Recall that the effective average action may be regarded as the standard effective action where
the bare action has been modified by the addition of the mode cutoff term
. Every given
(exact or truncated) renormalization group (RG) trajectory can be viewed as a collection of
effective field theories
. In this sense a single fundamental theory gives rise to a
double infinity of effective theories – one for each trajectory and one for each value of
. As
explained in Appendix C, the motivation for this construction is that one would like to be
able to ‘read off’ part of the physics contents of the theory simply by inspecting the effective
action relevant to the problem under consideration. If the problem has only one scale
, the
values of the running couplings and masses in
may be treated approximately as classical
parameters.
In a gravitational context this construction has been described in Section 4.1. The above effective
field theory aspect has also been used in Equation (2.62) of Section 2.4. Here we elaborate
on the significance of the solutions of the effective field equations that come with it. The two
distinct versions have been discussed in Item 6 of the previous Section 4.1.1; here we consider
We now select a state which favors geometries that are smooth and almost flat on large scales as in
Section 2.3. We can think of this state as a background dependent expectation functional
where the background has been self-consistently adjusted through the condition
(see
Equation (2.48
)). This switches off the source, and any fixed point of the map
gives a particular solution to Equation (4.26
) at
, implicitly referring to the underlying
state [156
].
In terms of the effective average action the state should implicitly determine a family of solutions
. Structures on a scale
are best described by
. In principle one
could also use
with
to describe structures at scale
but then a further
functional integration would be needed. It is natural to think of the family
as
describing aspects of a “quantum spacetime”. By a “quantum spacetime” we mean a manifold
equipped with infinitely many metrics; in general none of them will be a solution of the Einstein
field equations. One should keep in mind however that the quantum counterpart of a classical
spacetime is characterized by many more data than the metric expectation values
alone,
in particular by all the higher functional derivatives of
evaluated at
. The second
derivative for instance evaluated at
is the inverse graviton propagator in the background
. Note that all these higher multi-point functions probe aspects of the underlying quantum
state.
By virtue of the effective field theory properties of the interpretation of the metrics
is as
follows. Features involving a typical scale
are best described by
. Hence
is the average
metric detected in a (hypothetical) experiment which probes aspects of the quantum spacetimes with
typical momenta
. In more figurative terms one can think of
as a ‘microscope’ whose variable
‘resolving power’ is given by the energy scale
.
This picture underlies the discussions in [135, 134
] where the quantum spacetimes are viewed as
fractal-like and the qualitative properties of the spectral dimension (2.53
) have been derived. We refer to
Section 2.4 and [135
, 134
] for detailed expositions. The fractal aspects here refer to the generalized ‘scale’
transformations
, say. Moreover a scale dependent metric
associates a resolution dependent
proper length to any (
-independent) curve. The
-dependence of this proper length can be thought of
as analogous to the well-known example that the length of the coast line of England depends on the size of
the yardstick used.
Usually the resolving power of a microscope is characterized by a length scale defined
as the smallest distance of two points the microscope can distinguish. In the above analogy
between the effective average action and a “microscope” the resolving power is implicitly given
by the mass scale
and it is not immediate how
relates to a distance. One would like
to know the minimum proper distance
of two points which can be distinguished in a
hypothetical experiment with a probe of momentum
, effectively described by the action
.
Conversely, if one wants to ‘focus’ the microscope on structures of a given proper length
one
must know the
-value corresponding to this particular value of
. For non-gravitational
theories in flat Euclidean space one has
, but in quantum gravity the relation is more
complicated.
Given a family of solutions with the above interpretation the construction of a
candidate for
proceeds as follows [135
, 134
, 184
]: One considers the spectral problem of the (tensor)
Laplacian
associated with
. To avoid technicalities inessential for the discussion we assume
that all geometries in the family
are compact and closed. The spectrum of
will
then be discrete; we write
,
, for the spectral problem. As
indicated, both the spectral values
and the eigenfunctions
will now depend on
.
The collection of eigenfunctions
will be referred to as “cutoff modes at scale Given a wave function in Equation (4.27) one can now ask what a typical coordinate distance
is
over which it varies. Converting this into a proper distance with the metric
defines the proposed
resolving power
Since depends on the choice of the mode-cutoff scheme so will the solutions of Equation (4.26
),
and hence the resolving power
. It can thus not be identified with the resolving power of an actual
experimental set up, but is only meant to provide an order of magnitude estimate. The scheme
independence of the resolution which can be achieved in an actual experimental set up would in this picture
arise because the scheme dependence in the trajectory cancels against that in the
versus
relation.
This concludes our presentation of the effective average action formalism for gravity. In the next
Section 4.3 we will use the FRGE for as a tool to gain insight into the gravitational renormalization
flow.
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