### 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 are , i.e. .
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” indices never contract with
the “right” 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.