Hypersurface foliation approach to renormalization of ADM formulation of gravity
Eur. Phys. J. C
Hypersurface foliation approach to renormalization of ADM formulation of gravity
I. Y. Park 0 1
0 Department of Applied Mathematics, Philander Smith College , Little Rock, AR 72223 , USA
1 Department of Physics, Hanyang University , Seoul 133791 , Korea
We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measurezero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of the ADM formulation of 4D Einstein gravity for which a complete gaugefixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit timedependent backgrounds.

The quantization of 4D Einstein–Hilbert action has been
a longstanding problem. (See [1] for a review.) Oneloop
renormalizability of pure 4D gravity (i.e., gravity without
any matter field coupled) was established in [2]. However,
the presence of matter fields does not preserve the oneloop
renormalizability [3,4]. Also, it was subsequently shown [5]
that twoloop and higher order diagrams of the pure
gravity require proliferation of counter terms, thereby leading to
unrenormalizability. Needless to say, the lack of
renormalizability has been a serious obstacle that has delayed (or even
blocked) progress in many fundamental issues such as the
black hole information paradox.
There have been several approaches to the quantization of
gravity. The first was through the Hamiltonian formulation
of gravity [6,7]. (Earlier discussions can be found, e.g., in
[8–11].) The obstacle in this approach was the complexity
of the Hamiltonian constraint; a different approach based on
a different set of variables was proposed and is now known
as loop quantum gravity [12–14]. Still another approach is
based on adding higher derivative terms in the action [15,16].
In this work, we show by conducting the 3 + 1 splitting
in the Lagrangian formulation that the dynamics of the pure
4D gravity is effectively reduced.1 to the dynamics of pure
“3D gravity.” (As is well known, the genuine 3D gravity does
not have a graviton. Our “3D” theory – originating from 4D
gravity – is not the genuine 3D gravity; as we will see, it
has two propagating degrees of freedom that are inherited
from 4D. It holographically represents the original 4D
system as shown in Fig. 1.) Since pure 3D gravity is known to be
renormalizable [19,20], our result implies that the 4D
Einstein gravity is renormalizable: renormalizability of the 4D
gravity is achieved essentially by removing all of the
unphysical degrees of freedom in the particular manner which we
will describe in detail in the main body. Some of the
analyses in this work have been repeated from different angles in
[21–23].
It may be good to clearly state at this point the task
undertaken in the present work. The equivalence between the usual
formulation and ADM formulation of general relativity was
questioned in [24]. In our view, it is a legitimate concern
given, e.g., that the ADM formulation may not generally be
applicable to an arbitrary spacetime but is most useful in
dealing with a globally hyperbolic spacetime. The ADM
formulation should be applicable to a “locally hyperbolic” spacetime
1 To our pleasant surprise, we recently have found the work of [17]
where a Hamilatonain reduction led to lower dimensional volume. There
also is an earlier related work [18] We will comment on these further
in the conclusion.
Fig. 1 a 4D scattering; b projection onto 3D hypersurface
when the issue under consideration is local in nature such as
renormalizability. Furthermore, once one takes the gauge in
which the shift vector is set to zero, the analysis becomes
essentially that of the usual formulation in the synchronous
gauge. It is certainly possible to repeat all of the analysis
in this work without referring to the ADM formulation but
rather within the usual Lagrangian and Hamiltonian
formulations with the synchronous gauge. Since gaugefixing is a
subtle issue in general, we clearly state the task undertaken
in this work: quantization associated with the physical states
dictated by the ADM formulation with the synchronous type
gauge.
The focus of the present work is the possibility that the
alleged unrenormalizability may be due to the presence of
unphysical degrees of freedom in the 4D Einstein action.
There are eight unphysical degrees of freedom, and four of
them are associated with the 4D diffeomorphism. The other
four degrees of freedom are nondynamical, and they may
explicitly be removed. In principle, there is a chance that such
a removal may eliminate the need for some of the counter
terms, and we confirm this anticipation in a dramatic way.
Basically, we remove the unphysical fields by gauge
fixings. (It may appear that the procedure depends on the
gaugefixing too sensitively. We will have more to say on this in
the conclusion.) The field equations of the lapse function
and shift vector do not have any time derivative acting on
them. Therefore, it should be unnecessary to use the bulk
gauge symmetry to gaugefix them. Instead, one may use the
residual symmetry after the bulk fixing to gaugefix these
nondynamic fields. (This idea may be related to the one in
[10,11].) Gaugefixing of the bulk diffeomorphism will be
carried out in the standard manner; it is the gaugefixing of
the nondynamical fields that makes the difference.2 In terms
of Dirac’s terminology [25], the nondynamical fields
introduce the first class constraints in the Hamiltonian formalism.
In the spirit of the comment made in section 7.6 of [26] (in
which it was stated that the first class constraints can be
eliminated by a choice of gauge), we deal with the Lagrangian
analog of the first class constraints simply by additional
gaugefixings. In the Hamiltonian approach, the first class
constraints are associated with gauge symmetries. The
necessity of such a gaugefixing was discussed in [1,27]. Since
the nondynamical fields do not evolve in time and are thus
virtually threedimensional, they should be viewed as
generating threedimensional (as opposed to fourdimensional)
symmetries. Because of this, a measurezero symmetry will
be used to remove the nondynamical fields. (See [21] for an
explicit and quantitative analysis on the nature of the residual
symmetry and ghostterm related issues.)
In spite of the unphysical nature of the nondynamical
fields, the conventional procedure includes them in the
renormalization program: they appear as external lines of loop
diagrams in the Green function computations. They also run
around the loops of various loop diagrams. The main
reason for not gaugefixing them should be maintaining the
covariance. (Moreover, such a procedure was successful in
the cases of gauge theories.) However, if there exists a
(noncovariant) renormalization procedure that does not include
the nondynamical fields and is still controllable, it would
be worthwhile to examine the procedure. We show that there
indeed exists such a procedure that maintains 3D covariance.
