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4.4 Isotropy: Effective densities in phenomenological equations

The isotropic model is thus quantized in such a way that the operator ˆp has a discrete spectrum containing zero. This immediately leads to a problem, since we need a quantization of −3∕2 |p| in order to quantize a matter Hamiltonian such as (24View Equation), where not only the matter fields but also the geometry is quantized. However, an operator with zero in the discrete part of its spectrum does not have a densely defined inverse and does not allow a direct quantization of |p|−3∕2.

This leads us to the first main effect of the loop quantization: It turns out that despite the non-existence of an inverse operator of ˆp one can quantize the classical − 3∕2 |p| to a well-defined operator. This is not just possible in the model but also in the full theory, where it has even been defined first [291Jump To The Next Citation Point]. Classically, one can always write expressions in many equivalent ways, which usually results in different quantizations. In the case of |p |−3∕2, as discussed in Section 5.3, there is a general class of ways to rewrite it in a quantizable manner [49Jump To The Next Citation Point], which differ in details but all have the same important properties. This can be parameterized by a function d(p)j,l [55Jump To The Next Citation Point58Jump To The Next Citation Point], which replaces the classical |p|−3∕2 and strongly deviates from it for small p, while being very close at large p. The parameters j ∈ 1ℕ 2 and 0 < l < 1 specify quantization ambiguities resulting from different ways of rewriting. With the function

( 3- 1−l -1---( l+2 l+2) pl(q ) = 2lq l + 2 (q + 1) − |q − 1| (25 ) ( )) − --1--q (q + 1)l+1 − sgn (q − 1 )|q − 1|l+1 l + 1
we have
d(p)j,l := |p|−3∕2pl(3 |p|∕γjℓ2P )3∕(2− 2l), (26 )
which indeed fulfills − 3∕2 d(p)j,l ∼ |p| for 1 2 |p| ≫ p∗ := 3 jγℓP, but is finite with a peak around p∗ and approaches zero at p = 0 in a manner
d(p)j,l ∼ 33(3−l)∕(2−2l)(l + 1)− 3∕(2−2l)(γj)−3(2−l)∕(2− 2l)ℓ−P3(2−l)∕(1−l)|p|3∕(2−2l) (27 )
as it follows from 2− l pl(q) ∼ 3q ∕(1 + l). Some examples displaying characteristic properties are shown in Figure 9View Image in Section 5.3.

The matter Hamiltonian obtained in this manner will thus behave differently at small p. At those scales other quantum effects such as fluctuations can also be important, but it is possible to isolate the effect implied by the modified density (26View Equation). We just need to choose a rather large value for the ambiguity parameter j, such that modifications become noticeable even in semiclassical regimes. This is mainly a technical tool to study the behavior of equations, but can also be used to find constraints on the allowed values of ambiguity parameters.

We can thus use classical equations of motion, which are corrected for quantum effects by using the phenomenological matter Hamiltonian

(phen) 1- 2 3∕2 Hφ (p,φ, pφ) := 2d(p)j,lpφ + |p| V(φ ) (28 )
(see Section 6 for details on the relationship to effective equations). This matter Hamiltonian changes the classical constraint such that now
--3-- −2 2 2 ∘ --- (phen) H = − 8 πG (γ (c − Γ ) + Γ ) |p| + H φ (p,φ,p φ) = 0. (29 )
Since the constraint determines all equations of motion, they also change; we obtain the Friedmann equation from H = 0,
( ˙a)2 k 8 πG ( 1 ) -- + -2-= ----- --|p|−3∕2d(p)j,lp2φ + V (φ) (30 ) a a 3 2
and the Raychaudhuri equation from c˙= {c,H },
( ) ¨a 4 πG ∂Hmatter(p,φ, pφ) a-= − 3|p|3∕2- Hmatter(p, φ,pφ) − 2p-------∂p-------- (31 ) ( ( ) ) 8πG-- − 3∕2 −1 ˙2 1 d-log(|p|3∕2d(p)j,l) = − 3 |p| d(p)j,lφ 1 − 4a da − V (φ) . (32 )

Matter equations of motion follow similarly as

φ˙= {φ,H } = d(p)j,lpφ 3∕2 ′ p˙φ = {pφ,H } = − |p| V (φ),
which can be combined to form the Klein–Gordon equation
¨φ = φ˙˙ad-logd(p)j,l − |p|3∕2d(p) V′(φ). (33 ) da j,l

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