The problem of non-renormalizability of quantum gravity does
not mean that quantum mechanics is incompatible with gravity,
only that quantum gravity should be treated as an effective field
theory [140,
64
,
51
,
84
,
100
] for energies well below the Planck scale of
GeV (which is, of course, many orders of magnitude beyond the
reach of any conceivable experiment). In an effective field
theory, as one computes higher loop orders, new and usually
unknown couplings need to be introduced to absorb the
divergences. Generally, these new couplings are suppressed at low
energies by ratios of energy to the fundamental high energy
scale, but at sufficiently high energies the theory loses its
predictive power. In quantum gravity this happens at the Planck
scale.
Quantum gravity based on the Feynman diagram expansion allows
for a direct investigation of the non-renormalizability issue.
For a theory of pure gravity with no matter, amazingly, the
one-loop divergences cancel, as demonstrated by 't Hooft and
Veltman
[132
]. Unfortunately, this result is ``accidental'', since it does
not hold generically when matter is added to the theory or when
the number of loops is increased. Explicit calculations have
shown 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]. The two-loop calculations were performed using various
improvements to the Feynman rules such as the background field
method [130
,
46
,
1
].
Supersymmetric theories of gravity are known to have less
severe divergences. In particular, in any four-dimensional
supergravity theory, supersymmetry Ward identities [72,
71] forbid all possible one-loop [74] and two-loop [101,
134] divergences. There is a candidate divergence at three loops for
all supergravities including the maximally extended
N
=8 version [39,
81,
83,
80
]. However, no explicit three-loop (super)gravity calculations
have been performed to confirm the divergence. In principle it is
possible that the coefficient of a potential divergence obtained
by power counting can vanish, especially if the full symmetry of
the theory is taken into account. As described in Section
7, this is precisely what does appear to happen [19
,
128
] in the case of maximally supersymmetric supergravity.
The reason no direct calculation of the three-loop
supergravity divergences has been performed is the overwhelming
technical difficulties associated with multi-loop gravity Feynman
diagrams. In multi-loop calculations the number of algebraic
terms proliferates rapidly beyond the point where computations
are practical. As a particularly striking example, consider the
five-loop diagram in Fig.
4, which, as noted in Section
7, is of interest for ultraviolet divergences in maximal
N
=8 supergravity in
D
=4. In the standard de Donder gauge this diagram contains
twelve vertices, each of the order of a hundred terms, and
sixteen graviton propagators, each with three terms, for a total
of roughly
terms, even before having evaluated any integrals. This is
obviously well beyond what can be implemented on any computer.
The standard methods for simplifying diagrams, such as
background-field gauges and superspace, are unfortunately
insufficient to reduce the problem to anything close to
manageable levels. The alternative of using string-based methods
that have proven to be useful at one-loop and in certain two-loop
calculations [27
,
25
,
119,
55
,
56,
57,
120] also does not as yet provide a practical means for performing
multi-loop scattering amplitude calculations [112,
113,
47,
114,
63], especially in gravity theories.
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Perturbative Quantum Gravity and its Relation to Gauge
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |