertices will have factors of p G/ε not √ G and hence the perturbative expansion is really an expansion in G/ε ≪ G . Note that, since in the limit D → s is the one gauge invariant degree offreedom, all other contributions to the renormalization originate from the measure. Consequently there isno expansion in G itself.Now the important point to realise is that if we impose that the theory should be Weyl invariant in thelimit D → ψ where in two dimensions the reference observable O is invariant under g µν → Ω( x ) − g µν , ψ → Ω( x ) d ψ ψ , (7.4)where d ψ is the dimension of the field. Such an observable is provided by a four-fermion– n S -scalar interactionsince for fermions d ψ = ( D − / d ψ = ( D − / D → G/ε are encountered and the beta function obtained at one-loop will already tell us where thefixed point β ( G ∗ ) = 0 lies.In this section we will investigate the fixed point near two dimensions and calculated the critical exponents.We closely follow the previous work [66, 68] where dimensional regularisation was used. In these works itis pointed out that there occurs an over subtraction since the one-loop counter term for the conformalfluctuations is of order O ( ε ) i.e. ( k ) ε ε √ gR ∼ (1 + log( k ) ε ) ∂ µ s∂ µ s (7.5)where here k is the IR renormalization scale. Thus one subtracts a finite term rather than a pole 1 /ε .In [68] a non-standard counter term was included in order to evade this perceived issue and in [70] (andsubsequent works [71–77]) diffeomorphism invariance was sacrificed for the same reason. Here we do notperceive this as a problem since it is just a consequence of diffeomorphism invariance of action and therenormalization group invariance of the functional integral. Furthermore we want to renormalise gravity inhigher dimensions where of course these are real divergencies.5 A. Physical schemes and matter interactions near two dimensions
In order to determine the beta function for Newton’s constant near two dimensions and the scalingdimensions of various observables, we now consider a more general set of physical renormalization schemesbased matter self interactions (or masses) O [ g µν , ψ ] = Z d D x √ g L int ( ψ ) , (7.6)which appears in the action with a coupling constant g . If we denote by d the classical scaling dimensionof O then the scaling dimension will general receive an anomalous correction due to gravity d = d + η . We then consider the action S = S EH [ g µν ] + S ψ [ g µν , ψ ] + g Z d D x √ g L int ( ψ ) (7.7)where S ψ is the kinetic part of the matter field action which is conformally coupled: the action is given by S [ ψ, g µν ] = 12 Z d x √ g N S X n =1 Z S n ψ S, n ( −∇ ) ψ S, n + N D X n =1 iZ F n ¯ ψ F, n / ∇ ψ F, n (7.8)with / ∇ denoting the Dirac operator. The central charge of the matter is given by c ψ = N D + N S where N D is the number of Dirac fermions and N S is the number of scalars. We will not consider boundaries in thissection.The case we have been studying up to this point is L ( g µν , ψ ) = 1 where O = V and g = λ . If we nowconsider a path integral with the interaction O instead of the cosmological constant term we can generaliseour RG scheme. In particular we can consider the flow equation which takes the form (4.19) but where weimpose − ∂ t hOi = (cid:28) d a δδφ a O (cid:29) g =0 (7.9)where the field content φ = { g µν , ψ } now includes the matter fields ψ . It follows that we should impose ∂∂ g F = 0 (7.10)on the coarse graining part of the flow equation. Furthermore we impose that the Z F n = 1 = Z S n suchthat no wave function renormalization of the matter sector is generated by gravity (we will neglect therenormalization of the matter interactions when G = 0). This can be achieved by introducing dimensionlessmatter fields ψ = g − d ψ / (2 D ) ˆ ψ (7.11)such that L ( ψ ) = √ g − − d /D L ( ˆ ψ ) (7.12)where d = − D (1 − ∆ ) and imposing that ˆ ψ has dimensions d ˆ ψ = 0 also when quantum corrections areincluded along with the condition ∂∂Z F n F = 0 = ∂∂Z S n F . (7.13) Here we do not include a subscript for the dimensions corresponding to O to avoid clutter; it should be understood that d = d O , g = g O etc. φ = { g µν , ˆ ψ } we have that d a δδφ a = d g Z d D x g µν ( x ) δδg µν ( x ) , (7.14)and thus all scaling dimensions are encoded in the metric. We can then write down a one-loop flow equationclose to two dimensions using the proper-time regulator. As we show in appendix C, the only modificationto the flow equation near two dimensions is to replace F with F = Tr [ e − ∆ ] − [ e − ∆ ] + Tr [ e − ( −∇ − d R ) ] − Tr [ e − ( −∇ + R ) ] + matter contributions (7.15)where the extra terms follow from the modification of the way gravity couples to an operator of generaldimension d . This involves gauge invariant scalar s which couples to the matter interactions which pro-ducing the third term. For d = − s is cancelled by the fourth term which arisesfrom the measure. When matter with an interaction d = − η g ( G ) = − d η ( G ) . (7.16)One observes that for general d keeping η g small would require η/d ∝ G if the expansion was in G , on theother hand the expansion is in Gε and hence we expect η/d ∝ Gε . B. One-loop beta functions
