Before turning to renormalization aspects proper, let us describe the special role of Newton’s constant in a
diffeomorphism invariant theory with a dynamical metric. Let be any local action, where
is the metric and the “matter” fields are not scaled when the metric is. Scale
changes in the metric then give rise to a variation of the Lagrangian which vanishes on shell:
The physics interpretation of the inessential parameter is that it also sets the absolute momentum
or spectral scale. To see this we can think of
as a reference metric in the background field formalism.
For example for the spectral values
of the covariant Laplacian
associated with
one has
Being inessential the quantum field theoretical running of has significance only relative to the
running coefficient of some reference operator. The most commonly used choice is a cosmological constant
term
. Indeed
The second possibility is realized when inserting a singular solution of the equation for into the
equation for
. This naturally occurs when working in Planck units. One makes use of
the fact that an inessential parameter can be frozen at a prescribed value. Specifically fixing
By higher derivative theories we mean here gravitational theories whose bare action contains, in addition to the Einstein–Hilbert term, scalars built from powers of the Riemann tensor and its covariant derivatives. In overview there are two distinct perturbative treatments of such theories.
The first one, initiated by Stelle [206], uses
type propagators (in four dimensions) in which case a
higher derivative action containing all (three) quartic derivative terms can be expected to be power
counting renormalizable. In this case strict renormalizability with only
(or
, if Newton’s constant is
included) couplings can be achieved [206
]. However the
type propagators are problematic from the
point of view of unitarity.
An alternative perturbative treatment of higher derivative theories was first advocated by
Gomis–Weinberg [94]. The idea is to try to maintain a
type propagator and include all (infinitely
many) counterterms generated in the bare action. Consistency requires that quadratic counterterms (those
which contribute to the propagator) can be absorbed by field redefinitions. As shown by Anselmi [10
] this is
the case either in the absence of a cosmological constant term or when the background spacetime admits a
metric with constant curvature.
We now present both of these perturbative treatments in more detail. A putative matching to a
nonperturbative renormalization flow is outlined in Equation (2.32).
The general classical action in dimensions containing up to four derivatives of the metric reads
The perturbative quantization of Equation (2.26) proceeds as usual. Gauge fixing and ghost terms are
added and the total action is expanded in powers of
. Due to the crucial
term the
gauge-fixed propagator read off from the quadratic part of the full action has a characteristic
falloff
in
,
To describe the flow of the Newton and cosmological constants one switches to the dimensionless
parameters and
as in Section 2.3.1. The result obtained in Berrodo–Peixoto and Shapiro [59] via
dimensional regularization reads in our conventions
The flow equations (2.20, 2.30
) of course also admit the Gaussian fixed point
, and one
may be tempted to identify the ‘realm’ of perturbation theory (PT) with the ‘expansion’ around a Gaussian
fixed point. As explained in Section 2.1, however, the conceptual status of PT referring to a non-Gaussian
fixed point is not significantly different from that referring to a Gaussian fixed point. In other words there is
no reason to take the perturbative non-Gaussian fixed point (2.31
) any less serious than the
perturbative Gaussian one. This important point will reoccur in the framework of the
truncation in Section 3, where a non-Gaussian fixed point is also identified by perturbative
means.
The fact that a non-Gaussian fixed point can already be identified in PT is important for several
reasons. First, although the value of in Equation (2.31
) is always non-universal, the anomalous
dimension
is exactly
at the fixed point (2.31
). The general argument for the
dimensional reduction of the residual interactions outlined after Equation (1.5
) can thus already be based
on PT alone! Second the result (2.31
) suggests that the interplay between the perturbative
and the nonperturbative dynamics might be similar to that of non-Abelian gauge theories,
where the nonperturbative dynamics is qualitatively and quantitatively important mostly in the
infrared.
It is instructive [157] to compare the perturbative one-loop flow (2.20
, 2.30
) with the linearization of
the
flow obtained from the FRGE framework described in Section 4. In the so-called
Einstein–Hilbert truncation using an optimed cutoff and a limiting version of the gauge-fixing parameter,
the ‘beta’ functions
,
reduce to ratios of polynomials in
,
[136
]. Upon expansion to
quadratic order one finds
The most important drawback of the perturbatively renormalizable theories based on Equation (2.26)
are the problems with unitarity entailed by the propagator (2.28
). As already mentioned these problem
are absent in an alternative perturbative formulation where a
type propagator is used
throughout [94
]. We now describe this construction in slightly more detail following the presentation
in [10
].
