Kiefer C. Quantum gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 159
latter two are odd Grassmann variables, that is, they are anticommuting among
themselves. The action then reads (for Λ = 0)
S = 1 ∫
d 4 x (dete n
16πG
µ)R + 1 ∫ ( )
d 4 xɛ µνρσ ¯ψA ′
µ e AA′ νD ρ ψσ A +h.c. . (5.77)
2
The derivative D ρ acts on spinor-valued forms (i.e. acts on their spinor indices
only),
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 remark
that 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 the
action with respect to the connection forms; see van Nieuwenhuizen (1981). The
action (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 µ
with N AB = N (AB) .
3. Local coordinate transformations:
, δ¯ψ
A′
µ
¯N
A′
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 for
SUGRA. For this purpose, one splits e AA′
µ into e AA′
0 and e AA′
a to get the spatial
metric
where e AA′
a = e n aσn
AA′
vector n µ reads
h ab = −e AA′ ae AA′
b = g ab , (5.86)
in analogy to (5.74). The spinorial version of the normal