Within the schemes with reference observable of dimension d we obtain the one-loop beta function β G = ε (cid:18) ηd (cid:19) G −
23 (25 + 3 d − c ψ ) G (7.17)where the first term comes from first term of (4.19) and the second term comes from (7.15). The secondterm was first found in [66] using dimensional regularisation where η was set to zero. The beta function forthe interaction couplings are given by β g = ( d + η ) g . (7.18)Now we note that in any given scheme η is unfixed by the beta functions. However if we fix the anomalousdimension η in a single scheme corresponding to a particular value of d we will determine all over anomalousdimensions. Equivalently we can express the beta function for Newton’s constant in the form β G = εG + 23 ( c g + c ψ ) G (7.19)where c g can be thought of as the central charge for the gravitational degrees of freedom. Then comparingthe two beta functions we arrive at the one-loop anomalous dimensions given by η = 23 d ( c g + 3 d + 25) Gε (7.20)which depends on the number c g which remains undetermined. Comparing with (7.16) we see that theanomalous dimension for the metric is small only if either c g = − d −
25 or
G/ε is small. Since we knowthat in a generic scheme the expansion is in Gε this leaves the value of c g undetermined without furtherinsight.7 C. Higher loops and the UV fixed point
Now if we were to go to higher loop orders the beta function for Newton’s constant will be given by β G = ε (cid:16) − η g (cid:17) G −
23 (25 + 3 d − c ψ ) G + G ( b ( d ) Gε + b ( d ) G ε + ... ) (7.21)where the coefficients b n will depend on d . Let us now consider the case where the reference observable isWeyl invariant in two dimensions which means that d = aε + O ( ε ) (7.22)for some constant a . In this case all of the coefficients b n will vanish for ε →
0. This is seen most easily byexploiting the dimensionless parameterisation of the matter fields (7.11) and using the conformal gauge forthe gravitational degrees of freedom g µν = e √ − πG/εσ ˆ g µν (7.23)where ˆ g µν is gauge fixed up to topological fluctuations. In this parameterisation the Einstein-Hilbert actionbecomes the canonically normalised Liouville action : S [ φ ] = − πG Z d D x p ˆ g h ˆ R (cid:16) ε p − πG/εσ (cid:17) − πG ˆ g µν ∂ µ σ∂ ν σ i + g Z d D x p ˆ g L ( ˆ ψ ) + S ψ [ˆ g µν , ˆ ψ ] + O ( ε ) . (7.24)As such the integral over the gravitational degrees of freedom becomes gaussian , while decoupling fromthe Weyl invariant reference observable. Under the reasonable assumption that all to loop orders thethe evaluation of the functional integral is independent of the parameterisation of the physical degrees offreedom, (remember there is no source term present) this is just a convenient choice of coordinates. Theimportant point is that by using Weyl invariant reference observables to define the renormalization scheme,we guarantee that the higher-loop coefficients in (7.21) vanish.In this case the loop expansion is no longer an expansion in G/ε and hence to keep the anomalousdimension of the metric small we require that η g ∝ G . As a result the beta function for Newton’s constantis given by β G = ( D − G −
23 (25 − c ψ ) G (7.25)with all higher loop terms being zero in the limit ε →
0. This beta function agrees with the beta functioncomputed in exactly two dimensions using Liouville theory [79, 85, 86, 133]. From (7.25) we see that thereexists a UV fixed point at G ∗ = 32 D − − c ψ ) , (7.26)where we observe that G ∗ is positive for N S + N D <
25. Although the fixed point (7.26) has been foundpreviously [66], here we observe that (7.26) is not an approximation. In particular it is exact when weexploit dimensional regularisation which sets all non-universal terms in β G − εG , that vanish for ε →
0, tozero. As such the fixed point exists for all dimensions
D > ε -expansion for the fixed point onlyhas a linear term. More generally the two-dimensional limit of the Einstein-Hilbert action with G ∼ ε is related to the covariant Polyakovaction [80] which reduces to the Liouville action in the conformal gauge Strictly the measure makes the integral non-gaussian but the “vertices” which enter at two-loops and beyond lead only tonon-universal divergencies which are set to zero using dimensional regularisation. G in the weakly coupled phase 0 ≤ G ≤ G ∗ can be expressed in terms of the RG time t withthe reference scale taken to be the IR Planck mass Λ = M Pl > G ( t ) = e t ( D − /G ∗ e t ( D − . (7.27)Where for all values of M Pl > M − .It should be remarked that a consequence of the one-loop exactness of the beta function implies that theEinstein-Hilbert action scales canonically − ∂ t (cid:28)Z d D x √ gR (cid:29) = − ( D − (cid:28)Z d D x √ gR (cid:29) (7.28)and thus it is possible to consider the Einstein-Hilbert action itself as the reference observable. This wouldnot be the case if the higher loops did not vanish. Since R d D x √ gR vanishes on-shell and appears as aneigen-perturbation of the fixed point it is seen to be a redundant operator at the UV fixed point [125]. Thisreflects the fact that the fixed point action itself is not scheme independent and we have just chosen thenatural scheme, i.e. dimensional regularisation, where the action is of the Einstein-Hilbert form. In otherschemes the action can take a different form even though the universal critical exponents should be thesame. D. One-loop anomalous dimensions
