Kiefer C. Quantum gravity

PATH-INTEGRAL QUANTIZATION 41
canonical ensemble one has Z =tre −βH ). Fourth, the Euclidean formulation is
convenient for lattice gauge theory where one considers
∫
Z[U] = DU e −SW[U] , (2.70)
with U denoting the lattice gauge fields defined on the links and S W the Wilson
action, see also Chapter 6. The justification of performing Wick rotations in
quantum field theory relies on the fact that Euclidean Green functions can be
analytically continued back to real time while preserving their pole structure; cf.
Osterwalder and Schrader (1975).
The quantum-gravitational path integral, first formulated by Misner (1957),
would be of the form
∫
Z[g] = Dg µν (x) e iS[gµν (x)] , (2.71)
where the sum runs over all metrics on a four-dimensional manifold M divided
by the diffeomorphism group DiffM (see below). One might expect that an
additional sum has to be performed over all topologies, but this is a contentious
issue. As we shall see in Section 5.3.4, the path integral (2.71) behaves more like
an energy Green function instead of a propagator. The reason is the absence of
an external time as already emphasized above.
Needless to say that (2.71) is of a tremendously complicated nature, both
technically and conceptually. One might therefore try, for the reasons stated
above, to perform a Wick rotation to the Euclidean regime. This leads, however,
to problems which are not present in ordinary quantum field theory. First, not
every Euclidean metric (in fact, only very few) possesses a Lorentzian section,
that is, leads to a signature (–,+,+,+) upon τ → it. Such a section exists only for
metrics with special symmetries. (The Wick rotation is not a diffeomorphisminvariant
procedure.) Second, a sum over topologies cannot be performed even in
principle because four-manifolds are not classifiable (Geroch and Hartle 1986). 13
The third, and perhaps most severe, problem is the fact that the Euclidean gravitational
action is not bounded from below. Performing the same Wick rotation
as above (in order to be consistent with the matter part), one finds from (1.1)
for the Euclidean action, the expression
S E [g] =− 1
16πG
∫
M
d 4 x √ g (R − 2Λ) − 1
8πG
∫
∂M
d 3 x √ hK . (2.72)
To see the unboundedness of this action, consider a conformal transformation of
the metric, g µν → ˜g µν =Ω 2 g µν . This yields (Gibbons et al. 1978; Hawking 1979)
S E [˜g] =− 1 ∫
d 4 x √ g(Ω 2 R+6Ω ;µ Ω ;ν g µν −2ΛΩ 4 )− 1 ∫
d 3 x √ hΩ 2 K.
16πG M
8πG ∂M
(2.73)
13 Still, it is possible that topology change in quantum gravity is required for reasons of
consistency; cf. Sorkin (1997).