### 5.8 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. 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 [48] for the Bianchi I model
and in [62] for all other class A models.
Simpler cases arise in so-called 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]). 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.