Let us stress that although we exploited a particular scheme to obtain the exact beta function schemeindependence is restored as the exact level. In particular the exact one-loop beta functions can be made toagree and thus by comparing the expression for the beta functions (7.25) and (7.21) in different schemes wecan determine the anomalous for observables with d ∼ O (1). Comparing the one-loop expressions (7.17)and (7.25) we can then infer that the one-loop anomalous dimensions for an observable of dimension d aregiven by η = 2 d Gε + O ( G /ε ) (7.29)and that at the fixed point (7.26) we obtain η ∗ = 3 d − c ψ + O (1 / (25 − c ψ ) ) . (7.30)However, since this calculation involves breaking Weyl invariance we have to resum the loop-expansion in G/ε in order to obtain the leading order critical exponents in the ε -expansion. E. Two-dimensional quantum gravity
An important question is whether two-dimensional gravity can be obtained from the ε → G = − G ∗ and then taking the limit. In fact there is a goodreason for this since G = − G ∗ is nothing but the IR fixed point in two dimensions when c ψ <
25. Let usnow explain how this comes about.First we observe that the two-dimensional beta function for Newton’s coupling is given by β G = −
23 (25 − c ψ ) G (7.31)which for c ψ <
25 has a UV fixed point at G → +0 and an IR fixed point at G → −
0. On the otherhand for
D > G → +0 in the limit ε →
0. Now the key9point to realise is that, from the two-dimensional point of view, ε > G = − ε − c ψ )3 (7.32)such that the IR divergence occurs for ε →
0. Thus one observes that the bare coupling is sent to G → − G ∗ thus we find that the IR fixed point in D = 2 is at G IR2D = − G ∗ (7.33)in the limit ε →
0. Note that this is also a fixed point of (7.25) but only when the two-dimensional limit istaken. This suggests that we evaluate η at G = G IR2D to obtain the scaling exponents in two-dimensionalquantum gravity. As we showed in section II D the fact that G IR2D ∝ ε means that the measure in thislimit is not singular. On the other hand if we keep G fixed and take D → F. Non-perturbative calculation
We now wish to calculate the anomalous scaling dimensions η for observables with a non-vanishing classicaldimension d in the limit D → G/ ( D −
2) constant. This holds at the fixed points G = G ∗ and G = G IR2D where the scaling dimensions correspond to the scaling exponents at the UV fixed point in
D > d = 0 we break Weylinvariance the loop expansion is an expansion in G/ε and to obtain non-perturbative critical exponents onemust resum this series. On the other hand the critical exponents in two dimensions are known exactly andare given by [85, 86] d IR2D ≡ − β = − − c ψ ) 1 − q − c ψ ) (∆ − d ( D = 2) ≡ − − ∆ ) (7.35)for the classical dimensionality of the observable. Equivalently by denoting the scaling dimension of thevolume by α ≡ β | ∆ =0 the relative scaling dimension ∆ ≡ − β/α satisfies the KPZ relation [84]:∆ − ∆ = 6 α − c ψ ∆(1 − ∆) . (7.36)If we now take the one-loop approximation we evaluate (7.29) at G = G IR2D to obtain − β = − − ∆ ) −
12 (1 − ∆ ) − c ψ + O (1 / (25 − c ψ ) ) (7.37)which agrees with the exact result to this order. In [68] it was shown that the exact critical exponents(7.34) can be obtained by re-summing the loop expansion for G = G . However it is straightforward toperform the same calculation for general G/ε ∼ O (1) and therefore to obtain the critical exponents at theUV fixed point G = G ∗ . Since in dimensional regularisation all quantum corrections will be evaluated intwo dimensions we will use ∆ defined by (7.35) to express this two dimensional classical scaling dimensionand d for the D -dimensional classical scaling dimension which we retain only at tree-level. See [79] for a calculation of these exponents using the functional renormalization group. g µν = (cid:16) ε σ (cid:17) ε ˆ g µν (7.38)where ˆ g µν is a metric with unit determinant which we then gauge fix. This parameterisation takes the formof an exponential in the limit ε →
0. In terms of the field variables ˆ g µν and σ the Einstein-Hilbert actionis given by − πG Z d D x √ gR = − πG Z d D x p ˆ g (cid:20) ˆ R (cid:16) ε σ (cid:17) + ( D − D − g µν ∂ µ σ∂ ν σ (cid:21) , (7.39)where up to topological fluctuations ˆ g µν is pure gauge in the limit D →
2. To canonically normalise σ ,which plays the role of the gauge invariant scalar s , we perform the replacement σ → s − πG ( D − D − σ (7.40)which removes also these factors from the functional measure. Removing the pole from the propagator for σ when D →