Since we claim in this work that a different (and complete)
gaugefixing may render the 4D gravity renormalizable,3 one
may raise a question how that could happen given that physics
should not depend on a gauge choice. Firstly, let us recall that
only the correlator of gauge invariant quantities are
independent of the gaugefixing; in a renormalization procedure, one
considers correlators of the elementary fields – which may
or may not be gauge singlets (for example, the metric is not
a gauge singlet). Therefore, renormalization procedure does
depend on the gauge choice, and we have demonstrated this
in the Appendix. What we observe in this work is that the
dependence can be drastic. A second clue to the answer to the
question above can be found in [28]. We will come back to
this work later but briefly the authors noted that the presence
of an unphysical mode makes the path integral divergent and
illdefined. The divergence here is something that one should
worry about before one examines renormalizability of the
theory, and is worse in that sense than unrenormalizability.
Therefore, it seems reasonable to expect that removing all,
as opposed to four (as in, e.g., the de Donder gauge) out of
eight, of the unphysical modes can, in principle, make a
drastic difference. Therefore, gaugechoice independence of the
physics including the renormalizability would be expected
for different gauges as far as they are complete fixings.
2 In this work we adopt the approach in which all of the unphysical
degrees of freedom are explicitly fixed. Alternatively one may keep the
offshell unphysical degrees of freedom that are associated with the
lapse and shift. See [23] for this approach.
3 The analogous phenomenon does not occur in gauge theories since
gauge theories are renormalizable in the Lorenz gauge that keeps the
nondynamical time component. It seems that gravity is just different
from gauge theories in this aspect.
There exists a gauge, called the radiation gauge [29,30],
in which all of the unphysical fields are removed. The
gaugefixing procedure that we adopt keeps only the physical
degrees of freedom as in the radiation gauge, but is different
from the radiation gauge in that it leads to effective reduction
of the dynamics onto the hypersurface. The hypersurface
foliation approach4 in this work provides a convenient arena for
examining the renormalizability after removal of the
unphysical degrees of freedom. Let us consider the 3 + 1 splitting
of the 4D Einstein–Hilbert action. Compared with the
existing covariant approach, the ADM formulation employed in
this manuscript has two advantages: firstly, the ADM
formulation is effective in organizing the degrees of freedom for
easy isolation of the unphysical degrees of freedom. In other
words, the formulation readily identifies the nondynamical
fields thereby setting the stage for their removal. Secondly,
the ADM splitting brings out the utility of the measurezero
gauge symmetry for removal of the nondynamic fields. This
feature will play a crucial role in Sect. 2.
The rest of the paper is organized as follows. In Sect. 2,
we discuss the removal of unphysical degrees of freedom.
After starting off by recalling the gaugefixing procedure of
a YM type gauge theory, we note that it should be possible to
gaugefix the lapse function and shift vector by the
measurezero diffeomorphisms after fixing the bulk gauge symmetry
(the de Donder gauge will be adopted for the bulk fixing). As
is well known (see, e.g. [35]), the choice of the lapse and shift
is arbitrary, hence gaugefixing the lapse and shift should be
a legitimate procedure in any case. What is important is that
it should be possible to gaugefix them by using the
measurezero gauge symmetry, not the bulk ones. The result of these
gaugefixings is the projection of 4D dynamics onto the 3D
hypersurface: the 3D system has two physical degrees of
freedom inherited from 4D. (The reduction is limited to a pure
Einstein system; a matter field, if present, will not be reduced,
at least not in any simple way.) By invoking the logic of [5,20]
one arrives at renormalizability. This will be pointed out in
Sect. 3, in which we will also comment on precisely what
physics the reduced theory should describe. The procedure
should be viewed as a generalization of the holography idea
of [36].5 A flat spacetime will be considered throughout. It
should be possible to apply the procedure to a Schwarzschild
black hole background with relatively minor modifications.
However, it is not clear whether the procedure can be applied
to a more complex background such as an explicit
time
4 The hypersurface foliation approach combined with explicit dimen
sional reduction has been fruitful [31–34] Unlike these works, no
explicit dimensional reduction is carried out here: the projection onto
3D is dictated by removal of unphysical degrees of freedom through the
measurezero diffeomorphism. Thus the reduction is “spontaneous”.
5 ’t Hooft observed the Holography in the black hole context. If what we
propose here is true, gravity theory itself has the holographic property
through a lower dimensional gravity theory.
dependent black hole background, and this is potentially a
limitation of the procedure. For the case of globally
hyperbolic spacetimes, the reduction can easily be understood
from a different and more mathematical perspective [37].
(The condition of global hyperbolicity is not strictly required,
though.) This result is summarized in Sect. 3.2. In the
conclusion, we comment on several issues such as recovery of the
4D covariance in the present context or the fully 4Dcovariant
formulation of the quantization. (Progress has been recently
made in [23].) In the Appendix, we illustrate the gaugefixing
dependence of a renormalization procedure by taking a
system of metric coupled with a scalar. The sole purpose of
considering this system is to demonstrate the dependence:
we consider only the pure gravity system in the main body.
2 Removal of unphysical degrees of freedom
It will be useful for what follows to recall the quantization
procedure in Maxwell’s theory. The vector field has four
components to start with but only two of them are physical degrees
of freedom. The system has gauge symmetry; it reduces the
number of degrees of freedom to three. The time
component is nondynamical, leading to the further reduction of the
number of physical fields to two. Let us consider temporal
gauge to be specific. It turns out that temporal gauge does not
entirely use up the gauge freedom but leaves gauge
symmetry associated with the hypersurface of a fixed time [38]. For
the perspective of our gravity analysis, what is important is
that the nondynamical time component can be gaugefixed
without using the full bulk symmetry but instead by using
measurezero lower dimensional gauge symmetry. Below we
will show that there is an analogous procedure in general
relativity.
As in the Maxwell case, it is the close conspiracy between
nondynamism and gauge symmetry that brings complete
removal of the unphysical degrees of freedom. In due course
of the analysis below, the lapse function n will get fixed to
n = 1. (This choice is suitable because we are
considering expansion of the theory around a flat spacetime to be
specific.) This introduces a constraint that corresponds to
Hamiltonian constraint of the Hamiltonian formulation. As
we will see, the constraint can be solved, and its solution
implies, among other things, the effective projection of the
4D dynamics onto the 3D hypersurface.
