The results from the expansion were part of Weinberg’s original motivation to propose the
scenario. Since gravity in two and three dimensions is non-dynamical, however, the lessons for a genuine
quantum gravitational dynamics are somewhat limited. Higher derivative theories were known to be strictly
renormalizable with a finite number of couplings, at the expense of having unphysical propagating modes
(see [207
, 206
, 83
, 19
, 59
]). In hindsight one can identify a non-Gaussian fixed point for Newton’s
constant already in this setting (see [54
] and Section 2.3). The occurance of this non-Gaussian fixed point
is closely related to the
-type propagator that is used. The same happens when (Einstein or
a higher derivative) gravity is coupled to a large number
of matter fields and a
expansion is performed. A nontrivial fixed point is found that goes hand in hand with a
-type
progagator (modulo logs), which here arises from a resummation of matter self-energy bubbles,
however.
As emphasized before the challenge of Quantum Gravidynamics is not so much to achieve (perturbative or nonperturbative) renormalizability but to reconcile asymptotically safe couplings with the absence of unphysical propagating modes. Two recent developments provide complementary evidence that this might indeed be feasible. Both of these developments take into account the dynamics of infinitely many physical degrees of freedom of the four-dimensional gravitational field. In order to be computationally feasible the ‘coarse graining’ has to be constrained somehow. To do this the following two strategies have been pursued (which we label Strategies (c) and (d) according to the discussion below):
(c) The metric fluctuations are constrained by a symmetry requirement, but the full (infinite-dimensional) renormalization group dynamics is considered. We shall refer to this as the strategy via symmetry reductions.
(d) All metric fluctuations are taken into account but the renormalization group dynamics is projected onto a low-dimensional submanifold. Since this is done using truncations of functional renormalization group equations, we shall refer to this as the strategy via truncated functional flow equations.
Both strategies (truncation in the fluctuations but unconstrained flow and unconstrained quantum fluctuations but constrained flow) are complementary. Tentatively both results are related by the dimensional reduction phenomenon described before (see Section 2.4). The techniques used are centered around the background effective action, but are otherwise fairly different. For the reader’s convenience we included summaries of the relevant aspects in Appendices A and B. The main results obtained from Strategies (c) and (d) are reviewed in Sections 3 and 4, respectively.
For the remainder of this section we now first survey the pieces of evidence from all the computational settings (a – d):
(a) Evidence from expansions: In the non-gravitational examples of perturbatively
non-renormalizable field theories with a non-Gaussian fixed point the non-Gaussian fixed point can be
viewed as a ‘remnant’ of an asymptotically free fixed point in a lower-dimensional version of
the theory. It is thus natural to ask how gravity behaves in this respect. In
spacetime
dimensions Newton’s constant
is dimensionless, and formally the theory with the bare action
is power counting renormalizable in perturbation theory. However, as the
Einstein–Hilbert term is purely topological in two dimensions, the inclusion of local dynamical degrees of
freedom requires, at the very least, starting from
dimensions and then studying the
behavior near
. The resulting “
-expansion” amounts to a double expansion in the
number of ‘graviton’ loops and in the dimensionality parameter
. Typically dimensional
regularization is used, in which case the UV divergencies give rise to the usual poles in
.
Specific for gravity are however two types of complications. The first one is due to the fact that
is topological at
, which gives rise to additional “kinematical” poles of
order
in the graviton propagator. The goal of the renormalization process is to remove
both the ultraviolet and the kinematical poles in physical quantities. The second problem is
that in pure gravity Newton’s constant is an inessential parameter, i.e. it can be changed at
will by a field redefinition. Newton’s constant
can be promoted to a coupling proper by
comparing its flow with that of the coefficient of some reference operator, which is fixed to be
constant.
For the reference operator various choices have been adopted (we follow the discussion in Kawai et
al. [118, 116
, 117
, 3
] with the conventions of [117
]):
(i) a cosmological constant term ,
(ii) monomials from matter fields which are quantum mechanically non-scale invariant in ,
(iii) monomials from matter fields which are quantum mechanically scale invariant in ,
(iv) the conformal mode of the metric itself in a background field expansion.