2. In the limit D → g µν = η µν one hasjust a canonically normalised scalar field − πG Z d D x √ gR = 12 Z d D x p ˆ g ˆ g µν ∂ µ σ∂ ν σ , (7.41)and thus the theory is free which makes the perturbative treatment straight forward. The propagator forthe mode σ around flat spacetime is then just G ( p ) = 1 p (7.42)and thus when performing the loop expansion each momentum integral will be regularised to obtain Z d D p (2 π ) D G ( p ) = − π k ε ε + O ( ε ) (7.43)by dimensional regularisation with k the IR renormalization scale. It follows that we can write down a zerodimensional propagator G = − π k ε ε , (7.44)which then appears in place of the standard Feynman rule. Then the functional integral for the conformalfactor is reduced to Z σ ( k ) = N Z ∞−∞ d ( iσ ) e πk − ε εσ (7.45)where here we are working in units of the UV scale Λ and we note that in fact the we should reverse theWick rotation of σ by sending σ → − iσ such that the Gaussian integrals have the right sign. To normalisethe functional integral we should take Z σ ( k ) | ε → = 1 which determines that N = √ ε .Now to calculate the averages of observables (7.6), which in terms of the dimensionless fields (7.11) takethe form of composite operators , O = Z d D x √ g − d D L ( ˆ ψ ) , (7.46) See [134] for a study of composite operators in quantum gravity using the functional renormalization group. hOi k = √ ε Z σ ( k ) Z d D x p ˆ g − d D L ( ˆ ψ ) Z ∞−∞ dσ r πGε ε σ ! ε (1 − ∆ ) e − πk − ε εσ (7.47)where now if we expand in G/ε we will produce the loop expansion we want to resum. In particular wewant to take the limit ε → G/ε . To do so one then makes the change ofvariables σ → σ/ε (7.48)in order that we can apply the method of steepest descent where ε is the small parameter. The integral isgiven by hOi k = 1 √ ε Z σ ( k ) Z d D x p ˆ g − d D L ( ˆ ψ ) Z ∞−∞ dσ exp ( ε − ∆ ) log r πGε σ ! − πk − ε σ !) (7.49)Let us note that after performing all redefinitions of σ we have g µν = s πG ( D − D − σ ! ε ˆ g µν (7.50)which is a parameterisation which sets up an ε expansion i.e it ensures that the action is quadratic in thefield and proportional to 1 /ε . We can make a saddle point approximation by writing σ = σ + √ εδσ (7.51)where inside Z σ ( k ) we have σ = 0 and inside the integral over O the saddle point σ should minimises the‘potential’ ∂∂σ (cid:16) k − ε πσ − − ∆ ) log(1 + p πG/εσ ) (cid:17) = 0 (7.52)which has two solutions σ = − ± p − Gε − k ε (1 − ∆ )2 √ π √ Gε − . (7.53)Performing the saddle point approximation we then have the expression hOi k ≈ Z d D x p ˆ g − d D L ( ˆ ψ ) exp ( ε − ∆ ) log r πGε σ ! − πk − ε σ !) (7.54)from which we can extract the anomalous dimensions. Since here k is the IR cutoff the anomalous dimensioncan be obtain by k∂ k hOi = η hOi . (7.55)Equally we may take a derivative with respect to the UV cutoff scale Λ. In this case we should use thescaling laws − ∂ t ˆ g µν = − g µν , ∂ t σ = 0 = ∂ t ˆ ψ for the fields, − ∂ t k = k for the IR renormalization scale and ∂ t G = β G for Newton’s constant. Then we obtain the scaling dimension by the familiar expression − ∂ t hOi = d hOi (7.56)2with d = d + η agreeing with (7.55) provided we are at a fixed point β G = 0. Using either (7.55) or (7.56)yields the scaling dimension given by d ( G ) = d ( D ) + 1 − q − GD − (∆ − GD − + 2(1 − ∆ ) (7.57)where we choose the negative root solution (7.53) such that for G/ε → d → d .To obtain the critical exponents β at the IR fixed point in two dimensions we take G = G DIR andthen take ε → G = G ∗ we obtain the criticalexponents at the UV fixed point (7.26) defined by θ ≡ − d ∗ which are given by θ = −
16 (25 − c ψ ) − s − d − c ψ ! − − ∆ ) − d . (7.58)If we expand in 1 / (25 − c ψ ) we recover the one-loop result (7.30).Away from the fixed point the running of the couplings g is given by β g = d ( G ) g , (7.59)which is non-perturbative in G .An interesting outcome of this prediction is that, although the critical exponents of two-dimensionalquantum gravity at the IR fixed point and the critical exponents at the UV fixed point in higher dimensionsdiffer as D →
2, they are nonetheless related by analytical continuation G ∗ → − G ∗ . The theories obtainedin different limits for quantum gravity close to two dimensions are summarised in table I. Newton’s Constant Dimension Theory G → D >
D > G → G ∗ D >
D > G = 0 D → G = − G ∗ D → TABLE I: The table shows the which theories the various phases of quantum gravity in
D > G ∗ is finite. If one starts in higher dimensions and takesthe limit to two dimensions the functional integral becomes singular. However if we first go to the G = − G ∗ andthen take the limit D → G. Non-perturbative scheme independence
