A still very general truncated functional space consists of ‘all’ background invariant functionals
which neglect the evolution of the ghost action. The corresponding ansatz reads [179
]
The truncation ansatz (4.29) is still too general for practical calculations to be easily possible.
Computations simplify considerably with the choice
and a local curvature polynomial
for
. The first specialization includes in
only the wave function renormalization and gives
For two choices of local curvature polynomials have been considered in detail
The truncation (4.35) will be called the
-truncation. It likewise keeps the gauge fixing and ghost
sector classical as in Equation (4.29
) but includes a local curvature squared term in
. In this case the
truncated theory space is three-dimensional. Its natural (dimensionless) coordinates are
, where
We should mention that apart from familiarity and the retroactive justification through the
results described later on, there is no structural reason to single out the truncations (4.34, 4.35
).
Even the truncated coarse graining flow in Equation (4.31
) will generate all sorts of terms in
, the only constraint comes from general covariance. Both local and nonlocal terms are
induced. The local invariants contain monomials built from curvature tensors and their covariant
derivatives, with any number of tensors and derivatives and of all possible index structures. The
form of typical nonlocal terms can be motivated from a perturbative computation of
; an
example is
. Since
approaches the ordinary effective action
for
it is clear that such terms must generated by the flow since they are known
to be present in
. For an investigation of the non-ultraviolet properties of the theory, the
inclusion of such terms is very desirable but it is still beyond the calculational state of the art (see
however [180
]).
The main technical complication comes from evaluating the functional trace on the right-hand-side of
the flow equation (4.31) to the extent that one can match the terms against those occuring on the
left-hand-side. We shall now illustrate this procedure and its difficulties in the case of the Einstein–Hilbert
truncation (4.34
) in more detail.
Upon inserting the ansatz (4.33) into the partially truncated flow equation (4.31
) it should eventually
give rise to a system of two ordinary differential equations for
and
. Even in this simple case
their derivation is rather technical, so we shall focus on matters of principle here. In order to find
and
it is sufficient to consider (4.31
) for
. In this case the left-hand-side of
the flow equation becomes
. The right-hand-side
contains the functional derivatives of
; in their evaluation one must keep in mind that the
identification
can be used only after the differentiation has been performed at
fixed
. Upon evaluation of the functional trace the right-hand-side should then admit an
expansion in terms of invariants
, among them
and
. The projected flow
equations are obtained by extracting the
-dependent coefficients of these two terms and
discarding all others. Equating the result to the left-hand-side and comparing the coefficients of
and
, the desired pair of coupled differential equations for
and
is
obtained.
In principle the isolation of the relevant coefficients in the functional trace on the right-hand-side can be
done without ever considering any specific metric . Known techniques like the derivative
expansion and heat kernel asymptotics could be used for this purpose, but their application is extremely
tedious usually. For example, because the operators
and
are typically of a complicated
non-standard type so that no efficient use of the tabulated Seeley–deWitt coefficients can be made.
Fortunately all that is needed to extract the desired coefficients is to get an unambiguous signal for the
invariants they multiply on a suitable subclass of geometries
. The subclass of geometries should be
large enough to allow one to disentangle the invariants retained and small enough to really simplify the
calculation.
For the Einstein–Hilbert truncation the most efficient choice is a family of -spheres
, labeled
by their radius
. For those geometries
, so they give a vanishing value on all invariants
constructed from
containing covariant derivatives acting on curvature tensors. What remains
(among the local invariants) are terms of the form
, where
is a polynomial in the Riemann
tensor with arbitary index contractions. To linear order in the (contractions of the) Riemann tensor the
two invariants relevant for the Einstein–Hilbert truncation are discriminated by the
metrics as they scale differently with the radius of the sphere:
,
.
Thus, in order to compute the beta functions of
and
it is sufficient to insert an
metric with arbitrary
and to compare the coefficients of
and
. If one
wants to do better and include the three quadratic invariants
,
, and
, the family
is not general enough to separate them; all scale like
with the
radius.
Under the trace we need the operator , the Hessian of
at fixed
. It is calculated
by Taylor expanding the truncation ansatz,
, and
stripping off the two
’s from the quadratic term,
. For
a metric on
one obtains
At this point we can fix the coefficients which appear in the cutoff operators
and
of
Equation (4.14
). They should be adjusted in such a way that for every low–momentum mode the cutoff
combines with the kinetic term of this mode to
times a constant. Looking at Equation (4.34
,
4.35
) we see that the respective kinetic terms for
and
differ by a factor of
. This
suggests the following choice:
From now on we may set and for simplicity we have omitted the bars from the metric and the
curvature. Since we did not take into account any renormalization effects in the ghost action we set
in
and obtain similarly, with
,
At this point the operator under the first trace on the right-hand-side of Equation (4.31) has become
block diagonal, with the
and
blocks given by Equation (4.40
). Both block operators are
expressible in terms of the Laplacian
, in the former case acting on traceless symmetric tensor fields, in
the latter on scalars. The second trace in Equation (4.31
) stems from the ghosts; it contains (4.41
) with
acting on vector fields.
