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2.4 Dimensional reduction of residual interactions in UV

As highlighted in the introduction an important qualitative feature of an asymptotically safe functional integral can be inferred without actually evaluating it, namely that in the extreme ultraviolet the residual interactions appear two-dimensional. There are a number of interconnected heuristic arguments for this phenomenon which we present here.

2.4.1 Scaling of fixed point action

Consider a candidate for a quasilocal microscopic action

∑ Sk[g] = ui(k )Pi [g], (2.54 ) i
where the u(k ) i are running couplings of mass dimension d i and P [g] i are local invariants of mass dimension − di. By quasilocal we mean here that the sum may be infinite and off hand arbitrary high derivative terms may occur. For example such an action arises in the perturbative framework advocated by Gomis and Weinberg [94Jump To The Next Citation Point]. When viewed as a renormalized action perturbatively defined in the above sense (with the UV cutoff strictly removed) the running of the uPT (k) i is unknown but expectations are that PT − di PT gi (k) = k ui (k) are not uniformly bounded functions in k; then the dimensionless couplings are not asymptotically safe but blow up at various (i-dependent) intermediate scales. The situation is drastically different if all the couplings are assumed to be asymptotically safe. Then ui(k) = gi(k)kdi ∼ g∗ikdi as k → ∞ and if one uses the fact that si = di (see the discussion after Equation (2.36View Equation)) for all local invariants one gets
∑ ∗ 2 2 Sk[g] ∼ giPi[k g ] = S ∗[k g] (2.55 ) i
for k → ∞, with ∑ S∗[g ] = ig∗iSi[g] the candidate fixed point action. The overall scale of the metric is an inessential parameter (see Equations (1.1View Equation, A.8View Equation)), and as discussed in Section 2.1 a fixed point action always refers to an equivalence class modulo possibly running inessential parameters.

One sees that in the fixed point regime gi(k) ∼ gi∗ the k-dependence enters only through the combination k2gαβ, a kind of self-similarity. This simple but momentous fact eventually underlies all the subsequent arguments. It is ‘as if’ in the fixed point regime only a rescaled metric ˜g = k2g αβ αβ entered which carries dimension two. This has consequences for the ‘effective dimensionality’ of Newton’s constant: Recall that conventionally the Ricci scalar term, ∫ √ -- dx gR (g), has mass dimension 2 − d in d dimensions. Upon substitution gαβ ↦→ ˜gαβ one quickly verifies that ∫ √ -- dx ˜gR (˜g) is dimensionless. Its prefactor, i.e. the inverse of Newton’s constant, then can be taken dimensionless – as it is in two dimensions. Compared to the infrared regime it looks ‘as if’ Newton’s constant changed its effective dimensionality from d − 2 to zero, i.e. at the fixed point there must be a large anomalous dimension ηN = 2 − d.

Formally what is special about the Einstein–Hilbert term is that the kinetic (second derivative) term itself carries a dimensionful coupling. To avoid the above conclusion one might try to assign the metric a mass dimension 2 from the beginning (i.e. not just in the asymptotic regime). However this would merely shift the effect from the gravity to the matter sector, as we wish to argue now.

In addition to the dimensionful metric ˜gαβ := k2gαβ, we introduce a dimensionful vielbein by E˜ m := kEm α α, if g = E mE nη αβ α β mn is the dimensionless metric. With respect to a dimensionless metric ∫ √ -- dx gR (g) has mass dimension 2 − d in d dimensions, while the mass dimensions dχ of a Bose field χ and that dψ of a Fermi field ψ are set such that their kinetic terms are dimensionless, i.e. dχ = (d − 2)∕2 and d ψ = (d − 1)∕2. Upon substitution gαβ ↦→ ˜gαβ the gravity part ∫ dx √ ˜gR (˜g) becomes dimensionless, while the kinetic terms of a Bose and Fermi field pick up a mass dimension of d − 2 and d − 1, respectively. This means their wave function renormalization constants Z χ(k) and Z ψ(k) are now dimensionful and should be written in terms of dimensionless parameters as Zχ(k) = kd− 2∕g χ(k) and Z ψ(k) = kd−1∕gψ (k ), say. For the dimensionless parameters one expects finite limit values limk →∞ gχ(k) = g∗χ > 0 and limk → ∞ gψ(k) = g∗ > 0 ψ, since otherwise the corresponding (free) field would simply decouple. Defining the anomalous dimension as usual. ηχ = − k∂k lnZ χ and ηψ = − k ∂k lnZ ψ, the argument presented after Equation (1.5View Equation) can be repeated and gives that η∗ = 2 − d χ, η∗ = 1 − d ψ for the fixed point values, respectively. The original large momentum behavior 1∕p2 for bosons and 1∕p for fermions is thus modified to a d 1 ∕p behavior in the fixed point regime, in both cases.