The gaugefixings of the measurezero symmetries will
be carried out following the spirit of section 15.4 of [39]
in which the first class constraint was eliminated by fixing
the corresponding symmetry. In essence, what we do here is
fix the gauge and explicitly solve the resulting constraints.
Therefore the procedure does not introduce any ghosts at the
bulk level. (In [39], quantization in the axial gauge was
analyzed, and it was shown that the axial gauge quantization does
where L∂t denotes the Lie derivative along the time
coordinate t and ∇a is the 3D covariant derivative (namely, its
connection is constructed out of γab); n and Na denote the
lapse function and shift vector, respectively. Since the time
derivative does not act on Na or n in their field equations,
which read
not introduce any ghosts. For the actual perturbation
computations, a covariant gauge – which does introduce ghosts –
was used. See [21] for the ghostrelated issues.)
2.1 Isolation of unphysical degrees of freedom Consider the 4D Einstein–Hilbert action (see, e.g. [40] for a review)
S =
S =
−gˆ R.
To illustrate the procedure with a specific example, we
separate out the time coordinate and split the coordinates into
where μ = 0, . . . , 3 and a = 1, 2, 3. (The resulting 3D
system will be Euclidean, thus nondynamical. For the study of
dynamics, one should consider a different setup by
separating out, say, the y3 coordinate from the rest. We will come
back to this issue below.) By parameterizing the 4D metric
[6,35]
is also consistent with the observation made in [42].) As we
will see below, they can be combined and solved.
The action has the gauge symmetry of measurezero
(compared with the 4D gauge symmetry) left after imposing the
4D de Donder gauge. After we discuss gaugefixing of the
4D diffeomorphism, we will come back to this symmetry
to fix Na = 0 on the hypersurface. (As far as we can see,
this view is consistent with [10,11,43]. Again the use of the
ADM variables is crucial in isolating the unphysical fields:
as shown in [44] the first class constraints will generate the
bulk diffeomorphism in the Hamiltonian approach with the
full 4D metric chosen as the canonical variables.) Then by
invoking the nondynamical nature of Na , the bulk value will
be taken as its hypersurface value, namely, Na = 0 for the
entire bulk. Similarly, the lapse function can be fixed to n = 1
for the entire bulk.6
Let us fix the 4D diffeomorphism by imposing the de
Donder gauge. The full form of the de Donder gauge is given by
[29].7 We have
In the conventional perturbative analysis, one proceeds and
adds the corresponding ghost term in the action. However, the
measurezero gauge fixing will affect, as we will show now,
the bulk fixing simply because the bulk fixing (9) contains the
nondynamical fields that get fixed by the 3D gaugefixing. At
the end, only the 3D ghosts will be required after the system
is reduced. In terms of the ADM variables, (9) translates to
where bac denotes the 3D Christoffel symbol. As is well
known, the de Donder gauge leaves a residual symmetry (see
6 This choice will be good for quantization around a flat background.
If one has a black hole background in mind with the radial coordinate
separated out, one will have to introduce a radial coordinate
transformation such that grr does not depend on r. As we will further comment
later, this is where one may encounter difficulties when trying to apply
the procedure to an explicitly timedependent background because it is
not clear whether such a coordinate transformation will always be
available without subtleties. This also reveals the background dependence of
our procedure: although the methodology is background independent,
the detailed steps will depend on the background under consideration.
This aspect also makes a connection with precisely what physics the
reduced theory describes, a point that we will discuss in Sect. 3 and the
conclusion.
7 As a matter of fact, one only needs the μ = 0 part of this for the
reduction:
The rest will serve as the 3D de Donder gauge in the reduced theory.
Improved versions of the reduction can be found in [22,23]. The account
in [23] is especially simple.
R(3) + K 2 − Kab K ab = 0,
these fields are nondynamical: they do not have any time
derivative acting on them, and thus their bulk values can be
taken as the corresponding values on the hypersurface of the
fixed time once n and Na are specified on the hypersurface
of a given time. After the gaugefixings that we turn to now,
the equations above should be taken as the constraints that
determine the physical states. (This is in the same spirit as the
quantization of string (see, e.g., ch2 of [41].) The procedure
for example [45]; a nonlinear level discussion can be found
in [21]) although it is not manifest in (10). It is thus possible,
by using the residual diffeomorphism,8 to set
Na = 0
initially on the hypersurface of the fixed time. As mentioned
above, this equation can then be taken as valid in the entire
bulk due to the nondynamism of Na . Substituting Na = 0
into (6), which now serves as a constraint, one gets
a n1 (L∂t γab − γabγ cd L∂t γcd ) = 0.
When the covariant derivative acts on objects other than
n, it yields zero because the covariant derivative and the
Lie derivative commute in the present case [46]. This step
requires the advancedlevel knowledge in differential
geometry of Ch.6 of [46]; for further details see [22,37]. The
constraint (12) can be solved and implies
∂a n = 0.
n = 1
Since n is nondynamical, namely, n = n(ya ), (13) implies
that n should be a constant9: the Na constraint implies n =
n(t ), but since n is nondynamical, one can set
even in the bulk. This fixing of n should be supplemented by
its field equation which should now serves as a constraint,
system is not the genuine 3D gravity: it has two propagating
degrees of freedom, the physical degrees of freedom of the
original 4D gravity projected onto the hypersurface. (We will
have more to say on this below.)
Let us split the constraint (15) into two parts: the R(3) term
and the K terms. They vanish separately: to see that R(3)
vanishes we just need to recall that it is an onshell constraint. In
other words, we define the physical states to be annihilated by
(15), in which the fields are now viewed as the corresponding
operators acting on the Fock space. One may now consider
the full field equation for the 3D metric (i.e., the one that
follows from (17) and first obtain the mode expansion at the
linear level. Then the linear level expression can be
substituted into the full field equation, from which one can obtain
the full expansion by iteration. (For the definition of the
physical states in the actual perturbative computations, one may
expand the metric to the linear order and neglect the higher
interactions. This is in the same spirit as the comment in one
of the footnotes in section 15.7 of [39].) The full expansion
of γab should yield zero once substituted into the R(3) part of
the constraint (15). With this, only the K terms remain. As
can be seen from the first equation in (10) K vanishes,
K = 0,
Ka2b = 0.
