The quantum billiards near cosmological singularities of (super-)gravity theories

Transcription

1 The quantum billiards near cosmological singularities of (super-)gravity theories 8. Kosmologietag, IBZ, Universität Bielefeld 25. April 2013 Michael Koehn Max Planck Institut für Gravitationsphysik Albert Einstein Institut Potsdam

2 Overview 1 Introduction: arithmetic cosmological billiards 2 Hyperbolic billiards: shape and volume [ P. Fleig, MK, H.Nicolai Lett Math Phys 100, 2011] 3 Quantum cosmological billiards A.Kleinschmidt, MK, H.Nicolai [ Phys Rev D80, 2009 ] 4 Dynamics of relativistic quantum wavepackets [ MK Phys Rev D85, 2012] 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 0

3 Gravitational Models Action (bosonic sector) S[g µν, φ, A (p) ] d D x [ g R(g) µφ µ φ 1 ] 1 2 (p + 1)! eλpφ F µ (p) 1 µ p+1 F (p)µ 1 µ p p Examples: D = 4 pure gravity, D = 11 supergravity Hamiltonian description [ R.Arnowitt,S.Deser,C.Misner 1962 ] (d + 1)-decomposition of spacetime: Spatial metric h ij, lapse N(τ, x k ), shift vector N i (τ, x k ) Στ2 N n µ τ µ Στ1 ds 2 = N 2 dτ 2 + h ij(dx i + N i dτ)(dx j + N j dτ) N i Constraints Hamiltonian constraint H 0 Diffeomorphism constraints, Gauss constraint 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 1

4 Spacelike Singularities Singularity Theorems BKL Conjecture Cosmological Billiards Hawking, Penrose [ 1970 ] Belinskii, Khalatnikov, Lifshitz 1970,1982 ][ Misner 1969 ] [ Damour,Henneaux,Nicolai 2003 ] [ Decoupling of spatial points close to a space-like singularity: t t t t = t > t t = t > t t = t > 0 t = 0 (Proper time dt = Ndτ) Ultralocal dynamics ds 2 = N(τ) 2 dτ 2 + d i=1 e 2βi (τ) dx i 2 (Pseudo-Gaussian gauge: N i = 0) Degrees of freedom reduced to space of dynamic scale factors β i H = H 0 + V 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 2

5 Sharp Walls in the BKL limit H = H 0 + V β 1 Lorentzian signature in β-space Hyperbolic coordinates β µ = ργ µ ρ 2 = β µ β µ γ µ γ µ = 1 ρ 2 = 2 ρ 2 = 1 Singularity at ρ β 2,..., β d Effective potential V = Cρ 2 exp( ρw(γ)) 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 3

6 Sharp Walls in the BKL limit H = H 0 + V β 1 Lorentzian signature in β-space Hyperbolic coordinates β µ = ργ µ ρ 2 = β µ β µ γ µ γ µ = 1 ρ 2 = 2 ρ 2 = 1 Singularity at ρ β 2,..., β d Effective potential V = Cρ 2 exp( ρw(γ)) V γi 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 3

7 Sharp Walls in the BKL limit H = H 0 + V β 1 Lorentzian signature in β-space Hyperbolic coordinates β µ = ργ µ ρ 2 = β µ β µ γ µ γ µ = 1 ρ 2 = 2 ρ 2 = 1 Singularity at ρ β 2,..., β d Effective potential V = Cρ 2 exp( ρw(γ)) V V γi γi 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 3

8 Sharp Walls in the BKL limit H = H 0 + V β 1 Lorentzian signature in β-space Hyperbolic coordinates β µ = ργ µ ρ 2 = β µ β µ γ µ γ µ = 1 ρ 2 = 2 ρ 2 = 1 Singularity at ρ β 2,..., β d Effective potential V = Cρ 2 exp( ρw(γ)) V V V γi γi γi 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 3

9 Billiard dynamics in β-space Effective Lagrangian for β i (t) L = 1 2 n 1 G µν βµ βν + V Spatial inhomogeneities, off-diagonal metric elements, matter fields Free null motion interspersed by specular reflections Decoupled homogeneous dynamics at each spatial point Chaotic metric oscillations if billiard wedge inside light cone 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 4

