Kiefer C. Quantum gravity

182 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPS
δΨ
Ĝ i Ψ=0 −→ D a
δA i =0. (6.4)
a
It expresses the invariance of the wave functional with respect to infinitesimal
gauge transformations of the connection. The diffeomorphism constraints (4.127)
become
ˆ˜H a Ψ=0 −→ Fab
i δΨ
=0. (6.5)
Similar to the classical case, it expresses the invariance of the wave functional
under a combination of infinitesimal diffeomorphism and gauge transformations;
cf. Section 4.3.2.
The Hamiltonian constraint (4.126) cannot be treated directly in this way
because the Γ i a-terms contain the tetrad in a complicated non-linear fashion.
This would lead to similar problems as with the Wheeler–DeWitt equation discussed
in Chapter 5, preventing to find any solutions. In Section 6.3, we shall see
how a direct treatment of the quantum Hamiltonian constraint can at least be
attempted. Here, we remark only that (4.126) is easy to handle only for the value
β = i in the Lorentzian case or for the value β = 1 in the Euclidean case. In the
first case, the problem arises that the resulting formalism uses complex variables
and that one has to impose ‘reality conditions’ at an appropriate stage (which
nobody has succeeded in implementing). In the second case, the variables are
real but one deals with the unphysical Euclidean case. Nevertheless, for these
particular values of β, the second term in (4.126) vanishes, and the quantum
Hamiltonian constraint for Λ = 0 would simply read
δA i b
ɛ ijk δ 2 Ψ
F kab
δA i a δAj b
=0. (6.6)
Note that in (6.4)–(6.6), a ‘naive’ factor ordering has been chosen: all derivatives
are put to the right. Formal solutions to these equations have been found; see
for example, Brügmann (1994). Some solutions have been expressed in terms
of knot invariants. 1 Many of these solutions are annihilated by the operator
corresponding to √ h and may therefore be devoid of physical meaning, since
matter fields and the cosmological term couple to √ h.
In the case of vacuum gravity with Λ ≠ 0, an exact formal solution in the
connection representation was found by Kodama (1990). Using a factor ordering
different from (6.6), the Hamiltonian constraint reads for β =1
δ
δ
ɛ ijk
δA i a δA j b
(
F kab − iΛ
6 ɛ abc
δ
δA k c
)
Ψ[A] =0. (6.7)
We note that the second term in parentheses comes from the determinant of the
three-metric,
1 A knot invariant is a functional on the space of loops which assigns to loops in the same
knot class the same number.