Elementary particle states in homogeneous cosmology Zhirayr - PowerPoint PPT Presentation

Elementary particle states in homogeneous cosmology Zhirayr Avetisyan Department of Mathematics, UCL 40th LQP Workshop, Leipzig’17 Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 1 / 26

Contents 1. Wigner particle states 2. QM in homogeneous cosmology 3. Quantum field theory in homogeneous cosmology Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 2 / 26

Wigner elementary particle states Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 3 / 26

Wigner elementary particle states Unitary representations of groups G - locally compact type I group H - separable complex Hilbert space ρ : G → U ( H ) - strongly cont. unitary rep. ˆ G = { π : G → H π π unitary irrep } / [ ≃ ] Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 4 / 26

Wigner elementary particle states Unique decomposition µ measure on ˆ G , function m : ˆ ∀ ρ unirep, ∃ ˆ G → N 0 and unitary ∫ ⊕ ⊕ F : H → d ˆ µ ( π ) H π ˆ G n π such that ∫ ⊕ F ◦ ρ ◦ F − 1 = ⊕ d ˆ µ ( π ) π. ˆ G n π Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 5 / 26

Wigner elementary particle states Spectral theorem H : D → H self-adjoint, D ⊂ H dense Stone’s theorem: U t = e ı tH , strongly cont. unirep U : R → U ( H ) ˆ R = { π ( x ) = e ıπ x , π ∈ R } , H π = C ∫ ⊕ ∃ ˆ µ , ∃ n π , ∃F : H → R d ˆ µ ( π ) n π C such that ∫ ⊕ ∫ ⊕ F ◦ U t ◦ F − 1 = F ◦ H ◦ F − 1 = µ ( π ) e ıπ t , d ˆ d ˆ µ ( π ) π R R Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 6 / 26

Wigner elementary particle states Wigner’s elementary particle concept In quantum mechanics, ∀ f ∈ H , C f - proper (vector) state If ˆ µ discrete (e.g., G compact) then ∫ ⊕ ⊕ ⊕ ⊕ F H = d ˆ µ ( π ) H π = µ ( π ) ˆ H π , ˆ G n π n π ˆ G so that F − 1 H π ⊂ H subspace Then ∀ f ∈ H π , C f - proper elementary particle state with ’momentum’ π More generally, ∀ f ∈ H π , C f - improper elementary particle state with ’momentum’ π Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 7 / 26

Wigner elementary particle states Eigenfunction expansion D ⊂ H ⊂ D ′ Gelfand triple for H self-adjoint ν π ) and { ξ ω } ω ∈ Ω π ∈ D ′ such that ∀ π ∈ R , ∃ (Ω π , ˆ ∫ ⊕ ⊕ H π ≃ d ˆ ν π ( ω ) C ξ ω Ω π n π H ξ ω = πξ ω , ω ∈ Ω π . Improper eigenfunctions. For H = ∆ on H = L 2 ( R ) , D = C ∞ 0 ( R ) , ξ ω ( x ) = e ıω x ∈ C ∞ ⊂ C ∞ 0 ( R ) ′ , ω ∈ Ω π = {± π } Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 8 / 26

Wigner elementary particle states Paley-Wiener theorem M a G -homogeneous space, µ a left G -invariant measure Gelfand triple C ∞ 0 ( M ) ⊂ L 2 ( M , µ ) ⊂ C ∞ 0 ( M ) ′ How to decompose { } C ∞ π ∈ ˆ 0 ( M ) ⇔ D π G ? Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 9 / 26

Wigner elementary particle states Zonal functions M a G -homogeneous space, µ a left G -invariant measure H self-adjoint G -invariant operator with Gelfand triple C ∞ 0 ( M ) ⊂ L 2 ( M , µ ) ⊂ C ∞ 0 ( M ) ′ Invariance of H implies (formally) for ∀ π ∈ ˆ G , ∫ ⊕ ⊕ H π ≃ d ˆ ν π ( ω ) C ξ ω Ω π n π For M = G compact, ˆ G is discrete. Peter-Weyl theorem ⊕ ⊕ ν π ( ω ) C ξ ω ⊂ C ∞ 0 ( G ) = C ∞ ( G ) ˆ µ ( π ) ˆ π ∈ ˆ ω ∈ Ω π G is dense. Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 10 / 26

QM in homogeneous cosmology Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 11 / 26

QM in homogeneous cosmology Homogeneous cosmological spacetimes Assume that M - smooth connected 4-dim Lorentzian manifold G = Iso 0 ( M ) - connected type I Lie group ∀ m ∈ M , the orbit Gm ⊂ M an immersed 3-dim Riemannian submanifold Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 12 / 26

QM in homogeneous cosmology Homogeneous cosmological spacetimes It follows that M ≃ R × Σ , { t } × Σ = G · { t } × Σ an orbit ∀ t ∈ R Lorentzian metric g ( t , x ) = dt 2 − h t ( x ) (Σ , h t ) - Riemannian G -homogeneous space ∀ t ∈ R M - globally hyperbolic spacetime Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 13 / 26

