Kiefer C. Quantum gravity

CONNECTION AND LOOP VARIABLES 183
h =detE a i
= i3
6 ɛijk ɛ abc
δ
δA i a
δ
δ
δA j δA k b c
.
The solution for Ψ is given by
(
)
6
Ψ Λ =exp i
GΛ S CS[A]
, (6.8)
where S CS [A] denotes the ‘Chern–Simons action’
∫
S CS [A] =
This follows after one notes that
Σ
d 3 xɛ abc tr
(G 2 A a ∂ b A c − 2 )
3 G3 A a A b A c
. (6.9)
ɛ abc
δΨ Λ
δA k c
= − 6i
Λ F kab ,
that is, the term in parentheses in (6.7) by itself annihilates the state Ψ Λ .Due
to the topological nature of the Chern–Simons action, the state (6.8) is both
gauge- and diffeomorphism-invariant. In contrast to the states mentioned above
(for vanishing cosmological constant), (6.8) is not annihilated by the operator
corresponding to √ h and may thus have physical content. The Chern–Simons
action is also important for GR in 2+1 dimensions; cf. Section 8.1.3.
A state of the form (6.8) is also known from Yang–Mills theory. The state
(
Ψ g =exp − 1 )
2g 2 S CS[A]
is an eigenstate of the Yang–Mills Hamiltonian
H YM = 1 ∫
)
d 3 x tr
(−g 2 2 δ2
2
δA 2 + B2 a
a g 2 ,
where B a =(1/2)ɛ abc F bc , with eigenvalue zero (Loos 1969, Witten 2003). However,
this state has unpleasant properties (e.g. it is not normalizable), which
renders its physical significance dubious. It is definitely not the ground state of
Yang–Mills theory. The same reservation may apply to (6.8).
Since the ‘real’ quantum Hamiltonian constraint is not given by (6.6), see
Section 6.3, we shall not discuss further this type of solutions. What can generally
be said about the connection representation? Since (6.4) guarantees that
Ψ[A] =Ψ[A g ], where g ∈ SU(2), the configuration space after the implementation
of the Gauss constraints is actually given by A/G, whereA denotes the
space of connections and G the local SU(2) gauge group. Because the remaining