### 4.7 Isotropy: Phenomenological higher curvature corrections

In addition to the behavior of effective densities, there is a further consequence of a discrete triad
spectrum: Its conjugate cannot exist as a self-adjoint operator because in that case it would generate
arbitrary continuous translations in . Instead, only exponentials of the form can be quantized,
where the parameter is related to the precise form of the discreteness of . Corresponding operators
in a triad representation of wave functions are finite differences rather than infinitesimal differentials,
which also expresses the underlying discreteness. (This is analogous to quantum mechanics
on a circle, although in loop quantum cosmology the discreteness is not based on a simple
periodic identification but rather a more complicated compactification of the configuration
space 5.2.)
Any classical expression when quantized has to be expressed in terms of functions of , such as
, while common classical expressions, e.g., the Hamiltonian constraint (23), only
depend on directly or through powers. When occurs instead, the classical
expression is reproduced at small values of , i.e., small extrinsic curvature, but higher-order
corrections are present [52, 36, 146, 299]. This provides a further discreteness effect in loop
cosmology, which is part of the effective equations. The effect has several noteworthy properties:

- One should view the precise function as a simple placeholder for a perturbative
expansion in powers of , rather than a precise expansion with infinitely many terms. Even
if one would use the function to represent a whole perturbation expansion, which is rarely
allowed because there are additional quantum corrections of different forms dominating almost
all terms in the series, it would still be a perturbative quantum correction. This distinguishes
this type of correction from effective densities, which have a non-perturbative contribution
involving inverse powers of the Planck length. Thus these two types of corrections appear on
rather different footings.
- Seen as higher curvature corrections, higher power corrections in cannot be complete
because they do not provide higher derivative terms of . They represent only one
phenomenological effect, which originates from quantum geometry and provides some terms of
higher curvature corrections. But, for a reliable evaluation of effective equations, they need to
be combined with other corrections, whose origin we will describe in Section 6. This further
distinguishes these higher power corrections from effective densities.
- The parameter can be scale dependent, i.e., a function of the triad . As increases
and the universe expands, the discreteness realized in a quantum state in general changes; see
Section 3.7. Thus, must change with to describe the precise discreteness realized at
different volumes. More details are provided in Section 5.5 and 6.4.