Cosmological Effective Hamiltonian from LQG. from full Loop Quantum Gravity

Transcription

1 Cosmological Effective Hamiltonian from full Loop Quantum Gravity Friedrich Alexander University Erlangen-Nurnberg Based on arxiv: International Loop Quantum Gravity Seminar 26th September /26

2 Introduction The effective dynamics program [Ashtekar, Pawlowski, Singh 06] had much success by giving an excellent approximation to the exact quantum corrections of LQC, e.g. big bounce Its idea was the following: Take the classical cosmological Hamiltonian H cos (p, c) pc 2 (1) where (c, p) are obtained from the Ashetkar-Barbero variables (A, E) of the isotropic spacetime by fixing a fiducial Minkowski metric: A I a = cδ I a, E a I = pδ a I (2) By a loop-inspired quantization one promotes this to the LQC Hamiltonian constraint operator µ-scheme: λ 2 = l 2 P 2 3πβ): Ĥ LQC ˆN λ ˆN λ 2λ ˆν ˆN λ ˆN λ ˆν (3) 2λ 2/26

3 Introduction It was shown that the LQC coherent states ψ LQC coh (v) = 0 (k ko )2 dk e 2σ 2 e k (v), Ĥ LQC e k = 12πGk 2 e k are stable under the evolution produced by Ĥ LQC. When evaluating the expectation value of Ĥ LQC on ψ LQC coh, one obtains the effective Hamiltonian: (4) H eff (p, c) sin(c µ)2 p µ 2, µ = λ/ p (5) Quantum evolution of state (4) is well approximated by classical evolution by H eff. Moreover the framework of QRLG produces the same effective Hamiltonian [Alesci, Cianfrani 14] Can we repeat this effective dynamics program in the full theory? 3/26

4 Outline of the talk Can we repeat this effective dynamics program in the full theory? 1 Choose a suitable full-theory coherent state Ψ (c,p) which resembles isotropic cosmology peaked on (c, p) 2 Calculate the expectation values of basic polynomials in (ĥ, Ê) on said Ψ (c,p), e.g. kinematical observables like the volume 3 Plug everything together to obtain the expectation value of the physical Hamiltonian Ĥ (c,p), i.e. a candidate for the effective Hamiltonian 4 Show that Ψ (c,p) also stay peaked under evolution of Ĥ (OPEN) 5 Compare the effective dynamics of LQC with the one of LQG (or at least of a toy-model) 4/26

5 (brief) Recap of LQG We perform a canonical quantization of general relativity in its formulation of the canonical pair (A I a, EI a ) known as the Ashtekar-Barbero variables The kinematical Hilbertspace H kin is the set of all graphs γ, where on every edge e lives an square integrable function over SU(2), i.e. H kin = e γ H e, H e = L2 (SU(2), dµ H ) (6) Holonomy- and flux-operators act on them in the usual way: ĥ ab (e)f e (g) := D (1/2) ab (g)f e (g) (7) Ê k (S e )f e (g) := i κβ d 4 ds s=0 f e (e isσ k g) (8) 5/26

6 The graph To perform the effective dynamics program, we have to pick a suitable state: v A cubic lattice, i.e. every vertex has valence 6, with periodic boundaries, i.e. a 3-torus T 3 = S 1 S 1 S 1 We embed each edge e along the coordinate axes Associate with every edge the coordinate length µ Finally, the SU(2)-function on each edge is assigned to a semi-classical coherent state of the classical variables along this path. 6/26

7 Complexifier coherent states The complexifier coherent states ψ e,he for gauge group SU(2) are given on each edge e by [Thiemann, Winkler 00] ψ t e,h e (g) := 1 ψ e,he j (2j + 1)e j(j+1)t j=0 m= j e izm D (j) mm(n egn e) with the semiclassicality parameter t being any dimensionless number (e.g. t = l 2 p/a 2 ) and the SL(2, C) element: h e = n e e izeσ3/2 n e (9) For the cosmological coherent state we choose on every link n e = n e SU(2) and z = µc iµ 2 a 2 β 1 p Ψ (c,p) := (10) e Z 3 M ψ t e,n e exp ( izσ 3 /2)n e 7/26

