Consider a candidate for a quasilocal microscopic action
where the One sees that in the fixed point regime the
-dependence enters only through the
combination
, a kind of self-similarity. This simple but momentous fact eventually underlies all the
subsequent arguments. It is ‘as if’ in the fixed point regime only a rescaled metric
entered
which carries dimension two. This has consequences for the ‘effective dimensionality’ of Newton’s constant:
Recall that conventionally the Ricci scalar term,
, has mass dimension
in
dimensions. Upon substitution
one quickly verifies that
is dimensionless. Its
prefactor, i.e. the inverse of Newton’s constant, then can be taken dimensionless – as it is in two
dimensions. Compared to the infrared regime it looks ‘as if’ Newton’s constant changed its effective
dimensionality from
to zero, i.e. at the fixed point there must be a large anomalous dimension
.
Formally what is special about the Einstein–Hilbert term is that the kinetic (second derivative) term
itself carries a dimensionful coupling. To avoid the above conclusion one might try to assign the metric
a mass dimension from the beginning (i.e. not just in the asymptotic regime). However
this would merely shift the effect from the gravity to the matter sector, as we wish to argue
now.
In addition to the dimensionful metric , we introduce a dimensionful vielbein by
, if
is the dimensionless metric. With respect to a dimensionless metric
has mass dimension
in
dimensions, while the mass dimensions
of a
Bose field
and that
of a Fermi field
are set such that their kinetic terms are
dimensionless, i.e.
and
. Upon substitution
the gravity
part
becomes dimensionless, while the kinetic terms of a Bose and Fermi field
pick up a mass dimension of
and
, respectively. This means their wave function
renormalization constants
and
are now dimensionful and should be written in
terms of dimensionless parameters as
and
, say.
For the dimensionless parameters one expects finite limit values
and
, since otherwise the corresponding (free) field would simply decouple. Defining
the anomalous dimension as usual.
and
, the argument
presented after Equation (1.5
) can be repeated and gives that
,
for the
fixed point values, respectively. The original large momentum behavior
for bosons and
for fermions is thus modified to a
behavior in the fixed point regime, in both
cases.
This translates into a logarithmic short distance behavior which is universal for all (free) matter.
Initially the propagators used here should be viewed as “test propagators”, in the sense that one transplants
the information in the ’s derived from the gravitational functional integral into a conventional
propagator on a (flat or curved) background spacetime. Since the short distance asymptotics is the same on
any (flat or curved) reference spacetime, this leads to the prediction anticipated in Section 2.3: The short
distance behavior of the quantum gravity average of the geodesic two-point correlator (2.52
) of a scalar field
should be logarithmic.
On the other hand the universality of the logarithmic short distance behavior in the matter propagators also justifies to attribute the phenomenon to a modification in the underlying random geometry, a kind of “quantum equivalence principle”.
The “anomalous dimension argument” has already been sketched in the introduction. Here we present a few more details and relate it to Section 2.4.1.
Suppose again that the unkown microscopic action of Quantum Gravidynamics is quasilocal and
reparameterization invariant. The only term containing second derivatives then is the familiar
Einstein–Hilbert term of mass dimension
in
dimensions, if the metric is
taken dimensionless. As explained in Section 2.3.2 the dimensionful running prefactor multiplying it
(
for “Newton”) can be treated either as a wave function renormalization or as a
quasi-essential dimensionless coupling
, where
The fact that a large anomalous dimension occurs at a non-Gaussian fixed point was initially observed in
the context of the expansion [116
, 117] and later in computations based on the effective
average action [133
, 131
]. The above argument shows that no specific computational information
enters.
Let us emphasize that in general an anomalous dimension is not related to the geometry of field propagation and in a conventional field theory one cannot sensibly define a fractal dimension by looking at the high momentum behavior of a two-point function [125]. What is special about gravity is ultimately that the propagating field itself defines distances. One aspect thereof is the universal way matter is affected, as seen in Section 2.4.1. In contrast to an anomalous dimension in conventional field theories, this “quantum equivalence principle” allows one to attribute a geometric significance to the modified short distance behavior of the test propagators, see Section 2.4.4.
With hindsight the above patterns are already implicit in earlier work on strictly renormalizable gravity
theories. As emphasized repeatedly the benign renormalizability properties of higher derivative theories are
mostly due to the use of type propagator (in
dimensions). As seen in Section 2.3.2 this
type behavior goes hand in hand with asymptotically safe couplings. Specifically for the
dimensionless Newton’s constant
it is compatible with the existence of a nontrivial fixed point (see
Equation (2.31
)). This in turn enforces anomalous dimension
at the fixed point which links back
to the
type propagator.
Similarly in the expansion [216
, 217
, 203
] a nontrivial fixed point goes hand in hand with a
propagator whose high momentum behavior is of the form
in four dimensions, and
formally
in
dimensions. In position space this amounts to a
behavior, once
again.
