semiclassicality in Loop Quantum Cosmology

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semiclassicality in Loop Quantum Cosmology

The unforgiving universe(semiclassicality in Loop Quantum Cosmology)ILQG SEMINAR15.02.2011Tomasz PawłowskiDepartment of Mathematics and StatisticsUniversity of New Brunswick, Canadawork by:W. Kamiński, TP: arXiv:1001.2663A. Ashtekar, TP unpublished– p. 1


The problemLoop quantization of the isotropic/homogeneous cosmologicalmodels ⇒ changes of the dynamics at near-Planck densities causingthe Big Bounce.|Ψ(v,φ)|1.510.505*10 31.0*10 4v1.5*10 42.0*10 42.5*10 43.0*10 43.5*10 44.0*10 4Observation: States sharply peaked throughout the evolution.-0.2Problem: In numerical simulations one has to select an example ofthe state for evolution:-0.4is the preservation of the semiclassicality robust?Addressing on the quantum level: Monte Carlo methods – probingthe space of solutions via random samples (brute force approach).Need to run large number of time-costly simulations!Any Alternative approach?-0.6-0.8φ-1-1.21.61.41.210.80.60.40.20– p. 2


An example modelFlat isotropic universe with massless scalar field.Spacetime: manifold M ×R where M is topologically R 3 .M ×{t} (where t ∈ R) – homogeneous slices.Metric:g = −dt 2 +a 2 (t) o qo q - flat fiducial metric (dx 2 +dy 2 +dz 2 ).The treatment:The degrees of freedom (canonical variables):Geometry: (v,b), v - prop. to the volume of chosen fiducialregion.Matter: (φ,p φ ), φ - scalar field valueThe quantization:Matter: standard Schrödinger representationGeometry:· Wheeler-DeWitt: Schrödinger representation· LQC: methods of Loop Quantum GravityNontrivial Hamiltonian constraint: Dirac program.Scalar field used as an internal time– p. 5


Wheeler-DeWitt quantizationAll elements expressed in (c,p,φ,p φ ). Standard quantization.Kinematical Hibert space: (v ∝ sgn(p)|p| 3/2 )H kin = H grav ⊗H φ , H grav = L 2 (R,dv), H φ = L 2 (R,dφ).Basic operators: (ˆp, ĉ ∝ i∂ p , ˆφ, ˆpφ ∝ i∂ φ ).Basis: eigenstates (v|ˆp = v(v|,(φ|ˆφ = φ(φ|.Quantum constraint:[∂ 2 φ Ψ](v,φ) = −[ΘΨ](v,φ) := 12πG[(v∂ v) 2 +v∂ v +1/4]Ψ(v,φ).Physical states: Ψ(v,φ) = ∫ R dk˜Ψ(k)e k (v)e iωφ ,˜Ψ ∈ L 2 (R,dk), ω = √ 12πG|k|, e k (v) = (1/ √ 2πv)e ikln(v) .Observables:ˆp φ : ˜Ψ(k) ↦→ ω(k)˜Ψ(k),ln(v) φo : Ψ(v,φ) ↦→ e i√ Θ(φ−φ o ) ln(v)Ψ(v,φ o ).Dynamics:Two classes: ever contracting and ever expanding.The dispersions σ ln(v)φ are constant.– p. 6


WDW dynamics0outgoingincoming-0.2-0.4-0.6φ-0.8-1-1.20*10 0 1*10 4 2*10 4 3*10 4 4*10 4 5*10 4v– p. 7


LQC quantizationGeometry polymeric, matter standard, structure analogousKinematics:Geometry space: H grav = L 2 (R Bohr dµ Bohr ).Geometry basis: |v〉 : 〈v|v ′ 〉 = δ vv ′.Basic operators: holonomies h = eThe constraint: reexpressed in terms of ĥ, Ŝ∫γ Adx , fluxes S = ∫ S ⋆Edσ.[Θψ](v) = −f + (v)ψ(v +4)+f o (v)ψ(v)−f − (v)ψ(v −4)for large v: f o,± (v) ∝ v 2 .The physical states: Ψ(v,φ) = ∫ dk˜Ψ(k)eR + k (v)e iωφ˜Ψ ∈ L 2 (R + ,dk), ω = √ 12πG|k|, Θe k (v) = ω 2 (k)e k (v).Dirac observables: analogous to WDW.The dynamics:in distant future and past agreement with GR,bounce in the Planck regime (energy densities).– p. 8


LQC dynamics0LQCclassical-0.2-0.4φ-0.6-0.8-1-1.20 1*10 4 2*10 4 3*10 4 4*10 4 5*10 4v– p. 9


