Performing a standard expansion ,
, around the metric
of
Minkowski space the quadratic part of the Einstein–Hilbert action
reads
The 2-Killing vector reduction of the action (3.30) amounts to considering field configurations obeying
(which is equivalent to
having two Killing vectors). In
adapted coordinates where
,
, the metric components depend only on
the non-Killing coordinates
. Moreover
can be assumed to be block-diagonal
Entering with the ansatz (3.33) into Equation (3.30
) gives the reduced action
We add some remarks. As one might expect, the action (3.34) can also be obtained by “first reducing”
and then “linearizing”. Indeed, from Equation (3.50
) below one has
Further the operations “varying the action” and “reduction” are commuting, as expected from the
principle of symmetric criticality. Thus, the reduction of the field equations (3.32) coincides with the field
equations obtained by varying Equation (3.34
). The latter are
For the special case of the Einstein–Rosen waves an on-shell formulation suffices and a related study linking the ‘graviton modes’ to the ‘Einstein–Rosen modes’ can be found in [23].
The linearized theory is well suited to discuss the physics content of the 2-Killing vector
reduction. To this end one fixes the gauge and displays the independent on-shell degrees of freedom.
The most widely used gauge in the linearized theory (3.30) is the transversal-traceless gauge,
In summary we conclude that the 2-Killing vector subsector comprises the gravitational self-interaction of collinear gravitons, in the same sense as the full Einstein–Hilbert action describes the self-interaction of non-collinear gravitons.
The formulation of the perturbative functional integral in this subsector would now proceed in exact parallel to the non-collinear case: A gauge fixing term
implementing the harmonic gauge condition is added to the action. This renders the kinetic term in Equation (3.34
For the systems at hand we now want to argue that this is not the method of choice. To this end we return
to the covariant action (3.10) and decompose
into a conformal factor
and a two-parametric
remainder
to be adjusted later,
. Using
we can
rewrite Equation (3.10
) as a gravity theory for the two-parametric
The choice (3.47) is adapted to a covariant quantization. Here
is a generic (off-shell)
background metric with again the conformal mode
split off. The fluctuation field
is trace-free with respect to the background
. Then Equation (3.45
) describes a
unimodular gravity theory (
) and the original metric is parameterized by
and
as
. In a covariant formulation the degrees of freedom in
would be promoted to propagating ones by adding a gauge fixing term to the action (3.45
).
The associated Faddeev–Popov determinant is designed to cancel out their effect again. In
the case at hand this is clearly roundabout as the gauge-frozen Lagrangian (3.45
) is already
nondegenerate.
This setting can be promoted to a generalization of the one presented in Section 3.1.2 to generic backgrounds. In the terminology of Section B.2 one then gets a non-geodesic background-fluctuation split, which treats the nonpropagating lapse and shift degrees of freedom on an equal footing with the others. In order to contrast it with the geodesic background-fluctuation split for the propagating modes used later on, we spell out here the first few steps of such a procedure.
We wish to expand the Einstein–Hilbert action for the class of metrics (3.6) around a generic
background. Technically it is simpler to “first reduce” and then “expand”. This is legitimate since all
operations involved are algebraic. Recall that the reduced Lagrangian is defined by inserting the ansatz
The counterpart of Equation (3.49) for the lower
block in Equation (3.48
) is
Writing for Einstein–Hilbert Lagrangian with the blockdiagonal metric (3.48
) one arrives at
an expansion of the form
In all cases one sees that the procedure outlined has two drawbacks. First, the split (3.49) ignores the
special status of the lapse and shift degrees of freedom in
; all components are expanded. We know,
however, that there must be two infinite series built from the components of
and
that enter the
left-hand-side of Equation (3.53
) anyhow linearly. Concerning the lower
block in Equation (3.48
)
both the linear and the York-type decomposition will only keep the
symmetry more or less
manifest. The nonlinear realization of the
symmetry then has to be restored through
Ward identites, iteratively in a perturbative formulation or otherwise in a nonperturbative
one.
In the following we shall adopt the following remedies. The fact that the lapse and shift degrees of
freedom in enter the left-hand-side of Equation (3.53
) linearly of course just means that they are the
Lagrange multipliers of the constraints in a Hamiltonian formulation. The linearity can thus be exploited
either by a gauge fixing with respect to these variables before expanding, giving rise to a proper time
formulation, or by directly adopting a Dirac quantization prodecure. By and large both should be
eqaivalent; in [154
, 155
] a direct Dirac quantization was used, and we shall describe the results in the next
two Sections 3.3 and 3.4. With the lapse and shift in Equation (3.14
) ‘gone’ one only needs to
perform a background-fluctuation split only for the remaining propagating fields
,
,
,
.
