Quantum Gravity

50 COVARIANT APPROACHES TO QUANTUM GRAVITYFor the gravitational field, we have the following Lagrangian (for vanishingcosmological constant), see ’t Hooft and Veltman (1974):L g =√ −gR16πG = √ ( )¯R−ḡ16πG + L(1) g + L (2)g + ..., (2.86)where the ‘barred’ quantities refer to the background metric, see (2.76). We havefor L (1)g the expressionL (1)g = f µν√32πG(ḡµν¯R − 2 ¯Rµν ) . (2.87)This vanishes if the background is a solution of the (vacuum) Einstein equations.The expression for L (2)g readsL (2)g= 1 2 f µν;αf µν;α − 1 2 f ;αf ;α + f ;α f αβ ;β − f µβ;αf µα;β+ ¯R( 1 2 f µνf µν − 1 4 f 2 )+ ¯R µν ( ff µν − 2f αµ f να). (2.88)The first line corresponds to the Fierz–Pauli Lagrangian (2.20), while the secondline describes the interaction with the background (not present in (2.20) becausethere the background was flat); we recall that f ≡ f µ µ . For the gauge-fixing partL gf ,onechoosesL gf = √ −ḡ(f ;νµν − 1 2 f ;µ)(f µρ ;ρ − 1 2 f ;µ ) . (2.89)This condition corresponds to the ‘harmonic gauge condition’ (2.3) and turnsout to be a convenient choice. For the ghost part, one findsσL ghost = √ −ḡη ∗µ ( ηµ;σ; − ¯R µν η ν) . (2.90)The first term in the brackets is the covariant d’Alembertian, ✷η µ .Thefullaction in the path integral (2.84) then reads, with the background metric obeyingEinstein’s equations,∫S tot = d 4 x √ ( ¯R−ḡ16πG − 1 2 f µνD µναβ f αβ + f µν [ḡµν√ ¯R ]− 2 ¯Rµν32πG+η ∗µ (ḡ µν ✷ − ¯R)µν )η ν + O(f 3 ) , (2.91)where D µναβ is a shorthand for the terms occurring in (2.88) and (2.89). Thedesired Feynman diagrams can then be obtained from this action. The operatorD µναβ is—in contrast to the original action without gauge fixing—invertibleand defines both propagator and vertices (interaction with the background field).The explicit expressions are complicated, see ’t Hooft and Veltman (1974) andDonoghue (1994). They simplify considerably in the case of a flat background