semiclassicality in Loop Quantum Cosmology

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semiclassicality in Loop Quantum Cosmology

Hamiltonian constraint:Λ > 0 - WDW model−∂ 2 φ = Θ o −πGγ 2 ∆ΛI = Θ ΛΘ Λ admits 1d family of selfadjoint extensions, labeled by β ∈ U(1)Each extension Θ Λβ has continuous spectrum: Sp(|Θ Λβ |) = R +Physical states:Ψ(v,φ) = ∫ ∞dk˜Ψ(k)e β 0 k (v)eiωφ , ω = √ 12πGke β k (v) = √ 1 [c 1 (β,Λ,k)H (1)|v| ik (av)+c 2(β,Λ,k)H (2)ik (av)],√γwhere a =2 ∆Λ12πG and H(1) ,H (2) - Hankel functions.For each extension all e β khave common asymptoticse β k = N(Λ,β,k)|v|−1 cos(Ω(Λ)|v|+σ(Λ,β))+O(|v| −3/2 )Dynamics: follows analytically extended classical trajectory.– p. 18

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