Starting from the Lagrangian
without cosmological constant the one-loop
divergencies come out in dimensional regularization as [210]
Let us briefly recap the power counting and scaling dimensions of local curvature invariants. These are
integrals over densities
which are products of factors of the form
, suitably contracted to get a scalar and then multiplied by
. One easily checks
,
, with
, where
is the total power of the Riemann tensor
and
is the (necessarily even) total number of covariant derivatives. This scaling dimension matches
minus the mass dimension of
if
is taken dimensionless. For the mass dimension
of the
associated coupling
in a product
one thus gets
. For example, the
three local invariants in Equation (1.14
) have mass dimensions
,
,
, respectively. There are three other local invariants with mass dimension
, namely the ones with integrands
(the square
of the Weyl tensor),
(the generalized Euler density), and
. Then there is a set of dimension
local invariants, and so on. Note that in
the integrands of the last two of the dimensionless invariants are total divergencies
so that in
there are only
local invariants with non-positive mass dimension (see
Equation (2.26
)).
A generic term in will be symbolically of the form
, where all possible contractions of the
indices may occur. Since the Ricci tensor is schematically of the form
, the
piece in
quadratic in
is of the form
. The coefficient of
is a tensor with 4 free
indices and one can verify by inspection that the possible index contractions are such that the Ricci tensor
or Ricci scalar either occurs directly, or after using the contracted Bianchi identity. In summary, one may
restrict the sum in Equation (2.36
) to terms with
,
, and the propagator
derived from it will remain of the
type to all loop orders. This suggests that Equation (2.36
) will
give rise to a renormalizable Lagrangian. A proof requires to show that after gauge fixing and ghost terms
have been included all counter terms can be chosen local and covariant and has been given
in [94
].
Translated into Wilsonian terminology the above results then show the existence of a “weakly
renormalizable” but “propagator unitary” Quantum Gravidynamics based on a perturbative Gaussian
fixed point. The beta functions for this infinite set of couplings are presently unknown. If they
were known, expectations are that at least a subset of the couplings would blow up at some
finite momentum scale and would be unphysical for
. In this case the
computed results for physical quantities are likely to blow up likewise at some (high) energy scale
. In other words the couplings in Equation (2.36
) are presumably not all asymptotically
safe.
Let us add a brief comment on the relevant-irrelevant distinction in this context, if only to point out
that it is no longer useful. Recall from Section 1.3 that the notion of a relevant or irrelevant coupling
applies even to flow lines not connected to a fixed point. This is the issue here. All but a few of the
interaction monomials in Equation (2.36) are power counting irrelevant with respect to the
propagator. Equivalently all but a few couplings
have non-negative mass dimensions
. These are the only ones not irrelevant with respect to the stability matrix
computed at the
perturbative Gaussian fixed point. However in Equation (2.36
) these power counting irrelevant couplings
with
are crucial for the absorption of infinities and thus are converted into practically
relevant ones. In the context of Equation (2.36
) we shall therefore discontinue to use the terms
relevant/irrelevant.
Comparing both perturbative constructions one can see that the challenge of Quantum Gravidynamics
lies not so much in achieving renormalizability, but to reconcile asymptotically safe couplings with the
absence of unphysical propagating modes. This program is realized in Section 3 for the reduction;
the results of Section 4 for the
type truncation likewise are compatible with the absence of
unphysical propagating modes.
In order to realize this program without reductions or truncations, a mathematically controllable nonperturbative definition of Quantum Gravidynamics is needed. Within a functional integral formulation this involves the following main steps: definition of a kinematical measure, setting up a coarse graining flow for the dynamical measures, and then probing its asymptotic safety.
For a functional integral over geometries even the kinematical measure, excluding the action dependent
factor, is nontrivial to obtain. A geometric construction of such a measure has been given by Bern, Blau,
and Mottola [31] generalizing a similar construction in Yang–Mills theories [20]. It has the advantage of
separating the physical and the gauge degrees of freedom (at least locally in field space) in a way that
is not tied to perturbation theory. The functional integral aimed at is one over geometries,
i.e. equivalence classes of metrics modulo diffeomorphisms. For the subsequent construction the
difference between Lorentzian and Riemannian signature metrics is inessential; for definiteness we
consider the Lorenzian case and correspondingly have an action dependence
in
mind.
