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5.3 Isotropy: Matter Hamiltonian

We now know how the basic quantities p and c are quantized, and can use the operators to construct more complicated ones. Of particular importance, as well as for cosmology, are matter Hamiltonians, where not only the matter field but also geometry is quantized. For an isotropic geometry and a scalar, this requires us to quantize −3∕2 |p | for the kinetic term and 3∕2 |p| for the potential term. The latter can be readily defined as |ˆp|3∕2, but for the former we need an inverse power of p. Since ˆp has a discrete spectrum containing zero, a densely-defined inverse does not exist.

At this point, one has to find an alternative route to the quantization of − 3∕2 d(p) = |p|, or else one could only conclude that there is no well-defined quantization of matter Hamiltonians as a manifestation of the classical divergence. In the case of loop quantum cosmology it turns out, following a general scheme of the full theory [291], that one can reformulate the classical expression in an equivalent way such that quantization becomes possible. One possibility is to write, similar to Equation (13View Equation),

( 3 ( ))6 --1---∑ −1 √ -- d(p) = 3πγG tr τIhI{h I , V} , I=1

where we use holonomies of isotropic connections and the volume 3∕2 V = |p|. In this expression we can insert holonomies as multiplication operators and the volume operator, and turn the Poisson bracket into a commutator. The result

^ ( −1 − 2 1 ∘ -- 1 1 ∘ -- 1 )6 d(p) = 8iγ ℓP (sin 2c ˆV cos 2c − cos2c Vˆ sin 2c) (50 )
is not only a densely defined operator but even bounded, which one can easily read off from the eigenvalues [49]
^ ( − 1− 2∘ ----- ∘ ----- )6 d(p)|μ ⟩ = 4γ ℓP ( Vμ+1 − V μ−1) |μ ⟩ (51 )
with 3∕2 V μ = |pμ | and pμ from Equation (49View Equation).

Rewriting a classical expression in such a manner can always be done in many equivalent ways, which in general all lead to different operators. In the case of |p|−3∕2, we highlight the choice of the representation in which to take the trace (understood as the fundamental representation above) and the power of |p| in the Poisson bracket (√V-- = |p|3∕4 above). This freedom can be parameterized by two ambiguity parameters 1 j ∈ 2ℕ for the representation and 0 < l < 1 for the power such that

( )3∕(2− 2l) 3 ∑3 − 1 l d(p) = ---------------------- trj(τIhI{hI ,|p|} ) . 8π γGlj (j + 1)(2j + 1) I=1

Following the same procedure as above, we obtain eigenvalues [5558]

( )3 ∕(2−2l) (μ) 9 ∑j ^d(p)j,l = --2----------------- k|pμ+2k|l , γ ℓPlj(j + 1)(2j + 1) k= −j

which, for larger j, can be approximated by Equation (26View Equation) (see also Figure 9View Image). This provides the basis for effective densities in loop cosmology as described in Section 4.

View Image

Figure 9: Discrete subset of eigenvalues of ^d(p) (left) for two choices of j (and l = 3 4), together with the approximation d(p)j,l from Equation (26View Equation) and small-p power laws. The classical divergence at small p, where the behavior differs strongly from eigenvalues, is cut off. The right panel shows the dependence of the initial increase on l.

Notice that operators for the scale factor, volume or their inverse powers do not refer to observable quantities. It can thus be dangerous, though suggestive, to view their properties as possible bounds on curvature. The importance of operators for inverse volume comes from the fact that this appears in matter Hamiltonians, and thus the Hamiltonian constraint of gravity. Properties of those operators such as their boundedness or unboundedness can then determine the dynamical behavior (see, e.g., [62Jump To The Next Citation Point]).

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