Feynman Path Integral Formulation

12 1 Continuum FormulationL = − 1 4 ∂ μh αβ ∂ μ h αβ + 1 8 (∂ μh α α) 2 + 1 2 C2 μ + 1 2 κ h μν T μν + L gf + ... (1.67)where the dots indicate terms that are either total derivatives, or higher order in h.Asuitable gauge fixing term C μ is given byC μ ≡ ∂ α h α μ − 1 2 ∂ μh α α . (1.68)Without such a term the quadratic part of the gravitational Lagrangian of Eq. (1.7)would contain a zero mode h μν ∼ ∂ μ ε ν + ∂ ν ε μ , due to the gauge invariance ofEq. (1.11), which would make the graviton propagator ill defined.The gauge fixing contribution L gf itself will be written as the sum of two terms,L gf = − 1 2 C2 μ + L ghost , (1.69)with the first term engineered so as to conveniently cancel the + 1 2 C2 μ in Eq. (1.67)and thus give a well defined graviton propagator. Note incidentally that this gaugeis not the harmonic gauge condition of Eq. (1.12), and is usually referred to insteadas the DeDonder gauge. The second term is determined as usual from the variationof the gauge condition under an infinitesimal gauge transformation of the type inEq. (1.11)δC μ = ∂ 2 ε μ + O(ε 2 ) , (1.70)which leads to the lowest order ghost LagrangianL ghost = −∂ μ ¯η α ∂ μ η α + O(h 2 ) , (1.71)where η α is the spin-one anticommuting ghost field, with propagatorD (η)μν (k)= η μνk 2 . (1.72)In this gauge the graviton propagator is finally determined from the survivingquadratic part of the pure gravity Lagrangian, which isL 0 = − 1 4 ∂ μh αβ ∂ μ h αβ + 1 8 (∂ μh α α) 2 . (1.73)The latter can be conveniently re-written in terms of a matrix VwithL 0 = − 1 2 ∂ λ h αβ V αβμν ∂ λ h μν (1.74)V αβμν = 1 2 η αμη βν − 1 4 η αβη μν . (1.75)The matrix V can easily be inverted, for example by re-labeling rows and columnsvia the correspondence11 → 1, 22 → 2, 33 → 3,...12 → 5, 13 → 6, 14 → 7 ... (1.76)