(The same conclusion has been reached in a complementary
analysis in [22,23].) Since this is a positive definite metric,
this implies
once (11) and (14) are substituted into (5). The constraint
(15) allows one to rewrite the action
where the overall factor 2 has been absorbed. The form of
the action (17) does not yet imply that the system becomes
threedimensional. This is because one should still consider
the constraint (15). We will now argue that (17) with (15)
implies projection of the 4D dynamics onto 3D. But before
we proceed, a cautionary remark is in order. As is well known,
3D gravity does not have a graviton. In this sense, our reduced
8 The ADM form of the action has the manifest 3D gauge symmetry.
The (linear form of) the 4D de Donder gauge has the same symmetry in
the form of the residual symmetry that is parameterized by a(y). The
shift vector can be gaugefixed by using this 3D symmetry.
9 Here by a “constant” is meant a yaindependent expression. In the
case of expanding the 4D theory around a Schwarzschild background
with the radial coordinate separated out, this step implies that n should
be independent of (t, θ, φ).
R(3) + K 2 − Kab K ab = 0
where Kab takes the form
S =
K = hmn Kmn.
S =
In other words, the physical states of the original 4D system
are fully reduced to 3D. The projected system has two
physical degrees of freedom: the metric of the hypersurface has six
components. The first equation of (10) imposes a constraint
and the second equation of (10) imposes the 3D de Donder
gauge, effectively removing three degrees of freedom. (We
will have to say more on this in the next section.)
In the remainder of this section, we discuss the case of
separating out one of the spatial directions, say, y3:
x μ ≡ (zm , y3) where m = 0, 1, 2
The procedure goes through almost identically, except for
several sign changes. For example, the split form of the action
now takes the form
d4x n√−γ (R(3) + K 2 − Kmn K mn)
The only nontrivial difference compared with the t splitting
is the condition that corresponds to (19):
Unlike in (19), the contractions of indices in this equation are
done with the 3D metric with (−++) signature. In general, a
Wick rotation must be considered in field theories with
nonpositive definite metrics, since otherwise the path integral is
not well defined. In the gravitational case, the procedure has
a subtlety addressed in [28,47,48]. As far as we can see, it is
not a subtlety associated with the Wick rotation but with the
presence of an unphysical component of the metric. We will
take up this issue in the next section. Once the Wick rotation
is carried out, Eq. (24) leads to
Kmn = 0,
namely, an onshell reduction to a dynamical 3D system.
K m2n = 0.
3 Quantization and renormalization
The goal of the previous section was to remove
unphysical degrees of freedom. In this section we tackle the very
issue of renormalizability of 4D Einstein gravity. Afterwards,
we note that the holographic reduction admits a nice
mathematical perspective for the cases of globally hyperbolic
spacetimes.
For quantization, the t separation case and y3separation
case do not have an essential difference. Both cases are
subject to the subtlety observed in [28,47,48], namely, the
divergence associated with the trace part of the metric.
3.1 4D quantization through hypersurface
The upshot of the analysis in the previous section is that,
upon fixing n = 1, Na = 0, by using a lower dimensional
diffeomorphism after the bulk fixing, the dynamics of the
original 4D system is projected onto “3D”:
S =
The quotation mark around 3D above is written because of the
presence of the 4D integration and 4D coordinate dependence
of the offshell metric γab. We now show that the integral
over the 4D spacetime is effectively reduced to 3D. For the
discussion in this section, we consider separating out the
spatial coordinate y3. This action should be supplemented
by the nonlinear form of the 3D de Donder gauge,
With the genuine 3D gravity, one can use the residual
symmetry after imposing (27) to gauge away three nondynamical
components, thereby arriving at the wellknown absence of
propagating degrees of freedom. In the current case, however,
it is impossible to do the same simply because the metric in
(26) still has the fourth coordinate dependence offshell. The
residual symmetry after (27) should be the symmetry within
the hypersurface. This can be seen by considering the
residual symmetry condition stated in [45]. Namely, it is a partial
differential equation of the gauge parameter m on the
hypersurface and therefore cannot be used to gauge away additional
components. Alternatively, one can consider the entire
procedure of the gaugefixings in the pathintegral setup. The
result (26) with (27) has been obtained as a result of this
complete gaugefixing by implicitly following the steps given in
section 15.4 of [39], and therefore one should not further
gaugefix (26) other than by (27).
Remarks on precisely what physics the reduced system
(26) with (27) is supposed to describe are in order. The focus
of the present work is perturbative quantization around a flat
background, exactly the goal that we set out for. Although
the procedure should be applicable to a Schwarzschild
background, it is not clear whether the procedure can be used to
study more complex backgrounds. Related to this, there is
an issue of nonperturbative corrections. We will not pursue
these issues here.
For perturbation, one can use the usual linear form of the
3D de Donder gauge (27). The 3D gaugefixing and ghost
terms will have to be introduced, and with that one can carry
out the BRST quantization. Here we focus on the graviton
part of the computation. Since the procedure maintains only
the 3D covariance, one should worry about compatibility
with the (yet unavailable) fully 4D covariant approach. We
will further remark on this in the conclusion.
The perturbation series with (26) can be set up as follows.
(The following analysis has been substantially expanded in
[21].) For the perturbative analysis, the usual linearized
version of (27) can be imposed. The propagator is then given by
d4k eik·(x1−x2)
where k is the 3vector part of the 4momentum kμ.
The field qrs is the traceless part of the fluctuation around
a flat metric (use of the traceless metric qmn is not essential
for the reduction),
This form of the propagator implies that one can carry out the
renormalization program just as in the regular 4D case. After
performing various momentum integrals, thereby removing
momentum delta functions, one will be left with
integration over the loop momenta in which the propagators take
the threedimensional forms. The 3D integration can be
carried out by the standard techniques. The integration over the
reduced direction can be carried out by taking, e.g., a
momentum cutoff regularization. In general, the momentum cutoff
regularization does not respect gauge symmetry. However,
adopting it for the fourth direction does not pose a problem
here: dimensional regularization can be used for the 3D
integrals. The divergence factor arising from the k0 integration
can be absorbed by rescaling the fields (i.e., wavefunction
renormalization). In more detail, let us introduce a
rescaling of w by a dimensional parameter L (we have defined
w ≡ y3):
such that u is dimensionless. With this rescaling, one can use
a dimensionless momentum cutoff for the u direction. The
rescaling will lead to renormalization (i.e., rescaling by L)
of Newton’s constant, which has been suppressed, and one
achieves an effective reduction to 3D:
S =
Since the 4D system (4) with which we have started has been
reduced to (31) with the constraint (27), the former would be
renormalizable if the latter is.
