Gravity and gauge theory Feynman rules

2.3 Gravity and gauge theory Feynman rules

The heuristic relation (1

) suggests a possible way to deal with multi-loop diagrams such as the one in Figure 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 Figure 5

where the two indices of each graviton labeled by $i = 1,2,3$ are $μiνi$ , i.e. $hμiνi$ .

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 (3 ) 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 (4 ) is that the “left” $μi$ indices never contract with the “right” $νi$ indices. This is clearly violated by the three-vertex in Eq. (3 ). Indeed, the standard formulations of quantum gravity generate a plethora of terms that violate the heuristic relation (1 ).

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 (1 ) 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.