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3.6 Hamiltonian constraint

Similarly to matter Hamiltonians one can also quantize the Hamiltonian constraint in a well-defined manner [292Jump To The Next Citation Point]. Again, this requires us to rewrite triad components and to make other regularization choices. Thus, there is not just one quantization but a class of different possibilities.

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 γ = 1, 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 (13View Equation), and curvature components are obtained through a holonomy around a small loop α of coordinate size Δ and with tangent vectors a s1 and sa2 at its base point [259Jump To The Next Citation Point]:

sa1sb2Fiabτi = Δ −1(hα − 1) + O (Δ ). (14 )

Putting this together, an expression for the Euclidean part E H [N ] can then be constructed in the schematic form

HE [N ] ∝ ∑ N (v)εIJK tr (h h {h −1,V }) + O(Δ ), (15 ) αIJ sK sK v
where one sums over all vertices of a triangulation of space, whose tetrahedra are used to define closed curves αIJ and transversal edges sK.

An important property of this construction is that coordinate functions such as Δ 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 h α, 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 [170Jump To The Next Citation Point].

The second part of the constraint is more complicated since one has to use the function Γ (E) in i K a. As also developed in [292Jump To The Next Citation Point], extrinsic curvature can be obtained through the already-constructed Euclidean part via K ∼ {HE, V }. The result, however, is rather complicated, and in models one often uses a more direct way, exploiting the fact that Γ 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., [181Jump To The Next Citation Point]).

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