For renormalizability of (31) with the constraint (27), the
result of [20] can mostly be borrowed with several cautions
to which we will now turn. It was shown in [20] that 3D pure
gravity is renormalizable even though it is powercounting
nonrenormalizable. To use the result of [20] in the present
case, one should first understand that the theory that was
considered in [20] is the genuine 3D Einstein gravity that has no
propagating degrees of freedom. In the case of the 3D pure
gravity, the author relied on certain kinematic features of 3D
gravity instead of explicit computation in order to deduce
renormalizability. However, the author considered
propagating gravitons as can be seen in the gravity–matter coupling
case, where explicit computation was carried out. The
propagating degrees of freedom must be the nondynamical ones;
this is in line with the common practice in the literature before
and after [20], in which nondynamical fields are allowed to
propagate in loops. In the current setup, those nondynamical
fields are dynamical in the 4D standpoint, and they are
justified to be present in the loops.
The second cautionary step one should take is the fact that
the result of [20] was obtained up to the observation in [28],
which addressed the issue raised in [47,48]. The constraint
shows that the trace part of the original 4D metric is
nondynamical. Upon the fixing of n = 1, Na = 0, the constraint
implies that the trace part of the 3D metric is nondynamical.
It was shown in [28] that the 4D trace part is nondynamical
in the 4D sense. Since our reduced system should represent
L∂a n = 0.
the 4D system, the trace part of the metric (3D one after the
fixing) should be excluded from the dynamics.
3.2 Perspective of mathematics of foliation
The analysis of the previous section does not need any
assumption on the causality of the starting 4D manifold. It is
possible to have a complementary understanding of the
finding of the previous section through abstract foliation theory.
We will assume for simplicity, although it is not necessary,
that the background is globally hyperbolic. This section is a
brief summary of [37], in which more details can be found.
(See [22] for further development of the idea along these
lines.)
A globally hyperbolic spacetime – which covers most
of the cosmologically interesting spacetimes – admits a
codimension1 foliation through a family of hypersurfaces.
The condition (13) obtained by the shift vector constraint can
be written precisely as the condition for the foliation to be
Riemannian:
In mathematics, it is known that a codimension3
Riemannian foliation admits a “dual” totally geodesic foliation. The
duality involved is a mathematical one and operates between
two different foliations: the Riemannian and totally geodesic.
Now the starting manifold can be viewed as a principal
bundle of 1D abelian fibration over the 3D base through the
totally geodesic foliation. Then one may associate the action
of the 1D group fibration with the gauge symmetry;
modding out the gauge symmetry will correspond to casting the
4D manifold into its 3D base. The presence of the totally
geodesic foliation has an enlightening implication for the
apparent gaugechoice sensitivity, as we will discuss in the
conclusion.
4 Conclusion
A metric in four dimensions has ten components, four of
which are nondynamical, and another four components are
associated with 4D gauge symmetry. In the conventional
covariant renormalization program, only the modes
associated with 4D gauge symmetry are removed by a
gaugefixing such as the de Donder gauge; the 4D Einstein gravity
is powercounting unrenormalizable and indeed turned out
to be unrenormalizable. Even though it is only the
measurezero diffeomorphism that remains unfixed in the
conventional covariant approach, the nondynamical fields circulate
the loop acting as bulk fields, thereby apparently ruining the
renormalizability.
In this work, we have contemplated the possibility that
additional removal of the nondynamic fields may lead to
renormalizability of the ADM formulation of 4D Einstein
gravity. We have shown that there exists a way to remove all
(or most) of the unphysical degrees of freedom and at the
same time to set up a convenient stage for examining
renormalizability. (As stated in one of the footnotes in the
Introduction, one may keep the four of the eight unphysical fields at
the offshell level and carry out renormalization [23].) After
removal of the unphysical fields, the 4D dynamics admits the
effective description in terms of the 3D language: 4D
renormalizability is achieved based on the 3D renormalizability.
Compared with AdS/CFT, it is notable that the boundary
theory in the present case is not a gauge theory but takes
the form of a lower dimensional gravity theory. We do not
believe that this has anything to do the hypersurface
foliation approach, because the approach (combined with manual
dimensional reduction) has led to a boundary gauge theory
in the IIB supergravity setup [31,49].
There is an interesting implication of the present work for
the previous works of [32–34], wherein explicit dimensional
reduction was carried out in order to avoid the quantization
issues. The analyses there were for certain subsectors (i.e.,
the sectors associated with the hypersurfaces selected) of the
whole moduli space. The result of the present work implies
that the analyses in those works are much more complete
than originally believed.
We end with several directions for future research.
One of the subtle points is that the whole procedure of
reduction seems to heavily depend on the elaborate
gaugefixings. (It would thus be worthwhile to explore whether there
exists a bulk gauge other than the de Donder gauge that would
lead to the same conclusion. Perhaps an easier task that will
nevertheless shed light on this issue is to study how the form
of the de Donder gauge (9) should be modified in the case
of a Schwarzschild background.) Part of the reason for the
gauge sensitivity should be that the ratio of the physical
versus unphysical fields is much higher than in YM theory. It
seems, therefore, natural to a certain extent that certain things
depend on the gauge choice more sensitively than in YM
theory. The mathematical approach in [22,37] – which takes
advantage of the presence of the dual totally geodesic
foliation – strongly suggests that the reduction should be a rather
general phenomenon (associated with relatively simple
backgrounds). This is because in that approach the only thing one
needs is the very natural gaugefixing gˆ0a = 0 (which of
course is Na = 0). Judging from this and other pieces of
evidence, the gauge sensitivity of the present approach should
not be such a serious problem.
