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3.4 Non-Gaussian fixed point and asymptotic safety

As explained before the function h plays the role of a generating functional for an infinite set of essential couplings. As such it is subject to a flow equation
-d--- -- μdμ h = λβh (h∕λ ), (3.83 )
where μ is the renormalization scale and -- μ → h (⋅,μ) is the ‘running’ coupling function. The flow equation can be obtained by the usual procedure starting from the fact that the left-hand-side of Equation (3.58View Equation) is μ-independent. One finds [154Jump To The Next Citation Point]
∫ λ β (h∕ λ) = (2 − d )h(ρ) − h (ρ) du h(u)δH-(ρ)- h δh (u) h2 ( λ ) ( h ) = (2 − d )h(ρ) −--βG ∕H -- + Ξ˙ρ -- ∂ρh, (3.84 ) λ h λ
where we suppress the λ dependence of h. Inserting Equation (3.69View Equation) and setting d = 2 in the result gives
[ h(ρ)∫ ∞ du h (u) ( λ )] βh(h∕λ ) = ρ∂ρ ----- --------βG∕H ----- . (3.85 ) λ ρ u λ h(u )
Here βG∕H(λ ) = − λ ∑ lζl( λ-)l l≥1 2π is again the conventional (numerical) beta function of the G ∕H nonlinear sigma-model without coupling to gravity, computed in the minimal subtraction scheme. β (h) h can thus be regarded as a “gravitationally dressed” version of βG ∕H (λ). The term is borrowed from [120] where a similar phenomenon was found in a different context and to lowest order. In contrast to Equation (3.85View Equation) the effect of quantum gravity on the running of the couplings in these Liouville-type theories cannot be represented as a simple “dressing relation” beyond lowest order [168]. The flow equation resulting from Equation (3.85View Equation) will be studied in more detail below. We anticipate however that the appropriate boundary conditions are such that the solution ¯ h (ρ, λ) is stationary (constant in μ) for ρ → ∞. This guarantees that the renormalization flow is exclusively driven by the counterterms, as it should. We add some comments on the structure of the beta function (3.85View Equation).

An initially puzzling feature of βh(h ) is that it comes out as a total ln ρ-derivative. Restoring the interpretation of ρ = ρ(x) as a field on the 2D base space, however, it has a natural interpretation: An immediate consequence of Equations (3.83View Equation, 3.85View Equation) is that contour integrals of the form

∫ μ -- C dx ∂μ ln ρh (ρ,μ) (3.86 )
are μ-independent for any closed contour C in the base space, as can be seen by differentiating Equation (3.86View Equation) with respect to μ. They are thus invariants of the flow and can be used to discriminate the inequivalent quantum theories parameterized by the hl(ρ). With the initial condition -- h (ρ, μ0) = h(ρ) the μ-independence of Equation (3.86View Equation) is equivalent to -- ∂ μ[(h − h )∂ μlnρ] = 0. On the other hand the (classical and quantum) equations of motion for ρ with respect to the h-modified action are just ∂ μ(h∂μ ln ρ) = 0. Combining both we find that the significance of βh (h ) being a total lnρ-derivative is that this feature preserves the equations of motion for ρ under the μ-evolution of -- h (⋅,μ ):
∫ ρ du ∫ ρ du-- ---h (u) harmonic =⇒ ---h (u, μ) harmonic. (3.87 ) u u
This provides an important consistency check as Equation (3.87View Equation) is also required by the non-renormalization of the R μ Noether current in Equation (3.78View Equation).

Another intriguing property of Equation (3.85View Equation) can be seen from the second line in Equation (3.84View Equation). The first two terms correspond to the beta function of a G ∕H sigma-model without coupling to 2D gravity (i.e. with nondynamical ρ and σ). The last term is crucial for all the subsequent properties of the flow (3.83View Equation). Comparing with Equation (3.75View Equation) one sees that it describes a backreaction of the scale dependent area radius ρ on the coupling flow, which is mediated by the quantum dynamics of the other fields. We shall return to this point below.