8 Calculation of the expectation values In order to obtain the expectation values one needs knowledge of the SU(2)-calculus, e.g.: D (j 1) ab (g)d(j 2) cd (g) = (11) j 1 +j 2 ( ) ( ) = d j ( 1) m n j1 j 2 j j1 j 2 j D (j) a c m b d n m n (g) and j= j 1 j 2 dµ H (g)d (j) mn (g)d (j ) m n (g) = 1 d j δ jj δ mm δ nn (12) which allows us to calculate h e = n exp(ξ + iη)n ψ e,he, ĥ ab ψ e,he = e iξc D (1/2) acb (n)d(1/2) cb (n )e η(m+m ) (13) ( d j d j e t[j(j+1)+j (j +1)]/2 1/2 j j c m m j,j ) 2 8/26

9 Poisson-Summation formula Theorem: (Poisson Summation Formula) Let f L 1 (R, dx), then f (ns) = dxe i2πmx f (sx) (14) n Z m Z R Applying this changes e tn(n+1)/2 e 4π2 m 2 /t which decays for t 0 extremly fast unless m = 0. We conclude that for 1 t the sum over m collapses to only one term! Evaluating said one yields finally: ψ e,he, ĥ ab ψ e,he = D (1/2) ab (ne iµcσ 3/2 n ) (15) [1 + t 3 16 t a2 β 4µ 2 p tanh( µ2 p 2a 2 β ) + O(t2 )] 9/26

10 Polynomials of the fluxes I To continue with computing the expectation value of monomials of the fluxes Ê k, one needs two observations: Theorem: (Cyclicity Property) Be ψ h as coherent state with h = e iµcσ 3/2, then ψ h, Ê k 1... Ê k N ψ h = e 2k N µ 2 p a 2 β ψ h Ê k N Ê k 1... Ê k N 1 ψ h (16) Further, as Ê k is subject to the algebraic relations (k i 0) [Ê k 1, Ê k 2 ] i(k 1 k 2 )Ê 0, [Ê k, Ê 0 ] 2ikÊ k (17) we could also permute the Ê k N in (16) accordingly! 10/26

11 Polynomials of the fluxes II Combining (16) and (17) suitably one finds at the end: [ ψ he, Ê k 1... Ê k N ψ he = D (1) k 1 s 1 (n)... D (1) k N s N (n) δ s δs N 0 (i η ) N N i sinh(η) A<B=1 2e t/4 ( i κβ 4 δ s 1.. s A.. s B..s N 0 (δ s A 1 δ s B 1 e η + δ s A 1 δ s B 1 e η )(i η ) N 1] ) N π ηe η2/t t 3 sinh(η) η= µ 2 p a 2 β This gives peakedness on the classical phase space variable plus additional quantum correction for any N. E.g. N = 1: (18) ψ e,he, Ê k ψ e,he = µ 2 pd (1) k0 (n)[1 + t( a2 β µ 2 p + coth(µ2 p a 2 β )) + O(t2 )] (19) 11/26

12 The Volume Operator I Example: A kinematic operator, which is crucial for defining the dynamics, is the Volume! A definition which is consistent with the flux-operators is the Ashtekar-Lewandowski volume operator [ 98] ˆQ v (20) ˆV (R) := v R ˆQ v := i ˆV AL v e e e =v v R ɛ(e, e, e )ɛ ijk Ê i (e)ê j (e )Ê k (e ) (21) The square-root makes analytical treatment nearly impossible as long as we don t know the spectrum of ˆQ v! Is there a work-around? 12/26

13 The Volume Operator II Maybe a power-series of.? An alternative definition instead of Giesel-Thiemann volume operator 2k+1 ˆV k,v GT := ˆQ v [1 1/2 + n=1 ( 1) n+1 n! ˆV AL v = ˆQ v we call the ] (n )( ˆQ v 2 ˆQ v 2 1)n The amazing feature is, that for the expectation values of any coherent state Ψ (c,p) and any polynomial P: [Giesel, Thiemann 06] Ψ (c,p), P( AL ˆV v, {ĥ})ψ GT (c,p) = Ψ (c,p), P( ˆV k,v, {ĥ})ψ (c,p) + O(t k+1 ) This allows us to replace ˆV v AL with e.g. ˆV 1,v GT and to compute every expectation value correctly up to O(t 2 )! 13/26

