Our cosmology, however, is not close to a harmonic oscillator, which makes this solvable system
unsuitable as zeroth order for a perturbative analysis. There is, however, a solvable system available for
quantum cosmology: a spatially-flat isotropic model sourced by a free scalar [70]. The Friedmann equation,
written in isotropic Ashtekar variables, is
More remarkably, this solvability behavior even extends to loop quantum cosmology. In this case, there
is no operator for and thus one has to refer to non-canonical variables involving holonomies or the
exponentials
. For this reason, the Hamiltonian operator as constructed in Section 5.4 receives
higher-order corrections and takes the form
in a certain factor ordering. Since this is no longer
quadratic in canonical variables and we are forced to use a non-canonical basic operator algebra, it seems
difficult to relate this to the solvable free scalar cosmology. However, there is a change of basic
variables, which provides an exact solvable model even for loop quantum cosmology: using
and
(in this ordering) as basic operators, we obtain a closed algebra of basic
variables together with the Hamiltonian operator
. This provides a solvable
model for loop quantum cosmology as the starting point for a systematic derivation of effective
equations.
While we keep calling the geometrical variable , one should note that the solvable model is insensitive
to the precise form of refinement used in the underlying quantization. Instead of starting with the canonical
pair
, one can use
This model also provides an instructive example for the role of physical inner-product issues in effective
theory. By our definition of , we will have to deal with complex-valued expectation values
of our basic operators, in particular
. After solving the effective equations, we have to
impose reality conditions, which, due to the form of
, are not linear but derived from the
quadratic relation
. Taking an expectation value of this relation provides the reality
condition to be imposed, which not only involves expectation values but also fluctuations. Thus,
fluctuations can in principle play a significant role for admissible solutions, but this does not
happen for initially semiclassical states. Physical solutions for expectation values and higher
moments can then be found more conveniently than it would be for states in a Hilbert-space
representation.
Thanks to the solvability of this model it is clear that it does not receive quantum corrections from
backreaction. Thus, it is strictly justified in this model to use higher-power corrections as they are
implemented by . But inverse volume corrections have been neglected by construction, since
they would alter the Hamiltonian and destroy the solvability. In a complete treatment, such
corrections would have to arise, too, and with them quantum backreaction would play a role in the
absence of solvability. Nevertheless, for many applications, one can argue that the latter types of
corrections are small in an isotropic model because one usually requires a large matter content and
thus large
, which enhances higher-power corrections. However, the situation is not fully
clarified since the original versions, where the magnitudes of higher power and inverse volume
corrections were compared numerically [26
, 27
, 28], were based on formulations of isotropic loop
quantum cosmology, where inverse volume corrections are artificially suppressed due to the form
of the symmetry reduction [75]. Taking this into account makes inverse volume corrections
important for a larger parameter range. They are also much more prominent in inhomogeneous
models.
Independent of this question, quantum backreaction inevitably results in the presence of matter
interactions, anisotropies or inhomogeneities. In those cases, taking higher-power corrections, as in the free
isotropic model, is not as reliable, and a detailed analysis is required, which so far has been provided only
for a perturbative treatment of scalar potentials [87]. As anticipated, the resulting effective equations do
not seem to allow an adiabatic approximation since quantum variables in the underlying solvable models are
not constant. One thus has to deal with higher-dimensional effective systems, where some of the quantum
variables remain as independent variables. Effective equations have been derived to first order in the
perturbative potential and in a semiclassical expansion, but are rather involved and have not been analyzed
much yet.
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