As usual fixed points of the flow correspond to zeros of the generalized beta function. The flow has two fixed points, a degenerate Gaussian one corresponding to h = ∞ formally, and a non-Gaussian fixed point hbeta(⋅) which is of main interest here. We postpone the discussion of the Gaussian fixed point and focus on the non-Gaussian one here. The defining relation for such a nontrivial fixed point amounts by Equation (3.85View Equation) to the differential equation

λ h ( λ) ---ρ∂ρh = − C (λ)h2--βG∕H -- , (3.88 ) 2π λ h
where C (λ ) is a constant, which in principle could be λ-dependent. The Equation (3.88View Equation) is to be interpreted with Equation (3.67View Equation) inserted and expanded in powers of λ. This can be solved recursively for h1, h2, etc. We denote the solutions by hbeta(ρ) l. The unique solution adhering to the above boundary condition corresponds to a λ-independent C, for which we write p∕ζ1, where ζ1 is the first beta function coefficient. The solution then reads
( )2 hbeta(ρ,λ) = ρp − -λ-2ζ2-− -λ- 3ζ3ρ −p + ... (3.89 ) 2π ζ1 2π 2ζ1
This is a nontrivial fixed point, in the sense that gravity remains self-interacting and coupled to matter.

The renormalized action (at or away from the fixed point) has no direct significance in the quantum theory. It is however instructive to note that L h in Equation (3.56View Equation) can for any h(ρ) ⁄= ρp not be written in the form √--- LEH (g ) = − gR (g) for any metric with two commuting Killing vectors. In accordance with the general renormalization group picture one can of course write Lh ∼ LEH (g) + Lother(g). Interestingly, in the 2-Killing vector reduction one can write the renormalized Lagrangian Lh as a sum of two terms which are reductions of LEH. Since the parameter a in Lh can be changed at will by redefining σ ↦→ σ + a−˜aρ−1 2b, we may assume that b = − p, a = − 3p2 + 2p, for some p ⁄= 0. Then

( h (ρ ) ) λLh = LEH -----g + LEH (g0), (3.90 ) ρ
modulo total derivatives, where in the first term g is the metric with line element (3.8View Equation) and g0 is a metric with line element
( ∫ ) 2 −σ− k(ρ) 0 2 1 2 p ρ du 2 1 2 2 2 2 dS0 = e [− (dx ) + (dx ) ] + 1 − -- ---h(u) (dy ) − h(ρ) (dy ) , h u (3.91 ) [3 3 ] 2ρ∂ h ( p ∫ ρ du ) ∂ρk = ∂ρ --ln h + (--p − 1 )ln ρ − ------ρ---∂ ρ -- ---h(u) . 2 2 ρ ∂ρh − ph h u
The extra term L (g ) = 1(ρ∂ h − ph)[(3p − 2 + 3ρ∂ ln h)ρ−2∂ μρ∂ ρ + 2ρ−1∂ μρ∂ σ] EH 0 2 ρ ρ μ μ involves only the area radius ρ and the conformal factor σ. It vanishes iff h (ρ) is proportional to p ρ. From Equation (3.89View Equation) one sees that beta p h (ρ) ⁄= ρ always; by construction in the minimal subtraction scheme, but since the coefficients ζ1, ζ2 are universal, the same holds in any other scheme.

The origin of this feature is the seeming violation of scale invariance on the level of the renormalized action. Recall from after Equation (3.62View Equation) that μ ∂ C μ = ρ∂ρ ln h ⋅ Lh, so that for beta p h (ρ ) ⁄= ρ the action is no longer scale invariant. However this is precisely the property which allows one to cancel the (otherwise) anomalous term in the trace of the would-be energy momentum tensor, as discussed before, rendering the system conformally invariant at the fixed point. Due to the lack of naive scale invariance on the level of the renormalized action the dynamics of quantum gravidynamics is different from that of quantum general relativity, in the sector considered, even at the fixed point. The moral presumably generalizes: The form of the (bare and/or renormalized) action may have to differ from the Einstein–Hilbert action in order to incorporate the physics properties aimed at.

