Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 159latter two are odd Grassmann variables, that is, they are anticommuting amongthemselves. The action then reads (for Λ = 0)S = 1 ∫d 4 x (dete n16πGµ)R + 1 ∫ ( )d 4 xɛ µνρσ ¯ψA ′µ e AA′ νD ρ ψσ A +h.c. . (5.77)2The derivative D ρ acts on spinor-valued forms (i.e. acts on their spinor indicesonly),D µ ψν A = ∂ µ ψν A + ωBµψ A ν B , (5.78)where ωBµ A denotes the spinorial version of ωnm µ (see D’Eath 1984). We remarkthat the presence of gravitinos leads to torsion,D [µ e AA′ν] = Sµν AA′A′= −4πiG ¯ψ [µ ψA ν] , (5.79)where Sµν AA′ denotes the torsion, and the last step follows from variation of theaction with respect to the connection forms; see van Nieuwenhuizen (1981). Theaction (5.77) is invariant under the following infinitesimal local symmetry transformations:1. Supersymmetry (SUSY) transformations:δe AA′µ = −i √ 8πG(ɛ A ¯ψA ′µ +¯ɛ A′ ψµ A ) , (5.80)δψµ A = D µɛ A√ , δ¯ψ µ A′µ¯ɛ A′√ ,2πG 2πG(5.81)where ɛ A and ¯ɛ A′ denote anticommuting fields.2. Local Lorentz transformations:δe AA′µ = N A B e BA′µ +δψ A µ = N A B ψB µwith N AB = N (AB) .3. Local coordinate transformations:, δ¯ψA′µ¯NA′B ′ eAB′µ , (5.82)A′ B′= ¯N B ′ ¯ψ µ , (5.83)δe AA′µ = ξ ν ∂ ν e AA′µ + e AA′ν ∂ µ ξ ν , (5.84)δψµ A = ξ ν ∂ ν ψµ A + ψν A ∂ µ ξ ν , (5.85)where ξ ν are the parameters defining the (infinitesimal) coordinate transformation.The right-hand sides are just the Lie derivatives of these fields.In analogy to Chapter 4 for GR, one can develop a Hamiltonian formalism forSUGRA. For this purpose, one splits e AA′µ into e AA′0 and e AA′a to get the spatialmetricwhere e AA′a = e n aσnAA′vector n µ readsh ab = −e AA′ ae AA′b = g ab , (5.86)in analogy to (5.74). The spinorial version of the normal