A geometry can be described by picking a representative described by a
parametric
metric. Here
can be specified by picking an explicit parameterization or by imposing a gauge fixing
condition
. Typical choices are a harmonic gauge condition with respect to some reference
metric connection, or a proper time gauge
, for a fixed timelike co-vector
. Once
has been fixed, the push forward with a generic diffeomorphism
will generate the associated orbit,
On the tangent space the parameterization (2.37) amounts to
In summary one arrives at the following proposal for a kinematical measure over geometries:
Here we omitted the normalization factor and for illustration included the factor In the above discussion we did not split off the conformal factor in the geometries. Doing this however
only requires minor modifications and was the setting used in [31, 142, 149]. In Equation (2.37
) then
is written as
, where now
is subject to a gauge condition
. On the
cotangent space this leads to a York-type decomposition [235] replacing (2.41
), where the variations
of
the conformal factor and that of the tracefree part
of
describe the variations of the geometry,
while the tracefree part,
, and the trace part of the Lie derivative
describe the gauge variations. Writing
the computation of the
Jacobian proceeds as above and leads to Equation (2.47
) with the following replacements:
is replaced with
,
with
, and
with
in the integrand. By
studying the dependence of
on the conformal factor it has been shown
in [142
] that in the Gaussian approximation of the Euclidean functional integral the instability
associated with the unboundedness of the Euclidean Einstein–Hilbert action is absent, due to
a compensating contribution from the determinant. It can be argued that this mechanism is
valid also for the interacting theory. From the present viewpoint however the (Euclidean or
Lorentzian) Einstein–Hilbert action should not be expected to be the proper microscopic action.
So the “large field” or “large gradient” problem has to be readdressed anyhow in the context
of Quantum Gravidynamics. Note also that once the conformal mode of the metric has been
split off the way how it enters a microscopic or an effective action is no longer constrained
by power counting considerations. See [12] for an effective dynamics for the conformal factor
only.
Once a kinematical measure on the equivalence classes of metrics (or other dynamical variables) has been defined, the construction of an associated dynamical measure will have to rest on renormalization group ideas. Apart from the technical problems invoved in setting up a computationally useful coarse graining flow for the measure on geometries, there is also the apparent conceptual problem how diffeomorphism invariance can be reconciled with the existence of a scale with respect to which the coarse graining is done. However no problem of principle arises here. First, similar as in a lattice field theory, where one has to distinguish between the external lattice spacing and a dynamically generated correlation length, a distinction between an external scale parameter and a dynamically co-determined resolution scale has to be made. A convenient way to achieve compatibility of the coarse graining with diffeomorphism invariance is by use of the background field formalism. The initially generic background metric serves as a reference to discriminate modes, say in terms of the spectrum of a covariant differential operator in the background metric (see Section 4.1). Subsequently the background is self-consistenly identified with the expectation value of the quantum metric as in the discussion below.
The functional integral over “all geometries” should really be thought of as one over “all geometries subject to suitable boundary conditions”. Likewise the action is meant to include boundary terms which indirectly specify the state of the quantum system.
After a coarse graining flow for the dynamical measures has been set up the crucial issue will be whether or not it has a fixed point with a nontrivial finite-dimensional unstable manifold, describing an interacting system. In this case it would define an asymptotically safe functional measure in the sense defined in Section 1.3. For the reasons explained there the existence of an asymptotically safe functional integral masure is however neither necessary nor sufficient for a physically viable theory of Quantum Gravidynamics. For the latter a somewhat modified notion of a safe functional measure is appropriate which incorporates the interplay between couplings and observables:
Unfortunately, at present little is known about generic quantum gravity observables, so that the functional
averages whose expansion would define physical couplings are hard to come by. For the time being we
therefore adopt a more pragmatic approach and use as the central object to formulate the renormalization
flow the background effective action as described in Appendix B. Here
is interpreted
as an initially source-dependent “expectation value of the quantum metric”,
is an initially
independently prescribed “background metric”, and the dots indicate other fields, conjugate to sources,
which are inessential for the following discussion. For clarities sake let us add that it is not assumed that
the metric exists as an operator, or that the metric-like “conjugate sources”
,
are necessarily the
best choice.