In the present context the most important feature of the fixed point beta h is its ultraviolet stability: For μ → ∞ all linearized perturbations around the fixed point function (3.89View Equation) are driven back to the fixed point. Since the fixed point (3.89View Equation) has the form of a power series in the loop counting parameter λ, the proper concept of a “linearized perturbation” has the form

-- --- h(ρ,λ,μ ) = hbeta(ρ,λ) + δh(ρ,λ, μ), ( ) --- λ--- λ-- 2-- (3.92 ) δh(ρ,λ,μ ) = 2πs1(ϱ,t) + 2π s2(ϱ,t) + ...
where the -- sl(ϱ,t) are functions of ϱ := ρp and t = 12π-lnμ ∕μ0. Note that the perturbation involves infinitely many functions of two variables. The boundary condition mentioned before, which guarantees that the full h- flow is driven by the counterterms, only amounts to the requirement that all the s(ϱ,t) l vanish for ϱ → ∞ uniformly in t. Inserting the ansatz (3.92View Equation) into the flow equation -d-- -- μdμh = βh(h∕λ ) and linearizing in --- δh(ρ,λ, μ) gives a recursive system of inhomogeneous integro-differential equations for the sl,
∫ d--- ∞ du--- -- -- -- dtsl = ζ1ϱ u3 sl(u,t) − ζ1∂ϱsl + Rl [sl−1,...,s1], l ≥ 1. (3.93 ) ϱ
Here R1 = 0, while the Rl, l ≥ 2, are complicated integro-differential operators acting (linearly) on the s1,...,sl−1 (see [155Jump To The Next Citation Point] for the explicit expressions). The lowest order equation (3.93View Equation) is homogeneous and its solution is given by
-- ∫ ∞ du s1(ϱ,t) = ϱ ---r1(u − ζ1t), (3.94 ) ϱ u
where r1 is an arbitrary smooth function of one variable satisfying u r1(u) → 0 for u → 0. This function can essentially be identified with the initial datum at some renormalization time t = 0, as r1(ϱ) = − ϱ∂ϱ[s1(ϱ,t = 0)∕ϱ]. Evidently s1(ϱ,t) → 0 for t → ∞, if ζ1 < 0. This condition is indeed satisfied by all the systems (3.1View Equation, 3.56View Equation) considered, precisely because the coset space G ∕H is noncompact. Interestingly one has the simple formula [155Jump To The Next Citation Point]
k + 2 1 -- ζ1 = − -----, k = # Abelian vector fields = -(dim G − dim G − 3). (3.95 ) 2 4
It follows from the value of the quadratic Casimir in the appropriate representation and is consistent with [47]. Since ζ1 < 0 always, Equation (3.94View Equation) shows that the lowest order perturbation s1 will always die out for t → ∞, for arbitrary smooth initial data prescribed at t = 0. It can be shown that this continues to hold for all higher order -- sl irrespective of the signs of the coefficients ζl, l ≥ 2.

Result (UV stability):
Given smooth initial s-(ϱ,0) l with s-(∞, t) = 0 l, l ≥ 1, the solution of the linearized flow equations (3.93View Equation) is unique and satisfies

-- sl(ϱ,t) −→ 0 for t −→ ∞ if ζ1 < 0,

where the convergence is uniform in ϱ.

The situation is illustrated in Figure 1View Image. The proof of this result is somewhat technical and can be found in [155Jump To The Next Citation Point].

View Image

Figure 1: Schematic form of the linearized flow of the s- l, l ≥ 1, perturbations; t = -1 ln μ∕μ 2π 0, p ϱ = ρ.

Often the stability properties of fixed points are not discussed by solving the linear flow equations directly, but by studying the spectral properties of the linearized perturbation operator (the “stability matrix” in Equation (A.10View Equation)). Since the generalized couplings here are functions, the linearized perturbation operator Θ is a formal integral operator,

