On the algebraic quantization of a scalar field in anti-de Sitter - PowerPoint PPT Presentation

On the algebraic quantization of a scalar field in anti-de Sitter spacetime Hugo Ferreira Joint work with Claudio Dappiaggi Phys. Rev. D94 (2016) 12, 125016; arXiv:1610.01049 [gr-qc] arXiv:1701.07215 [math-ph] INFN Pavia / Università degli Studi di Pavia 23 June 2017, Leipzig 40th LQP

0. Introduction We discuss the algebraic quantization of a Klein-Gordon field in anti-de Sitter (AdS), a simple example of a non-globally hyperbolic spacetime, extending the work of Avis, Isham, Storey (1978), Allen & Jacobson (1986) and others. We consider Robin boundary conditions at infinity, by treating the system as a Sturm-Liouville problem, complementing the work of Wald & Ishibashi (2004). We show that it is possible to associate an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. We characterize the wavefront set of the ground state and propose a natural generalization of the definition of Hadamard states in AdS.

1 Anti-de Sitter spacetime 2 Klein-Gordon equation and causal propagator 3 AQFT and Hadamard condition for AdS 4 Conclusions

Outline 1 Anti-de Sitter spacetime 2 Klein-Gordon equation and causal propagator 3 AQFT and Hadamard condition for AdS 4 Conclusions

1. Anti-de Sitter spacetime Definition: Anti-de Sitter AdS d +1 ( d ≥ 2 ) is the maximally symmetric solution to the vacuum Einstein’s equations with a negative cosmological constant Λ < 0 . It is defined as the hypersurface in R d +2 with line element d +1 � d s 2 = − d X 2 0 − d X 2 d X 2 1 + i i =2 given by the relation d +1 � = − d ( d − 1) ℓ . i = − ℓ 2 , − X 2 0 − X 2 X 2 1 + . Λ i =2 Anti-de Sitter spacetime is not globally hyperbolic: it possesses a timelike boundary at spatial infinity.

1. Anti-de Sitter spacetime Poincaré patch ( t, z, x i ) , t ∈ R , z ∈ R > 0 and x i ∈ R , i = 1 , . . . , d − 1 , d s 2 = ℓ 2 � � − d t 2 + d z 2 + δ ij d x i d x j . z 2 The region covered by this chart is the Poincaré fundamental domain , PAdS d +1 .

1. Anti-de Sitter spacetime H d +1 . = R > 0 × R d via a conformal rescaling PAdS d +1 can be mapped to ˚ d s 2 �→ z 2 ℓ 2 d s 2 = − d t 2 + d z 2 + δ ij d x i d x j . We can attach a conformal boundary as the locus z = 0 and obtain H d +1 . = R ≥ 0 × R d , the half Minkowski spacetime . Remark: From now on, we set ℓ = 1 .

Outline 1 Anti-de Sitter spacetime 2 Klein-Gordon equation and causal propagator 3 AQFT and Hadamard condition for AdS 4 Conclusions

2. Klein-Gordon equation and causal propagator 2.1. Klein-Gordon equation and boundary conditions Klein-Gordon equation. Poincaré domain ( PAdS d +1 , g ) , φ : PAdS d +1 → R , � � � g − m 2 Pφ = 0 − ξR φ = 0 . 1 − d H d +1 → R is a solution of Lemma: In (˚ 2 φ : ˚ H d +1 , η ) , Φ = z � � � η − m 2 P η Φ = Φ = 0 , z 2 with m 2 . 0 − ( ξ − d − 1 = m 2 4 d ) R .

2. Klein-Gordon equation and causal propagator 2.1. Klein-Gordon equation and boundary conditions Klein-Gordon equation. Poincaré domain ( PAdS d +1 , g ) , φ : PAdS d +1 → R , � � � g − m 2 Pφ = 0 − ξR φ = 0 . 1 − d H d +1 → R is a solution of Lemma: In (˚ 2 φ : ˚ H d +1 , η ) , Φ = z � � � η − m 2 P η Φ = Φ = 0 , z 2 with m 2 . 0 − ( ξ − d − 1 = m 2 4 d ) R . Fourier expansion. Fourier representation of Φ : � x . k . R d d d k e ik · x � Φ = Φ k , = ( t, x 1 , . . . , x d − 1 ) , = ( ω, k 1 , . . . , k d − 1 ) , where � Φ k are solutions of the ODE � � d − 1 � − d 2 d z 2 + m 2 Φ k ( z ) . λ . = ω 2 − L � Φ k ( z ) = λ � � k 2 = Φ k ( z ) , i z 2 i =1 This is a Sturm-Liouville problem on z ∈ (0 , + ∞ ) with spectral parameter λ .

