semiclassicality in Loop Quantum Cosmology

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semiclassicality in Loop Quantum Cosmology

InstantiationsThe continuous limit:We identified the relations between H phy , H phy and theappropriate limit spaces.To build H phy ↔ H phy we need H ± ↔ H ± .Problem: Identification of Φ n and Φ(k) will produce zero normWDW states!The instantiations:Fix the moment φ = φ o . By rotation ˜Ψ(k) ↦→ ˜Ψ(k)e iωφ oone canbring it to φ = 0.On H ± the operator ˆx := ˆθ a −π/2 takes the form ˆx = ia2Aω ∂ ωOn H ± one can build “sin(cx)/c” oper. via h : [h˜Φ] n = ˜Φ n−1aisin(cx)/c =2A[∆ω](ω+[∆ω]/2 [h−h−1 ]+O(e −aω ),where ∆ω = lim n→∞ [ω n −ω n−1 ].If at φ o state sharply peaked about x = 0: sin(cx)/c - goodreplacement of ˆxInstantiation: Interpolation of ˜Φ n s.th. actions of ˆx n and[sin(cx)/c] n agree up to n = 2.– p. 27

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