So far we have identified the fixed point for Newton’s constant based on the physical scheme whichpreserved two-dimensional Weyl invariance. On the other hand universal results should not depend on thischoice which is just a scheme allowing us to compute the non-perturbative beta function with ease. With thenon-perturbative beta functions at hand let us now write out the full beta function for Newton’s constantin an general physical scheme. We write first that the exact beta function in a general scheme is given by β exact G = ε G − η g εG + ˜ β ( G ) (7.60)where we determine ˜ β ( G ) for a scheme based on an observables with dimension d = − − ∆ ) in twodimensions by comparing to the beta function obtained in the Weyl invariant scheme and using the physicalrenormalization condition (7.16). This then leads to the identity − εG − − q − Gε (∆ − Gε + 2(1 − ∆ ) + ˜ β ( G ) = −
23 (25 − c ψ ) G . (7.61)3Thus for a general scheme the exact beta functions is given by β G = ε G −
23 (25 − c ψ ) G − η g ε G − εG − − q − Gε (∆ − Gε + 2(1 − ∆ ) . (7.62)Now if we were to expand in G we would get the loop expansion β G = ε G − η g ε G + 23 G ( c ψ − − −
32 (∆ − G ε −
320 (∆ − G ε (7.63) − − G ε − − G ε − − G ε − − G ε + O (cid:0) G (cid:1) and come to the conclusion that taking the limit ε → ε for the exact expression we have β G ( ε →
0) = −
23 (25 − c ψ ) G . (7.64)It then follows that within dimensional regularisation the exact beta function in dimensions D >
H. Non-perturbative renormalization
At the asymptotically safe fixed point G = G ∗ an observable is relevant if the real part of the exponent θ is positive, ℜ ( θ ) > ℜ ( θ ) < g = 0. For an n S -scalar– n F -fermion interaction: L int ( ψ ) = ψ n S S ( ¯ ψ F ψ F ) nF (7.65)we have d ( D ) = − D + n S ( D − / n F ( D − / θ = −
16 (25 − c ψ ) − s − (cid:0) n F − (cid:1) − c ψ + 12 ( D − − n F − n S ) . (7.66)We observe that for all c ψ there is always a finite number of relevant interactions in integer dimensions D > D −
2, decreases as the number of powers of the fields in the interactions increases.
VIII. DISCUSSION
In this paper we have sought to carefully refine the application of the renormalization group to gravity inorder to study the asymptotic safety by means of the ε -expansion. This is motivated by the problem thatbeta functions can appear to depend on the parameterisation of physical degrees of freedom. The dependenceis understood more generally as a dependence on the renormalization scheme and can be compensated by arenormalization of the fields. Since neither the parameterisation of the fields, nor the renormalization of thefields, is physical, this suggests we take a different approach. Here we have defined physical renormalizationschemes where the unphysical dependencies are replaced by the dependence on scaling dimensions of physicalobservables.Working directly with physical observables, rather than local correlation functions, also a great technicalconvenience since the equations become reparameterisation invariant. As such one can use the choice ofparameterisation and gauge fixing to one’s advantage, i.e. to simplify the problem at hand, safe in the4knowledge that one is not implicitly modifying the renormalization scheme. Of course this hinges onthe regularisation scheme being reparameterisation and diffeomorphism invariant. At the classical level,diffeomorphism invariant and background independent flow equations have been derived in [135]. Here theflow equations we have used achieve this already at one-loop (notwithstanding the issue of finding suitableboundary conditions), however the proper-time regularisation breaks down at the two-loop level. As suchwe have then used dimensional regularisation to achieve a non-perturbative result. As advocated in [135]constructing an exact diffeomorphism invariant flow equation could be achieved by using supersymmetricPauli-Villars fields. It would also be desirable if such an equation was reparameterisation invariant.Here we have seen that adopting a reparameterisation, diffeomorphism and background independentapproach bears many fruits. Exploiting dimensional regularisation a UV fixed point can be identified sincethe non-perturbative beta function in the limit D → D = c ψ +1 dimensions. In addition, the fact that the critical exponentsare obtained almost directly from two-dimensional quantum gravity indicates that the fractal dimension ofspacetime may be close to two. This observation was first made in causal dynamical triangulation simulations[136] and has since also been observed in other approaches to quantum gravity [137–139]. Taking theradically conservative view that Nature is indifferent to how we parameterise her, it could be the case thatquantum gravity is described both by string theory and a genuine non-perturbative quantisation of generalrelativity. Acknowledgements
I would like to thank Stanley Deser, Michael Duff, Mikhail Kalmykov, and Aron Wall for constructive com-ments on [106]. This work has benefited from discussions with Dario Benedetti, Alessandro Codello, NicolaiChristiansen, Astrid Eichhorn, John Gracey, Stefan Lippoldt, Tim Morris, Carlo Pagani, Jan Pawlowskiand Masatoshi Yamada. I also thank Alejandro Satz for correspondence on the diffeomorphism invariantboundary conditions. This work was supported by the European Research Council grant ERC-AdG-290623.
