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1.3 Some working definitions

Here we attempt to give working definitions for some of the key terms used before.

Quantum Gravidynamics:
We shall use the term “Quantum Gravidynamics” to highlight a number of points in the present circle of ideas. First, that one aims at relating the micro- and the macro-physics of the gravitational field through a renormalization flow defined conceptually in terms of a functional integral. In contrast to “Quantum General Relativity” the microscopic action is allowed to be very different from the Einstein–Hilbert action or a discretization thereof. Plausibly it should be still quasilocal, i.e. have a well-defined derivative expansion, and based on perturbatively renormalizable higher derivative theories one would expect it to contain at least quartic derivative terms. This means that also the number of physical propagating degrees of freedom (with respect to a background) may be different from the number entailed by the Einstein–Hilbert action. The second motivation for the term comes from the analogy with Quantum Chromodynamics. Indeed, the premise is that the self-interaction for the quantized gravitational field is predominantly “anti-screening” in the ultraviolet in a similar sense as in Quantum Chromodynamics, where it is responsible for the characteristic high energy behavior of physical quantities. As in Quantum Chromodynamics the proper identification of the antagonistic degrees of freedom (screening versus anti-screening) may well depend on the choice of field variables.

As with “Quantum General Relativity” we take the term “Gravidynamics” in a broad sense, allowing for any set of field variables (e.g. vielbein and spin connection, Ashtekar’s variables, Plebanski and BF type formulations, teleparallel, etc.) that can be used to recast general relativity (see e.g. the review [167]). For example the coupling of fermions might be a good reason to use a vielbein formulation. If the metric is taken as dynamical variable in four dimensions we shall also use the term “Quantum Einstein Gravity” as in [154Jump To The Next Citation Point133Jump To The Next Citation Point131Jump To The Next Citation Point]. It is of course not assumed from the outset that the quantum gravidynamics based on the various set of field variables are necessarily equivalent.

Gaussian fixed point:
A fixed point is called Gaussian if there exists a choice of field variables for which the fixed point action is quadratic in the fields and the functional measure is Gaussian. This includes the local case but also allows for nonlocal quadratic actions. The drawback of this definition is that the proper choice of field variables in which the measure reveals its Gaussian nature may be hard to find. (For example in the correlation functions of the spin field in the two-dimensional Ising model the underlying free fermionic theory is not visible.)

A non-Gaussian fixed point is simply one where no choice of fields can be found in which the measure becomes Gaussian. Unfortunately this, too, is not a very operational criterion.

Unstable manifold:
The unstable manifold of a fixed point with respect to a coarse graining operation is the set of all points that can be reached along flow lines emanating from the fixed point, the so-called renormalized trajectories. Points on such a flow line correspond to perfect actions. The stable manifold is the set of points attracted to the fixed point in the direction of coarse graining.

Strict (weak) renormalizability:
We call a field theory strictly (weakly) renormalizable with respect to a fixed point and a coarse graining operation if the dimension of its unstable manifold is finite (infinite). It is implied that if a field theory has this property with respect to one coarse graining operation it will have it with respect to many others (“universality”). Strict or weak renormalizability is believed to be a sufficient condition for the existence of a genuine continuum limit for observables.

Relevant coupling:
Given an expansion “sum over couplings times interaction monomials”, a coarse graining operation, and a fixed point of it, a coupling is called relevant (irrelevant) if it is driven away from (towards) the value the corresponding coordinate has at the fixed point, under a sufficient number of coarse graining steps. Note that this distinction makes sense even for trajectories not connected to the fixed point (because they terminate). It is however an explicitly ‘coordinate dependent’ notion. The same terms are used for the interaction monomials associated with the couplings. The dimension of the unstable manifold equals the maximal number of independent relevant interaction monomials ‘connected’ to the fixed point. All points on the unstable manifold are thus parameterized by relevant couplings but not vice versa.

Couplings which are relevant or irrelevant in a linearized analysis are called linearly relevant or linearly irrelevant, respectively. A coupling which is neither linearly relevant nor linearly irrelevant is called (linearly) marginal.