10 Cosmological Kac-Moody billiards Dictionary [ T.Damour,M.Henneaux 2000 ] Cartan matrix A ij = 2 (α i,α j ) (α j,α j, det A > 0, det A = 0, det A < 0 ) Root space has Lorentzian signature for det A < 0 gravity Kac-Moody algebra g ++ β-space billiard bounces billiard table root space Weyl reflections Weyl chamber KMA of hyperbolic type chaotic billiard motion [ T.Damour,M.Henneaux, B.Julia,H.Nicolai, 2001 ] cf. hidden symmetries revealed upon dimensional reduction D = 4 pure gravity a 1 ++ e 10 e 8 ++ D = 11 supergravity April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 5

11 Shape Billiard table on hyperboloid F(PSL 2(O A )) on H(A) z j u (j) + iv (j) = λ j n j + i 1 λ2 j n 2 j z 0 = i z 1 = +i Hyperbolic billiard volume and shape P. Fleig,MK,H.Nicolai [ Lett Math Phys, 2011] v Volume λ1 λ2 Complication: vol(f 0) = 1 N dvol(u, v) = dn udv v n+1 N dt 1 dt N (1 det S t i S ij t j ) N 2 u2 a2 Standard simplex N := { (t 1,..., t N ) t i 0, t i 1 } S ij = (2m im j) 1 (B 1 ) ij B ij = A ij(α j, α j) λ2 λ1 θ u1 Low rank: L functions, Higher rank: numerical agreement a1 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 6

12 Canonical Dirac quantization L β µ Wheeler-DeWitt equation ĤΨ({β µ }) = 0 µ µψ = ( 2 Quantum Cosmological Billiards = π µ i i β µ µ Ω 2 [ A.Kleinschmidt,MK,H.Nicolai Phys Rev D, 2009 ] ) d 2 i=2 β Ψ = 0 i2 Problem: Ω-dependent boundary conditions hyperbolic coordinates (ρ, z) [ ( ] ρ 1 d ρ ρ d 1 ρ) ρ 2 LB Ψ(ρ, z) = 0 set Ψ(ρ, z) = R(ρ)F (z) ( Radial equation ρ 3 d ρ ρ d 1 ρ) R(ρ) = k 2 R(ρ) Angular equation LB F (z) = k 2 F (z) β1 v ρ 2 = 2 ρ 2 = 1 β2,..., βd u 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 7

13 Solutions for radial part: R ±(ρ) = ρ 2 d ±i k 2 e 2 ( d 2 2 )2 log ρ Singularity Avoidance [ A.Kleinschmidt,MK,H.Nicolai Phys Rev D, 2009 ] = Ψ = R(ρ)F (z) 0 for ρ Dirichlet b.c. on angular part = k 2 ( d 2 2 Ψ generically complex and oscillating Analytic continuation beyond singularity impossible Positive norm with R +(ρ) ) April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 8

14 Solutions for radial part: R ±(ρ) = ρ 2 d ±i k 2 e 2 ( d 2 2 )2 log ρ Singularity Avoidance [ A.Kleinschmidt,MK,H.Nicolai Phys Rev D, 2009 ] = Ψ = R(ρ)F (z) 0 for ρ Dirichlet b.c. on angular part = k 2 ( d 2 2 Ψ generically complex and oscillating Analytic continuation beyond singularity impossible Positive norm with R +(ρ) ) 2 Ψ(ρ, z) = ±Ψ(ρ, w I(z)) Ψ is superposition of odd automorphic Maass waveforms for PSL 2(O A ) Discrete spectrum for LB (cf. [Forte 2008] for D = 4) [Sarnak 1995] 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 8

15 Relativistic wavepackets as quantum billiards [ MK Phys Rev D85, 2012] Flat Space: Moving Domain walls Hyperbolic space: Complicated shape Analytic solutions for spectral problem out of reach Numeric investigation of quantum wavepackets 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 9

16 D = 4 pure gravity quantum billiards: Flat Space Classical trajectory 18 β 12 a 6 c Ω 0 Ω 1 b β + Diagonal coordinates in β-space: (Ω, β +, β ) 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 10

17 D = 4 pure gravity quantum billiards: Flat Space Classical trajectory 18 β 12 a 6 c Ω 0 Ω 1 b β + Initial conditions 1) Ψ Ω0 = 2π Ae β 2 c 2 2c 2 +i p 0( β β 0 ) 2) ( Ω Ψ) Ω0 = ia d2 p[ω( p) p β] e c 2 Diagonal coordinates in β-space: (Ω, β +, β ) 2 ( p p 0) 2 +i p ( β β 0 ) 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 10

