Feynman Path Integral Formulation

1.4 Feynman Rules 11One can exploit the freedom under general coordinate transformations x ′μ =f (x μ ) to impose a suitable coordinate condition, such asΓ λ ≡ g μν Γ λμν = 0 , (1.62)which is seen to be equivalent to the following gauge condition on the metric∂ μ ( √ gg μν )=0 , (1.63)and therefore equivalent, in the weak field limit, to the harmonic gauge conditionintroduced previously in Eq. (1.12).1.4 Feynman RulesThe Feynman rules represent the standard way to do perturbative calculations inquantum gravity. To this end one first expands again the action out in powers of thefield h μν and separates out the quadratic part, which gives the graviton propagator,from the rest of the Lagrangian which gives the O(h 3 ),O(h 4 )... vertices. To definethe graviton propagator one also requires the addition of a gauge fixing term and theassociated Faddeev-Popov ghost contribution (Feynman, 1962; Faddeev and Popov,1967). Since the diagrammatic calculations are performed using dimensional regularization,one first needs to define the theory in d dimensions; at the end of thecalculations one will be interested in the limit d → 4.So first one expands around the d-dimensional flat Minkowski space-time metric,with signature given by η μν = diag(−1,1,1,1,...). The Einstein-Hilbert action ind dimensions is given by a generalization of Eq. (1.35)I E = 1 ∫d d x √ g(x)R(x) , (1.64)16πGwith again g(x) =−det(g μν ) and R the scalar curvature; in the following it willbe assumed, at least initially, that the bare cosmological constant λ 0 is zero. Thesimplest form of matter coupled in an invariant way to gravity is a set of spinlessscalar particles of mass m, with action∫I m = 1 2d d x √ g(x) [ −g μν (x)∂ μ φ(x)∂ ν φ(x) − m 2 φ 2 (x) ] . (1.65)In Feynman diagram perturbation theory the metric g μν (x) is expanded around theflat metric η μν , by writing againg μν (x) =η μν + √ 16πGh μν (x) . (1.66)The quadratic part of the Lagrangian [see Eq. (1.7)] is then