Despite these differences, string theory teaches us that gravity and gauge theories can, in fact, be unified. The Maldacena conjecture [94, 2], for example, relates the weak coupling limit of a gravity theory on an anti-de Sitter background to a strong coupling limit of a special supersymmetric gauge field theory. There is also a long history of papers noting that gravity can be expressed as a gauging of Lorentz symmetry [135, 82, 78], as well as examples of non-trivial similarities between classical solutions of gravity and non-Abelian gauge theories [126]. In this review a different, but very general, relationship between the weak coupling limits of both gravity and gauge theories will be described. This relationship allows gauge theories to be used directly as an aid for computations in perturbative quantum gravity.
The relationship discussed here may be understood most easily
from string perturbation theory. At the semi-classical or
``tree-level'', Kawai, Lewellen, and Tye (KLT) [85
] derived a precise set of formulas expressing closed string
amplitudes in terms of sums of products of open string
amplitudes. In the low-energy limit (i.e.
anywhere well below the string scale of
GeV) where string theory effectively reduces to field theory,
the KLT relations necessarily imply that similar relations must
exist between amplitudes in gravity and gauge field theories: At
tree-level in field theory, graviton scattering must be
expressible as a sum of products of well defined pieces of
non-Abelian gauge theory scattering amplitudes. Moreover, using
string based rules, four-graviton amplitudes with one quantum
loop in Einstein gravity were obtained in a form in which the
integrands appearing in the expressions were given as products of
integrands appearing in gauge theory [25,
55]. These results may be interpreted heuristically as
This remarkable property suggests a much stronger relationship between gravity and gauge theories than one might have anticipated by inspecting the respective Lagrangians.
The KLT relations hold at the semi-classical level,
i.e.
with no quantum loops. In order to exploit the KLT relations in
quantum gravity, one needs to completely reformulate the
quantization process; the standard methods starting either from a
Hamiltonian or a Lagrangian provide no obvious means of
exploiting the KLT relations. There is, however, an alternative
approach based on obtaining the quantum loop contributions
directly from the semi-classical tree-level amplitudes by using
D
-dimensional unitarity [15,
16,
28,
20,
115]. These same methods have also been applied to non-trivial
calculations in quantum chromodynamics (see
e.g.
Refs. [28,
21,
12]) and in supersymmetric gauge theories (see
e.g.
Refs. [15,
16,
29,
19]). In a sense, they provide a means for obtaining collections of
quantum loop-level Feynman diagrams without direct reference to
the underlying Lagrangian or Hamiltonian. The only inputs with
this method are the
D
-dimensional tree-level scattering amplitudes. This makes the
unitarity method ideally suited for exploiting the KLT
relations.
An interesting application of this method of perturbatively
quantizing gravity is as a tool for investigating the
ultra-violet behavior of gravity field theories. Ultraviolet
properties are one of the central issues of perturbative quantum
gravity. The conventional wisdom that quantum field theories of
gravity cannot possibly be fundamental rests on the apparent
non-renormalizability of these theories. Simple power counting
arguments strongly suggest that Einstein gravity is not
renormalizable and therefore can be viewed only as a low energy
effective field theory. Indeed, explicit calculations have
established that non-supersymmetric theories of gravity with
matter generically diverge at one loop [132,
43,
42], and pure gravity diverges at two loops [66,
136]. Supersymmetric theories are better behaved with the first
potential divergence occurring at three loops [39,
81,
80]. However, no explicit calculations have as yet been performed
to directly verify the existence of the three-loop supergravity
divergences.
The method described here for quantizing gravity is well
suited for addressing the issue of the ultraviolet properties of
gravity because it relates overwhelmingly complicated
calculations in quantum gravity to much simpler (though still
complicated) ones in gauge theories. The first application was
for the case of maximally supersymmetric gravity, which is
expected to have the best ultra-violet properties of any theory
of gravity. This analysis led to the surprising result that
maximally supersymmetric gravity is less divergent [19] than previously believed based on power counting
arguments [39,
81,
80]. This lessening of the power counting degree of divergence may
be interpreted as an additional symmetry unaccounted for in the
original analysis [128]. (The results are inconsistent, however, with an earlier
suggestion [73] based on the speculated existence of an unconstrained covariant
off-shell superspace for
N
=8 supergravity, which in
D
=4 implies finiteness up to seven loops. The non-existence of
such a superspace was already noted a while ago [80].) The method also led to the explicit construction of the
two-loop divergence in eleven-dimensional supergravity [19,
40,
41,
17]. More recently, it aided the study of divergences in
type I supergravity theories [54] where it was noted that they factorize into products of gauge
theory factors.