This translates into a logarithmic short distance behavior which is universal for all (free) matter. Initially the propagators used here should be viewed as “test propagators”, in the sense that one transplants the information in the η’s derived from the gravitational functional integral into a conventional propagator on a (flat or curved) background spacetime. Since the short distance asymptotics is the same on any (flat or curved) reference spacetime, this leads to the prediction anticipated in Section 2.3: The short distance behavior of the quantum gravity average of the geodesic two-point correlator (2.52View Equation) of a scalar field should be logarithmic.

On the other hand the universality of the logarithmic short distance behavior in the matter propagators also justifies to attribute the phenomenon to a modification in the underlying random geometry, a kind of “quantum equivalence principle”.

2.4.2 Anomalous dimension at non-Gaussian fixed point

The “anomalous dimension argument” has already been sketched in the introduction. Here we present a few more details and relate it to Section 2.4.1.

Suppose again that the unkown microscopic action of Quantum Gravidynamics is quasilocal and reparameterization invariant. The only term containing second derivatives then is the familiar Einstein–Hilbert term ∫ √ -- ZN dx gR (g) of mass dimension 2 − d in d dimensions, if the metric is taken dimensionless. As explained in Section 2.3.2 the dimensionful running prefactor multiplying it ZN (k) (N for “Newton”) can be treated either as a wave function renormalization or as a quasi-essential dimensionless coupling gN, where

− 1 2−d cdGNewton = ZN (k) = gN (k)k . (2.56 )
Here we treat gN as running, in which case its running may also be affected by all the other couplings (gravitational and non-gravitational, made dimensionless by taking out a suitable power of k). The short distance behavior of the propagator will now be governed by the “anomalous dimension” ηN = − k ∂k lnZN (k) by general field theoretical arguments. On the other hand the flow equation for gN can be expressed in terms of η N as k∂ g = [d − 2 + η (g ,other)]g k N N N, where we schematically indicated the dependence on the other dimensionless couplings. If this flow equation now has a nontrivial fixed point ∞ > g∗N > 0, the only other way how the right-hand-side can vanish is for
∗ ηN (g N,other) = 2 − d, (2.57 )
irrespective of the detailed behavior of the other couplings as long as no blow-up occurs. This is a huge anomalous dimension. We can now transplant this anomalous dimension into a “test graviton propagator” on a flat background. The characteristic property of ηN then is that it gives rise to a a high momentum behavior of the form 2− 1+ η ∕2 (p ) N modulo logarithms, or a short distance behavior of the form √ --- ( x2)2− d−ηN modulo logarithms. This follows from general field theoretical principles: a Callan–Symanzik equation for the effective action, the vanishing of the beta function at the fixed point, and the decoupling of the low momentum modes. Keeping only the leading part the vanishing power at ηN = 2 − d translates into a logarithmic behavior, 2 lnx, formally the same as for a massless Klein–Gordon field in a two-dimensional field theory.

The fact that a large anomalous dimension occurs at a non-Gaussian fixed point was initially observed in the context of the 2 + ε expansion [116Jump To The Next Citation Point117] and later in computations based on the effective average action [133Jump To The Next Citation Point131Jump To The Next Citation Point]. The above argument shows that no specific computational information enters.