∫ δβ (h∕λ)(ϱ)|| (Θs )(ϱ) = du ---h--------|| s(u), (3.96 ) δh (u) h=hbeta
with a distributional kernel Θ(ϱ,u ) = ∑ ( λ-)l−1Θl (ϱ,u) l 2π, which can be computed from the explicit formula (3.85View Equation). For example
Θ1(ϱ,u ) = ζ1[∂uδ(u − ϱ) + ϱu− 3θ(u − ϱ)], (3.97 )
where θ(u) is the step function. Writing s(ϱ) = ∑ (-λ)l−1s(ϱ) l≥1 2π l, ϑ = ∑ (-λ)l−1ϑ l≥1 2π l, ϑ ∈ ℂ, the spectral problem Θs = ϑs decomposes into a sequence of integro-differential equations for the Θ1sl − ϑ1sl = rhs, where the right-hand-side is determined by the solution of the lower order equations. Only the l = 1 equation is a spectral problem proper, Θ1s1 = ϑ1s1, ϑ1 ∈ ℂ. The relevant and irrelevant perturbations have spectra with negative and positive real parts, respectively. Remarkably all (nontrivial) eigenfunctions of Θ1 are normalizable; the spectrum is “purely discrete” and consists of the entire halfplane {ϑ1 ∈ ℂ | Re ϑ1 < 0}. Indeed, the general solution to Θ1s1 = ϑ1s1 is ∫∞ s1(ϱ) = aϱ + bϱ ϱ duu-e−ϑ1u∕ζ1, with a,b ∈ ℂ. The first term merely corresponds to a change of normalization of hbeta(ϱ) = ϱ + O (λ) and we may set a = 0, b = 1 without loss of generality. The second term corresponds to Equation (3.94View Equation) with r1(u) = e− ϑ1u∕ζ1. This clearly confirms the above result from a different perspective. For l > 1 the parameters ϑ l are not spectral values for Θ l. Moreover, since the kernels Θ l are distributions it is not quite clear which precise functional analytic setting one should choose for the full spectral problem Θs = ϑs. This is why above we adopted the direct strategy and determined the solutions of the linearized flow equations. Their asymptotic decay shows the ultraviolet stability of the fixed point unambiguously and independent of functional analytical subtleties.

We can put this result into the context the general discussion in Section 2 and arrive at the following conclusion:

With respect to the non-Gaussian fixed point beta h (⋅) all couplings in the generating functional h(⋅) are asymptotically safe. All symmetry reduced gravity systems satisfy the Criteria (PTC1) and (PTC2) to all loop orders of sigma-model perturbation theory. As explained in Section 3.2 from the viewpoint of the graviton loop expansion the distinction between a perturbative and a non-perturbative treatment is blurred here.

It is instructive to compare these properties to that of the Gaussian fixed point. The Gaussian fixed point of the flow (3.83View Equation) is best understood in analogy to the Gaussian fixed-point of a conventional nonlinear sigma-model. For a G ∕H nonlinear sigma-model with Lagrangian -1- μ i j L = − 2g0𝔪ij(ϕ) ∂ ϕ ∂μϕ (with 𝔪 satisfying Equation (3.12View Equation)) the beta function ∑ g βG∕H (g0) = g0 l≤1lζl(20π)l has only the trivial zero g∗0 = 0. As g0 → 0 the renormalized Lagrangian blows up, but in an expansion around 𝔪ij(ϕ ) = δij one can see that for g0 → 0 the interaction terms vanish. In this sense the fixed point ∗ g0 = 0 is Gaussian. This holds irrespective of the sign of ζ1, which however determines the stability properties of the flow. The stability ‘matrix’ vanishes so that the linearized stability analysis is empty. By direct inspection of the differential equation one sees that the unstable manifold of g∗0 is one-dimensional for ζ1 > 0 (typical for G ∕H compact) and empty for ζ1 < 0 (typical for G ∕H noncompact). Indeed, − μ d-¯g = ζ1¯g2+ O (¯g3) dμ 0 2π 0 0, and if one insists on ¯g ≥ 0 0 for positivity-of-energy reasons, the flow will be attracted to ∗ g0 = 0 for μ → ∞ iff ζ1 > 0. In particular for ζ1 < 0 these models are, based on the Criterion (PTC2) of Section 2, not expected to have a genuine continuum limit.

The Gaussian fixed point of the symmetry reduced gravity theories can be analyzed similarly. In terms of

