Since the microscopic action is likely to contain higher derivative terms, don’t the problems with non-unitarity notorious in higher derivative gravity theories reappear?
In brief, the unitarity issue has not much been investigated so far, but the presumed answer is No.
First, the problems with perturbatively strictly renormalizable higher derivative theories stem mostly from the -type propagator used. The alternative perturbative framework already mentioned, namely to use a -type propagator at the expense of infinitely many essential (potentially ‘unsafe’) couplings, avoids this problem [94, 10]. The example of the reduction shows that the reconcilation of safe couplings with the absence of unphysical propagating modes can be achieved in principle. Also the superrenormalizable gravity theories with unitary propagators proposed in [218] are intriguing in this respect.
Second, when the background effective action is used as the central object to define the quantum theory, the ‘background’ is not a solution of the classical field equations. Rather it is adjusted self-consistenly by a condition involving the full quantum effective action (see Appendix B). If the background effective action is computed nonperturbatively (by whatever technique) the intrinsic notion of unitarity will not be related to the ‘propagator unitarity’ around a solution of the classical field equations in any simple way.
One aspect of this intrinsic positivity is the convexity of the background effective action. In the flow equation for the effective average action one can see, for example, that the wrong-sign of the propagator is not an issue: If is of the type, the running inverse propagator when expanded around flat space has ghosts similar to those in perturbation theory. For the FRG flow, however, this is irrelevant since in the derivation of the beta functions no background needs to be specified explicitly. All one needs is that the RG trajectories are well defined down to . This requires that is a positive operator for all . In the untruncated functional flow this is believed to be the case. A rather encouraging first result in this direction comes from the truncation [131].
More generally, the reservations towards higher derivative theories came from a loop expansion near the perturbative Gaussian fixed point. In contrast in Quantum Gravidynamics one aims at constructing the continuum limit nonperturbatively at a different fixed point. In the conventional setting one quantizes as the bare action, while in Quantum Gravidynamics the bare action, defined by backtracing the renormalized trajectory to the non-Gaussian fixed point, may in principle contain all sorts of curvature invariants whose impact on the positivity and causality of the theory is not even known in perturbation theory.
In the previous discussion we implicitly assumed that generic physical quantities are related
in a rather simple way to the interaction monomials entering the microscopic action. For Dirac
observables however this is clearly not the case. Assuming that the physically correct notion of
unitarity concerns such observables it is clear that the final word on unitarity issues can only be
spoken once actual observables are understood.
Doesn’t the very notion of renormalizability presuppose a length or momentum scale? Since the latter is absent in a background independent formulation, the renormalizability issue is really an artifact of the perturbative expansion around a background.
No. Background independence is a subtle property of classical general relativity (see e.g. [199] for a discussion) for which it is unclear whether or not it has a compelling quantum counterpart. As far as the renormalization problem is concerned it is part of the physics premise of a functional integral type approach that there is a description independent and physically relevant distinction between coarse grained and fine grained geometries. On a classical level this amounts to the distinction, for example, between a perfect fluid solution of the field equations and one generated by its or so molecular constituents. A sufficiently large set of Dirac observables would be able to discriminate two such spacetimes. Whenever we shall refer later on to “coarse grained” versus “fine grained” geometries we have a similar picture in mind for the ensembles of off-shell geometries entering a functional integral.
Once such a physics premise is made, the renormalization in the Kadanoff–Wilson sense is clearly relevant for the computation of observable quantities and does not just amount to a reshuffling of artifacts. Renormalization in this sense is, for example, very likely not related to the regularization ambiguities [151, 173] appearing in loop quantum gravity. A minimal requirement for such an interpretation of the regularization ambiguites would be that reasonable coarse graining operations exist which have a preferred discretization of the Einstein–Hilbert action as its fixed point. This preferred discretization would have to be such that the observables weakly commuting with the associated Hamiltonian constraint reproduce those of loop quantum gravity.
