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 Euclidean part of the constraint.) Triad components and their inverse determinant
are again expressed as a Poisson bracket using the 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 [176]:
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 a graph the 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 [189]. 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 the construction is not required to be the fundamental one
[116
].
The second part of the constraint is more complicated since one has to use the function
in
. As also developed in [194
], 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 can always be decided by a direct comparison of different constructions (see, e.g.,
[125]).
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