The scaling (2.55) of the fixed point action also allows one to estimate the behavior of the
spectral dimension in the ultraviolet. This leads to a variant [157
] of the argument first used
in [135
, 134
]).
Consider the quantum gravity average over the trace of the heat kernel
in a class of
states to be specified later. Morally speaking the functional average is over compact closed
-dimensional
manifolds
, and the states are such that they favor geometries which are smooth and
approximately flat on large scales.
Let us briefly recapitulate the definition of the heat kernel and some basic properties. For a smooth
Riemannian metric on a compact closed
-manifold let
be
the Laplace–Beltrami operator. The heat kernel
associated with it is the
symmetric (in
) bi-solution of the heat equation
with initial condition
. Since
is compact,
has purely discrete spectrum with
finite multiplicities. We write
,
, for the spectral problem and assume that
the eigenfunctions
are normalized and the eigenvalues monotonically ordered,
. We
write
for the volume of
and
For one has an asymptotic expansion
, where the
are the Seeley–deWitt coefficients. These are local curvature invariants,
, etc. The
series can be rearranged so as to collect terms with a fixed power in the curvature or with a fixed number of
derivatives [225
, 17
]. Both produces nonlocal curvature invariants. The second rearrangement
is relevant when the curvatures are small but rapidly varying (so that the derivatives of the
curvatures are more important then their powers). The leading derivative terms then are given by
, where
is a known nonlocal quadratic
expression in the curvature tensors (see e.g. [225, 17] for surveys). The
behavior is more subtle
as also global information on the manifold enters. For compact manifolds a typical behavior is
, where the rate of decay
of the subleading term is governed by
the smallest non-zero eigenvalue.
Returning now to the quantum gravity average , one sees that on any state on which all local
curvature polynomials vanish the leading short distance behavior of
will always be
, as
on a fixed manifold. The same will hold if the nonlocal invariants occurring in the derivative expansion all
have vanishing averages in the state considered. A leading short distance behavior of the form
We assume now that the states considered are such that the behavior of
is like that
in flat space, i.e.
for
. This is an indirect characterization of a
class of states which favor geometries that are smooth and almost flat on large scales [157
].
(A rough analogy may be the way how the short-distance Hadamard condition used for free
QFTs in curved spacetime selects states with desirable stability properties.) Recall that in a
functional integral formulation the information about the state can be encoded in suitable
boundary terms added to the microscopic action. The effective action used in a later stage of the
argument is supposed to be one which derives from a microscopic action in which suitable (though
not explicitly known) boundary terms encoding the information about the state have been
included.
Since one can give the stipulated
asymptotics an
interpretation in terms of the spectrum
of the Laplacian of a ‘typical’ reference metric
which is smooth and almost flat at large scales. The spectrum of
must be such that the small
spectral values can be well approximated by
for some constant
. Its
unknown large eigenvalues will then determine the short distance behavior of
. We
can incorporate this modification of the spectrum by introducing a function
which
tends to
for
, and whose large
behavior remains to be determined. Thus
The argument for for
goes as follows: We return to discrete description
for
compact, and consider the average of one term in the sum
,
with
being large. The computation of this average is a single scale problem in the terminology of
Appendix A. As such it should allow for a good description via an effective field theory at scale
. One
way of doing this is in terms of the effective average action
as described in Section 4.1. Here only
the fact is needed that the average
can approximately be evaluated as [135
, 134
]
In summary, the asymptotic safety scenario leads to the specific (theoretical) prediction that the
(normally powerlike) short distance singularities of all free matter propagators are softened to logarithmic
ones – normally a characteristic feature of massless Klein–Gordon fields in two dimensions. In quantum
gravity averages like Equation (2.52) this leads to the expectation that they should scale like
, for
. On the other hand this universality allows one to shuffle the effect from
matter to gravity propagators. This justifies to attribute the effect to a modification in the underlying
random geometry. The average of heat of the heat kernel,
in Equation (2.53
), then scales like
. This means the spectral dimension of the random geometries probed by a certain
class of “macroscopic” states equals
, which (notably!) equals
precisely in
dimensions.
Accepting this dimensional reduction in the extreme ultraviolet as a working hypothesis one is led to the following question: Is there a two-dimensional field theory which provides an effective description of this regime? “Effective” can mean here “approximate” but quantitatively close, or a system which lies in the same universality class as the original one in the relevant regime. “Effective” is of course not meant to indicate that the theory does not make sense beyond a certain energy scale, as in another use of the term “effective field theory”. We don’t have an answer to the above question but some characteristics of the putative field theory can easily be identified:
Note that in principle the identification of such a UV field theory is a well-posed problem. Presupposing that the functional integral has been made well-defined and through suitable operator insertions data for its extreme UV properties have been obtained, for any proposed field theory with the Properties 1 – 3 one can test whether or not these data are reproduced.
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