To cope with the second of the before-mentioned drawbacks we equip – following deWitt and
Vilkovisky – this space of propagating fields with a pseudo-Riemannian metric and perform a
normal-coordinate expansion around a (‘background’) point with respect to it. This leads to
the formalism summarized in Section B.2.2. The pseudo-Riemannian metric on the space of
propagating fields can be read off from Equation (3.16) and converts the gauge-frozen but
nondegenerate Lagrangian into that of a (pseudo-)Riemannian nonlinear-sigma model. The
renormalization theory of these systems is well understood and we summarize the aspects needed here in
Section B.3. In exchange for the gauge-freezing one then has to define quantum counterparts of the
constraints (3.15
) as renormalized composite operators. This will be done in Equations (3.100
)
ff.
For the sake of comparison with Equations (3.49, 3.51
) we display here the first two terms of the
resulting geodesic background-fluctuation split:
In summary, we find the following differences to standard perturbation theory:
The first point entails that the sigma-model perturbation theory we are going to use is partially non-perturbative from the viewpoint of a standard graviton loop expansion.
Before turning to the quantum theory of these warped product sigma-models we briefy discuss the status
of the conformal factor instability in a covariant formulation of the truncation. As
emphazised by Mazur and Mottola [142] in linearized Euclidean quantum Einstein Gravity (based on
the Euclidean version of the action (3.30
)) there is really no conformal factor instability. The
kinetic term in the second part of Equation (3.30
) with the wrong sign receives an extra
contribution from the measure which after switching to gauge invariant variables renders both the
Gaussian functional integral over the conformal factor and that for the physical degrees of
freedom well-defined. They also gave a structural argument why this should be so even on a
nonlinear level: As one can see from a canonical formulation the conformal factor in Einstein
gravity is really a constrained degree of freedom and should not have a canonically conjugate
momentum.
In the truncation we shall use a Lorentzian functional integral defined through the sigma-model
perturbation theory outlined above. So a conformal factor instability proper associated with a Euclidean
functional integral anyhow does not arise. Nevertheless it is instructive to trace the fate of the incriminated
term.
From the York-type decomposition (3.49, 3.50
) one sees that
plays the
role of the (gauge-variant) conformal factor. The wrong sign kinetic term is indeed still present in the
second part of Equation (3.34
) and
also appears through a dilaton type coupling in the
term. In 2D however
has no propagating degrees of freedom and the term could be
taken care of promoting
to a dynamical degree of freedom via gauge fixing and then cancelling the
effect by a Faddeev–Popov determinant. As already argued before it is better to avoid this and look at the
remaining propagating degrees of freedom directly. They simplify when reexpressed in terms of
and
, viz.
. This
occurs here on the linearized level but comparing with Equation (3.16
) one sees that the same structure is
present in the full gauge-frozen action. We thus consider from now on directly the corresponding terms
proportional to
. By a local redefinition of
one can eliminate the term
quadratic in
and in dimensional regularization used later no Jacobian arises. One is left with
a
term which upon diagonalization gives rise to one field whose kinetic term has
the wrong sign. However
is a dilaton type field which multiplies all of the self-interacting
positive energy scalars in the first term of Equation (3.16
), and the dynamics of this mode
turns out to be very special (see Section 3.3). Heuristically this can be seen by viewing the
field in the Lorentzian functional integral simply as a Lagrange multiplier for a
insertion. The remaining Lorentzian functional integral would allow for a conventional Wick
rotation with a manifestly bounded Euclidean action. We expect that roughly along these lines a
non-perturbative definition of the functional integral for Equation (3.16
) could be given, which would
clearly be one without any conformal factor instability. Within the perturbative construction
used in Section 3.3 the special status of the
field, viewed as a renormalized operator,
can be verified. Since the system is renormalizable only with infinitely many couplings, the
functional dependence on
in the renormalized Lagrangian and in the
field has to be
‘deformed’ in a systematic way; however this does not affect the principle aspect that no instability
occurs.
Finally, let us briefly comment on the role of Newton’s constant and of the cosmological constant in the
truncations. The gravity part of the action (3.10
) or (3.45
) arises from evaluating the
Einstein–Hilbert action
on the class of metrics (3.6
). The constant
in Equation (3.10
, 3.45
)
can be identified with
, i.e. with Newton’s constant per unit volume of the orbits. As such
is
an inessential parameter and its running is defined only relative to a reference operator. For the
truncations it turns out that the way how the action (3.10
) depends on
has to be modified in a
nontrivial and scale dependent way by a function
(see Equation (3.56
) below) in order to achieve
strict cut-off independence. This modification amounts to the inclusion of infinitely many essential
couplings, only the overall scale of
remains an inessential parameter. It is thus convenient
not to renormalize this overall scale and to treat
in Equation (3.56
) as a loop counting
parameter.
A similar remark applies to the cosmological constant. Adding a cosmological constant term to the Ricci
scalar term results in a type addition to Equation (3.56
) below. In the quantum theory one is again
forced to replace
with an scale dependent function
in order to achieve strict cutoff
independence [156
]. The cosmological constant proper can be identified with the overall scale of the
function
. The function
is subject to a non-autonomous flow equation, triggered by
, but if its
initial value is set to zero it remains zero in the course of the flow [156
]. To simplify the exposition we thus
set
from the beginning and omit the cosmological constant term in the following. It is however a
nontrivial statement that this can be be done in a way compatible with the renormalization
flow.
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