Effective Dynamics of Loop Quantum Gravity: Bouncing Black Holes and Gravitational Phonons Andrea Dapor* Louisiana State University Jurekfest, Warsaw, September 20th 2019 Phys. Lett. B 785, 506-510 (2018) Phys. Rev. Lett. 121, 081303 (2018) arXiv:1906.05315 arXiv:1908.05756 *with Mehdi Assanioussi, Wojciech Kami´ nski, Klaus Liegener and Tomasz Paw� lowski This work is supported by the National Science Foundation and the Hearne Institute for Theoretical Physics at LSU
introduction QG on a graph cosmology BH’s GW’s conclusion outline introduction: what is effective dynamics? 1 semiclassical states in quantum gravity on a graph 2 example: homogeneous cosmology 3 example: spherical black hole interior 4 example: linearized Einstein equations 5 conclusion 6 2 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion outline introduction: what is effective dynamics? 1 semiclassical states in quantum gravity on a graph 2 example: homogeneous cosmology 3 example: spherical black hole interior 4 example: linearized Einstein equations 5 conclusion 6 3 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion A “semiclassical state” in QM: e − ( x − xo )2 1 + ip o ( x − x o ) ψ ( x o , p o ) ( x ) = ǫ √ π 2 ǫ 2 � Peakedness: � ψ ( x o , p o ) | ˆ � ψ ( x o , p o ) | ˆ X | ψ ( x o , p o ) � = x o , P | ψ ( x o , p o ) � = p o and � � ∆ X 2 ∆ P 2 1 ǫ δ X := X � 2 = √ ≪ 1 , δ P := P � 2 = √ ≪ 1 � ˆ � ˆ 2 x o 2 ǫ p o which is achieved if p o ≫ ǫ − 1 ≫ 1. For any polynomial operator ˆ A = A ( ˆ X , ˆ P , [ · , · ]), we have � ψ ( x o , p o ) | ˆ A | ψ ( x o , p o ) � = A ( x o , p o , i {· , ·} ) [1 + O ( δ X + δ P )] 4 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion What do we meen by “effective dynamics”? Quantum dynamics: � ψ t ( x o , p o ) | ˆ A | ψ t ( x o , p o ) � where ( x o , p o ) := e − i ˆ ψ t Ht ψ ( x o , p o ) “Classical” dynamics on phase space: a ( t ) = { H eff , a ( t ) } , ˙ a (0) = A ( x o , p o ) where H eff ( x , p ) := � ψ ( x , p ) | ˆ H | ψ ( x , p ) � Effective Dynamics � ψ t ( x o , p o ) | ˆ A | ψ t ( x o , p o ) � = a ( t )[1 + O ( δ X + δ P )] 5 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion a good example (free particle) and a bad one (just everything else) 6 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion outline introduction: what is effective dynamics? 1 semiclassical states in quantum gravity on a graph 2 example: homogeneous cosmology 3 example: spherical black hole interior 4 example: linearized Einstein equations 5 conclusion 6 7 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Effective dynamics in LQG so far: Successfull in LQC [Ashtekar, Pawlowski and Singh, 2006; Taveras, 2008] Conjectured in QRLG [Alesci, Bahrami, Botta, Cianfrani, Luzi, Pranzetti, Stagno] What about the full theory (at least on a fixed graph)? Fix Γ (e.g., cubic lattice) with N edges: kinematical Hilbert space H = L 2 ( SU (2) N , d g 1 .. d g N ) with operators, ˆ h e (multiplication by g e ) and ˆ E I e (right-invariant v.f. R I e ) discrete “geometry”: Collection ( u e , ξ e ) ∈ SU (2) × su 2 for every edge e . Inspired on complexifier coherent states [Sahlmann, Thiemann and Winkler, 2002; Bahr and Thiemann, 2007] , we construct a class of “generalized coherent states”. 8 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Generalized coherent state: Ψ ( u ,ξ ) ∈ H given by N ψ ( u e ,ξ e ) ( g ) = 1 � N e f e ( g ) e − S e ( g ) /ǫ Ψ ( u ,ξ ) ( g 1 , .., g N ) = ψ ( u e ,ξ e ) ( g e ) , e =1 where S e satisfies: Re( S e ) has single minimum at g e , o Hessian R I e R J e S e is non-degenerate at g e , o ⇒ “approximate peakedness” wrt ˆ h e and ˆ E I e : � � �� 1 � Ψ ( u ,ξ ) | ˆ � Ψ ( u ,ξ ) | ˆ E I e | Ψ ( u ,ξ ) � = ξ I � � h e | Ψ ( u ,ξ ) � = u e 1 + O ( ǫ ) , 1 + O e ǫ | ξ e | 2 δ h e ∝ √ ǫ [1 + O ( ǫ )] , 1 � � 1 �� δ E I e ∝ √ ǫ | ξ e | 1 + O √ ǫ | ξ e | e = 1 where u e = g e , o and ξ I ǫ Im( R I e S e )( g e , o ). So, if we take | ξ e | 2 ≫ ǫ − 1 ≫ 1, we find that δ h e , δ E I e ≪ 1, and hence Ψ ( u ,ξ ) is peaked on the discrete “geometry” ( u e , ξ e ). 