14 The Volume Operator III We are now in the situation that we have on six edges a polynomial of Ê and can thus use (18): Ψ (c,p), ˆQ N v Ψ (c,p) = ( ) N = Ψ (c,p), i N 6(Ê1 I + Ê 1)(Ê I 2 J + Ê 2)(Ê J 3 K + Ê 3)ɛ K IJK Ψ(c,p) = ( ) 2ηi 3N (6i) N [2 N + t t 2η 2 (N2 + 3N)2 N 2 t 2η N2N coth(η)] 3 and finally using the definition of the Giesel-Thiemann volume we approximate the Ashetkar-Lewandowski volume operator (with N 3 being the number of vertices in σ) Ψ (c,p), ˆV (σ)ψ (c,p) = N 3 (µ 2 p) 3/2 [1 + t 3a4 β 2 4µ 4 p 2 ( 7 8 µ2 p a 2 β coth(µ2 p a 2 β ) ) + O(t 2 )] (22) 14/26

15 Kinematics: Recap Ψ (c,p) is a state in the full theory, peaked in holonomy and flux of flat RW metric, (c, p): ĥab = D (1/2) ab (ne iµcσ3/2 n )[1 + O(t)] Ê k = µ 2 pd (1) k0 (n)[1 + O(t)] We can compute expectation value of polynomials in holonomy and polynomials in flux. E.g., relative dispersions: δh ab := ĥ ab ĥ ab ĥ ab 2 1 = t f (p, c, n) δe k := Ê k Ê k 1 = t g(p, n) Ê k 2 We can compute expectation value of (G-T) volume operator: ˆV (σ) = N 3 (µ 2 p) 3/2 [1 + O(t)] 15/26

16 Dynamics: The Quantum Hamiltonian We take a non-graph-changing regularization [Giesel, Thiemann, 06] of the scalar-constraint as quantized by Thiemann [ 98] Ĥ = Ĥ E (β 2 + 1)Ĥ L where Ĥ E (N v ) Ĥ L (N v ) v V (γ) N v e e e =v ɛ(e, e, e ) ( Tr (ĥ( ee ) ĥ( ee ) 1)ĥ(e ) [ĥ(e ) 1, v V (γ) N v e e e =v ɛ(e, e, e ) ( ) Tr ˆK(e) ˆK(e )ĥ(e )[ĥ(e ) 1, ˆV GT 1,v ] ]) ˆV GT 1,v with the extrinsic curvature ˆK ab (e) given by ˆK(e) = ĥ(e)[ĥ(e) 1, [Ĥ E (1), ˆV GT 1,v ]] 16/26

17 Action of the Euclidean part Ĥ E (N v ) Tr v V (γ) N v e e e =v ɛ(e, e, e ) ( (ĥ( ee ) ĥ( ee ) 1)ĥ(e ) [ĥ(e ) 1, ]) GT ˆV 1,v v 17/26

18 Effective Hamiltonian H eff (p, c) := Ĥ E (β 2 + 1) Ĥ L + O(t) Isotropic cosmology choose lapse function N v = 1. One finds: Ĥ E = N 3 µ ( p sin(cµ) [c 2 coth(c 3 µ 2 ) ] p) t c 2 µ 2 c 4 p (µ 2 p) 2 + O(t 2 ) Identify the number of vertices with the inverse coordinate volume (N = µ 1 ): ĤE p sin(cµ)2 µ 2 [1 + O(t)] In the classical limit t 0 we obtain a lattice theory, which equals the effective one of LQC (in what scheme? Depends on what µ is...) In the classical&continuum-limit t, µ 0 we obtain the classical cosmological Hamiltonian pc 2 18/26

19 The Effective Lorentzian Hamiltonian Recall that we also have Lorentzian part! H eff (p, c) p sin(cµ)2 µ 2 (β 2 + 1) Ĥ L + O(t) Due to limited space, we only present the leading order: Ĥ L 1 4β 2 p sin(2cµ) 2 µ 2 + O(t) Again we find a term, that reduce to the classical function for t, µ 0 HOWEVER: it is different from the one euclidean one, and not accounted for in LQC! 19/26