Other applications include the construction of infinite
sequences of amplitudes in gravity theories. Given the complexity
of gravity perturbation theory, it is rather surprising that one
can obtain compact expressions for an arbitrary number of
external legs, even for restricted helicity or spin
configurations of the particles. The key for this construction is
to make use of previously known sequences in quantum
chromodynamics. At tree-level, infinite sequences of maximally
helicity violating amplitudes have been obtained by directly
using the KLT relations [10,
14] and analogous quantum chromodynamics sequences. At one loop, by
combining the KLT relations with the unitarity method, additional
infinite sequences of gravity and super-gravity amplitudes have
also been obtained [22,
23]. They are completely analogous to and rely on the previously
obtained infinite sequences of one-loop gauge theory
amplitudes [11,
15,
16]. These amplitudes turn out to be also intimately connected to
those of self-dual Yang-Mills [143,
53,
93,
92,
4,
30,
33] and gravity [108,
52,
109]. The method has also been used to explicitly compute two-loop
supergravity amplitudes [19] in dimension
D
=11, that were then used to check M-theory dualities [68].
Although the KLT relations have been exploited to obtain
non-trivial results in quantum gravity theories, a derivation of
these relations from the Einstein-Hilbert Lagrangian is lacking.
There has, however, been some progress in this regard. It turns
out that with an appropriate choice of field variables one can
separate the space-time indices appearing in the Lagrangian into
`left' and `right' classes [124,
123,
125,
26], mimicking the similar separation that occurs in string theory.
Moreover, with further field redefinitions and a non-linear gauge
choice, it is possible to arrange the off-shell three-graviton
vertex so that it is expressible in terms of a sum of squares of
Yang-Mills three-gluon vertices [26]. It might be possible to extend this more generally starting
from the formalism of Siegel [124,
123,
125], which contains a complete gravity Lagrangian with the required
factorization of space-time indices.
This review is organized as follows. In Section 2 the Feynman diagram approach to perturbative quantum gravity is outlined. The Kawai, Lewellen, and Tye relations between open and closed string tree amplitudes and their field theory limit are described in Section 3 . Applications to understanding and constructing tree-level gravity amplitudes are also described in this section. In Section 4 the implications for the Einstein-Hilbert Lagrangian are presented. The procedure for obtaining quantum loop amplitudes from gravity tree amplitudes is then given in Section 5 . The application of this method to obtain quantum gravity loop amplitudes is described in Section 6 . In Section 7 the quantum divergence properties of maximally supersymmetric supergravity obtained from this method are described. The conclusions are found in Section 8 .
There are a number of excellent sources for various subtopics
described in this review. For a recent review of the status of
quantum gravity the reader may consult the article by
Carlip [31]. The conventional Feynman diagram approach to quantum gravity
can be found in the Les Houches lectures of Veltman [138]. A review article containing an early version of the method
described here of using unitarity to construct complete loop
amplitudes is ref. [20]. Excellent reviews containing the quantum chromodynamics
amplitudes used to obtain corresponding gravity amplitudes are
the ones by Mangano and Parke [99], and by Lance Dixon [48]. These reviews also provide a good description of helicity
techniques which are extremely useful for explicitly constructing
scattering amplitude in gravity and gauge theories. Broader
textbooks describing quantum chromodynamics are Refs. [107,
141,
58]. Chapter 7 of
Superstring Theory
by Green, Schwarz, and Witten [70] contains an illuminating discussion of the relationship of
closed and open string tree amplitudes, especially at the
four-point level. A somewhat more modern description of string
theory may be found in the book by Polchinski [110,
111]. Applications of string methods to quantum field theory are
described in a recent review by Schubert [120].
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Perturbative Quantum Gravity and its Relation to Gauge
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