First the standard effective action is not a gauge invariant functional of its argument. For
example if in a Yang–Mills theory one gauge-fixes the functional integral with an ordinary gauge
fixing condition like , couples the Yang–Mills field
to a source, and constructs
the ordinary effective action, the resulting functional
is not invariant under the gauge
transformations of
. Although physical quantities extracted from
are expected to be
gauge invariant, the noninvariance is cumbersome for renormalization purposes. The second
problem is related to the fact that in a gauge theory a “coarse graining” based on a naive Fourier
decomposition of
is not gauge covariant and hence not physical. In fact, if one were
to gauge transform a slowly varying
with a parameter function
with a fast
-variation, a gauge field with a fast
-variation would arise, which however still describes the same
physics.
Both problems can be overcome by using the background field formalism. The background effective
action generally is a gauge invariant functional of its argument (see Appendix B). The second problem is
overcome by using the spectrum of a covariant differential operator built from the background
field configuration to discrimate between slow modes (small eigenvalues) and fast modes (large
eigenvalues) [187]. This sacrifices to some extent the intuition of a spatial coarse graining, but it produces a
gauge invariant separation of modes. Applied to a non-gauge theory it amounts to expanding the field in
terms of eigenfunctions of the (positive) operator
and declaring its eigenmodes ‘long’ or ‘short’
wavelength depending on whether the corresponding
is smaller or larger than a given
.
This is the strategy adopted to define the effective average action for gravity [179]. In short: The
effective average action for gravity is a variant of the background effective action
described in Appendix B (see Equations (B.48
, B.51
)), where the bare action is modified by mode cutoff
terms as in Appendix C, but with the mode cutoff defined via the spectrum of a covariant differential
operator built from the background metric. For convenience we quickly recapitulate the main features of
the background field technique here and then describe the modifications needed for the mode
cutoff.
The initial bare action is assumed to be a reparameterization invariant functional of the metric
. Infinitesimally the invariance reads
, where
is the Lie
derivative of
with respect to the vector field
. The metric
(later the integration variable in
the functional integral) is decomposed into a background
and a fluctuation
, i.e.
.
The fluctuation field
is then taken as the dynamical variable over which the functional integral is
performed; it is not assumed to be small in some sense, no expansion in powers of
is implied by the
split. Note however that this linear split does not have a geometrical meaning in the space of
geometries. The symmetry variation
can be decomposed in two different ways
In the next step the initial bare action should be replaced by one involving a mode cutoff term. In the
background field technique the mode cutoff should be done in a way that preserves the invariance under the
background gauge transformations (4.2). We now first present the steps leading to the scale
dependent effective average action
in some detail and then present the
FRGE for it. The functional integrals occuring are largely formal; for definiteness we consider
the Euclidean variant where the integral over Riemannian geometries is intended. The precise
definition of the generating functionals is not essential here, as they mainly serve to arrive at the
gravitional FRGE. The latter provides a novel tool for investigating the gravitational renormalization
flow.
We begin by introducing a scale dependent variant of the generating functional of the connected
Greens functions. The cutoff scale is again denoted by
, it has unit mass dimension, and no physics
interpretation off hand. The defining relation for
reads
The construction of the effective average action now parallels that in the scalar case. We quickly run
through the relevant steps. The Legendre transform of at fixed
is
Usually one is not interested in correlation functions involving Faddeev–Popov ghosts and it is sufficient to know the reduced functional
As indicated we shall simply write The precise form of the gauge condition is inessential, only the invariance under
Equation (4.2
) is important. It ensures that the associated ghost action is invariant under Equation (4.2
)
and
,
. We shall ignore the problem of the global existence of gauge slices
(“Gribov copies”), in accordance with the formal nature of the construction. For later reference let us briefly
describe the most widely used gauge condition, the “background harmonic gauge” which reads
The last ingredient in Equations (4.3, 4.4
) to be specified are the mode cutoff terms, not present in the
usual background effective action. Their precise form is arbitrary to some extent. Naturally they will be
taken quadratic in the respective fields, with a kernel which is covariant under background gauge
transformations. These requirements are met if
Consider the Laplacian of the Riemannian background metric
with
being its
torsionfree connection. We assume
to be such that
has a non-negative spectrum and a complete
set of (tensorial) eigenfunctions. The spectral values
of
will then be functionals of
and one can choose
and
such that only eigenmodes with spectral values
(
being the mass dimension of the operator) enter the
functional integral unsuppressed. Here one
should think of the
functional integral as being replaced by one over the (complete system of)
eigenfunctions of
, for a fixed
. Concretely, for
and
we take expressions of the
form
The essential ingredient in Equation (4.14) is a function
interpolating smoothly
between
and
; for example
This concludes the definition of the effective average action and its various specializations. We now present its key properties.
The conceptual status and the use of the gravitational FRGE (4.20) is the same in the
scalar case discussed in Section 2.2. Its perturbative expansion should reproduce the
traditional non-renormalizable cutoff dependencies starting from two-loops. In the context of
the asymptotic safety scenario the hypothesis at stake is that in an exact treatment of
the equation the cutoff dependencies entering through the initial data get reshuffled in a
way compatible with asymptotic safety. The Criterion (FRGC1) for the existence of a
genuine continuuum limit discussed in Section 2.3 also applies to Equation (4.20
). In
brief, provided a global solution of the FRGE (4.20
) can be found (one which exists for
all
), it can reasonably be identified with a renormalized effective average
action
constructed by other means. The intricacies of the renormalization
process have been shifted to the search for fine-tuned initial functionals for which a global
solution of Equation (4.20
) exists. For such a global solution
then is the full
quantum effective action and
is the fixed point action. As already noted in
Section 2.3 the appropriate positivity requirement (FRGC2) remains to be formulated; one
aspect of it concerns the choice of
factors in Equation (4.13
) and will be discussed
below.