Let us emphasize that in general an anomalous dimension is not related to the geometry of field propagation and in a conventional field theory one cannot sensibly define a fractal dimension by looking at the high momentum behavior of a two-point function [125]. What is special about gravity is ultimately that the propagating field itself defines distances. One aspect thereof is the universal way matter is affected, as seen in Section 2.4.1. In contrast to an anomalous dimension in conventional field theories, this “quantum equivalence principle” allows one to attribute a geometric significance to the modified short distance behavior of the test propagators, see Section 2.4.4.

2.4.3 Strict renormalizability and 4 1∕p propagators

With hindsight the above patterns are already implicit in earlier work on strictly renormalizable gravity theories. As emphasized repeatedly the benign renormalizability properties of higher derivative theories are mostly due to the use of 1∕p4 type propagator (in d = 4 dimensions). As seen in Section 2.3.2 this 1∕p4 type behavior goes hand in hand with asymptotically safe couplings. Specifically for the dimensionless Newton’s constant gN it is compatible with the existence of a nontrivial fixed point (see Equation (2.31View Equation)). This in turn enforces anomalous dimension ηN = − 2 at the fixed point which links back to the 1∕p4 type propagator.

Similarly in the 1∕N expansion [216Jump To The Next Citation Point217Jump To The Next Citation Point203Jump To The Next Citation Point] a nontrivial fixed point goes hand in hand with a propagator whose high momentum behavior is of the form 4 2 1∕ (p ln p ) in four dimensions, and formally d 1∕p in d dimensions. In position space this amounts to a 2 lnx behavior, once again.

2.4.4 Spectral dimension and scaling of fixed point action

The scaling (2.55View Equation) of the fixed point action also allows one to estimate the behavior of the spectral dimension in the ultraviolet. This leads to a variant [157Jump To The Next Citation Point] of the argument first used in [135Jump To The Next Citation Point134Jump To The Next Citation Point]).

Consider the quantum gravity average ⟨Pg(T)⟩ over the trace of the heat kernel Pg(T ) in a class of states to be specified later. Morally speaking the functional average is over compact closed d-dimensional manifolds (ℳ, g), and the states are such that they favor geometries which are smooth and approximately flat on large scales.

Let us briefly recapitulate the definition of the heat kernel and some basic properties. For a smooth Riemannian metric g on a compact closed d-manifold let αβ √ -− 1 √ --αβ Δg := g ∇ α∇ β = g ∂ α( gg ∂β) be the Laplace–Beltrami operator. The heat kernel ′ exp (T Δg )(x,x ) associated with it is the symmetric (in x,x′) bi-solution of the heat equation ∂TK = ΔgK with initial condition limT →0exp (TΔg )(x,x′) = δ(x,x′). Since (ℳ, g) is compact, Δg has purely discrete spectrum with finite multiplicities. We write − Δ φ (g) = ℰ (g )φ (g) g n n n, n ≥ 0, for the spectral problem and assume that the eigenfunctions φn are normalized and the eigenvalues monotonically ordered, ℰn(g) ≤ ℰn+1 (g ). We write ∫ √ -- V (g) = dx g for the volume of (ℳ, g) and

∫ --1-- √ -- --1--∑ −ℰn(g)T Pg (T) = V (g) dx g exp(T Δg )(x, x) = V (g) e , (2.58 ) n
for the trace of the heat kernel. In the random walk picture Pg(T ) can be interpreted as the probability of a test particle diffusing away from a point x ∈ ℳ and to return to it after the fictitious diffusion time T has elapsed. In flat Euclidean space d (ℳ, g) = (ℝ ,η) for example −d∕2 P η(T ) = (4πT ) for all T. For a generic manifold the trace of the heat kernel cannot be evaluated exactly. However the short time and the long time asymptotics can to some extent be described in closed form. Clearly the T → ∞ limit probes the large scale structure of a Riemannian manifold (small eigenvalues ℰn(g)) while the T → 0 limit probes the small scales (large eigenvalues ℰn (g )).

