It is more direct to quantize the first part of the constraint containing only the Ashtekar curvature.
(This part agrees with the constraint in Euclidean signature and Barbero–Immirzi parameter , and
so is sometimes called the Euclidean part of the constraint.) Triad components and their inverse
determinant are again expressed as a Poisson bracket using identity (13
), and curvature components are
obtained through a holonomy around a small loop
of coordinate size
and with tangent vectors
and
at its base point [259
]:
Putting this together, an expression for the Euclidean part can then be constructed in the
schematic form
An important property of this construction is that coordinate functions such as disappear from the
leading term, such that the coordinate size of the discretization is irrelevant. Nevertheless, there are
several choices to be made, such as how a discretization is chosen in relation to the graph a
constructed operator is supposed to act on, which in later steps will have to be constrained
by studying properties of the quantization. Of particular interest is the holonomy
, since
it creates new edges to a graph, or at least new spin on existing ones. Its precise behavior is
expected to have a strong influence on the resulting dynamics [283]. In addition, there are
factor-ordering choices, i.e., whether triad components appear to the right or left of curvature
components. It turns out that the expression above leads to a well-defined operator only in the
first case, which, in particular, requires an operator non-symmetric in the kinematical inner
product. Nevertheless, one can always take that operator and add its adjoint (which in this full
setting does not simply amount to reversing the order of the curvature and triad expressions) to
obtain a symmetric version, such that the choice still exists. Another choice is the representation
chosen to take the trace, which for this construction is not required to be the fundamental
one [170
].
The second part of the constraint is more complicated since one has to use the function
in
. As also developed in [292
], extrinsic curvature can be obtained through the
already-constructed Euclidean part via
. The result, however, is rather complicated, and
in models one often uses a more direct way, exploiting the fact that
has a more special
form. In this way, additional commutators in the general construction can be avoided, which
usually does not have strong effects. Sometimes, however, these additional commutators can be
relevant, which may always be determined by a direct comparison of different constructions (see,
e.g., [181
]).
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