Kiefer C. Quantum gravity

276 QUANTUM COSMOLOGY
holonomies into multiplication operators and the Poisson bracket into a commutator,
one indeed finds a densely defined bounded operator. Its spectrum is
given by
where
( 1
ˆd(p)|µ〉 =
2πβlP
2 ( √ V µ+1 − √ 6
V µ−1 ))
|µ〉 , (8.90)
( 4πβl
V µ ≡|p µ | 3/2 2 3/2
= P
|µ|)
; (8.91)
3
cf. (8.87). There are, of course, ambiguities in defining such an operator. A more
general class of functions, d(p) j,l , is obtained by introducing the parameters
j ∈ 1 2N and l, 0< l < 1, where j arises from the freedom to use different
representations of SU(2), and l from the classical freedom in writing V −1 =
(V l−1 ) 1/(1−l) . This, then, leads to an effective matter Hamiltonian
H m (eff) = 1 (
)
d(p) j,l p 2 φ
2
+ |p|3/2 V(φ) . (8.92)
The new term d(p) j,l appearing here has various consequences. It gives rise to
modified densities in the effective cosmological equations (Friedmann equation,
Raychaudhuri equation) and to a modified ‘damping term’ in the effective Klein–
Gordon equation for the scalar field. This leads to qualitative changes at small
a, sinced does not go to infinity as a approaches zero. One finds an effective
repulsion which can potentially prevent the big bang. Effectively, this corresponds
to the presence of a Planck potential as has been introduced by Conradi and Zeh
(1991); cf. Section 8.3.5. The new term can also enhance the expansion of the
universe at small scales, providing a possible mechanism for inflation from pure
quantum-gravitational effects. If one takes into account inhomogeneities, it is also
imaginable to find observable effects in the anisotropy spectrum of the cosmic
microwave background.
What about the gravitational part of the Hamiltonian constraint operator?
It contains the term c 2 and thus is not an almost-periodic function. One may
use instead, for example, a function proportional to δ −2 sin 2 δc, whereδ labels a
quantization ambiguity (there are many more such ambiguities). This reproduces
c 2 in the limit for small c, which is why the classical limit follows only for small c.
The situation here is thus much more involved than for the matter Hamiltonian
where one just has to address modified densities. Expanding the general solution
of the full constraint in terms of volume eigenstates,
|ψ〉 = ∑ µ
ψ µ |µ〉 , (8.93)
one arrives at the following difference equation for the coefficients of this expansion,
9
9 Restriction is being made to a flat Friedmann universe.