The present work was carried out for fluctuations around
a flat spacetime. For example, it would definitely be
worthwhile to carry out a similar analysis for a Schwarzschild
black hole background. Although the methodology would
be basically the same, the detailed steps would have to be
modified: the present approach has a moderate background
dependence. To this end, it would be better to consider
separating out the radial coordinate than the time coordinate.
We believe that this is a task that can be carried out with
only relatively minor changes in the present analysis. With
this achieved, one would be in a good position to tackle
the Firewall [50] (see [57] for an earlier related work) and
related issues, which was the main motivation for the present
work.
At some point, one must face the difficult issue of
nonperturbative contributions to the path integral. There are two
kinds of nonperturbative contributions. The first kind is the
contribution coming from summing over all orders of the
coupling constant in the perturbation theory. A 3D version
of the techniques of the asymptotically safe gravity [51–55]
may be useful along this line. There will be several important
issues such as the Gribov problem (see, e.g. [56] and
references therein for a recent account in the context of gravity)
that would have to be addressed eventually. The other kind
is the contribution analogous to the instanton contribution in
YM theory. It is not clear whether the present procedure is
adequate for that, given the limitation mentioned in the main
text. (Unlike the instantons in YM theory, much less is known
about the gravitational analogs anyway.)
One of the utmost important directions of research
concerns the 4D covariance, since the present approach
maintains only the 3D covariance. There exist several
different ways of ensuring equivalence between a noncovariant
approach and a covariant approach. (See [23] for recent
progress.) Let us recall by a few examples how the
equivalence is accomplished in quantum field theory and string
theory. One may attempt to construct a 4Dcovariant
operator formulation without using any constraint directly in
the action. In other words, one would attempt 4Dcovariant
removal of all or most of the unphysical degrees of freedom.
Perhaps it could be done along the line of the gauge
invariant quantization [39]. Some of the observations of [24] may
be useful. While these programs could succeed, an easier
approach will presumably be along the line of the
illuminating analysis in section 9.6 of [26]. It was shown that the
Coulomb gauge canonical quantization of electrodynamics
can be related to the Lorenz gauge path integral. When both
the noncovariant and the covariant formulations were
independently available, one could establish the equivalence by
simply comparing the amplitudes of the physical states (and
this is the way the equivalence between the lightcone
quantization and covariant quantization was achieved in string
theory). Computations of various Feynman diagrams must
be preceded to this end. Both of these approaches were
discussed in [21], at least to some extent.
The observation on the renormalizability in this work is,
to some extent, up to the issue of reduced space
quantization vs. Dirac quantization. In general, those two approaches
do not lead to the same physics [58,59]; see also the
referS =
component, the trace, is fixed.11 As we will see, whether
or not one gauges away the trace part makes the procedures
drastically different. What we observed in the main body was
that the difference can be so drastic as to render the 4D pure
Einstein action renormalizable.
The gaugefixing dependence of a renormalization
procedure can easily be illustrated by a coupled system of a metric
and scalar.12 Consider the 4D Einstein action coupled with a
scalar,
and the weakfield expansion around the Minkowski metric,
ences therein. This was demonstrated by taking an example
in which the first constraint was not associated with the gauge
symmetry. It was shown that, in general, both quantization
methods lead to different but consistent theories. We are not
aware of any work in which a similar conclusion was drawn
for a theory with a gauge symmetry inducing a first class
constraint. We believe that those two approaches are more
likely than not to lead to the same results at the end as was
the case in string quantization: lightcone, old covariant, and
modern BRST quantizations all led to the same results.
Finally, we have recently become aware of the work
of York [18], Moncrief [60], and Fischer–Moncrief [17]
wherein certain Hamiltonian reductions were carried out on
a certain class of 4D manifolds. It was a pleasant surprise
to discover that the reduced Hamiltonians were given by the
volumes of the hypersurfaces. Given that the action for a
particle is given by its length and that of a string by its surface, the
appearance of a volume for the hypersurface seems natural,
except that the volume appears as the Hamiltonian instead
of the Lagrangian. Presumably this is due the fact that the
reduced direction is the timedirection, so that the
hypersurface is Euclidean. In our case, it was the 3D Einstein action
that has emerged. Perhaps, the 3D hypersurfaces admit dual
descriptions, one through the volume and the other through
the Einstein action. It would be interesting to understand the
potential relation between the work of [17, 18, 60] and the
present one.
We will report on progress made with some of these issues
in the near future.10
Acknowledgments I acknowledge the hospitality of S.J. Sin and
B.H. Lee at Hanyang University and CQUeST during my stay. I thank
the members of Hanyang University and KIAS for their interest and
criticisms. This work benefited from the discussions I had with them. I
thank V. Moncrief, G. ’t Hooft, and R. Woodard for their
communications. I also thank H. Lee, E. Poisson, S. Weinberg for reading the main
idea of the work and making encouraging comments.
Open Access This article is distributed under the terms of the Creative
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and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
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Appendix: Gaugefixing dependence of renormalization
Let us demonstrate the complications caused by the presence
of the unphysical modes in carrying out a renormalization
program. We show that the renormalization procedure
substantially depends on whether or not the undynamical metric
10 Some of the issues have been recently addressed in [21–23].
Let us impose the de Donder gauge by adding the following
gaugefixing term to the action:
L =
− 21 gαβ ∂α gγ δ ∂β gγ δ + 2gαβ ∂α gγ δ ∂δ gβγ
+ 41 gαα ∂ δ gβγ ∂δ gβγ − gαβ ∂δ g ∂β gαδ + 21 gαβ ∂α g ∂β gρρ
−gαβ ∂δ gα ∂ gβγ + gαβ ∂δ gα ∂γ gβ
γ δ γ δ
−∂ ν C¯ μ∂μC ρ gνρ − ∂ ν C¯ μ∂ν C ρ gμρ − ∂ ν C¯ μC ρ ∂ρ gμν
1 ν
+∂μC¯ μ∂ ν C ρ gνρ + 2 ∂μC¯ μC ρ ∂ρ gν
+ 81 ημν (g2 − 2gρσ gρσ ) − 21 ggμν
+ · · ·
where C is the ghost field that corresponds to the gaugefixing
(A.3). The trace part is not a dynamical degree of freedom
11 Of course, keeping the trace part makes the path integral illdefined as
observed in [28,47,48]; for the sake of the discussion in this section, we
set this observation aside and formally proceed for now. (This problem
has been addressed recently in [23].)
