Kiefer C. Quantum gravity

164 QUANTUM GEOMETRODYNAMICS
H AA ′Ψ = 0 must hold automatically. This could lead to considerable simplification
because (5.115)–(5.118) involve at most first-order derivatives.
Canonical quantum SUGRA can be applied, for example, in the context of
quantum cosmology (Section 8.1). One can also study some general properties.
One of them is the fact that pure bosonic states cannot exist (see e.g. Moniz 1996
for discussion and references). This can easily be shown. Considering a bosonic
state Ψ[e AA′
a ], one has δΨ/δψ A a
= 0 and recognizes immediately that (5.118) is
solved, that is, S A Ψ = 0. Assuming that Ψ[e AA′
a ] is Lorentz invariant, that is,
that the Lorentz constraints are already fulfilled, it is clear that a state with
¯S A′ Ψ = 0 satisfies all constraints. However, such a state cannot exist. This can
be seen as follows (Carroll et al. 1994). One multiplies ¯S A′ Ψ=0by[Ψ] −1 and
integrates over space with an arbitrary spinorial test function ¯ɛ A′ (x) toget
∫
(
I ≡ d 3 x ¯ɛ A′ (x) ɛ abc e AA′ a s D b ψc
A
+4πGψ A a
δ(ln Ψ)
δe AA′ a
)
=0. (5.119)
This must hold for all fields and all ¯ɛ A′ (x). If one now replaces ¯ɛ A′ (x) by
¯ɛ A′ (x)exp(−φ(x)) and ψa A(x) byψA a (x)exp(φ(x)), where φ(x) is some arbitrary
function, the second term in (5.119) cancels out in the difference ∆I between
the old and the new integral, and one is left with
∫
∆I = − d 3 xɛ abc e AA′ a¯ɛ A′ ψc A ∂ bφ =0, (5.120)
which is independent of the state Ψ. It is obvious that one can choose the fields as
well as ¯ɛ A′ (x) andφ(x) in such a way that the integral is non-vanishing, leading
to a contradiction. Therefore, no physical bosonic states can exist, and a solution
of the quantum constraints can be represented in the form
Ψ[e AA′
a
(x),ψ A a (x)] = ∞ ∑
n=1
Ψ (n) [e AA′
a (x),ψa A (x)] , (5.121)
where the expansion is into states with fermion number n. In fact, one can
show with similar arguments that any solution of the quantum constraints must
have infinite fermion number. An explicit solution of a peculiar type was found
(without any regularization) by Csordás and Graham (1995).
5.4 The semiclassical approximation
5.4.1 Analogies from quantum mechanics
The semiclassical approximation to quantum geometrodynamics discussed here
uses, in fact, a mixture of two different approximation schemes. The full system
is divided into two parts with very different scales. One part is called the
‘heavy part’—for it the standard semiclassical (WKB) approximation is used.
The other part is called the ‘light part’—it is treated fully quantum and follows
adiabatically the dynamics of the heavy part. A mixed scheme of this kind is