QM in homogeneous cosmology Time separation in KG field Klein-Gordon field ( � + m 2 ) ϕ ( x , t ) = 0 � = D t − ∆ t + m 2 H t = − ∆ t + m 2 - a G -invariant Hamiltonian ∀ t ∈ R µ t = det h t = A ( t ) µ 0 Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 14 / 26

QM in homogeneous cosmology Time separation in KG field [ H t , H t ′ ] = 0 on C ∞ 0 (Σ) C ∞ 0 (Σ) ⊂ L 2 (Σ , µ 0 ) ⊂ C ∞ 0 (Σ) Σ ⊂ C ∞ (Σ) , Eigenfunctions { ξ ω ( x ) } ω ∈ ˆ H t ξ ω = λ ω ( t ) ξ ω Mode decomposition † ∫ ϕ ( x , t ) = d ˆ µ ( ω ) ˆ ϕ ( ω ) T ω ( t ) ξ ω ( x ) ˆ Σ † Z.A., ’A unified mode decomposition method for physical fields in homogeneous cosmology’, Rev. Math. Phys. 26(3), 2014 Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 15 / 26

QM in homogeneous cosmology Time separation in vacuum EM field Vacuum Maxwell’s equations F ∈ Ω 2 ( M ) dF = 0 , δ F = 0 , If H 1 (Σ) = 0 then F = dA , A ∈ Ω 1 ( M ) and δ dA = 0 Lorentz gauge δ A = 0, � A = − ( δ d + d δ ) A = 0 Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 16 / 26

QM in homogeneous cosmology Time separation in Dirac field Vacuum Dirac equations ( − ı / ∇ + m 0 ) ( ϕ ) ( ϕ, ψ ) ∈ C ∞ ( DM ⊕ D ∗ M ) = 0 , ı / 0 ∇ + m ψ It follows ( � L + m 2 ) ( ϕ ) 0 = 0 � L + m 2 0 ψ � L = � + 1 4 R - Lichnerowicz formula Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 17 / 26

QM in homogeneous cosmology Dynamical elementary particle states H t ξ ω = λ ω ( t ) ξ ω If ξ ω ∈ L 2 (Σ , µ 0 ) then proper eigenstate C ξ ω - dynamical elementary particle state Consistent in time Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 18 / 26

QM in homogeneous cosmology Symmetry elementary particle states U : G → U ( L 2 (Σ , µ t )) unirep U g f ( x ) = f ( g − 1 x ) , ∀ f ∈ L 2 (Σ , µ t ) ∀ g ∈ G , ∀ x ∈ Σ , Fourier transform ∫ ⊕ ⊕ F L 2 (Σ , µ t ) = d ˆ µ t ( π ) H π ˆ G n π ∀ f ∈ H π , C f - symmetry elementary particle state with ’momentum’ π Consistent in time: ˆ µ t = A ( t )ˆ µ 0 , n π = const Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 19 / 26

QM in homogeneous cosmology Consistency of dynamics and symmetry If G compact (e.g., SU ( 2 ) or SO ( 3 ) ) then Peter-Weyl theorem: ⊕ ω ∈ Ω π } = C ∞ ( G ) C { ξ ω π ∈ ˆ G If G = E + ( 3 ) = SO ( 3 ) ⋊ R 3 then ∫ ⊕ ξ ω ( x ) = e ıω · x , d ω C e ıω · x H π ≃ | ω | = π Results known for symmetric spaces (Harish-Chandra, Helgason etc.) and Bianchi models † † Z.A., R. Verch, ’Explicit harmonic and spectral analysis in Bianchi I-VII type cosmologies’, Class. Quant. Grav. 30(15), 2013 Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 20 / 26

QFT in homogeneous cosmology Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 21 / 26

QFT in homogeneous cosmology Symplectic representations Cauchy data V . For KG field V = C ∞ 0 (Σ) ⊕ C ∞ 0 (Σ) Conserved symplectic form σ on V (Bosonic) Dynamics: V t : R → Sp ( V , σ ) Symmetry: V g : G → Sp ( V , σ ) Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 22 / 26

QFT in homogeneous cosmology Covariant quantization CCR quantization ( V , σ ) → A Sp ( V , σ ) → Aut ( A ) Dynamics: α t : R → Aut ( A ) Symmetry: α g : G → Aut ( A ) Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 23 / 26

QFT in homogeneous cosmology Invariant equilibrium states Equilibrium state ω ◦ α t = ω , ∀ t ∈ R Invariant state ω ◦ α g = ω , ∀ g ∈ G In the GNS rep ( π ω , H ω , Ω ω ) α t ( A ) = U ∗ α g ( A ) = U ∗ t AU t , g AU g , ∀ A ∈ A , U t : R → U ( H ω ) , U g : G → U ( H ω ) Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 24 / 26

QFT in homogeneous cosmology Elementary particle states U t = e ı tH , H self-adjoint on H ω [ α t , α g ] = 0 hence [ U g , H ] = 0 Decomposition ∫ ⊕ ⊕ F H ω = d ˆ µ ( π ) H π ˆ G n π No natural Gelfand triple for H Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 25 / 26

Thank you. Zhirayr Avetisyan (UCL) Particles in cosmology 40th LQP Workshop, Leipzig’17 26 / 26