All choices lead to a flow equation of the form
but the coefficient Technically the non-universality of arises from the before-mentioned kinematical poles. In the early
papers [86, 53, 227
] the Choice i was adopted giving
, or
if free
matter of central charge
is minimally coupled. A typical choice for Choice ii is a mass term of
a Dirac fermion, a typical choice for Choice iii is the coupling of a four-fermion (Thirring)
interaction. Then
comes out as
, where
, respectively.
Here
is the scaling dimension of the reference operator, and again free matter of central
charge
has been minimally coupled. It has been argued in [118
] that the loop expansion
in this context should be viewed as double expansion in powers of
and
, and that
reference operators with
are optimal. The Choice iv has been pursued systematically in a
series of papers by Kawai et al. [116
, 117
, 3]. It is based on a parameterization of the metric
in terms of a background metric
, the conformal factor
, and a part
which is
traceless,
. Specifically
is inserted into the Einstein–Hilbert action;
propagators are defined (after gauge fixing) by the terms quadratic in
and
, and vertices
correspond to the higher order terms. This procedure turns out to have a number of advantages.
First the conformal mode
is renormalized differently from the
modes and can be
viewed as defining a reference operator in itself; in particular the coefficient
comes out as
. Second, and related to the first point, the system has a well-defined
-expansion (absence of poles) to all loop orders. Finally this setting allows one to make contact
to the exact (KPZ [122]) solution of two-dimensional quantum gravity in the limit
.
(b) Evidence from perturbation theory and large : Modifications of the
Einstein–Hilbert action where fourth derivative terms are included are known to be perturbatively
renormalizable [206
, 83
, 19
, 59
]. A convenient parameterization is
The action (1.7) can be supplemented by a matter action, containing a large number,
, of free
matter fields. One can then keep the product
fixed, retain the usual normalization of the matter
kinetic terms, and expand in powers of
. Renormalizability of the resulting ‘large
expansion’
then amounts to being able to remove the UV cutoff order by order in the formal series in
. This type
of studies was initiated by Tomboulis where the gravity action was taken either the pure Ricci scalar [216
],
Ricci plus cosmological term [203
], or a higher derivative action [217
], with free fermionic matter in all
cases. More recently the technique was reconsidered [169
] with Equation (1.7
) as the gravity action
and free matter consisting of
scalar fields,
Dirac fields, and
Maxwell
fields.
Starting from the Einstein–Hilbert action the high energy behavior of the usual -type propagator
gets modified. To leading order in
the modified propagator can be viewed as the graviton propagator
with an infinite number of fermionic self-energy bubbles inserted and resummed. The resummation changes
the high momentum behavior from
to
, in four dimensions. In
dimensions
the resulting
expansion is believed to be renormalizable in the sense that the UV cutoff
can
strictly be removed order by order in
without additional (counter) terms in the Lagrangian. In
the same is presumed to hold provided an extra
term is included in the bare Lagrangian, as in
Equation (1.7
). After removal of the cutoff the beta functions of the dimensionless couplings can be
analyzed in the usual way and already their leading
term will decide about the flow
pattern.
The qualitative result (due to Tomboulis [216] and Smolin [203
]) is that there exists a nontrivial fixed
point for the dimensionless couplings
, and
. Its unstable manifold is three dimensional, i.e. all
couplings are asymptotically safe. Repeating the computation in
dimensions the fixed point still
exists and (taking into account the different UV regularization) corresponds to the large
(central
charge) limit of the fixed point found the
expansion.
These results have recently been confirmed and extended by Percacci [169] using the heat kernel
expansion. In the presence of
scalar fields,
Dirac fields, and
Maxwell fields, the flow
equations for
and
come out to leading order in
as
As a caveat one should add that the -type propagators occuring both in the perturbative and in
the large
framework are bound to have an unphysical pole at some intermediate momentum scale. This
pole corresponds to unphysical propagating modes and it is the price to pay for (strict) perturbative
renormalizability combined with asymptotically safe couplings. From this point of view, the main challenge
of Quantum Gravidynamics lies in reconciling asymptotically safe couplings with the absence of unphysical
propagating modes. Precisely this can be achieved in the context of the
reduction.