----λ---- g (ρ, λ) := 2 πh(ρ,λ) (3.98 )
the flow equation (3.83View Equation) reads
[ ] d 1∫ ∞ du ∑ μ---¯g = ¯g2ρ∂ ρ -- --- lζl¯g(u )l d μ ¯g ρ u l≥1 [ ∫ ∞ du ] = − ζ1 ¯g2 + ρ ∂ρ¯g ---¯g(u) + O (¯g3). (3.99 ) ρ u
Clearly g (ρ) ≡ 0 ∗ is a fixed point (function) and in a similar sense as before it can be interpreted as a Gaussian fixed point. In contrast to the non-Gaussian fixed point the linearized stability analysis is now empty (just as it is for the G ∕H sigma-model flow). One thus has to cope with the nonlinear flow equation (3.99View Equation) at least to quadratic order. This is cumbersome but the qualitative feature of interest here can readily be understood: With respect to the Gaussian fixed point g∗(ρ) ≡ 0 not all couplings contained in the generating functional g( ⋅) are asymptotically safe. That is, there exists initial data g0(ρ) = ¯g0(ρ,μ0 ) (with g0 (ρ ) → 0, for ρ → ∞) for which the μ → ∞ asymptotics does not vanish identically in ρ. To see this, it suffices to note that the right-hand-side of Equation (3.99View Equation) to quadratic order reads ζ1¯g2 ρ∂ ρI (¯g ), with ∫ I(g) := g−1 ρ∞ duu g(u). Since ζ < 0 1 always, initial data g 0 for which I (g ) 0 is a strictly increasing function of ρ will give rise to solutions having the tendency to be driven towards larger values (pointwise in ρ) as μ increases. Conversely, only initial data g0 for which I(g0) is strictly decreasing in ρ can be expected to give rise to a solution g0 which vanishes identically in ρ as μ → ∞. A rigorous theorem describing the stable and the unstable manifold is presumably hard to come by, but for our purposes it is enough to know that there exist initial data which do not give rise to solutions decaying to g∗(ρ) ≡ 0 for μ → ∞. For example −aρ g0(ρ) = ρe or √ --------- g0(ρ) = exp (− a + bln ρ), a,b > 0, have this property. The upshot is that the Gaussian fixed point of the symmetry truncated gravity theories is not UV stable, the Condition (PTC2) is not satisfied, and one can presumably not use it for the construction of a genuine continuum limit.

At this point it may be instructive to contrast the quantum properties of the dimensionally reduced gravity theories with those of the same noncompact G ∕H sigma-model without coupling to gravity (which effectively amounts to setting ρ constant in Equation (3.16View Equation)). The qualitative differences are summarized in Table 2.

G ∕H sigma-model dimensionally reduced gravity with G∕H

renormalizable non-renormalizable
one essential coupling ∞ essential couplings
g0 function h(⋅)

flow is formally infrared free flow is asymptotically safe

formal trivial fixed point non-trivial fixed point
g0 = 0 hbeta(⋅)
formally IR stable UV stable
trace anomalous trace anomaly vanishes

Table 2: Comparison: noncompact G∕H sigma-model vs. dimensionally reduced gravity theory with G/H coset.

The comparison highlights why the above conclusion is surprising and significant. While the noncompact G ∕H sigma-models are renormalizable with just one relevant coupling (denoted by g in the table), at least in the known constructions they do not have a fixed point at which they are conformally invariant. Their gravitational counterparts require infinitely many relevant couplings for their UV renormalization. This infinite coupling flow has a nontrivial UV fixed point at which the theory is conformally invariant. Most importantly the stability properties of the renormalization flow are reversed (compared to the flow of g) for all of the infinitely many relevant couplings. As there appears to be no structural reason for this surprising reversal in the reduced theory itself, we regard it as strong evidence for the existence of an UV stable fixed point for the full renormalization group dynamics.

In the table we anticipated that at the fixed point the trace anomaly of the would-be energy momentum tensor vanishes for the symmetry truncated gravity theories. This allows one to construct quantum counterparts of the constraints ℋ0 and ℋ1 as well-defined composite operators. In detail this comes about as follows. Taking the trace in Equation (3.81View Equation) gives μ μ 2 [[T μ(h; φ)]] = [[t μ(h; φ)]] − ∂ [[V ]], again modulo the equations of motion operator. The first term has a nonzero trace anomaly given by

[[λ ( h) ]] 1 [[tμμ(h;φ )]] = --βh -- L + --[[ℒW − ˙Ξ𝔥ij(φ )∂μφi ∂μφj]]. (3.100 ) h λ 2λ
Here Ξ˙ is the field vector in Equation (3.76View Equation); furthermore W i = (0, ...,0, ρ-W (ρ,λ )) bh, where W (ρ,λ) is a functional of h which receives contributions only at three and higher loop orders (see Section B.3 and [154Jump To The Next Citation Point]). The improvement term is determined by the function f = f [h ] in Equation (3.81View Equation), which in turn is largely determined by h and thus cannot be freely chosen as a function of ρ.

