3 The Kawai-Lewellen-Tye Relations2 Outline of the Traditional 2.2 Divergences in quantum gravity

2.3 Gravity and gauge theory Feynman rules

The heuristic relation (1Popup Equation) suggests a possible way to deal with multi-loop diagrams such as the one in Fig.  4 by somehow factorizing gravity amplitudes into products of gauge theory ones. Since gauge theory Feynman rules are inherently much simpler than gravity Feynman rules, it clearly would be advantageous to re-express gravity perturbative expansions in terms of gauge theory ones. As a first step, one might, for example, attempt to express the three-graviton vertex as a product of two Yang-Mills vertices, as depicted in Fig.  5 :

  equation158

where the two indices of each graviton labeled by i = 1, 2, 3 are tex2html_wrap_inline2428, i.e. tex2html_wrap_inline2454 .

  

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Figure 5: String theory suggests that the three-graviton vertex can be expressed in terms of products of three-gluon vertices.

Such relations, however, do not hold in any of the standard formulations of gravity. For example, the three-vertex in the standard de Donder gauge (3Popup Equation) contains traces over gravitons, i.e. a contraction of indices of a single graviton. For physical gravitons the traces vanish, but for gravitons appearing inside Feynman diagrams it is in general crucial to keep such terms. A necessary condition for obtaining a factorizing three-graviton vertex (4Popup Equation) is that the ``left'' tex2html_wrap_inline2418 indices never contract with the ``right'' tex2html_wrap_inline2420 indices. This is clearly violated by the three-vertex in Eq. (3Popup Equation). Indeed, the standard formulations of quantum gravity generate a plethora of terms that violate the heuristic relation (1Popup Equation).

In Section  4 the question of how one rearranges the Einstein action to be compatible with string theory intuition is returned to. However, in order to give a precise meaning to the heuristic formula (1Popup Equation) and to demonstrate that scattering amplitudes in gravity theories can indeed be obtained from standard gauge theory ones, a completely different approach from the standard Lagrangian or Hamiltonian ones is required. This different approach is described in the next section.



3 The Kawai-Lewellen-Tye Relations2 Outline of the Traditional 2.2 Divergences in quantum gravity

image Perturbative Quantum Gravity and its Relation to Gauge Theory
Zvi Bern
http://www.livingreviews.org/lrr-2002-5
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