The use of an initially generic background geometry has the advantage that one can define
propagation and covariant mode-cutoffs with respect to it. A background effective action of this type has an
interesting interplay with the notion of a state [156
, 157
]. An effective action implicitly specifies an
expectation functional
(“a state”) which depends parameterically on the background metric.
The background metric is then self-consistently identified with the expectation value of the metric
The condition (2.49) is equivalent to the vanishing of the extremizing sources
in the
definition of Legendre transform (see Appendix B). Evidently Equation (2.49
) also amounts to the
vanishing of the one-point functions in Equation (2.51
). Usually the extremizing sources
are
constructed by formal inversion of a power series in
. Then
always is a solution
of Equation (2.49
) and the functional
is simply the identity. In this case the self-consistent
background coincides with the naive prescribed background. To find nontrivial solutions of
Equation (2.49
) one has to go beyond the formal series inversions and the uniqueness assumptions usually
made.
Due to the highly nonlocal character of the effective action the identification of physical solutions of
Equation (2.49) is a nontrivial problem. The interpretation via Equation (2.48
) suggests an indirect
characterization, namely those solutions of Equation (2.49
) should be regarded as physical which come from
physically acceptable states [157
].
The notion of a state is implicitly encoded in the effective action. Recall that the standard effective
action, when evaluated at a given time-independent function , is proportional to the minimum
value of the Hamiltonian
in that part of the Hilbert space spanned by normalizable states
satisfying
. A similar interpretation holds formally for the various background effective
actions [50
]. In conventional quantum field theories there is a clear-cut notion of a ground state and of the
state space based on it. In a functional integral formulation the information about the state can be encoded
in suitable boundary terms for the microscopic action. Already in quantum field theories on
curved but non-dynamical spacetimes a preferred vacuum is typically absent and physically
acceptable states have to be selected by suitable conditions (like, for example, the well-known
Hadamard condition in the case of a Klein–Gordon field). In quantum gravity the formulation of
analogous selection criteria is an open problem. As a tentative example we mention the condition
formulated after Equation (2.53
) below. On the level of the effective action one should think of
as a functional of both the selected state and of the fields. The selected state will indirectly
(co-)determine the space of functionals on which the renormalization flow acts. For example the type of
nonlocalities which actually occur in
should know about the fact that
stems from a
microscopic action suited for the appropriate notion of positivity and from a physically acceptable
state.
Finally one will have to face the question of what generic physical quantities are and how to compute them.
Although this is of course a decisive issue in any approach to quantum gravity, surprisingly little work has
been done in this direction. In classical general relativity Dirac observables do in principle encode all
intrinsic properties of the spacetimes, but they are nonlocal functionals of the metric and implicitly refer to
a solution of the Cauchy problem. In a canonical formulation quantum counterparts thereof
should generate the physical state space, but they are difficult to come by, and a canonical
formulation is anyhow disfavored by the asymptotic safety scenario. S-matrix elements with
respect to a self-consistent background (2.48) or similar objects computed from the vertex
functions (2.51
) might be candidates for generic physical quantities, but have not been studied so
far.
For the time being a pragmatic approach is to consider quantities which are of interest
in a quantum field theory on a fixed but generic geometry and then perform an average over
geometries with the measure previously constructed. On a perturbative level interesting possible
effects have been studied in [202, 215]. On a nonperturbative level this type of correlations
have been discussed mostly in discretized formulations but the principle is of course general.
To fix ideas we note the example of a geodesic two point correlator of a scalar field [58],
If one wants to probe the functional measure over geometries only, an interesting operator insertion is
the trace of the heat kernel [115, 118
, 8
],
In a lattice field theory the discretized functional measure typically generates an intrinsic scale, the
(dimensionless) correlation length , which allows one to convert lattice distances into a physical standard
of length, such that say,
lattice spacings equal
. A (massive) continuum limit is eventually
defined by sending
to infinity in a way such that physical distances
are kept fixed and a ‘nonboring’ limit arises. In a functional measure over geometries
, initially
defined with an UV cutoff
and an external scale parameter
, it is not immediate how to
generalize the concept of a correlation length. Exponents extracted from the decay properties of
Equation (2.52
) or Equation (2.53
) are natural candidates, but the ultimate test of the fruitfulness of
such a definition would lie in the successful construction of a continuum limit. In contrast to a
conventional field theory it is not even clear what the desired/required properties of such a continuum
system should be. The working definitions proposed in Section 1.3 tries to identify some salient
features.
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