Appendix A: Factoring out the gauge modes
Here we take a geometrical approach to the functional integral viewing the fields φ a as coordinates ona manifold Φ which can be thought of as a product of the physical space Φ / G and the gauge orbits G with coordinates ξ α . As an alternative to the usual Faddeev-Popov gauge fixing we take the geometricalapproach to factoring out the gauge modes [98, 118–120]) which keeps gauge independence manifest. Firstwe take the measure over the fluctuation fields δφ a Z Y a dφ a (2 π ) / p det C ab = Z Y a dδφ a (2 π ) / p det C ab (A1)and decompose the fluctuation as: δφ a = L a ¯ a f ¯ a + L aα ξ ′ α (A2)where the second term is a diffeomorphisms with ξ α parameterising the gauge orbit and f ¯ a are gaugeinvariant fields. The prime here indicates that zero modes of L aα must be left out of the spectrum of ξ ′ which for gravity corresponds to Killing vectors. In the case of gravity it is not possible to diagonalise5the field space metric in these coordinates, however one can make an additional shift, corresponding to thefreedom to fix the gauge f ¯ a → f ¯ a + t ¯ aα ξ ′ α , ξ ′ α → ξ ′ α (A3)which has unit Jacobian and hence does not alter the measure. Choosing t aα the DeWitt metric can be madeblock diagonal: δφ a C ab δφ b = f ¯ a C ¯ a ¯ b f ¯ b + ξ ′ β C αβ ξ ′ α (A4)where here ¯ a, ¯ b, .. are a set of DeWitt indices ¯ a = { x, ¯ A } for the gauge invariant fields and α, β, ... is a setof DeWitt indices for the diffeomorphisms e.g ξ α = ǫ µ ( x ). Next we write the gauge volume as V diff = Z Y α dξ α (2 π ) / p det G αβ ≡ Ω Z Y α dξ ′ α (2 π ) / q det ′ G αβ (A5)which comes with its own metric G αβ . Here Ω is the volume of the subgroup of diffeomorphisms H whichare zero modes of L aα and the prime indicates that these modes are removed from the determinant. Thetotal measure is then given by R Q a dδφ a √ det C ab V diff = 1Ω Z Y ¯ a df ¯ a (2 π ) / q det ′ ( G − ) αβ p det ′ C ab . (A6)where all gauge modes have been factored out apart from the zero modes that must be accounted from bydetermining Ω explicitly [121] (a similar factor is needed for Maxwell theory [140]).It is important to bare in mind that different choice of the fundamental degrees of freedom φ A ( x ) canlead to unphysical configuration spaces of different dimensionality. For example if we choose the metric φ A ( x ) = g µν or the Dirac matrices ˜ φ ˜ A ( x ) = γ µ then the number of ‘flavours’, i.e values A and ˜ A can take,is different but the same is true for the gauge orbits parameterised by ξ α and ˜ ξ ˜ α . If the physical degrees offreedom are the same then one expects that two different configuration space Φ and ˜Φ will lead to one andthe same physical configuration space Φ / G ∼ = ˜Φ / ˜ G . Here we consider only the ’metric’ configuration spacefor definiteness. The equivalence of the path integrals based on g µν and γ µ has been argued in [141]. Appendix B: Gaussian integrals and determinants
The one-loop formula for the generating function W is given by Z = 1Ω Z Y ¯ a df ¯ a √ π q det ′ ( G − ) αβ q det ′ C αβ p det C ¯ a ¯ b e − S [ ¯ φ ] − f · S (2) · f = 1Ω KV q det ′ ( G − ) αγ C γβ r det( C − ) ¯ a ¯ c (cid:16) S (2) J (cid:17) ¯ c ¯ b e − S J [ ¯ φ ] (B1)with all quantities evaluated on an Einstein space with Ricci curvature (5.3) and extrinsic curvature de-termined by (6.2) and (6.11) in case of a boundary. It is clear that Z is reparameterisation invariantsince it transforms as a scalar on configuration space Φ. To compute this integral we can pick any fieldparameterisation and then from there determine a decomposition (A2) which satisfies (A4) after a shift(A3).Taking the field to be given by the metric tensor φ A = g µν the metric C on Φ is of the DeWitt form C ab = C µν,ρσ δ ( x − y ) = µ πG √ g (cid:18)