The background gauge invariance of expressed in Equation (4.16
) is of great practical
importance. It ensures that if the initial functional does not contain non-invariant terms, the flow
implied by the above FRGE will not generate such terms. In contrast locality is not preserved of
course; even if the initial functional is local the flow generates all sorts of terms, both local and
nonlocal, compatible with the symmetries.
For the derivation of the flow equation it is important that the cutoff functionals in Equation (4.13)
are quadratic in the fluctuation fields; only then a flow equation arises which contains only second
functional derivatives of
, but no higher ones. For example using a cutoff operator involving the
Laplace operator of the full metric
would result in prohibitively complicated flow
equations which could hardly be used for practical computations.
For most purposes the reduced effective average action (4.9) is suffient and it is likewise background
invariant,
. Unfortunately
does not satisfy an exact FRGE,
basically because it contains too little information. The actual RG evolution has to be
performed at the level of the functional
. Only after the evolution one may set
,
. As a result, the actual theory space of Quantum Einstein Gravity in
this setting consists of functionals of all four variables,
, subject to
the invariance condition (4.9
). Since
involves derivatives with respect to
at
fixed
it is clear that the evolution cannot be formulated in terms of
alone.
An equivalent set of vertex functions should in analogy to the Yang–Mills case [68, 42, 67] be
obtained by differentiating
with respect to
. Specifically
for
one gets multipoint functions
, with the shorthand (4.9
). The
solutions
of
The precise physics significance of the multipoint functions and
remains to be
understood. One would expect them to be related to S-matrix elements on a self-consistent
background but, for example, an understanding of the correct infrared degrees of freedom is missing.
This concludes our summary of the key properties of the gravitational effective average action. Before
turning to applications of this formalism, we discuss the significance and the proper choice of the
factors in Equation (4.14
), which is one aspect of the positivity issue (FRGC2) of Section 2.2. The
significance of these factors is best illustrated in the scalar case. As discussed in Appendix C in scalar
theories with more than one field it is important that all fields are cut off at the same
. This is achieved
by a cutoff function of the form (C.21
) where
is in general a matrix in field space. In the sector of
modes with inverse propagator
the matrix
is diagonal with entries
. In a
scalar field theory these
factors are automatically positive and the flow equations in the various
truncations are well-defined.
In gravity the situation may be more subtle. First, consider the case where is some normal mode of
and that it is an eigenfunction of
with eigenvalue
, where
is a positive eigenvalue of
some covariant kinetic operator, typically of the form
-terms. If
the situation is clear,
and the rule discussed in the context of scalar theories applies: One chooses
because this
guarantees that for the low momentum modes the running inverse propagator
becomes
, exactly as it should be.
More tricky is the question how should be chosen if
is negative. If one continues to use
, the evolution equation is perfectly well defined because the inverse propagator
never vanishes, and the traces of Equation (4.19
) are not suffering from any infrared problems. In fact, if we
write down the perturbative expansion for the functional trace, for instance, it is clear that all propagators
are correctly cut off in the infrared, and that loop momenta smaller than
are suppressed. On the other
hand, if we set
, then
introduces a spurios singularity at
, and the
cutoff fails to make the theory infrared finite. This choice of
is ruled out therefore. At first sight the
choice
might have appeared more natural because only if
the factor
is a damped exponentially which suppresses the low momentum
modes in the usual way. For the other choice
the factor
is a
growing exponential instead and, at least at first sight, seems to enhance rather than suppress the
infrared modes. However, as suggested by the perturbative argument, this conclusion is too naive
perhaps.
In all existing RG studies using this formalism the choice
has been adopted, and there is little doubt that, within those necessarily truncated calculations, this is the correct procedure. Besides the perturbative considerations above there are various arguments of a more general nature which support Equation (4.25In fact, as we shall discuss in more detail later on, the to date best truncation used for the investigation of
asymptotic safety (“ truncation”) has only positive
factors in the relevant regime.
On the other hand, the simpler “Einstein–Hilbert truncation” has also negative
’s. If one
applies the rule (4.25
) to it, the Einstein–Hilbert truncation produces almost the same results
as the
truncations [131
, 132
]. This is a strong argument in favor of the
rule.
In [133, 131
] a slightly more general variant of the construction described here has
been employed. In order to facilitate the calculation of the functional traces in the
FRGE (4.19
) it is helpful to employ a transverse-traceless (TT) decomposition of the metric:
. Here
is a transverse
traceless tensor,
a transverse vector, and
and
are scalars. In this framework it is natural
to formulate the cutoff in terms of the component fields appearing in the TT decomposition
. This cutoff is referred to as a cutoff of “type B”, in
contradistinction to the “type A” cutoff described above,
. Since covariant derivatives do
not commute, the two cutoffs are not exactly equal even if they contain the same shape function
.
Thus, comparing type A and type B cutoffs is an additional possibility for checking scheme
(in)dependence [133
, 131
].
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