One can restrict the ambiguities to some degree by modeling the expression on that of the full theory.
This means that one does not simply replace by an almost periodic function, but uses holonomies
tracing out closed loops formed by symmetry generators [50
, 54
]. Moreover, the procedure can be
embedded in a general scheme that encompasses different models and the full theory [292
, 50, 68], further
reducing ambiguities. In particular models with non-zero intrinsic curvature on their symmetry orbits, such
as the closed isotropic model, can then be included in the construction. (There are different
ways to do this consistently, with essentially identical results; compare [111
] with [28
] for the
closed model and [300
] with [285
] for
.) One issue to keep in mind is the fact that
“holonomies” are treated differently in models and the full theory. In the latter case they are ordinary
holonomies along edges, which can be shrunk and then approximate connection components. In
models, on the other hand, one sometimes uses direct exponentials of connection components
without integration. In such a case, connection components are reproduced in the corrections
only when they are small. Alternatively, a scale-dependent
can provide the suppression
even if connection components remain large in semiclassical regimes. The requirement that this
happens in an acceptable way provides restrictions on refinement models, especially if one goes
beyond isotropy. Selecting refinement models, on the other hand, leads to important feedback
for the full theory. The difference between the two ways of dealing with holonomies can be
understood in inhomogeneous models, where they are both realized for different connection
components.
In the flat case the construction is easiest, related to the Abelian nature of the symmetry group. One
can directly use the exponentials in (44
), viewed as 3-dimensional holonomies along integral
curves, and mimic the full constraint where one follows a loop to get curvature components of
the connection
. Respecting the symmetry, this can be done in the model with a square
loop in two independent directions
and
. This yields the product
, which
appears in a trace as in Equation (15
), together with a commutator
, using the
remaining direction
. The latter, following the general scheme of the full theory reviewed in
Section 3.6, quantizes the contribution
to the constraint, instead of directly using the simpler
.
Taking the trace one obtains a diagonal operator
in terms of the volume operator, as well as the multiplication operator
as the only term resulting from One can symmetrize this operator and obtain a difference equation with different coefficients, which we
do here after multiplying the operator with for reasons that will be discussed in the context of
singularities in Section 5.16. The resulting difference equation is
Since , the difference equation is of higher order, even formulated on
an uncountable set, and thus has many independent solutions if
is constant. Most of them, however,
oscillate on small scales, i.e., between
and
with small integer
. Others oscillate only on
larger scales and can be viewed as approximating continuum solutions. For non-constant
, we have a
difference equation of non-constant step size, where it is more complicated to analyze the general form of
solutions. (In isotropic models, however, such equations can always be transformed to equidistant form up
to factor ordering [75
].) As there are quantization choices, the behavior of all the solutions leads to
possibilities for selection criteria of different versions of the constraint. Most importantly, one chooses the
routing of edges to construct the square holonomy, again the spin of a representation to take the
trace [170
, 299], and factor-ordering choices between quantizations of
and
. All these
choices also appear in the full theory, such that one can draw conclusions for preferred cases
there.
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