WDW limit of LQC statesf o,± - real: e k are standing waves.Observation: converge to WDW standing waves.e k (v) = ψ k(v)+O(|e k (v)|(k/v) 2 )ψ k(v) := r(k)[e iα(k) e k (v)+e −iα(k) e −k (v)]Comp. of LQC and WDW norms + self-adjointness of Θ: r(k) = 2Analytical proof:reformulation of the diff. equation in the 1st order form,(local) decomp. of the LQC eigenf. in terms of WDW ones,asymptotic properties of the resulting transfer matrix.Properties of the phase shifts:Analytic results for sLQC: simple analytic form of theeigenfunctions in the momentum of v + stationary phase methodα(k) = −k(ln|k|−1)−(3/4)π +o(k 0 )α ′ (k) = −ln|k|+O(k −1 ln|k|)Numerical results for: APS, sLQC, MMO:α ′ (k) = −ln|k|+O(k −2 )|kα ′′ (k)| ≤ 1– p. 10


The phase shiftsProperties analyzed analytically and numerically:Analytic results for sLQC: simple analytic form of the eigenfunctionsin the momentum of v + stationary phase methodα(k) = −k(ln|k|−1)−(3/4)π +o(k 0 )α ′ (k) = −ln|k|+O(k −1 ln|k|)Numerical results for: APS, sLQC, MMO:α ′ (k) = −ln|k|+O(k −2 )|kα ′′ (k)| ≤ 1Very regular behavior!– p. 11


Phase shift propertiesα ′ (k).420APSsLQCMMOlimit-2-4α’(k)-6-8-10-12-1410 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6k– p. 12


Phase shift propertieskα ′′ (k).0.5APSsLQCMMO-10kα"(k)-0.5-1-1.50.1 1 10k– p. 13


WDW limit: e k (v) → ψ k(v)The scattering picture˜Ψ(|k|) ↦→ ˜Ψ(k) = 2e isgn(k)α(|k|) sgn(k)˜Ψ(|k|)Two components: ˜Ψ ± (k) = θ(±k)˜Ψ(k)The limits of observables:lim φ→±∞ 〈Ψ|ln(v) φ |Ψ〉 = 〈Ψ ± |ln(v) φ |Ψ ± 〉,lim φ→±∞ 〈Ψ|∆ln(v) φ |Ψ〉 = 〈Ψ ± |∆ln(v) φ |Ψ ± 〉 =: σ ± .Interpretation as a scattering process:|Ψ〉 in↦→ |Ψ〉 out= ˆρ|Ψ〉 in,˜Ψ in (k) := ˜Ψ + (k), ˜Ψ out (k) := ˜Ψ − (k)The scattering matrix:ρ(k,k ′ ) = (e k |ˆρ|e k ′) = e −isgn(k′ )α(|k ′ |) δ(k +k ′ ).The transformation: total reflection˜Ψ(k) ↦→ U˜Ψ(k) := e 2isgn(k)α(|k|)˜Ψ(−k) – p. 14


The scattering1.41.2LQCincomingoutgoing10.8φ0.60.40.2010 1 10 2 10 3 10 4v– p. 15


The dispersion growthAn application: comparizon of the dispersions of ln(v) φ .Action on H phy :ln|ˆv| φ˜Ψ = [−i∂k −(∂ k ω(k))φÎ]˜ΨThe relation between action on limits: 〈Ô〉 ± := 〈Ψ ± |Ô|Ψ ±〉〈−i∂ k 〉 − = 〈U −1 [−i∂ k ]U〉 + , 〈∆[−i∂ k ]〉 − = 〈∆U −1 [−i∂ k ]U〉 + .where U −1 [−i∂ k ]U = −i∂ k −2α ′ I.The effects on dispersions: (σ A+B ≤ σ A +σ B - Schwartz ineq.)σ − ≤ σ + +2〈∆α ′ I〉 +Estimate via dispersion in ω:General inequality:〈∆α ′ I〉 2 + = 〈(α ′2 −〈α ′ 〉 + ) 2 I〉 + ≤ 〈(α ′2 −α ′⋆ ) 2 I〉 + .The choice: α ′⋆ = α ′ (exp(λ ⋆ )), λ ⋆ := 〈ln(k)〉 + and props. of α ′give:The final inequality:〈∆α ′ I〉 2 + ≤ 〈(ln(ˆk)−λ ⋆ I) 2 〉 + = 〈∆ln(ˆk)〉 2 + =: σ 2 ⋆σ − ≤ σ + +2σ ⋆– p. 16


