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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 (29View Equation) contains the connection component c, we have to use holonomy operators. In quantum algebra we have only almost-periodic functions at our disposal, which does not include polynomials such as c2. 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 2 sin c. Again, the procedure is not unique since there are many such possibilities, e.g., −2 2 δ sin δc, and more quantization ambiguities ensue. In contrast to the density |p|− 3∕2, 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 c2 by an almost periodic function, but uses holonomies tracing out closed loops formed by symmetry generators [50Jump To The Next Citation Point54Jump To The Next Citation Point]. Moreover, the procedure can be embedded in a general scheme that encompasses different models and the full theory [292Jump To The Next Citation Point5068], 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 [111Jump To The Next Citation Point] with [28Jump To The Next Citation Point] for the closed model and [300Jump To The Next Citation Point] with [285Jump To The Next Citation Point] for k = − 1.) 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 δ(μ) 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 hI in (44View 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 i A a. 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 hIhJh− 1h−1 I J, which appears in a trace as in Equation (15View Equation), together with a commutator −1 ˆ hK [hK ,V ], using the remaining direction K. The latter, following the general scheme of the full theory reviewed in Section 3.6, quantizes the contribution ∘ --- |p| to the constraint, instead of directly using the simpler ∘ --- |ˆp|.

Taking the trace one obtains a diagonal operator

1 ˆ 1 1 ˆ 1 sin(2δc)V cos(2δc) − cos(2δc)V sin(2δc)

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

sin2(1δc) cos2( 1δc) = sin2(δc), (52 ) 2 2
as the only term resulting from − 1 −1 hI hJhI h J in −1 − 1 − 1 tr(hIhJ hI hJ hK[hK , ˆV]). In the triad representation, where instead of working with functions ⟨c|ψ⟩ = ψ (c) one works with the coefficients ψ μ in an expansion ∑ |ψ ⟩ = ψμ|μ⟩ μ, this operator is the square of a difference operator. The constraint equation thus takes the form of a difference equation
ik 2 2 (V μ+5δ − V μ+3δ)e ψμ+4δ(φ ) − (2 + kγ δ )(Vμ+ δ − V μ−δ)ψμ(φ) −ik 16π 3 3 2 + (Vμ−3δ − Vμ−5δ)e ψ μ−4δ(φ) = − -3--Gγ δ ℓP ˆHmatter(μ )ψμ(φ) (53 )
for the wave function ψμ, which can be viewed as an evolution equation in internal time μ. Thus, discrete spatial geometry implies a discrete internal time [51Jump To The Next Citation Point]. The equation above results in the most direct path from a non-symmetric constraint operator with gravitational part acting as
ˆH |μ ⟩ = -----3------(Vμ+δ − Vμ−δ)(e−ik|μ + 4δ⟩ − (2 + k2γ2 δ2)|μ⟩ + eik|μ − 4δ ⟩) . (54 ) 16πG γ3δ3ℓ2P
Operators of this form are derived in [54Jump To The Next Citation Point16Jump To The Next Citation Point] for a spatially-flat model (k = 0), in [11128Jump To The Next Citation Point286Jump To The Next Citation Point] for positive spatial curvature (k = 1), and in [300285Jump To The Next Citation Point] for negative spatial curvature (k = − 1), which is not included in the forms of Equations (53View Equation) and (54View Equation). Note, however, that not all these articles used the same quantization scheme and thus operators even for one model appear different in details, although the main properties are the same. Some of the differences of quantization schemes will be discussed below. The main option in alternative quantizations relates to a possible scale dependence δ (μ ), which we leave open at this stage.

One can symmetrize this operator and obtain a difference equation with different coefficients, which we do here after multiplying the operator with sign(^p) for reasons that will be discussed in the context of singularities in Section 5.16. The resulting difference equation is

ik 2 2 2 (|Δ δV |(μ + 4 δ) + |Δ δV|(μ))e ψμ+4δ(φ) − 2(2 + k γ δ )|Δ δV |(μ)ψμ(φ) −ik 32π- 3 3 2 ˆ + (|Δ δV |(μ − 4δ) + |ΔδV |(μ))e ψμ−4δ(φ) = − 3 G γ δ ℓPHmatter(μ)ψμ(φ) (55 )
where |Δ δV |(μ ) := sgn(μ )(V μ+δ − Vμ−δ) = |Vμ+δ − Vμ−δ|.

Since sin c|μ⟩ = − 1i(|μ + 2⟩ − |μ − 2 ⟩) 2, the difference equation is of higher order, even formulated on an uncountable set, and thus has many independent solutions if δ is constant. Most of them, however, oscillate on small scales, i.e., between μ and μ + m δ with small integer m. Others oscillate only on larger scales and can be viewed as approximating continuum solutions. For non-constant δ, 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 [75Jump To The Next Citation Point].) 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 [170Jump To The Next Citation Point299], and factor-ordering choices between quantizations of c2 and ∘ --- |p|. All these choices also appear in the full theory, such that one can draw conclusions for preferred cases there.

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