Continuum limit:
By a genuine continuum limit we mean here a limit in which physical quantities become:
(C1) strictly independent of the UV cutoff,

(C2) independent of the choice of the coarse graining operation (within a certain class), and

(C3) invariant under point transformations of the fields.

Usually one stipulates Properties (C1) and (C2) for the functional measure after which Property (C3) should be a provable property of physical quantities like the S-matrix. The requirement of having also Properties (C1) and (C2) only for observables is somewhat weaker and in the spirit of the asymptotic safety scenario.

Typically the Properties (C1, C2, C3) cannot be rigorously established, but there are useful criteria which render the existence of a genuine continuum limit plausible in different computational frameworks. In Sections 2.1 and 2.2 we discuss in some detail such criteria for the perturbative and for the FRGE approach, respectively. For convenience we summarize the main points here.

In renormalized perturbation theory the criterion involves two parts:
(PTC1) Existence of a formal continuum limit. This means, the removal of the UV cutoff is possible and the renormalized physical quantities are independent of the scheme and of the choice of interpolating fields – all in the sense of formal power series in the loop counting parameter. The perturbative beta functions always have a trivial (Gaussian) fixed-point but may also have a nontrivial (non-Gaussian) fixed point.

The second part of the criterion is:
(PTC2) The dimension of the unstable manifold of the (Gaussian or non-Gaussian) fixed point as computed from the perturbative beta functions equals the number of independent essential couplings.

For example φ44 and QED meet Criterion (PTC1) but not (PTC2) while QCD satisfies both Criterion (PTC1) and (PTC2). In the framework of the functional renormalization group equations (FRGE) similar criteria for the existence of a genuine continuum limit can be formulated. Specifically for the FRGE of the effective average action one has:
(FRGC1) The solution of the FRG equation admits (for fine tuned initial data Γ initial at some k = k initial) a global solution Γ k, i.e. one that can be extended both to k → ∞ and to k → 0 (where the latter limit is not part of the UV problem in itself).

(FRGC2) The functional derivatives of limk→0 Γ k (vertex functions) meet certain requirements which ensure stability/positivity/unitarity.

In Criterion (FRGC1) the existence of the k → 0 limit in theories with massless degrees of freedom is nontrivial and the problem of gaining computational control over the infrared physics should be separated from the UV aspects of the continuum limit as much as possible. However the k → 0 limit is essential to probe stability/positivity/unitarity. For example, to obtain a (massive) Euclidean quantum field theory the Schwinger functions constructed from the vertex functions have to obey nonlinear relations which ensure that the Hilbert space reconstructed via the Osterwalder–Schrader procedure has a positive definite inner product.

Perturbative (weak) renormalizability:
We call a theory perturbatively (weakly) renormalizable if Criterion (PTC1) can be achieved with finitely (infinitely) many essential couplings. A theory were neither can be achieved is called perturbatively non-renormalizable. Perturbative (weak) renormalizability is neither necessary nor sufficient for (weak or strict) renormalizability in the above nonperturbative sense. It is only in combination with Criterion (PTC2) that perturbative results are indicative for the existence of a genuine continuum limit.

Asymptotically free coupling:
A non-constant coupling in the unstable manifold of a Gaussian fixed point.

The “non-constant” proviso is needed to exclude cases like a trivial φ4 4 coupling. In a nonperturbative lattice construction of 4 φ4 theory only a Gaussian fixed point with a one-dimensional unstable manifold (parameterized by the renormalized mass) is thought to exist, along which the renormalized φ44 coupling is constant and identically zero. The Gaussian nature of the fixed-point, on the other hand, is not crucial and we define:

Asymptotically safe coupling:
A non-constant coupling in the unstable manifold of a fixed point.