It is now a matter of straightforward algebra to compute the first two terms in the derivative expansion
of those traces, proportional to and
. Considering the trace of an
arbitrary function of the Laplacian,
, the expansion up to second order derivatives of the metric
is given by
In terms of the dimensionless couplings and
the RG equations become a system of autonomous
differential equations
The beta function for is given by
With the derivation of the system (4.45) we managed to find an approximation to a two-dimensional
projection of the FRGE flow. Its properties and the domain of applicability or reliability of the
Einstein–Hilbert truncation will be discussed in Section 4.4. It will turn out that there are
important qualitative features of the truncated coupling flow (4.45
) which are independent
of the cutoff scheme, i.e. independent of the function
. The details of the flow pattern
on the other hand depend on the choice of the function
and hence have no intrinsic
significance.
By construction the normalized cutoff function ,
, in Equation (C.21
)
describes the “shape” of
in the transition region where it interpolates between the
prescribed behavior for
and
, respectively. It is referred to as the shape function
therefore.
In the literature various forms of ’s have been employed. Easy to handle, but disadvantageous for
high precision calculations is the sharp cutoff [181
] defined by
, where
the limit is to be taken after the
integration. This cutoff allows for an evaluation of the
and
integrals in closed form. Taking
as an example, Equations (4.45
) boil down to the following simple
system then
Also the cutoff with allows for an analytic evaluation of the integrals; it has
been used in the Einstein–Hilbert truncation in [136
]. In order to check the scheme (in)dependence of
certain results it is desirable to perform the calculation, in one stroke, for a whole class of
’s. For this
purpose the following one parameter family of exponential cutoffs has been used [205
, 133
, 131
]:
The form of the expression (4.46) for the anomalous dimension illustrates the nonperturbative character
of the method. For
Equation (4.46
) can be expanded as
Above we illustrated the general ideas and constructions underlying truncated gravitational RG flows by
means of the simplest example, the Einstein–Hilbert truncation (4.34). The flow equations for the
truncation are likewise known in closed form but are too complicated to be displayed here. These ordinary
differential equations can now be analyzed with analytical and numerical methods. Their solution reveals
important evidence for asymptotic safety. Before discussing these results in Section 4.4 we comment here on
two types of possible generalizations.
Concerning generalizations of the ghost sector truncation, beyond Equation (4.29) no results are
available yet, but there is a partial result concerning the gauge fixing term. Even if one makes the
ansatz (4.33
) for
in which the gauge fixing term has the classical (or more appropriately, bare)
structure one should treat its prefactor as a running coupling:
. After all, the actual “theory space”
of functionals
contains “
-type” and “gauge-fixing-type” actions on a completely symmetric
footing. The beta function of
has not been determined yet from the FRGE, but there is a simple
argument which allows us to bypass this calculation.
In nonperturbative Yang–Mills theory and in perturbative quantum gravity is known to
be a fixed point for the
evolution. The following heuristic argument suggests that the same should be
true beyond perturbation theory for the functional integral defining the effective average action for gravity.
In the standard functional integral the limit
corresponds to a sharp implementation of the gauge
fixing condition, i.e.
becomes proportional to
. The domain of the
integration
consists of those
’s which satisfy the gauge fixing condition exactly,
. Adding the infrared
cutoff at
amounts to suppressing some of the
modes while retaining the others. But
since all of them satisfy
, a variation of
cannot change the domain of the
integration. The delta functional
continues to be present for any value of
if it was there
originally. As a consequence,
vanishes for all
, i.e.
is a fixed point of the
evolution [137].
In other words we can mimic the dynamical treatment of a running by setting the gauge fixing
parameter to the constant value
. The calculation for
is more complicated than at
,
but for the Einstein–Hilbert truncation the
-dependence of
and
, for arbitrary constant
,
has been found in [133
]. The
truncations could be analyzed only in the simple
gauge, but the
results from the Einstein–Hilbert truncation suggest the UV quantities of interest do not change much
between
and
[133
, 131
].
Up to now we considered pure gravity. As far as the general formalism is concerned, the inclusion of
matter fields is straightforward. The structure of the flow equation remains unaltered, except that now
and
are operators on the larger space of both gravity and matter fluctuations. In practice the
derivation of the projected FRG equations can be quite formidable, the main difficulty being the decoupling
of the various modes (diagonalization of
) which in most calculational schemes is necessary for the
computation of the functional traces.
Various matter systems, both interacting and non-interacting (apart from their interaction with gravity)
have been studied in the literature. A rather detailed analysis of the fixed point has been performed by
Percacci et al. In [72, 171, 170] arbitrary multiplets of free (massless) fields with spin
and
were included. In [170] a fully interacting scalar theory coupled to gravity in the Einstein–Hilbert
approximation was analyzed, with a local potential approximation for the scalar self-interaction.
A remarkable finding is that in a linearized stability analysis the marginality of the quartic
self-coupling is lifted by the quantum gravitational corrections. The coupling becomes marginally
irrelevant, which may offer a new perspective on the triviality issue and the ensued bounds on
the Higgs mass. Making the number of matter fields large
, the matter interactions
dominate at all scales and the nontrivial fixed point of the
expansion [216, 217, 203] is
recovered [169].
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