9 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Effective dynamics conjecture [AD, Kami´ nski and Liegener] Two observations: A is a pdo 1 with principal symbol a , then If ˆ � Ψ ( u ,ξ ) | ˆ A | Ψ ( u ,ξ ) � = a ( u , ξ )[1 + O ( δ h + δ E )] Egorov’s Theorem: Let ˆ B be a positive, self-adjoint, elliptic pdo. Then B ˆ B is a pdo A t := e it ˆ Ae − it ˆ ˆ ˆ A is a pdo = ⇒ If true, then � Ψ t ( u ,ξ ) | ˆ A | Ψ t ( u ,ξ ) � = � Ψ ( u ,ξ ) | ˆ � � A t | Ψ ( u ,ξ ) � = a t ( u , ξ ) 1 + O ( δ h + δ E ) with a t principal symbol of ˆ A t . It follows dt a t ( u , ξ ) ≈ d d ( u ,ξ ) | ˆ ( u ,ξ ) � = i � Ψ ( u ,ξ ) | [ˆ B , ˆ dt � Ψ t A | Ψ t A t ] | Ψ ( u ,ξ ) � ≈ −{ b ( u , ξ ) , a t ( u , ξ ) } with b ≈ � Ψ ( u ,ξ ) | ˆ B | Ψ ( u ,ξ ) � principal symbol of ˆ B . 1 See e.g. L. H¨ ormander, The Analysis of Linear Partial Differential Operators , (1987). 10 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Summary of the conjecture a t given by � Ψ t ( u ,ξ ) | ˆ A | Ψ t � � ( u ,ξ ) � = a t ( u , ξ ) 1 + O ( δ h + δ E ) satisfies a 0 = � Ψ ( u ,ξ ) | ˆ � � a t = { a t , b } , ˙ A | Ψ ( u ,ξ ) � 1 + O ( δ h + δ E ) with b = � Ψ ( u ,ξ ) | ˆ � � B | Ψ ( u ,ξ ) � 1 + O ( δ h + δ E ) This is exactly effective dynamics with effective Hamiltonian b ! Expectation value of ˆ H LQG [Giesel, Thiemann, 2006] on generalized coherent state: H eff ( u , ξ ) := � Ψ ( u ,ξ ) | ˆ H LQG | Ψ ( u ,ξ ) � = � Ψ ( u ,ξ ) | H LQG (ˆ h , ˆ E , [ · , · ]) | Ψ ( u ,ξ ) � = = H µ GR ( u , ξ, i {· , ·} )[1 + O ( δ h + δ E )] 11 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Examples: 1 homogeneous spacetimes (cosmology) 2 spherical black hole interior (BH’s) 3 linearized Einstein equations (GW’s) 12 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion outline introduction: what is effective dynamics? 1 semiclassical states in quantum gravity on a graph 2 example: homogeneous cosmology 3 example: spherical black hole interior 4 example: linearized Einstein equations 5 conclusion 6 13 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion ds 2 = − dt 2 + p ( t )[ dx 2 + dy 2 + dz 2 ] Ashtekar-Barbero variables: A I a = c δ I a and E a I = p δ a I fix the graph: cubic lattice embedded in space along the coordinate axes read off the classical holonomy and flux on each edge: u e = e − c µτ e , ξ I e = δ I e αµ 2 p with µ the coordinate length of each edge By construction, Ψ ( u ,ξ ) is peaked on this “geometry”: � Ψ ( u ,ξ ) | ˆ h e | Ψ ( u ,ξ ) � ≈ e − c µτ e , � Ψ ( u ,ξ ) | ˆ E I e | Ψ ( u ,ξ ) � ≈ δ I e µ 2 p Note – Scale µ is independent of ( c , p ): we are in µ o -scheme. 14 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Effective Hamiltonian [AD and Liegener, 2017] : 3 √ p sin 2 ( µ c ) − (1 + γ 2 ) sin 4 ( µ c ) + O ( δ h + δ E ) � � H eff = − 8 π G γ 2 µ 2 ≈ H eff 1 − (1 + γ 2 ) sin 2 ( µ c ) � � LQC If it were ¯ µ -scheme, then: example of general LQC effective Hamiltonian [Engle and Vilensky, 2018] Equations of motion analytically solvable [Assanioussi, AD, Liegener and Paw� lowski, 2018 and 2019] Volume: √ 2 ( φ ) ∝ 1 + γ 2 cosh 2 ( 12 π G φ ) 3 √ p | sinh( 12 π G φ ) | 15 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion V 10 8 6 4 2 ϕ - 1.0 - 0.5 0.5 1.0 Pre-bounce branch: contracting de Sitter with effective cosmological constant 3 Λ eff = ∆(1 + γ 2 ) Notes: quantum LQC-like model first proposed in [Yang, Ding and Ma, 2009] effective dynamics consistent with quantum dynamics [Assanioussi, AD, Liegener and Paw� lowski, 2018] inclusion of inflaton [Agullo, 2018; Li, Singh and Wang, 2018] 16 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion Generalization to Bianchi I [Garc´ ıa-Quismondo and Mena Marug´ an, 2019] 1 − 1 + γ 2 � p 2 p 3 � � 1 sin( c 2 µ 2 ) sin( c 3 µ 3 ) H eff = A ( c 3 , c 1 ) A ( c 1 , c 2 ) + cyclic 8 π G 4 γ 2 p 1 µ 2 µ 3 where A ( c i , c j ) := cos( c i µ i ) + cos( c j µ j ). k 1 k 1 2 0.7 0.6 1 0.5 0.4 t - 200 - 100 100 200 0.3 0.2 - 1 0.1 t - 20 - 10 - 2 10 20 Notes: Vacuum case: only small differences between LQC and the new model. Matter case: no Kasner transition, pre-bounce dS phase (with same Λ eff ) 17 / 29
introduction QG on a graph cosmology BH’s GW’s conclusion outline introduction: what is effective dynamics? 1 semiclassical states in quantum gravity on a graph 2 example: homogeneous cosmology 3 example: spherical black hole interior 4 example: linearized Einstein equations 5 conclusion 6 18 / 29
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