SummaryThe results: Devised method of comparing the properties of distantfuture and distant past statesdoes not require exact solvabilityuses only asymptotic properties of phys. Hilbert space basisis general (without restriction to particular types/shapes of states),in genuinely quantum (no semiclassical approximations of anykind),application to FRW with massless scalar:general triangle inequalities on dispersions. Strict upper boundon eventual dispersion growth.Just the upper bound, the actual dispersion may even shrink.Universe’s memory has to be indeed very sharp !generalization:isotropic sector of Bianchi Islightly weaker version: vacuum Bianchi I: arXiv:0906.3751further extension: Λ > 0 (see 2nd part).– p. 17


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


WDW: observablesObservables: analogous to Λ = 0 case.Problem: the analog of |v| φ leads outside of H phy !Failure of semiclassical treatment:If one selects canonical pair v,b the requirement〈(|ˆv| φ −〈|v| φ 〉) n 〉 < ∞ on some open φ ∈ O implies∫dk˜Ψ(k)N(Λ,β,k)e iωφ = 0, ∀φ ∈ OSince N(Λ,β,k) ∝ √ k we have: ˜Ψ(k) ∼ k −3/2 ⇒ 〈∆p φ 〉 = ∞.For |v| ≪ 1 e β k ≈ eikln|v| , thus at early times〈∆p φ 〉 = ∞ ⇒ 〈∆b| φ 〉 = ∞Impossible to built states well behaving in both v and b even for ashort time!Solution:Compactify v, for example use θ a = arctan(|v|/a) orUse truly measurable quantities, like Hubble parameter H orenergy density ρ.– p. 19


WDW dynamics for Λ > 00.80.6WDWclass.π/20.40.2φ0-0.2-0.4-0.6-0.80 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6|θ a | φ– p. 20


WDW dynamics for Λ > 00.80.6WDWclass.0.40.2φ0-0.2-0.4-0.6-0.810 -3 10 -2 10 -1 10 0 10 1ρ| φ– p. 21


LQC dynamics for Λ > 02.52LQCeff.class.π/21.5φ10.500.2 0.4 0.6 0.8 1 1.2 1.4 1.6|θ α | φ– p. 23


LQC dynamics for Λ > 02.521.5φ10.5LQCeff.class.010 -3 10 -2 10 -1 10 0ρ– p. 24


2nd order asymptoticsAnalysis of the transfer matrix asymptotics:e β n(v) = N n [e iα e + n(v)+e −iα e − n(v)]+O(v −3 ) (⋆)e ± n = |v| −1 e ±iΩ|v| ·e ±iκ(n,Λ,β)/|v| , 1/|v| ≈ (1/a)(θ a −π/2)where: cos(4Ω(Λ)) = 1−2Λ/Λ c =: 1−2λ,Limit spaces:κ(n,Λ,β) = 3πG(1−2λ)+ω2 n12πG √ λ(1−λ) =: Aω2 n +B“Standing wave form” of e β n:split onto incoming and outgoing components.For each comp. e ±iΩ|v| is a global rotation, only e ± n relevant.e ± n form wave packet regular in θ a . Schrödinger type rather thatKlein-Gordon.Transformation into limit spaces:Define spaces H ± spanned by e ± n with IP inherited from H phythrough the limit (⋆).H phy ∋ ˜Ψ n ↦→ Φ ± n := ˜Ψ n M n ∈ H ± – p. 25


WDW limit statesTaking the limit of the mass gap ∆ → 0 one can derive analogous2nd order limit in WDW theory.As Ω(Λ) depends implicitly on ∆ to ensure uniform convergenceone has to allow for flow Λ = Λ(∆).Result: LQC wave packet has the WDW limit spanned by e k suchthat:wheree β k (v) = N(k)[eiα e + k (v)+e−iα e − k (v)]+O(v−3 )e ± k = |v|−1 e ±iΩ|v| ·e ±iκ(k,Λ,β)/|v|with the same Ω and κ = κ(ω = √ 12πGk)however the WDW model corresponds to the cosmologicalconstantΛ/Λ c = λ = arccos(1−2λ).Construction of the limit spaces H ± analogous to LQC.Again Schrödinger wave packets nearθ = π/2.If the wave packet sharply peaked about θ = π/2 then (up tohigher order corrections)〈∆|ˆθ a | φ 〉 WDW = 〈∆|ˆθ a | φ 〉 lim– p. 26


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


Instantiations + scattering2.52LQCeff.WDWπ/2– p. 281.5φ10.500.2 0.4 0.6 0.8 1 1.2 1.4 1.6|θ α | φ


