Kiefer C. Quantum gravity

158 QUANTUM GEOMETRODYNAMICS
where D ab is the covariant derivative with respect to the DeWitt metric,
D ab ≡ DΨ .
Dh ab
The big open problem is of course to see whether this result survives the transition
to the field-theoretic case, that is, whether no anomalies are present after a
consistent regularization has been performed.
5.3.6 Canonical quantum supergravity
We have seen in Section 2.3 that the quantization of supergravity instead of GR
exhibits interesting features. Thus, it seems worthwhile to discuss the canonical
quantization of supergravity (SUGRA). This will be briefly reviewed here; more
details can be found in D’Eath (1984, 1996) and Moniz (1996). The situation in
three space–time dimensions is addressed in Nicolai and Matschull (1993). Here,
we will only consider the case of N = 1 SUGRA given by the action (2.142).
The classical canonical formalism was developed by Fradkin and Vasiliev
(1977), Pilati (1977), and Teitelboim (1977). Working again with tetrads (cf.
Section 1.1), one has
g µν = η nm e n µ em ν . (5.73)
For the quantization it is more convenient to use two-component spinors according
to
e AA′
µ = e n µσn AA′ , (5.74)
where A runs from 1 to 2, A ′ from 1 ′ to 2 ′ , and the van der Waerden symbols
denote the components of the matrices
σ AA′
n
σ 0 = − 1 √
2
I , σ a = 1 √
2
× Pauli matrix , (5.75)
with raising and lowering of indices by ɛ AB , ɛ AB , ɛ A′ B ′ , ɛ A′ B ′, which are all given
in matrix form by ( )
0 1
,
−1 0
see for example, Wess and Bagger (1992), Sexl and Urbantke (2001) for more
details on this formalism. The inverse of (5.74) is given by
e n µ = −σn AA ′eAA′ µ , (5.76)
where σAA n is obtained from ′
σAA′ n by raising and lowering indices. One can go
from tensors to spinors via e AA′
µ and from spinors to tensors via e µ AA ′.
One can now rewrite the action (2.142) in two-component language. Instead
of e n µ (vierbein) and ψµ α (gravitino), one works with the spinor-valued one-form
e AA′
µ and the spinor-valued one-form ψ A µ
plus its Hermitian conjugate ¯ψ
A′
µ .The