2. Klein-Gordon equation and causal propagator 2.1. Klein-Gordon equation and boundary conditions Definition: For any z 0 ∈ (0 , ∞ ) we call maximal domain associated to L � � D max ( L ; z 0 ) . Ψ : (0 , z 0 ) → C | Ψ , dΨ d z ∈ AC loc (0 , z 0 ) and Ψ , L (Ψ) ∈ L 2 (0 , z 0 ) = , where AC loc (0 , z 0 ) is the collection of all complex-valued, locally absolutely continuous functions on (0 , z 0 ) . Fundamental pair of solutions of L Φ = λ Φ . = q 2 Φ as � π 2 q − ν √ z J ν ( qz ) , Φ 1 ( z ) =  � π 2 q ν √ z J − ν ( qz ) ,    − ν ∈ (0 , 1) , � π � � Φ 2 ( z ) = √ z Y 0 ( qz ) − 2   − π log( q ) ν = 0 ,  , 2 √ where ν . 1 + 4 m 2 ≥ 0 . Only Φ 1 ∈ L 2 (0 , z 0 ) for ν ≥ 1 . = 1 2

2. Klein-Gordon equation and causal propagator 2.1. Klein-Gordon equation and boundary conditions Definition: Ψ α : (0 , ∞ ) → C satisfies an α -boundary condition at the endpoint 0, or equivalently that Ψ α ∈ D max ( L ; α ) , if the following two conditions are satisfied: 1 there exists z 0 ∈ (0 , ∞ ) such that Ψ α ∈ D max ( L ; z 0 ) ; 2 there exists α ∈ (0 , π ] such that � � lim cos( α ) W z [Ψ α , Φ 1 ] + sin( α ) W z [Ψ α , Φ 2 ] = 0 , z → 0 where W z [Ψ α , Φ i ] . = Ψ α dΦ i d z − Φ i dΨ α d z , i = 1 , 2 , is the Wronskian.

2. Klein-Gordon equation and causal propagator 2.1. Klein-Gordon equation and boundary conditions Definition: Ψ α : (0 , ∞ ) → C satisfies an α -boundary condition at the endpoint 0, or equivalently that Ψ α ∈ D max ( L ; α ) , if the following two conditions are satisfied: 1 there exists z 0 ∈ (0 , ∞ ) such that Ψ α ∈ D max ( L ; z 0 ) ; 2 there exists α ∈ (0 , π ] such that � � lim cos( α ) W z [Ψ α , Φ 1 ] + sin( α ) W z [Ψ α , Φ 2 ] = 0 , z → 0 where W z [Ψ α , Φ i ] . = Ψ α dΦ i d z − Φ i dΨ α d z , i = 1 , 2 , is the Wronskian. If ν ∈ [0 , 1) , Ψ α may then be written as Ψ α = cos( α )Φ 1 + sin( α )Φ 2 . If ν ≥ 1 , no boundary conditions at z = 0 are imposed and we may take Ψ ≡ Ψ π . These boundary conditions are also commonly known as Robin boundary conditions .

2. Klein-Gordon equation and causal propagator 2.2. Causal propagator The building block necessary for the algebraic quantization is the causal propagator G α ∈ D ′ ( PAdS d +1 × PAdS d +1 ) . The propagator in PAdS d +1 can be reconstructed via d − 1 G α = ( zz ′ ) 2 G H ,α . with G H ,α ∈ D ′ � ˚ H d +1 � H d +1 × ˚ . The latter satisfies ( P η ⊗ I ) G H ,α = ( I ⊗ P η ) G H ,α = 0 , � ˚ H d +1 � ∀ f, f ′ ∈ C ∞ G H ,α ( f, f ′ ) = − G H ,α ( f ′ , f ) , 0 d − 1 � δ ( x i − x ′ i ) δ ( z − z ′ ) . G H ,α | t = t ′ = 0 , ∂ t G H ,α | t = t ′ = ∂ t ′ G H ,α | t = t ′ = i =1

