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 133-791 , 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 measure-zero 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 gauge-fixing 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 time-dependent backgrounds.
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The quantization of 4D Einstein–Hilbert action has been
a long-standing problem. (See [1] for a review.) One-loop
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 one-loop
renormalizability [3,4]. Also, it was subsequently shown [5]
that two-loop 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 gauge-fixing 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 4 (...truncated)