Kiefer C. Quantum gravity

190 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPS
Ë
« ¾
È
« ½
«´×µ « ½ « ¾
Fig. 6.4. Example of an intersection of a link α with a surface S.
∂x b (⃗σ) ∂x c (⃗σ)
n a (⃗σ) =ɛ abc
∂σ 1 ∂σ 2 (6.25)
is the usual vectorial hypersurface element. The operator defined in (6.24) corresponds
to the flux of Ei
a through a two-dimensional surface. The canonical
variables of loop quantum gravity are thus the holonomy and this flux. They
obey the commutator relation
[Û[A, α], Ê i [S]]
=ilPβι(α, 2 S)U[α 1 ,A]τ i U[α 2 ,A] ,
where ι(α, S) =±1, 0 is the ‘intersection number’ that depends on the orientation
of α and S. We assume here the presence of only one intersection of α with S;
cf. Fig. 6.4 where α 1 refers to the part of α below S and α 2 to its part above
S. The intersection number vanishes if no intersection takes place. We want to
emphasize that in the following procedure, the diffeomorphism constraint is not
yet implemented.
We now want to calculate the action of Êi(S) on spin-network states Ψ S [A].
For this one needs its action on holonomies U[A, α]. This was calculated in detail
in Lewandowski et al. (1993) by using the differential equation (4.142) for the
holonomy. In the simplest case of one intersection of α with S (cf. Fig. 6.4), one
obtains
δU[A, α]
δA i a [x(⃗σ)] =
( [ ∫
])
δ
δA i a [x(⃗σ)] P exp G ds ˙α a A i a (α(s))τ i
α
∫
= G ds ˙α a δ (3) (x(⃗σ),α(s))U[A, α 1 ]τ i U[A, α 2 ] . (6.26)
α
One can now act with the operator Êi[S], (6.24), on U[A, α]. This yields