### 5.3 Isotropy: Matter Hamiltonian

We now know how the basic quantities and are quantized, and can use the operators to
construct more complicated ones. Of particular importance, as well as for cosmology, are matter
Hamiltonians, where not only the matter field but also geometry is quantized. For an isotropic geometry
and a scalar, this requires us to quantize for the kinetic term and for the potential
term. The latter can be readily defined as , but for the former we need an inverse power
of . Since has a discrete spectrum containing zero, a densely-defined inverse does not
exist.
At this point, one has to find an alternative route to the quantization of , or else one
could only conclude that there is no well-defined quantization of matter Hamiltonians as a manifestation
of the classical divergence. In the case of loop quantum cosmology it turns out, following a
general scheme of the full theory [291], that one can reformulate the classical expression in an
equivalent way such that quantization becomes possible. One possibility is to write, similar to
Equation (13),

where we use holonomies of isotropic connections and the volume . In this expression we can
insert holonomies as multiplication operators and the volume operator, and turn the Poisson bracket into a
commutator. The result

is not only a densely defined operator but even bounded, which one can easily read off from the
eigenvalues [49]
with and from Equation (49).
Rewriting a classical expression in such a manner can always be done in many equivalent ways, which in
general all lead to different operators. In the case of , we highlight the choice of the representation
in which to take the trace (understood as the fundamental representation above) and the power of
in the Poisson bracket ( above). This freedom can be parameterized by
two ambiguity parameters for the representation and for the power such
that

Following the same procedure as above, we obtain eigenvalues [55, 58]

which, for larger , can be approximated by Equation (26) (see also Figure 9). This provides the basis
for effective densities in loop cosmology as described in Section 4.

Notice that operators for the scale factor, volume or their inverse powers do not refer to observable
quantities. It can thus be dangerous, though suggestive, to view their properties as possible bounds on
curvature. The importance of operators for inverse volume comes from the fact that this appears in matter
Hamiltonians, and thus the Hamiltonian constraint of gravity. Properties of those operators
such as their boundedness or unboundedness can then determine the dynamical behavior (see,
e.g., [62]).