For T → 0 one has an asymptotic expansion ∑ ∫ √ -- Pg(T ) ∼ (4 πT )− d∕2 n≥0 Tn dx gan (x), where the an are the Seeley–deWitt coefficients. These are local curvature invariants, a0 = 1, a1 = 1R(g) 6, etc. The series can be rearranged so as to collect terms with a fixed power in the curvature or with a fixed number of derivatives [225Jump To The Next Citation Point17Jump To The Next Citation Point]. Both produces nonlocal curvature invariants. The second rearrangement is relevant when the curvatures are small but rapidly varying (so that the derivatives of the curvatures are more important then their powers). The leading derivative terms then are given by Pg (T ) ∼ (4πT )−d∕2[V (g) + T ∫ dx√ga1 + T 2N2 (T ) + ...], where N2 (T ) is a known nonlocal quadratic expression in the curvature tensors (see e.g. [22517] for surveys). The T → ∞ behavior is more subtle as also global information on the manifold enters. For compact manifolds a typical behavior is Pg (T ) ∼ (4πT )−d∕2[1 + O (exp(− cT))], where the rate of decay c of the subleading term is governed by the smallest non-zero eigenvalue.

Returning now to the quantum gravity average ⟨P (T )⟩ g, one sees that on any state on which all local curvature polynomials vanish the leading short distance behavior of ⟨Pg(T )⟩ will always be −d∕2 ∼ T, as on a fixed manifold. The same will hold if the nonlocal invariants occurring in the derivative expansion all have vanishing averages in the state considered. A leading short distance behavior of the form

⟨Pg (T )⟩ ∼ T− ds∕2, T → 0, (2.59 )
with ds ⁄= d will thus indicate that either the operations “taking the average” and “performing the asymptotic expansion for T → 0” no longer commute, or that the microscopic geometry is very rough so that the termwise averages no longer vanish, or both. Whenever well-defined the quantity ds(T ) := − 2dln⟨Pg (T )⟩∕dln T is known as the spectral dimension (of the micro-aspects of the random geometries probed by the state 𝒪 ↦→ ⟨𝒪 ⟩). See [1151188] for earlier uses in random geometry, and [27] for an evaluation of the spectral dimension for diffusion on the Sierpinski gasket based on a principle similar to Equation (2.62View Equation) below.

We assume now that the states considered are such that the T → ∞ behavior of ⟨Pg(T )⟩ is like that in flat space, i.e. ⟨Pg(T)⟩ ∼ T −d∕2 for T → ∞. This is an indirect characterization of a class of states which favor geometries that are smooth and almost flat on large scales [157Jump To The Next Citation Point]. (A rough analogy may be the way how the short-distance Hadamard condition used for free QFTs in curved spacetime selects states with desirable stability properties.) Recall that in a functional integral formulation the information about the state can be encoded in suitable boundary terms added to the microscopic action. The effective action used in a later stage of the argument is supposed to be one which derives from a microscopic action in which suitable (though not explicitly known) boundary terms encoding the information about the state have been included.

Since ∫ ddp (4πT )− d∕2 = (2π)d exp (− p2T ) one can give the stipulated T → ∞ asymptotics an interpretation in terms of the spectrum {p2, p ∈ ℝd} of the Laplacian of a ‘typical’ reference metric qαβ which is smooth and almost flat at large scales. The spectrum of Δq must be such that the small spectral values can be well approximated by 2 d {p < C, p ∈ ℝ } for some constant C > 0. Its unknown large eigenvalues will then determine the short distance behavior of ⟨Pg(T )⟩. We can incorporate this modification of the spectrum by introducing a function Fq(p2) which tends to 1 for p2 → 0, and whose large p2 behavior remains to be determined. Thus

∫ ddp 2 2 ⟨Pg (T )⟩ ≈ ----d-exp{− p Fq (p )T }. (2.60 ) (2π)
The following argument now suggests that within the asymptotic safety scenario Fq(p2) ∼ p2 for p2 → ∞. Before turning to the argument let us note that this property of Fq(p2) entails
−d∕4 ⟨Pg(T )⟩ ∼ T for T → 0, i.e.ds = d∕2. (2.61 )
The “microscopic” spectral dimension equals half the “macroscopic” d. Notably this equals 2, as suggested by the “anomalous dimension argument” precisely in d = 4 dimensions.