(c) Evidence from symmetry reductions: Here one considers the usual gravitational functional
integral but restricts it from “4-geometries modulo diffeomorphisms” to “4-geometries constant along a
foliation modulo diffeomorphisms”. This means that instead of the familiar
foliation of
geometries one considers a foliation in terms of two-dimensional hypersurfaces
and performs the
functional integral only over configurations that are constant as one moves along the stack of
two-surfaces. Technically this constancy condition is formulated in terms of two commuting
vectors fields
,
, that are Killing vectors of the class of geometries
considered,
. For definiteness we consider here only the case where both Killing
vectors are spacelike. From this pair of Killing vector fields one can form the symmetric
matrix
. Then
(with
the components of
and
) defines a metric on the orbit space
which obeys
and
. The functional integral is eventually performed over metrics of the form
In the context of the asymptotic safety scenario the restriction of the functional integral to metrics of
the form (1.9) is a very fruitful one:
Two additional bonus features are: In this sector the explicit construction of Dirac observables is feasible (classically and presumably also in the quantum theory). Finally a large class of matter couplings is easily incorporated.
As mentioned the effective dynamics looks two-dimensional. Concretely the classical action describing
the dynamics of the 2-Killing vector subsector is that of a non-compact symmetric space sigma-model
non-minimally coupled to 2D gravity via the “area radius” , of the two Killing
vectors. To avoid a possible confusion let us stress, however, that the system is very different from most
other models of quantum gravity (mini-superspace, 2D quantum gravity or dilaton gravity, Liouville theory,
topological theories) in that it has infinitely many local and self-interacting dynamical degrees of freedom.
Moreover these are literally (an infinite subset of) the degrees of freedom of the four-dimensional
gravitational field, not just analogues thereof. The corresponding classical solutions (for both
signatures of the Killing vectors) have been widely studied in the general relativity literature,
c.f. [98
, 26
, 121
]. We refer to [45
, 46
, 56
] for details on the reduction procedure and [197
] for a canonical
formulation.
Technically the renormalization is done by borrowing covariant background field techniques from
Riemannian sigma-models (see [84, 110
, 201
, 57
, 220
, 162
]). In the particular application here the
sigma-model perturbation theory is partially nonperturbative from the viewpoint of a graviton loop
expansion as not all of the metric degrees of freedom are Taylor expanded in the bare action (see
Section 3.2). This together with the field reparameterization invariance blurs the distinction between a
perturbative and a non-perturbative treatment of the gravitational modes. The renormalization can be done
to all orders of sigma-model perturbation theory, which is ‘not-really-perturbative’ for the gravitational
modes. It turns out that strict cutoff independence can be achieved only by allowing for infinitely many
essential couplings. They are conveniently combined into a generating functional
, which is a
positive function of one real variable. Schematically the renormalized action takes the form [154
]
This “coupling functional” is scale dependent and is subject to a flow equation of the form
where In summary, in the context of the reduction an asymptotically safe coupling flow can be
reconciled with the absence of unphysical propagating modes. In contrast to the technique on which
Evidence (d) below is based the existence of an infinite cutoff limit here can be shown and does
not have to be stipulated as a hypothesis subsequently probed for self-consistency. Since the
properties of the
truncation qualitatively are the ones one would expect from an ‘effective’
field theory describing the extreme UV aspects of Quantum Gravidynamics (see the end of
Section 2.4), its asymptotic safety is a strong argument for the self-consistency of the scenario.
(d) Evidence from truncated flows of the effective average action: The effective
average action is a generating functional generalizing the usual effective action, to which it
reduces for
. Here
depends on the UV cutoff
and an additional scale
,
indicating that in the defining functional integral roughly the field modes with momenta
in the
range
have been integrated out. Correspondingly
gives back the bare action
and
is the usual quantum effective action, in the presence of the UV cutoff
.