18 Data Sheets: Flat space Classical trajectory Wavepacket trajectory β 12 6 β β β April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 11

19 Data Sheets: Flat space Classical trajectory Wavepacket trajectory β 12 6 β β + Supremum β + Variance sup ψ σ Ω Ω 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 11

20 D = 4 pure gravity quantum billiards: Hyperbolic space v 2 v 1.5 a 1 b c u u 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 12

21 D = 4 pure gravity quantum billiards: Hyperbolic space [ ] ρ ρ ρ 2 ρ LB Ψ(ρ, u, v) = 0 ( ) LB = v u 2 v 2 3 Initial conditions v ) Ψ(ρ 0, u, v) = A d2 pe c 2 2) ρ 0 2 ( ρψ)(ρ 0, u, v) = iρ 0 2 A 2 ( p p 0) 2 e iρ 0f( p,u,v) d2 pe c 2 2 ( p p 0) 2 f( p, u, v) [ i piγi ] ω( p)γ0 v 1.5 a 1 b c u u 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 12

22 Data sheets: Hyperbolic space Classical trajectory 4 Wavepacket trajectory v 2.5 v a b 1 c u u 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 13

23 Data sheets: Hyperbolic space Classical trajectory Wavepacket trajectory sup ψ Supremum Ω Energy v 2.5 v 2.5 H Ω a b 1 c u u ( H) Ω 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 13

24 Interpretation of the results Flat hyperbolic: identical results Wavefunction vanishes as singularity is approached Loss of localization of relativistic wave packet: Transversal spreading Sudden spreading via successive redshifts Wavepacket follows classical trajectory and then deviates 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 14

25 Interpretation of the results Flat hyperbolic: identical results Wavefunction vanishes as singularity is approached Loss of localization of relativistic wave packet: Transversal spreading Sudden spreading via successive redshifts Wavepacket follows classical trajectory and then deviates Energy expectation value and variance remain constant does a highly-excited semiclassical state remain highly-excited and semiclassical all the way into the singularity? [ C.Misner 1969 ] Suggestion: loss of localization and vanishing of Ψ are better indicators for a quantum resolution of the classical singularity 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 14

26 Summary Singularities: incompleteness of the classical relativity theory Ultralocal billiard dynamics at the singularity reveals algebraic and arithmetic structures 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 15

27 Summary Singularities: incompleteness of the classical relativity theory Ultralocal billiard dynamics at the singularity reveals algebraic and arithmetic structures Direct method for shape and hyperbolic volumes 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 15

28 Summary Singularities: incompleteness of the classical relativity theory Ultralocal billiard dynamics at the singularity reveals algebraic and arithmetic structures Direct method for shape and hyperbolic volumes Canonical quantization entails Ψ 0 at the singularity Ψ is an odd Maass waveform automorphic under PSL 2(O A ) Emergence of space through algebraic concepts? 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 15

29 Summary Singularities: incompleteness of the classical relativity theory Ultralocal billiard dynamics at the singularity reveals algebraic and arithmetic structures Direct method for shape and hyperbolic volumes Canonical quantization entails Ψ 0 at the singularity Ψ is an odd Maass waveform automorphic under PSL 2(O A ) Emergence of space through algebraic concepts? Complexity and positive frequencies: arrow of time? Analytic solutions not accessible except in 1d hyperbolic space Numerical solutions reveal detailed insights about the possible avoidance of the classical singularity through quantum effects 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 15

30 The quantum billiards near cosmological singularities in (super-)gravity theories 8. Kosmologietag, IBZ, Universität Bielefeld 25. April 2013 Michael Koehn Max Planck Institut für Gravitationsphysik Albert Einstein Institut Potsdam

31 Algebras and Roots Algebra sl 2(C): Set of commutation relations amongst the generators of the algebra [L 0, L ±] = ±2L ± [L +, L ] = L 0 with L ± = L 1 ± il 2 L 0 = 2L 3 [L i, L j] = iε ijk L k A simple root α of the algebra is defined as: ad L0 (L +) [L 0, L +] = αl + α = 2 Simple roots contain complete information about the algebra 25. April 2013 Michael Koehn: The quantum billiards near cosmological singularities of (super-)gravity theories 17