12 The reduction to 3D that we observed in the main body was for the
pure Einstein gravity; we do not claim the same for the coupled system
that we consider in this section.
[61], and the traceless condition can explicitly be enforced
by using the traceless propagator as follows. Let us explicitly
introduce the traceless metric
as the kinetic term. The propagator then is traceless:
d4k eik·(x−y)
It follows from this that
= 0 =
any vertex term that contains a factor h will lead to zero in a
correlator computation. All the quadratic terms containing a
hfactor will not contribute to any correlator, and thereby be
effectively dropped; the action becomes substantially
simplified:
hαβ ∂α hγ δ ∂β hγ δ + 2hαβ ∂α hγ δ ∂δ hβγ
−hαβ ∂δ hα ∂ hβγ + hαβ ∂δ hα ∂γ hβ −hαβ ∂γ hαβ ∂δ hγ δ
γ δ γ δ
−∂ ν C¯ μ∂μC ρ hνρ − ∂ ν C¯ μ∂ν C ρ hμρ − ∂ ν C¯ μC ρ ∂ρ hμν
One can check that h = 0 is a legitimate gauge. At one loop
(see Fig. 2a), φ (x1)φ (x2)φ (x3)φ (x4) , which is a gauge
singlet, is the same whether one uses (A.4) or (A.9).
The see the gauge dependence, let us consider the scalar–
scalar–metric amplitudes drawn in Fig. 2b, c. One can easily
see that the treelevel correlators φφgμν with the action
(A.4) and φφhμν with the action (A.9) lead to different
results. The same is true for the oneloop correlators. The
computations are simplified in the latter case because the
number of relevant vertices is smaller. Although we have not
Fig. 3 Diagram with cubic
metric coupling
explicitly carried out, the diagram in Fig. 3 clearly shows the
drastic simplification in the computation with hμν as can be
seen by counting the number of relevant vertices: 13 in (A.4)
vs. 5 in (A.9).
Carrying out a normalization procedure for gμν means that
we consider the amplitudes in which the unphysical
component g appears, e.g., as an external line. It seems highly likely
that the exclusion of g will reduce the number of counter
terms. We have confirmed this expectation in the main text
in a very dramatic way.
1. S. Carlip , Quantum gravity: a progress report , Rept. Prog. Phys . 64 , 885 ( 2001 ) arXiv:grqc/0108040
2. G. 't Hooft, M.J.G. Veltman , One loop divergencies in the theory of gravitation . Annales Poincare Phys. Theor. A 20 , 69 ( 1974 )
3. S. Deser , P. van Nieuwenhuizen, Nonrenormalizability of the quantized DiracEinstein system . Phys. Rev. D 10 , 411 ( 1974 )
4. S. Deser , P. van Nieuwenhuizen, One loop divergences of quantized EinsteinMaxwell fields . Phys. Rev. D 10 , 401 ( 1974 )
5. M.H. Goroff , A. Sagnotti , The ultraviolet behavior of einstein gravity . Nucl. Phys. B 266 , 709 ( 1986 )
6. R.L. Arnowitt , S. Deser , C.W. Misner , The Dynamics of general relativity, Gen. Rel. Grav . 40 , 1997 ( 2008 ). arXiv:grqc/0405109
7. E. Gourgoulhon , 3 + 1 formalism and bases of numerical relativity . arXiv:grqc/0703035
8. P.A.M. Dirac , Fixation of coordinates in the Hamiltonian theory of gravitation . Phys. Rev . 114 , 924 ( 1959 )
9. P.G. Bergmann , A. Komar , The coordinate group symmetries of general relativity . Int. J. Theor. Phys . 5 , 15 ( 1972 )
10. C.J. Isham , K.V. Kuchar , Representations of spacetime diffeomorphisms . 1. Canonical parametrized field theories . Ann. Phys . 164 , 288 ( 1985 )
11. C.J. Isham , K.V. Kuchar , Representations of spacetime diffeomorphisms. 2. Canonical geometrodynamics. Ann. Phys . 164 , 316 ( 1985 )
12. A. Ashtekar , New variables for classical and quantum gravity . Phys. Rev. Lett . 57 , 2244 ( 1986 )
13. C. Rovelli , Loop quantum gravity, Living Rev. Rel . 1 , 1 ( 1998 ). arXiv:grqc/9710008
14. T. Thiemann , Modern Canonical Quantum General Relativity (Cambridge Univ. Pr., Cambridge, 2007 ). arXiv:grqc/0110034
15. K.S. Stelle , Renormalization of higher derivative quantum gravity . Phys. Rev. D 16 , 953 ( 1977 )
16. I. Antoniadis , E.T. Tomboulis , Gauge invariance and unitarity in higher derivative quantum gravity . Phys. Rev. D 33 , 2756 ( 1986 )
17. A.E. Fischer , V. Moncrief , Hamiltonian reduction of Einstein's equations of general relativity . Nucl. Phys. Proc. Suppl . 57 , 142 ( 1997 )
18. J.W. York Jr, Role of conformal three geometry in the dynamics of gravitation . Phys. Rev. Lett . 28 , 1082 ( 1972 )
19. E. Witten , ( 2 + 1 ) dimensional gravity as an exactly soluble system . Nucl. Phys . B 311 , 46 ( 1988 )
20. D. Anselmi , Renormalization of quantum gravity coupled with matter in threedimensions . Nucl. Phys. B 687 , 143 ( 2004 ). arXiv:hepth/0309249
21. I.Y. Park , Lagrangian constraints and renormalization of 4D gravity . JHEP 1504 , 053 ( 2015 ). arXiv:1412.1528 [hepth]
22. I.Y. Park , Foliation, Jet Bundle and Quantization of Einstein Gravity. arXiv:1503 . 02015 [hepth]
23. I.Y. Park , 4D Covariance of Holographic Quantization of Einstein Gravity. arXiv:1506 .08383 [hepth]