Asymptoticaly safe functional measure:
The functional measure of a statistical field theory is said to be asymptotically safe if is perturbatively weakly renormalizable or non-renormalizable, but possesses a fixed point with respect to which it is strictly renormalizable. Subject to the regularity assumption that the space of actions can in the vicinity of the fixed point be decomposed into a stable and an unstable manifold, this is equivalent to the following requirement: All relevant couplings are asymptotically safe and there is only a finite number of them. Note that unitarity or other desirable properties that would manifest itself on the level of observables are not part of this definition.

In a non-gravitational context the functional measure of the 3D Gross–Neveu model is presently the best candidate to be asymptotically safe in the above sense (see [10160198105] and references therein). Also 5D Yang–Mills theories (see [93148] and references therein) are believed to provide examples. In a gravitational context, however, there are good reasons to modify this definition.

First the choice of couplings has to be physically motivated, which requires to make contact to observables. In the above nongravitational examples with a single coupling the ‘meaning’ of the coupling is obvious; in particular it is clear that it must be finite and positive at the non-Gaussian fixed point. In general however one does not know whether ill behaved couplings are perverse redefinitions of better behaved ones. To avoid this problem the couplings should be defined as coefficients in a power series expansion of the observables themselves (Weinberg’s “reaction rates”; see the discussion in Section 1.1). Of course painfully little is known about (generic) quantum gravity observables, but as a matter of principle this is how couplings should be defined. In particular this will pin down the physically adequate notion of positivity or unitarity.

Second, there may be good reasons to work initially with infinitely many essential or potentially relevant couplings. Recall that the number of essential couplings entering the initial construction of the functional measure is not necessarily equal to the number eventually indispensable. In a secondary step a reduction of couplings might be feasible. That is, relations among the couplings might exist which are compatible with the renormalization flow. If these relations are sufficiently complicated, it might be better to impose them retroactively than to try to switch to a more adapted basis of interaction monomials from the beginning.

Specifically in the context of quantum gravity microscopic actions with infinitely many essential couplings occur naturally in several ways. First, when starting from the Gomis–Weinberg picture [94Jump To The Next Citation Point] of perturbative quantum gravity (which is implemented in a non-graviton expansion in Section 3 for the 2 + 2 reduction). Second, when power counting considerations are taken as a guideline one can use Newton’s constant (frozen in Planck units) to build dimensionless scalars (dilaton, conformal factor) and change the conformal frame arbitrarily. The way how these dimensionless scalars enter the (bare versus renormalized) action is not constrained by power counting considerations. This opens the door to an infinite number of essential couplings. The effective action for the conformal factor [149Jump To The Next Citation Point] and the dilaton field in the 2 + 2 reduction [154Jump To The Next Citation Point] provide examples of this phenomenon.

Third, the dimension of the unstable manifold is of secondary importance in this context. Recall that the dimension of the unstable manifold is the maximal number of independent relevant interaction monomials ‘connected’ to the fixed point. This maximal number may be very difficult to determine in Quantum Gravidynamics. It would require identification of all renormalized trajectories emanating from the fixed point – which may be more than what is needed physicswise: The successful construction of a subset of renormalized trajectories for physically motivated couplings may already be enough to obtain predictions/explanations for some observables. Moreover, what matters is not the total number of relevant couplings but the way how observables depend on them. Since generic observables (in the sense used in Section 1.1) are likely to be nonlinearly and nonlocally related to the metric or to the usual basis of interaction monomials (scalars built from polynomials in the curvature tensors, for instance) the condition that the theory should allow for predictions in terms of observables is only indirectly related to the total number of relevant couplings.

In summary, the interplay between the microscopic action, its parameterization through essential or relevant couplings, and observables is considerably more subtle than in the presumed non-gravitational examples of asymptotically safe theories with a single coupling. The existence of an asymptotically safe functional measure in the above sense seems to be neither necessary nor sufficient for a physically viable theory of Quantum Gravidynamics. This leads to our final working definition.

Asymptotically safe Quantum Gravidynamics:
 A quantum theory of gravity based on a notion of an asymptotically safe functional integral measure which incorporates the interplay between couplings and observables described above. In brief:

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