12 ( g µρ g νσ + g µσ g νρ ) − g µν g ρσ (cid:19) δ ( x − y ) (B2)6For a gauge parameter ξ α = ǫ µ ( x ) the corresponding metric is given by (2.20). Proceeding as outlined inAppendix A we can first decompose δg µν into the gauge modes and the gauge invariant fields: δg µν = h TT µν + 1 D g µν s + ∇ µ ǫ ′ ν + ∇ ν ǫ ′ µ (B3)where h TT µν is transverse and traceless. The Killing vectors are removed from ǫ ′ µ since these are the zeromodes of L aα . Furthermore if the background involves modes ∇ µ ∇ ν s = 1 D g µν ∇ s (B4)which, satisfy the eigen-problem −∇ s = RD − s these must be removed from the spectrum of s . This followssince g µν s ∝ ∇ µ ǫ CKV ν + ∇ ν ǫ CKV µ where ǫ CKV µ is a conformal Killing vector which is not a Killing vector(CKV) and is included in the spectrum of ǫ µ . The line element is given by C ab δφ a δφ b = µ πG Z d D x √ g (cid:18) h T Tµν h T T µν − D − D s + 2 ǫ ′ µ ∆ ǫ ′ µ − D − D ∇ µ ǫ ′ µ s (cid:19) (B5)where we have exploited the boundary conditions (6.3) to integrate by parts finding that all boundary termsvanish. Since the metric C ab is not diagonal in these coordinates we makes the shift s → s − ∇ µ ǫ ′′ µ (B6)to calculate the determinant where the double prime indicates that we do not include the CKV’s or KV’sin the transformation. We then have C ab δφ a δφ b = Z d D x µ πG √ g h T Tµν h T T µν − D − D s + 2 ǫ ′′ µ ˜∆ ǫ ′′ µ + 2 X CKV ǫ µ (cid:18) D − − D (cid:19) R ǫ µ ! where we have included explicitly the contribution from the CKV’s and introduced the differential operator˜∆ ǫ µ = ∆ ǫ µ − D − D ∇ µ ∇ ν ǫ ν . (B7)For the line element on the space of diffeomorphisms (2.20) we have q det ′ ( G − ) αγ C γβ = s(cid:18) R/µ D − − R/µ D (cid:19) N CKV det ′′ ˜∆ /µ . (B8)We then note that the spectrum of ˜∆ may be decomposed into transverse and longitudinal modes suchthat we obtain q det ′ ( G − ) αγ C γβ = s(cid:18) R/µ D − − R/µ D (cid:19) N CKV det ′ T [∆ /µ ]det ′′ (cid:20) D − D ∆ /µ (cid:21) (B9)where 1 T indicates that is the operator ∆ acts on transverse vectors and∆ = −∇ + RD − f ¯ a = { h Tµν , s } . Taking the secondvariation of the action we have δ S = Z d d x √ g (cid:18) − πG F µ ( h ) F µ ( h ) + 132 πG ¯ h µν ∆ h µν (cid:19) (B11)7where h µν = δg µν , ¯ h µν = h µν − g µν h λλ and F λ ( h ) = g µν ∇ µ ¯ h νλ . Inserting (B3) one readily finds thehessians for the gauge invariant fields h T T · S (2) h TT h TT · h T T = 132 πG Z d D x √ g h T Tµν ∆ h µνT T (B12) s · S (2) ss · s = − πG ( D − D − D Z d D x √ g s ∆ s (B13)while ǫ µ components of the hessian are zero. Here we see that the hessian for s has the wrong sign for allmodes where ∆ is positive which corresponds to all modes apart from the constant mode s when R > s → s − πGD ( D − D − s (B14)Such that hessian for s is then given by s · S (2) T T · s = Z d D x √ g s (cid:18) −∇ − RD − (cid:19) s (B15)additionally we canonically normalise h T Tµν via h T Tµν → √ πGh T Tµν . (B16)Then the metric on the gauge invariant field space becomes C ¯ a ¯ b f ¯ a f ¯ b = Z d D xµ √ g (cid:18) h T Tµν h T T µν + D D − s (cid:19) (B17)and we must remember that the constant mode of s must be Wick rotated back (since the gaussian integraloriginally had the correct sign). We then have q det | ( C − ) ¯ a ¯ c (cid:0) S (2) (cid:1) ¯ c ¯ b | = s det T [∆ /µ ]det ′ (cid:12)(cid:12)(cid:12)(cid:12) D − D ∆ /µ (cid:12)(cid:12)(cid:12)(cid:12) . (B18)Comparing this expression with (B9) we observe that integral over the scalar modes cancels with the thedeterminant from factoring out the longitudinal diffeomorphisms apart from the CKVs and the constantmode. Here 2 T means the determinate is over transverse-traceless modes.To check that the final result will not depend on the choice of field parameterisation φ (or equivalentlythe coefficients (2.6)) we note that terms involving δ g µν are not present since we expand around the saddlepoint and any dependence on T Aµν cancels between the determinates in (B1). We therefore have: − log Z = S [ ¯ φ ] + 12 Tr T log ∆ /µ + 12 log 2 D | R | /µ − N CKV log (cid:0) (1 / ( − D ) − /D ) R/µ (cid:1) −