24. N. Kiriushcheva , S.V. Kuzmin , The Hamiltonian formulation of general relativity: myths and reality , Central Eur. J. Phys . 9 , 576 ( 2011 ). arXiv:0809.0097 [grqc]
25. P.A.M. Dirac , Lectures on Quantum Mechanics (Dover publications, New York , 2001 )
26. S. Weinberg , The quantum theory of fields , vol I (Cambridge University Press, Cambridge, 1995 )
27. R.P. Woodard, Enforcing the Wheelerde Witt constraint the easy way . Class. Quant. Grav . 10 , 483 ( 1993 )
28. P.O. Mazur , E. Mottola , The gravitational measure, solution of the conformal factor problem and stability of the ground state of quantum gravity . Nucl. Phys. B 341 , 187 ( 1990 )
29. L. Smarr , J.W. York , Jr., Radiation gauge in general relativity , Phys. Rev. D 17 ( 8 ), 1945 ( 1978 )
30. R.M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984 )
31. M. Sato , A. Tsuchiya , BornInfeld action from supergravity . Prog. Theor. Phys . 109 , 687 ( 2003 ). arXiv:hepth/0211074
32. I.Y. Park , ADM reduction of Einstein action and black hole entropy . Fortsch. Phys . 62 , 950 ( 2014 ). arXiv:1304.0014 [hepth]
33. I.Y. Park , Reduction of BTZ spacetime to hypersurfaces of foliation . JHEP 1401 , 102 ( 2014 ). arXiv:1311.4619 [hepth]
34. I.Y. Park , Indication for unsmooth horizon induced by quantum gravity interaction , Eur. Phys. J. C 74(11) , 3143 ( 2014 ). arXiv:1401.1492 [hepth]
35. E. Poisson , A Relativists' Toolkit (Cambridge University Press, Cambridge, 2004 )
36. G. 't Hooft, Dimensional Reduction in Quantum Gravity. arXiv:grqc/9310026
37. I.Y. Park , Quantization of Gravity Through Hypersurface Foliation. arXiv:1406 .0753 [grqc]
38. H.O. Girotti , K.D. Rothe , Quantization of QED and QCD in a fully fixed temporal gauge . Z. Phys. C 27 , 559 ( 1985 )
39. S. Weinberg , The Quantum Theory of Fields, vol II (Cambridge University Press, Cambridge, 1995 )
40. N. Straumann , General Relativity (Springer, Berlin, 2013 )
41. M.B. Green , J.H. Schwarz , E. witten, Superstring Theory , vol. 1 (Cambridge Univesrsity Press, Cambridge, 1987 )
42. N. Kiriushcheva , S.V. Kuzmin , Dirac and Lagrangian reductions in the canonical approach to the first order form of the EinsteinHilbert action . Ann. Phys . 321 , 958 ( 2006 ). arXiv:hepth/0507074
43. J. Lee , R.M. Wald , Local symmetries and constraints . J. Math. Phys . 31 , 725 ( 1990 )
44. N. Kiriushcheva , S.V. Kuzmin , C. Racknor , S.R. Valluri , Diffeomorphism invariance in the hamiltonian formulation of general relativity , Phys. Lett. A 372 , 5101 ( 2008 ). arXiv:0808.2623 [grqc]
45. A. Zee , Einstein Gravity in a Nutshell (Princeton University Press, Princeton, 2013 )
46. S. Kobayashi , K. Nomizu , Foundations of Differential Geometry , vol. I (Interscience Publisher, New York, 1963 )
47. G.W. Gibbons , S.W. Hawking , M.J. Perry , Path integrals and the indefiniteness of the gravitational action . Nucl. Phys. B 138 , 141 ( 1978 )
48. K. Schleich , Conformal rotation in perturbative gravity . Phys. Rev. D 36 , 2342 ( 1987 )
49. E. Hatefi , A.J. Nurmagambetov , I.Y. Park , ADM reduction of IIB on √, H to dS braneworld. JHEP 1304 , 170 ( 2013 ). arXiv:1210.3825 [hepth]
50. A. Almheiri , D. Marolf , J. Polchinski , J. Sully , Black holes: complementarity or firewalls? JHEP 1302 , 062 ( 2013 ). arXiv:1207.3123 [hepth]
51. S. Weinberg , General Relativity, an Einstein Centenary Survey . in S. Hawking , W. Israel (eds.) (Cambridge University Press, Cambridge, 1979 )
52. M. Reuter , Nonperturbative evolution equation for quantum gravity . Phys. Rev. D 57 , 971 ( 1998 ). arXiv:hepth/9605030
53. M. Niedermaier , The Asymptotic safety scenario in quantum gravity: An Introduction, Class . Quant. Grav. 24 , R171 ( 2007 ). arXiv:grqc/0610018
54. D. F. Litim , Fixed Points of Quantum Gravity and the Renormalisation Group , PoS QG Ph , 024 ( 2007 ) arXiv:0810.3675 [hepth]
55. R. Percacci , A Short Introduction to Asymptotic Safety. arXiv:1110 .6389 [hepth]
56. A. Eichhorn , FaddeevPopov ghosts in quantum gravity beyond perturbation theory , Phys. Rev. D 87(12) , 124016 ( 2013 ). arXiv:1301.0632 [hepth]
57. S.L. Braunstein , S. Pirandola , K. Zyczkowski , Better late than never: information retrieval from black holes , Phys. Rev. Lett . 110 (10), 101301 ( 2013 ). arXiv: 0907 .1190 [quantph]
58. A. Ashtekar , G.t. Horowitz, On the canonical approach to quantum gravity , Phys. Rev. D 26 , 3342 ( 1982 )
59. K. Schleich , Is reduced phase space quantization equivalent to Dirac quantization? Class . Quant. Grav. 7 , 1529 ( 1990 )
60. V. Moncrief , Reduction of the Einstein equations in (2+1) dimensions to a Hamiltonian system over Teichmuller space . J. Math. Phys . 30 , 2907 ( 1989 )
61. K. Kuchar , Ground state functional of the linearized gravitational field . J. Math. Phys . 11 , 3322 ( 1970 )