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5.4 Isotropy: Hamiltonian constraint

Dynamics is controlled by the Hamiltonian constraint, which classically gives the Friedmann equation. Since the classical expression (28View Equation) contains the connection component c, we have to use holonomy operators. In the quantum algebra we only have almost periodic functions at our disposal, which does not include polynomials such as 2 c. Quantum expressions can therefore only coincide with the classical one in appropriate limits, which in isotropic cosmology is realized for small extrinsic curvature, i.e., small c in the flat case. We thus need an almost periodic function of c, which for small c approaches c2. This can easily be found, e.g., the function sin2 c. Again, the procedure is not unique since there are many such possibilities, e.g., d-2sin2dc, and more quantization ambiguities ensue. In contrast to the density - 3/2 |p|, where we also used holonomies in the reformulation, the expressions are not equivalent to each other classically but only in the small curvature regime. As we will discuss shortly, the resulting new terms have the interpretation of higher order corrections to the classical Hamiltonian.

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 2 c by an almost periodic function, but uses holonomies tracing out closed loops formed by symmetry generators [42Jump To The Next Citation Point46Jump To The Next Citation Point]. Moreover, the procedure can be embedded in a general scheme that encompasses different models and the full theory [194Jump To The Next Citation Point4258], 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. 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 approximated only when they are small; if they are not, the corresponding objects such as the Hamiltonian constraint receive infinitely many correction terms of higher powers in curvature (similarly to effective actions). The difference between both 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 hI in (41View Equation), 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 Aia. Respecting the symmetry, this can be done in the model with a square loop in two independent directions I and J. This yields the product h h h-1h -1 I J I J, which appears in a trace, as in (15View Equation), together with a commutator -1 hK [h K , ^V ] using the remaining direction K. The latter, following the general scheme of the full theory reviewed in Section 3.6, quantizes the contribution V~ --- |p| to the constraint, instead of directly using the simpler V~ --- |^p|.

Taking the trace one obtains a diagonal operator

sin(12dc)^V cos(12dc) - cos(12dc)^V sin(12dc)

in terms of the volume operator, as well as the multiplication operator

sin2(1dc)cos2(1dc) = sin2(dc). 2 2

In the triad representation where instead of working with functions <c|y > = y(c) one works with the coefficients ym in an expansion sum |y > = mym |m >, this operator is the square of a difference operator. The constraint equation thus takes the form of a difference equation [46Jump To The Next Citation Point7715Jump To The Next Citation Point]

(Vm+5d - Vm+3d)eikym+4d(f) - (2 + k2g2d2)(Vm+d - Vm -d)ym(f) -ik 16p 3 3 2 + (Vm-3d- Vm-5d)e ym -4d(f) = - ----Gg d lP ^Hmatter(m)ym(f) (49) 3
for the wave function ym, which can be viewed as an evolution equation in internal time m. (Note that this equation is not valid for k = - 1 since the derivation via a Hamiltonian formulation is not available in this case.) Thus, discrete spatial geometry implies a discrete internal time [43Jump To The Next Citation Point]. The equation above results in the most direct way from a non-symmetric constraint operator with gravitational part acting as
^ -----3------ -ik 2 2 2 ik H |m > = 16pGg3d3l2 (Vm+d - Vm-d)(e | m + 4d >- (2 + k g d )|m> + e |m - 4d>). P

One can symmetrize this operator and obtain a difference equation with different coefficients, which we do here after multiplying the operator with sgn p for reasons that will be discussed in the context of singularities in Section 5.15. The resulting difference equation is

ik 2 2 2 (| DdV |(m + 4d) + |DdV |(m))e ym+4d(f)- 2(2 + k g d )| DdV |(m)ym(f) + (|D V |(m - 4d) + |D V |(m))e-iky (f) = - 32p-Gg3d3l2 H^ (m)y (f) (50) d d m-4d 3 P matter m
where |D V |(m) := sgn(m)(V - V ) = |V - V | d m+d m-d m+d m-d.

Since 1 sin c| m > = - 2 i(|m + 2>- |m - 2 >), the difference equation is of higher order, even formulated on an uncountable set, and thus has many independent solutions. Most of them, however, oscillate on small scales, i.e., between m and m + md with small integer m. Others oscillate only on larger scales and can be viewed as approximating continuum solutions. The behavior of all the solutions leads to possibilities for selection criteria of different versions of the constraint since there are quantization choices. Most importantly, one chooses the routing of edges to construct the square holonomy, again the spin of a representation to take the trace [116Jump To The Next Citation Point201], and factor ordering choices between quantizations of c2 and V~ --- |p|. All these choices also appear in the full theory such that one can draw conclusions for preferred cases there.

When the symmetry group is not Abelian and there is non-zero intrinsic curvature, the construction is more complicated. For non-Abelian symmetry groups integral curves as before do not form a closed loop and one needs a correction term related to intrinsic curvature components [46Jump To The Next Citation Point62Jump To The Next Citation Point]. Moreover, the classical regime is not as straightforward to specify since connection components are not necessarily small when there is intrinsic curvature. A general scheme encompassing intrinsic curvature, other symmetric models and the full theory will be discussed in Section 5.14.

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