### 5.9 Homogeneity: Dynamics

The Hamiltonian constraint can be constructed in the standard manner and its matrix elements can be
computed explicitly thanks to the simple volume spectrum after diagonalization. There are
holonomy operators for all three directions, and so in the triad representation the constraint
equation becomes a partial difference equation for in three independent variables. Its
(lengthy) form can be found in [56] for the Bianchi I model and in [82] for all other class A
models.
Simpler cases arise in locally rotationally-symmetric (LRS) models, where a non-trivial isotropy
subgroup is assumed. Here, only two independent parameters and remain, where only one, e.g.,
can take both signs if discrete gauge freedom is fixed, and the vacuum difference equation is, e.g., for
Bianchi I,

from the non-symmetric constraint and
from the symmetric version (see also [14] and [75] for the correction of a typo). This leads to a
reduction between fully anisotropic and isotropic models with only two independent variables, and
provides a class of interesting systems to analyze effects of anisotropies. In [88], for instance,
anisotropies are treated perturbatively around isotropy as a model for the more complicated of
inhomogeneities. This shows that non-perturbative equations are essential for the singularity
issue, while perturbation theory is sufficient to analyze the dynamical behavior of semiclassical
states.
Also here, can be scale dependent, resulting in non-equidistant difference equations, which in the
case of several independent variables are only rarely transformable to equidistant form [75]. A special case
where this is possible is studied in [126].