Kiefer C. Quantum gravity

128 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
where τ i =iσ i /2withσ i as the Pauli matrices. Under an SU(2)-transformation
g, one then has
E a → gE a g −1 , A a → g(A a + ∂ a )g −1 . (4.125)
The next step is to rewrite the genuine gravitational constraints (4.69) and
(4.70) in terms of the new variables. Introducing ˜H ⊥ = −8πGβ 2 H g ⊥
(plus terms
proportional to the Gauss constraints) and ˜H a = −8πGβHa g (plus terms proportional
to the Gauss constraints), the new form of the constraints is
˜H ⊥ = − σ 2
ɛ ijk F abk
√
|detE
a
i | Ea i E b j
+ β2 σ − 1
β 2√ |detE a i |Ea [i Eb j] (GAi a − Γ i a)(GA j b − Γj b ) ≈ 0 , (4.126)
and
˜H a = F i abE b i ≈ 0 . (4.127)
Equation (4.127) has the form of the cyclic identity (4.117) with Rab i replaced
by Fab i . If applied on Ai a , the constraint (4.127) yields a transformation that can
be written as a sum of a gauge transformation and a pure diffeomorphism.
As in Section 4.1, one has σ = −1 for the Lorentzian and σ =1forthe
Euclidean case. One recognizes that (4.126) can be considerably simplified by
choosing β =i(orβ = −i) for the Lorentzian and β =1(orβ = −1) for the
Euclidean case, because then the second term vanishes. The potential term has
disappeared, leading to a situation resembling the strong-coupling limit discussed
at the end of Section 4.2.3 (see also section III.4 of Ashtekar (1988)). In fact, the
original choice was β = i for the relevant Lorentzian case. Then,
√
2 |detEi a| ˜H ⊥ = ɛ ijk F abk Ei a Ej b ≈ 0 .
This leads to a complex connection A i a, see (4.119), and makes it necessary to
implement reality conditions in order to recover GR—a task that seems impossible
to achieve in the quantum theory. However, this choice is geometrically
preferred (Rovelli 1991a); A i a is then the three-dimensional projection of a fourdimensional
self-dual spin connection A IJ
µ ,
A IJ
µ = ωIJ µ − 1 2 iɛIJ MN ωMN µ . (4.128)
(It turns out that the curvature Fµν IJ of the self-dual connection is the selfdual
part of the Riemann curvature.) To avoid the problems with the reality
conditions, however, it is better to work with real variables. Barbero (1995) has