12 Tr ′ T log ∆ /µ − log Ω (B19)independently of the gauge or field parameterisation. Finally, using the relations between traces of con-strained fields and unconstrained fields on an arbitrary Einstein space (see e.g appendix B of [44]):Tr T f (∆ ) = Tr f (∆ ) − Tr ′ f (∆ ) − Tr f ( −∇ − D R ) + N CKV f ((1 / ( − D ) − /D ) R ) , (B20)Tr ′ T f (∆ ) = Tr ′ f (∆ ) − Tr f ( −∇ − D R ) + f ( − D R ) , (B21)8we can then arrive at (5.4) where the traces are for unconstrained symmetric tensors and vectors . It isstraightforward to check that using the gauge fixing (3.2) with α = 1 gives the same result since the gaugefixing action cancels the first term in (B11) and the corresponding Faddeev-Popov determinant gives thevector trace. Upon replacing the C ab and G αβ with the regulated forms (5.6) and (5.7) we then obtain thetraces (5.9) which are free from divergencies. Appendix C: The gauge invariant hessian for general schemes near two dimensions
Close to two dimensions we are interested in matter with an interaction O = Z d D x √ g L ( ψ ) (C1)by writing going to dimensionless matter fields rescaled by the determinant of g µν to the appropriate power.Then the interaction becomes O [ φ ] = Z d D x √ g − d /D ( φ ) L ( ˆ ψ ( φ )) (C2)and the Kinetic terms become invariant under conformal transformations of the metric holding ˆ ψ fixed.Then the set of conformal gauges g µν = f ( σ )ˆ g µν (C3)with the determinate of ˆ g µν fixed, become useful since the kinetic terms are then in dependent of σ . We thenreplace in the last section λ V → g O and repeat the analysis. The calculation is simplest in the conformalgauges however since we only need the on-shell hessian to find the divergencies of Z all terms that dependon this choice vanish once we use the equations of motion. This results the operator ∆ in (B13) by beingreplaced by ∆ → −∇ − d R (C4)for D → D →
2. Thisagrees with (B10) in the case d = −
2. Since the kinetic term for the matter fields is conformally invariantthere is no mixing between ˆ ψ and σ from this term. From O there is a component of the hessian that mixes σ and ˆ ψ however this term only contributes to irrelevant power law divergencies and not the universal betafunctions. It follows that (C4) is the only significant difference between the the on-shell Hessians for thecase V = O and the general case. As such we arrive at the flow equation where the one-loop coarse grainingcontribution is given by (7.15). Appendix D: Volume of the stability group H Non-perturbatively the the volume Ω of the stability group H takes the form [121]Ω( µ ) = N KV Y ℓ =1 Z d M ( ǫ ℓ ) µ √ π || k ℓ || (D1)involving the Haar measure on H where || k ℓ || ≡ p h k ℓ | k ℓ i is the square root of the norm h k ℓ | k ℓ ′ i = 132 πG Z d D x √ g k µℓ k νℓ ′ g µν (D2) Here we have neglected a constant imaginary part which is needed to correct the contribution of the zero mode ensuring W is real for R > k µℓ = ∂x µ ∂ǫ ℓ are Killing vectors where h k ℓ | k ℓ ′ i = 0 for ℓ = ℓ ′ where we have decomposed ǫ µKV = P N KV ℓ =1 ǫ ℓ k µℓ .The volume Ω has been calculated explicitly for both S and S × S space-times in [121] (denoted thereby Ω ). We note that the proper-time regularisation replaces µ with Λ e − γ E in (D1) such thatΛ ∂ Λ Ω = 2 N KV Ω . (D3)which is important to obtain background independent beta functions.0 Appendix E: Boundaries
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