2. Klein-Gordon equation and causal propagator 2.2. Causal propagator The building block necessary for the algebraic quantization is the causal propagator G α ∈ D ′ ( PAdS d +1 × PAdS d +1 ) . The propagator in PAdS d +1 can be reconstructed via d − 1 G α = ( zz ′ ) 2 G H ,α . with G H ,α ∈ D ′ � ˚ H d +1 � H d +1 × ˚ . The latter satisfies ( P η ⊗ I ) G H ,α = ( I ⊗ P η ) G H ,α = 0 , � ˚ H d +1 � ∀ f, f ′ ∈ C ∞ G H ,α ( f, f ′ ) = − G H ,α ( f ′ , f ) , 0 d − 1 � δ ( x i − x ′ i ) δ ( z − z ′ ) . G H ,α | t = t ′ = 0 , ∂ t G H ,α | t = t ′ = ∂ t ′ G H ,α | t = t ′ = i =1 We consider a mode expansion for the integral kernel of G H ,α , � d d k e ik · ( x − x ′ ) � G H ,α ( x − x ′ , z, z ′ ) = G k,α ( z, z ′ ) , d (2 π ) R d 2 where � G k,α ( z, z ′ ) is a symmetric solution of = − d 2 d z 2 + m 2 L . G k,α ( z, z ′ ) = q 2 � q 2 = k · k . ( L ⊗ I ) � G k,α ( z, z ′ ) = ( I ⊗ L ) � G k,α ( z, z ′ ) , z 2 ,

2. Klein-Gordon equation and causal propagator 2.2. Causal propagator = � d − 1 � x i − x ′ i � 2 and Let r 2 . i =1 �� � � ∞ � k � d − 3 k 2 + q 2 ( t − t ′ − iǫ ) sin I ǫ ( q, r, t, t ′ ) . 2 = d k k 2 ( kr ) q � J d − 3 . 2 π ( k 2 + q 2 ) r 0 Proposition: The causal propagator G H ,α ∈ D ′ � ˚ H d +1 � H d +1 × ˚ for different values of ν ∈ [0 , ∞ ) has integral kernel given by the following expressions. 1 If ν ∈ [1 , ∞ ) , � ∞ √ G H ,π ( x, x ′ ) = lim d q I ǫ ( q, r, t, t ′ ) J ν ( qz ) J ν ( qz ′ ) . zz ′ ǫ → 0 + 0 2 If ν ∈ (0 , 1) and c α . = cot( α ) ≤ 0 , that is, α ∈ [ π 2 , π ] , � ∞ √ ψ c α ( z ) ψ c α ( z ′ ) G H ,α ( x, x ′ ) = lim d q I ǫ ( q, r, t, t ′ ) zz ′ α − 2 c α q 2 ν cos( νπ ) + q 4 ν , c 2 ǫ → 0 + 0 where ψ c α ( z ) = c α J ν ( qz ) − q 2 ν J − ν ( qz ) . Remark: There is no ground state for Robin boundary conditions with c > 0 and for ν = 0 , so these cases will not be further considered.

2. Klein-Gordon equation and causal propagator 2.2. Causal propagator Proposition: Let: � �� − 2 �   � � √ 2 σ ǫ d 2 + ν, 1 F 2 + ν ; 1 + 2 ν ; cosh   2 G (D) ( x, x ′ ) = lim   − ( ǫ ↔ − ǫ )  , � � √ 2 σ ǫ �� d  2 + ν ǫ → 0 + cosh 2 � �� − 2 �   � � √ 2 σ ǫ d 2 − ν, 1 2 − ν ; 1 − 2 ν ; F cosh   2 G (N) ( x, x ′ ) = lim   − ( ǫ ↔ − ǫ )  , � � √ 2 σ ǫ �� d  2 − ν ǫ → 0 + cosh 2 where σ ǫ . = σ + 2 iǫ ( t − t ′ ) + ǫ 2 and F is the Gaussian hypergeometric function. The integral kernel of the causal propagator on PAdS d +1 is � � cos( α ) G (D) ( u ) + sin( α ) G (N) ( u ) G α ( u ) = N α where N α is a normalization constant, ν ∈ (0 , 1) and α ∈ [ π 2 , π ] .