Kiefer C. Quantum gravity

274 QUANTUM COSMOLOGY
where in the case of the scalar field,
H m = 1 (
)
|p| −3/2 p 2 φ + |p| 3/2 V(φ) . (8.82)
2
The starting point is now set for the quantization.
8.4.2 Quantization
In accordance with the spirit of full loop quantum gravity we shall not quantize
c directly, but the related holonomy. For this purpose one has to consider special
holonomies constructed from isotropic connections. This introduces a parameter
length µ (chosen to be dimensionless) into the formalism, which captures
information about the edges and the spin labels of the spin network (µ is not
a physical length). That such a parameter appears is connected with the fact
that the reduced model still possesses a background structure, in spite of the
background independence of the full theory. This is because the spatial metric
is unique up to the scale factor, which thus introduces a conformal space as a
background.
Instead of turning the classical Poisson-bracket relation (8.80) into a commutator
acting on the standard Hilbert space, one makes use here of the Bohr
compactification of R. 7 One starts with the algebra of almost-periodic functions
of the form
f(c) = ∑ f µ e iµc/2 , (8.83)
µ
where the sum runs over a countable subset of R. The reduction procedure from
the holonomies of the full theory just leads to the factor e iµc/2 . This algebra is
isomorphic to the Bohr compactification of R, ¯R Bohr , which is a compact group
and contains R densely. It can be obtained as the dual group of the real line
endowed with the discrete topology. Representations of ¯R Bohr are labelled by
real numbers µ and are given by
ρ µ : ¯R Bohr → C , c ↦→ e iµc .
If we had decided to quantize c directly, we would have chosen the space of square
integrable functions, L 2 (Rdc), as the Hilbert space. Here, instead, we choose the
space L 2 (¯R Bohr , dµ(c)), where dµ(c) is the Haar measure, defined by
∫
∫
1 T
dµ(c) f(c) = lim dc f(c) .
¯R T →∞
Bohr
2T −T
The basis states are chosen to be
〈c|µ〉 =e iµc/2 , (8.84)
7 This concept is named after Harald August Bohr (1887–1951), the younger brother of Niels
Bohr, who contributed much to the theory of almost-periodic functions.