It is therefore a very nontrivial match that (i) upon insertion of the fixed point beta h = h the function beta -- beta f (ρ ) := f[h ](ρ) becomes stationary (μ-independent), (ii) the equation Ff [hbeta]f beta = 0 turns out to determine fbeta(ρ ) completely, and (iii) the so-determined function has the property that ∂2[[V ]]|f=fbeta precisely cancels the second term in Equation (3.100View Equation) evaluated for hbeta. Thus the trace anomaly vanishes precisely at the fixed point of the coupling flow

μ beta [[T μ(h ;φ)]] = 0. (3.101 )
As should be clear from the derivation this is a nontrivial property of the system, rather than one which is used to define the renormalized target space metric 𝔥ij(φ). The latter is already determined by the warped product structure (3.59View Equation, 3.60View Equation) and the renormalization result (3.57View Equation). At the fixed point hbeta the first term in Equation (3.100View Equation) vanishes, but the second then is completely determined. On the other hand by Equation (3.81View Equation) one is not free to choose the improvement potential as a function of ρ-independent of h, so the cancellation is not built in. One can also verify that the only solution for f [h ] such that μ μ [[T μ(h;φ]] = ∂ K μ(h ) is beta h = h with the above beta f, such that Equation (3.101View Equation) holds. That is, scale invariance implies here conformal invariance. Since the target space metric (3.60View Equation) has indefinite signature this does not follow from general grounds [176194].

Due to Equation (3.101View Equation) we can now define the quantum constraints by

[[ℋ0 ]] := [[T00]], [[ℋ1 ]] := [[T01]]. (3.102 )
The linear combinations [[ℋ0 ± ℋ1 ]] are thus expected to generate two commuting copies of a Virasoro algebra with formal central charge c = 2 + dim G ∕H. This central charge is only formal because it refers to a state space with indefinite norm (see [127128] for an anomaly-free implementation of the Virasoro constraints in essentially noninteracting systems). In the case at hand the proper positivity requirement will be determined by the quantum observables commuting with the constraints. Their construction and the exploration of the physical state space is a major desideratum. In summary, the systems should at the fixed point be described by one whose physical states can be set into correspondence to the above quantum observables. The infrared problem has not been investigated so far, but based on results in the polarized subsector [153] one might expect it to be benign.

Despite the fact that the system is conformally invariant at the fixed point there are still scale dependent running parameters. At first sight this seems paradoxical. However, already the example of the massless continuum limit in a 3D scalar field theory exhibits this behavior. The remaining scale parameter is related to the direction of instability within the critical manifold, pointing from the Gaussian to the Fisher–Wilson fixed point, in the direction of coarse graining. The systems considered here provide an intriguing other example of this phenomenon. The critical manifold can be identified with the subset of parameter values where the system is scale (and here conformally) invariant. This fixes beta h (⋅) = h ( ⋅), but the inessential parameters contained in the renormalized fields are left unconstrained. This allows one to introduce a running parameter as follows. One evaluates the running coupling function -- h at the ‘comoving’ field ρ- and sets