The scattering, decoherenceThe picture:Instantiations provide identification of quasi-periodic state of LQCwith non-periodic WDW one at given time φ o .Once can build a sequence of instantiations at φ n where〈θ〉 ≈ π/2.Transfer φ n → φ n+1 : scattering of WDW state into another.Evolution: sequence of scatterings (determined by instantiations).Since ω n = (nπ −β)/f 2 (Λ)+δω n one cycle corresponds toApplication:˜Ψ n ↦→ e 2πiδω n˜Ψn =: U ˜Ψ n〈∆sin(cx)/c〉 H phy = 〈∆sin(cx)/c〉 H± +O(v −3 ) .After N ≫ 1 cycles〈∆sin(cx)/c〉 φo +N∆φ ≤ 〈∆sin(cx)/c〉 φo +N〈∆ 2π∂ ωδωω〉where the last term 〈∆ 2π∂ ωδωω〉 ≤ C〈∆ 2πe−aωω〉.– p. 29


SummaryScattering picture successfully extended to the model with Λ > 0.Evolution between pure deSitter epochs - chain of Wheeler-DeWittuniverses consecutively scattered one into another.The instantiation procedure allowed to relate the dispersion incompactified volume θ a of the LQC state and its WDW limit at givenmoment φ o (in pure deSitter epoch).The scattering corresponds to unitary rotation by e 2πiδω n, where thedeviation δω n decays exponentially.Consequence: The decoherence of the state between large numberN of pure deSitter epochs is bounded from above by (up to a knownconstant) the dispersion of the operator Ne −aω /ω infinitesimal for thestates peaked about large p ⋆ φ . – p. 30


Appendix: The transfer matrix methodf o,± - real: e k are standing waves.Observation: converge to WDW standing waves.Verification: transfer matrices1st order form of the difference equation:[ ]e k (v)⃗e k (v) = , A(v) =e k (v −4)[fo (v)−ω 2 (k)f + (v)− f −(v)f + (v)1 0⃗e k (v +4) = A(v)⃗e k (v)Expressing in WDW basis:[e⃗e k (v) = B(v)⃗χ k (v), B(v) := k (v +4) e −k (v +4)e k (v) e −k (v)Final form: ⃗χ k (v +4) = B −1 (v)A(v)B(v −4)⃗χ k (v) =: M(v)⃗χ k (v)Limit of the transfer matrix: M(v) = I+O(v −3 ) ⇒e k (v) = ψ k(v)+O(|e k (v)|(k/v) 2 )ψ k(v) := r(k)[e iα(k) e k (v)+e −iα(k) e −k (v)]Comparizon of norms: r(k) = 2]].– p. 31


Appendix: Comparizon between the normsEvolution: mappingInner products:〈ψ|χ〉 = ∑ L + 0ψ(v)χ(v),R ∋ φ ↦→ ψ(·) := Ψ(·,φ) ∈ H grav〈ψ|χ〉 = ∫ R + dvψ(v)χ(v)Distributional estimates:splitting the domain:X(k,k ′ ) := 〈e k ′|e k 〉 = ∑ L + 0 ∩[1,∞]e k ′ (v)e k (v)extracting WDW limits:X(k,k ′ ) = + ∑ L + 0 ∩[1,∞][ψ k ′ (v)ψ k(v)+O(v −5/2 )]estimating the sum by the integral:∑ψk ′(v)ψ k(v) = (1/4) ∫ [][1,∞[ dv ψ k ′(v)ψ k(v)+O(v −3/2 )relation: ∫ (∫dx e ikx = 1 R + 2 R dx )eikx − iπkFinal relation: X(k,k ′ ) = (r 2 (k)/8)〈ψ k|ψ k ′〉+F(k,k ′ )Orthonormality: F(k,k ′ ) = 0, r(k) = 2.– p. 32


Appendix: The limits of observablesArgumentation for past limit, problem symmetric in time.Assumed localization in ln|v| φ of WDW limit: 〈ln|v| φ 〉 < ∞,〈∆ln|v| φ 〉 < ∞.Past wave packet follows the trajectory ¯x(φ) = x o −β(φ−φ o ),where β = √ 12πG.f(v) φ - multiplication operator on ID’s Ψ(·,φ) ∈ H grav ,expectations - local sums: 〈f(v) φ 〉 = ∑ L + f(v)|Ψ(v,φ)| 2 ,0analogous situation in WDW.introduce ˜x(φ) = x o −(β/2)(φ−φ o ) and split the local sums along it.the following properties:falloff conditions due to localization: parts for x > ˜x converge(sufficiently fast) to complete sums.relation between LQC and WDW norms: convergence of WDWand LQC partial sums for states localized in k.imply that:lim φ→−∞ 〈Ψ|ln(v) φ |Ψ〉 = 〈Ψ − |ln(v) φ |Ψ − 〉,lim φ→−∞ 〈Ψ|∆ln(v) φ |Ψ〉 = 〈Ψ − |∆ln(v) φ |Ψ − 〉.– p. 33

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