The modes in the momentum range
are omitted or suppressed by a mode cutoff
‘action’
, and one can think of
as being the conventional effective action
but
computed with a bare action that differs from the original one by the addition of
; specifically
The effective average action has been generalized to gravity [179] and we shall describe it and its
properties in more detail in Sections 4.1 and 4.2. As before the metric is taken as the dynamical variable
but the bare action
is not specified from the outset. In fact, conceptually it is largely determined by
the requirement that a continuum limit exists (see the criterion in Section 2.2).
can be expected to
have a well-defined derivative expansion with the leading terms roughly of the form (1.7
). Also the
gravitational effective average action
obeys an ‘exact’ FRGE, which is a new computational
tool in quantum gravity not limited to perturbation theory. In practice
is replaced in
this equation with a
independent functional interpreted as
. The assumption that
the ‘continuum limit’
for the gravitational effective average action exists is of course
what is at stake here. The strategy in the FRGE approach is to show that this assumption,
although without a-priori justification, is consistent with the solutions of the flow equation
(where right-hand-side now also refers to the Hessian of
). The structure of
the solutions
of this cut-off independent FRGE should be such that they can plausibly
be identified with
. Presupposing the ‘infrared safety’ in the above sense, a necessary
condition for this is that the limits
and
exist. Since
the first
limit probes whether
can be made large; the second condition is needed to have all modes
integrated out. In other words one asks for global existence of the
flow obtained by solving
the cut-off independent FRGE. Being a functional differential equation the cutoff independent
FRGE requires an initial condition, i.e. the specification of a functional
which coincides
with
at some scale
. The point is that only for very special ‘fine tuned’ initial
functionals
will the associated solution of the cutoff independent FRGE exist globally [157
].
The existence of the
limit in this sense can be viewed as the counterpart of the UV
renormalization problem, namely the determination of the unstable manifold associated with
the fixed point
. We refer to Section 2.2 for a more detailed discussion of this
issue.
In practice of course a nonlinear functional differential equation is very difficult to solve. To make the FRGE computationally useful the space of functionals is truncated typically to a finite-dimensional one of the form
where the The impact of matter has been studied by Percacci et al. [72, 171
, 170
]. Minimally coupling free fields
(bosons, fermions, or Abelian gauge fields) one finds that the non-Gaussian fixed point is robust,
but the positivity of the fixed point couplings
,
puts certain constraints on
the allowed number of copies. When a self-interacting scalar
is coupled non-minmally via
, one finds a fixed point
,
(whose values are with matched normalizations the same as
in the pure gravity
computation) while all self-couplings vanish,
,
. In the vicinity of the
fixed point a linearized stability analysis can be performed; the admixture with
and
then
lifts the marginality of
, which becomes marginally irrelevant [171
, 170
]. The running
of
and
is qualitatively unchanged as compared to pure gravity, indicating that the
asymptotic safety property is robust also with respect to the inclusion of self-interacting scalars.
Both Strategies (c) and (d) involve truncations and one may ask to what extent the results are
significant for the (intractable) full renormalization group dynamics. In our view they are significant.
This is because even for the truncated problems there is no a-priori reason for the asymptotic
safety property. In the Strategy (c) one would in the coupling space considered naively expect
a zero-dimensional unstable manifold rather than the co-dimension zero one that is actually
found! In Case (d) the ansatz (1.13, 1.14
) implicitly replaces the full gravitational dynamics by
one whose functional renormalization flow is confined to the subspace (1.13
, 1.14
) (similar to
what happens in a hierarchical approximation). However there is again no a-priori reason why
this approximate dynamics should have a non-Gaussian fixed point with positive fixed point
couplings and with an unstable manifold of co-dimension zero. Both findings are genuinely
surprising.
Nevertheless even surprises should have explanations in hindsight. For the asymptotic safety property of the truncated Quantum Gravidynamics in Strategies (c) and (d) the most natural explanation seems to be that it reflects the asymptotic safety of the full dynamics with respect to a nontrivial fixed point.
Tentatively both results are related by the dimensional reduction of the residual interactions in the ultraviolet. Alternatively one could try to merge both strategies as follows. One could take the background metrics in the background effective action generic and only impose the 2-Killing vector condition on the integration variables in the functional integral. Computationally this is much more difficult; however it would allow one to compare the lifted 4D flow with the one obtained from the truncated flows of the effective average action, presumably in truncations far more general than the ones used so far. A better way to relate both strategies would be by trying to construct a two-dimensional UV field theory with the characteristics to be described at the end of Section 2.4 and show its asymptotic safety.
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