20 Full Effective Hamiltonian I Effective Hamiltonian from full theory: H eff (p, c) p sin(cµ)2 µ 2 β β 2 p sin(2cµ) 2 µ 2 + O(t) 1 β 2 p sin(cµ) 2 µ 2 [ 1 (1 + β 2 ) sin(cµ) 2] (23) Genuinely different from LQC effective dynamics. It can be reproduced within LQC (with µ µ) from the following operator: Ĥ new LQC ˆN λ ˆN λ 2λ ˆν ˆN λ ˆN λ ˆν β λ 4 ˆN 2λ ˆN 2λ 2λ ˆν ˆN 2λ ˆN 2λ ˆν 2λ 20/26

21 Full Effective Hamiltonian II Action of Ĥ new LQC on Ψ(ν) H LQC is quartic equation: Ĥ new LQC Ψ(ν) C +(ν)ψ(ν + 4λ) + D 0 (ν)ψ(ν) + C (ν)ψ(ν 4λ)+ + D + (ν)ψ(ν + 8λ) + D (ν)ψ(ν 8λ) Similar operator obtained within LQC [Yang, Ding, Ma 09] by a more Thiemann-like quantization of H L. passes von Neumann test on numerical stability suited for numerical evolution by LQC methods. ĤLQC new Will the quantum evolution be well-approximated by the effective Hamiltonian H eff (p, c)? We do not know yet. 21/26

22 Toy Model Take seriously this effective Hamiltonian: what evolution does it generate? Add a scalar field φ: H = H φ + H eff = p2 φ 2p 3/2 3 sin(c µ) 2 [ p 1 (1 + β 2 8πβ 2 µ 2 ) sin(c µ) 2] 0 Hamilton equations solved numerically (plot for β = 0.12, p φ = 1): v v t ϕ (red = toy model; blue = LQC; green = GR) 22/26

23 Conclusions: what was shown Tools to compute expectation values on complexifier CS s Proposal for such CS s representing Robertson-Walker geometry (with k = 0) Application of tools to geometric operators (e.g., Volume) Application of tools to scalar constraint cosmological effective Hamiltonian H eff (p, c) = c 1 p sin(cµ) 2 µ 2 c 2 p sin(2cµ) 2 µ 2 + O(t) If lattice length µ 0, H eff = H class. If instead µ = µ, then H eff = H LQC [1 (1 + β 2 ) sin(c µ) 2 ]. Proposal of operator ĤLQC new on H LQC that corresponds to H eff. It produces a numerically stable 4th order difference equation. The evolution generated by H eff coincides with effective LQC (and GR) today, but in the early universe we have difference big bounce scenario is modified. 23/26

24 Conclusions: what to do next As for volume: ideas to obtain matrix elements from expectation values. As for Hamiltonian: modification to LQC effective dynamics is due to the Lorentzian term, which we regularize à la Thiemann. Different regularization (i.e., Warsaw Hamiltonian) will lead to different result, even at leading order! Must fix this arbitrariness (e.g., by renormalization group). Study quantum evolution generated by Ĥnew LQC. Extend to: k 0; anisotropic cases. Extend beyond cosmology (e.g., spherical symmetry). 24/26

25 THANK YOU! 25/26

26 Why sin(2cµ) 2? The extrinsic curvature ˆK(e) = ĥ(e)[ĥ(e) 1, [ĤE (1), ˆV v ]] e cµτ {e cµτ, {H E (1), p 3/2 }} Since H E (1) p sin(cµ) 2 /µ 2, one finds K(e) = e cµτ {e cµτ, {H E (1), p 3/2 }} 3 2µ 2 ecµτ {e cµτ, p}{sin(cµ) 2, p} 3 2 sin(cµ) cos(cµ)τ 2 sin(2cµ)τ So H L = ɛ(e, e, e ( )Tr K(e)K(e )h(e ){h(e ) 1, V v } ) v e e e N 3 µ p sin(2cµ) 2 ɛ(e, e, e )Tr(τ e τ e τ e ) e e e p sin(2cµ)2 µ 2 26/26