--1---- ¯gh(μ) := h(ρ,μ) . (3.103 )
This quantity carries a two-fold μ-dependence, one via the running coupling -- μ ↦→ h(⋅,μ) and one because now the argument at which the function is evaluated is likewise μ-dependent. Since ρ is a field on the base manifold, the quantity ¯g (μ) h depends parametrically on the value of h(ρ(x)) – and hence on x. Combining Equation (3.83View Equation) with Equation (3.75View Equation) one finds the following flow equation:
d μ ---λ¯gh = βG ∕H (λ¯gh). (3.104 ) dμ
These are the usual flow equations for the single coupling G ∕H sigma-model without coupling to gravity! In other words the ‘gravitationally dressed’ functional flow for -- h has been ‘undressed’ by reference to the scale dependent ‘rod field’ ρ- (the term is adapted from H. Weyl’s “Maßstabsfeld”). The Equations (3.104View Equation) are not by themselves useful for renormalization purposes – which requires determination of the flow of -- h(⋅,μ) with respect to a fixed set of field coordinates. Moreover in the technical sense ¯gh is an “inessential” parameter. The fact that the scale dependence of ¯gh is governed by the beta function βG ∕H means that for increasing μ it will be driven away from the fixed point gh = 0. The condition gh(x, λ) ≡ 0 can be traded for the specification of the Gaussian fixed point g(ϱ;λ ) ≡ 0. Thus the parameter flow ¯gh may be viewed as a coupling flow emanating (in the direction of increasing μ) from the Gaussian fixed point. At the non-Gaussian fixed point, on the other hand, ¯gh governs the scale dependence of the ‘rod field’ ¯ρbeta(x,μ ) := ¯ρ(x,μ )|h=hbeta via
d λ [ ] μ --ρ¯beta = ζ1---¯ρbeta¯ghbeta ¯ρbeta . (3.105 ) dμ 2π
This follows from Equation (3.75View Equation) and the relation hΞ˙3 [h ]|h=hbeta = ρζ1-λ 2π. Here we indicated the functional dependence of ¯g h on ¯ρ, which at the non-Gaussian fixed point gives a dependence of ¯ghbeta on beta ¯ρ. Since ζ1 < 0, one sees from Equation (3.105View Equation) that beta ¯ρ is pointwise for all x a decreasing function of μ, at least locally in μ. In addition Equation (3.87View Equation) implies
∫ ρ¯beta du-hbeta(u ) = χ+(x+, μ) + χ− (x − ,μ). (3.106 ) u
Here χ±( ⋅,μ) are functions of one variable which by Equation (3.105View Equation) are locally decreasing in μ and ± 0 1 x = (x ± x )∕2 are lightcone coordinates. Since the theory is conformally invariant at the fixed point (of the couplings) one can change coordinates x± ↦→ χ± to bring scaling operators into a standard form. The upshot is, as anticipated in Section 3.2, that the rod field ρ¯beta describes the local scale changes dynamically induced by quantum gravity and defines a resolution scale for the geometries.

For orientation we summarize here our results on the renormalization of the symmetry reduced Quantum Einstein Gravity theories (3.1View Equation):

  1. The systems inherit the lack of standard perturbative renormalizability from the full theory. A cut-off independent quantum theory can be achieved at the expense of introducing infinitely many couplings combined into a generating function h(⋅) of one variable.
  2. The argument of this function is the ‘area radius’ field ρ associated with the two Killing vectors. The field ρ is (nonlinearly) renormalized but no extra renormalizations are needed to define arbitrary powers thereof.
  3. A universal formula for the beta functional for h and hence for the infinitely many couplings contained in it can be given. The flow possesses a Gaussian as well as a non-Gaussian fixed point. With respect to the non-Gaussian fixed point all couplings in h are asymptotically safe.
  4. At the fixed point the trace anomaly vanishes and the quantum constraints (well-defined as composite operators) [[ℋ ]] 0, [[ℋ ]] 1 can in principle be imposed. The linear combinations [[ℋ0 ± ℋ1 ]] are expected to generate commuting copies of a centrally extended conformal algebra acting on an indefinite metric Hilbert space.
  5. Despite the conformal invariance at the fixed point there is a scale dependent local parameter, whose scale dependence is governed by the beta function of the G ∕H sigma-model without coupling to gravity.

So far we considered the renormalization of the symmetry reduced theories in its own right, leaving the embedding into the full Quantum Gravidynamics open. The proposed relation to qualitative aspects of the Quantum Gravidynamics in the extreme UV has already been mentioned. Here we offer some tentative remarks on the embedding otherwise. The constructions presented in this section can be extended to 2 + ε dimensions in the spirit of an ε-expansion. At the same time this mimics quantum aspects of the 1-Killing vector reduction. One finds that the qualitative features of the renormalization flow – non-Gaussian fixed point and asymptotic safety – are still present [156Jump To The Next Citation Point]. A cosmological ‘constant’ term can likewise be included and displays a similar pattern as outlined at the end of Section 3.2. The advantage of this setting is that the UV cutoff can strictly be removed, which is hard to achieve with a nonperturbative technique. The extension of these results from a quasi-perturbative analysis to a nonperturbative one, ideally via controlled approximations, is an important open problem. The same holds for the analysis of the 1-Killing vector reduction, which holds the potential for cosmological applications. These truncations can be viewed as complementary to the ‘hierarchical’ truncations used in Section 4: A manifest truncation is initially imposed on the functional integral, but the infinite coupling renormalization flow can then be studied in great detail, often without further approximations.

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