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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 ψ μ1,μ2,μ3 in three independent variables. Its (lengthy) form can be found in [56Jump To The Next Citation Point] for the Bianchi I model and in [82Jump To The Next Citation Point] 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 μ and ν remain, where only one, e.g., ν can take both signs if discrete gauge freedom is fixed, and the vacuum difference equation is, e.g., for Bianchi I,

∘ -------- 2δ |ν + 2δ|(ψμ+2δ,ν+2δ − ψ μ−2δ,ν+2δ) (∘ ------- ∘ ------) + 12 |ν + δ| − |ν − δ| ((μ + 4δ)ψμ+4δ,ν − 2μ ψμ,ν + (μ − 4δ)ψ μ−4δ,ν) ∘ -------- − 2δ |ν − 2δ|(ψμ+2δ,ν− 2δ − ψ μ− 2δ,ν−2δ) = 0 (57 )
from the non-symmetric constraint and
( ) 2δ ∘ |ν-+-2δ|-+ ∘ |ν| (ψ − ψ ) μ+2δ,ν+2δ μ−2δ,ν+2δ ( ∘ ------- ∘ ------) + |ν + δ| − |ν − δ| ((μ + 2δ)ψμ+2δ,ν − μψ μ,ν + (μ − 2δ)ψ μ− 2δ,ν) (∘ -------- ∘ --) − 2δ |ν − 2δ| + |ν| (ψμ+2 δ,ν−2δ − ψ μ−2δ,ν−2δ) = 0 (58 )
from the symmetric version (see also [14Jump To The Next Citation Point] and [75Jump To The Next Citation Point] 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 [88Jump To The Next Citation Point], 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, δ(μ) 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 [75Jump To The Next Citation Point]. A special case where this is possible is studied in [126].


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