The argument for Fq(p2) ∼ p2 for p2 → ∞ goes as follows: We return to discrete description ∑ Pg (T ) = n e−ℰn(g)T for (ℳ, g) compact, and consider the average of one term in the sum ⟨e− ℰn(g)T⟩, with ℰn (g ) being large. The computation of this average is a single scale problem in the terminology of Appendix A. As such it should allow for a good description via an effective field theory at scale k. One way of doing this is in terms of the effective average action ¯ Γ k[g] as described in Section 4.1. Here only the fact is needed that the average ⟨e−ℰn(g)T ⟩ can approximately be evaluated as [135Jump To The Next Citation Point134Jump To The Next Citation Point]

δ¯Γ ⟨e− ℰn(g)T ⟩ ≈ e−ℰn(ˇgk)T, where ---k-[ˇgk] = 0. (2.62 ) δgαβ
As indicated (ˇg ) k αβ is a stationary point of the effective action ¯Γ [g] k at a certain scale k. Since the only scale available is ℰn itself, the relevant scale k is for given n determined by the implicit equation 2 k = ℰn (ˇgk). Next we consider how these spectral values scale in the fixed point regime where the dimensionless couplings are approximately constant, gi(k) ≈ gi. Recall from Equation (2.55View Equation) the limiting behavior ¯Γ k[g] → S∗[k2g] as k → ∞. Two stationary points (ˇgk)αβ for ¯Γ k and (ˇg ) k0 αβ for ¯Γ k0 will thus in the fixed point regime be simply related by k2ˇg = k2ˇg k 0 k0. Since 2 k Δk2g = Δg this means for the spectral values 2 2 k ℰn(ˇgk) = k0 ℰn(ˇgk0). In order to make contact to the continuum parameterization in Equation (2.60View Equation) we now identify for given p the n’s such that for the typical metric qαβ entering Equation (2.60View Equation) one has ℰn (q) ∼ p2 for large n. After this reparameterization ℰn = ℰp, ∘ --- p = p2 one can identify the Fq(p2) in Equation (2.60View Equation) with 2 Fq(p ) = ℰp(ˇgk=p)∕ℰp(ˇgk0). This scales for p → ∞ like 2 p, which completes the argument.

In summary, the asymptotic safety scenario leads to the specific (theoretical) prediction that the (normally powerlike) short distance singularities of all free matter propagators are softened to logarithmic ones – normally a characteristic feature of massless Klein–Gordon fields in two dimensions. In quantum gravity averages like Equation (2.52View Equation) this leads to the expectation that they should scale like G (R ) ∼ ln R, for R → 0. On the other hand this universality allows one to shuffle the effect from matter to gravity propagators. This justifies to attribute the effect to a modification in the underlying random geometry. The average of heat of the heat kernel, G (T ) in Equation (2.53View Equation), then scales like T −d∕4. This means the spectral dimension of the random geometries probed by a certain class of “macroscopic” states equals d∕2, which (notably!) equals 2 precisely in d = 4 dimensions.

Accepting this dimensional reduction in the extreme ultraviolet as a working hypothesis one is led to the following question: Is there a two-dimensional field theory which provides an effective description of this regime? “Effective” can mean here “approximate” but quantitatively close, or a system which lies in the same universality class as the original one in the relevant regime. “Effective” is of course not meant to indicate that the theory does not make sense beyond a certain energy scale, as in another use of the term “effective field theory”. We don’t have an answer to the above question but some characteristics of the putative field theory can easily be identified:

  1. It should be two-dimensional and self-interacting, the latter because of the non-Gaussian nature of the original fixed point.
  2. It should not be a conformal field theory in the usual sense, as the extreme UV regime in the original theory is reached from outside the critical surface (“massive continuum limit”).
  3. It should have degrees of freedom which can account for the antiscreening behavior presumed to be responsible for the asymptotically safe stabilization of the UV properties.

Note that in principle the identification of such a UV field theory is a well-posed problem. Presupposing that the functional integral has been made well-defined and through suitable operator insertions data for its extreme UV properties have been obtained, for any proposed field theory with the Properties 13 one can test whether or not these data are reproduced.


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