2.3 Gravity and gauge theory Feynman rules

The heuristic relation (1View Equation) suggests a possible way to deal with multi-loop diagrams such as the one in Figure 4View Image 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 5View Image:
μ ν ,μ ν,μ ν μ μ μ νν ν Gf1act1ori2zin2g 33(k1,k2,k3) ∼ V Y1M 2 3(k1, k2,k3)VY1M 2 3(k1, k2,k3), (4 )
where the two indices of each graviton labeled by i = 1,2,3 are μiνi, i.e. hμiνi.
View Image

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


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