For clarity’s sake let us add that the geometries entering a functional integral are expected to
be very rough on the cutoff scale (or of a distributional type without a cutoff) but superimposed
to this ‘short wavelength zigzag’ should be ‘long wavelength’ modulations (defined in terms of
dimensionless ratios) to which different observables are sensitive in different degrees. In general it
will be impractical to base the distinction between ensembles of fine grained and coarse grained
geometries directly on observables. In the background field formalism the distinction is made with
respect to an initially prescribed but generic background geometry which after the functional
integral is performed (entirely or in a certain mode range) gets related to the expectation value
of the quantum metric by a consistency condition involving the full quantum dynamics.
Doesn’t such a non-perturbative renormalizability scenario require a hidden enhanced symmetry?
Improved renormalizability properties around a given fixed point are indeed typically rooted
in symmetries. A good example is QCD in a lightfront formulation where gauge invariance
is an ‘emergent phenomenon’ occuring only after an infinite reduction of couplings [174]. In
the case of Quantum Gravidynamics, the symmetry in question would be one that becomes
visible only around the non-Gaussian fixed point. If it exists, its identification would constitute
a breakthrough. From the Kadanoff–Wilson view of renormalization it is however the fixed point
which is fundamental – the enhanced symmetry properties are a consequence (see the notion of
generalized symmetries in [236, 160]).
Shouldn’t the proposed anti-screening be seen in perturbation theory?
Maybe, maybe not. Presently no good criterion for antiscreening in this context is known. For the reasons explained in Section 1.1 it should not merely be identified with the sign of the dominant contribution to some beta function. The answer to the above question will thus depend somewhat on the identification of the proper degrees of freedom and the quantitiy considered.
As an example one can look at quantum gravity corrections to the Newton potential, which have been considered in some detail. The result is always of the form
Interpreted as a modification of Newton’s constant , one sees that roughly corresponds to screening and to anti-screening behavior. The value of is unambigously defined in 1-loop perturbation theory and is a genuine prediction of quantum gravity viewed as an effective field theory (as stressed by Donoghue). However will depend on the precise definition of the nonrelativistic potential and there are various options for it.
One is via the scattering amplitude. The coefficient was computed initially by Donoghue and later by Khriplovich and Kirilin; the result considered definite in [32] is . It decomposes into a negative vertex and triangle contributions , and a just slightly larger positive remainder coming from box, seagull, and vacuum polarization diagrams.
Another option is to consider corrections to the Schwarzschild metric. Different sets of diagrams have been used for the definition [119, 33] and affect the parameterization (in-)dependence and other properties of the corrections. Both choices advocated lead to , which amounts to antiscreening.
Let us also mention alternative definitions of an effective Newton potential via Wilson lines in Regge
calculus [100] or by resummation of scalar matter loops [226]. The latter gives rise to an
“antiscreening” Yukawa type correction of the form , with
. Via it can be interpreted as a running Newton constant
.
There are several thought experiments suggesting a fundamental limit to giving an operational meaning to spacetime resolutions, for example via generalized uncertainty relations of the form (see [85, 163, 143] and references therein)
These relations are sometimes taken as hinting at a “fundamental discretum”. If so, does this not contradict the asymptotic safety scenario, where in the fixed point regime the microscopic spacetimes become self-similar?
No, the arguments assume that Newton’s constant is constant. (We momentarily write for in .) If is treated as a running coupling the derivations of the uncertainty relations break down. As an example consider a photon-electron scattering process as in [143, 163]: refers to gravity in the (‘photon’ – ‘electron’ ) interaction region with a pointlike ‘electron’. If viewed as running one expects in the fixed point regime. Hence in the above relation one should replace by . This gives
and there is no limit on the spatial resolution. One can of course decide to choose units in which is constant by definition (see [172]) in which case the derivations go through. Our conclusion is that the perceived dichotomy between a fundamentally ‘discrete’ versus ‘continuum’ geometry may itself not be fundamental.
Each of the issues raised clearly deserves much further investigation. For the time being we conclude however that the asymptotic safety scenario is conceptually self-consistent. It remains to assemble hard computational evidence for the existence of a non-Gaussian fixed point with a nontrival and regular unstable manifold. This task will be taken up in Sections 3 and 4.
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