Generalized Wentzell boundary conditions and holography Jochen Zahn - PowerPoint PPT Presentation

Generalized Wentzell boundary conditions and holography Jochen Zahn Universit¨ at Leipzig based on arXiv:1512.05512 LQP38, May 2016

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ).

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 M B M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ).

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 M B M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ § Counterterms in the AdS/CFT correspondence [Balasubramanian & Kraus 99] : ? 1 ? g 1 ª ª d 5 x ´ d 4 x . ` R g ´ 12 ˘ ` Θ ´ ` 4 R h ` 3 ˘ S “ ´ h ` 2 16 ⇡ G 8 ⇡ G ` M B M

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 M B M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ § Counterterms in the AdS/CFT correspondence [Balasubramanian & Kraus 99] : ? 1 ? g 1 ª ª d 5 x ´ d 4 x . ` R g ´ 12 ˘ ` Θ ´ ` 4 R h ` 3 ˘ S “ ´ h ` 2 16 ⇡ G 8 ⇡ G ` M B M

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ § Counterterms in the AdS/CFT correspondence [Balasubramanian & Kraus 99] : ? 1 ? g 1 ª ª d 5 x ´ d 4 x . ` R g ´ 12 ˘ ` Θ ´ ` 4 R h ` 3 ˘ S “ ´ h ` 2 16 ⇡ G 8 ⇡ G ` M B M § Holographic renormalization [Skenderis et al] ? g S “ ´ 1 ª ´ ´ d 2 ¯ � 2 ¯ d d ` 1 x g µ ⌫ B µ � B ⌫ � ` 4 ´ 1 2 ⇢ • " ? ´ ` ª ` d ´ � 2 ¯ 2 log " h µ ⌫ B µ � B ⌫ � ` d d x . 1 ˘ h 2 ´ 1 2 B M ε

Introduction We study a scalar field � subject to the action S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 M B M Specifically, M “ R ˆ Σ with Σ “ R d ´ 1 ˆ r´ S , S s (but also general Σ Ä R d ). This is similar to § The Nambu-Goto string with masses at the ends [Chodos & Thorn 74] : ª ª a | g | d 2 x ´ m a S “ ´ � | h | d x . B Ξ Ξ § Counterterms in the AdS/CFT correspondence [Balasubramanian & Kraus 99] : ? 1 ? g 1 ª ª d 5 x ´ d 4 x . ` R g ´ 12 ˘ ` Θ ´ ` 4 R h ` 3 ˘ S “ ´ h ` 2 16 ⇡ G 8 ⇡ G ` M B M § Holographic renormalization [Skenderis et al] ? g S “ ´ 1 ª ´ ´ d 2 ¯ � 2 ¯ d d ` 1 x g µ ⌫ B µ � B ⌫ � ` 4 ´ 1 2 ⇢ • " ? ´ ` ª ` d ´ � 2 ¯ 2 log " h µ ⌫ B µ � B ⌫ � ` d d x . 1 ˘ h 2 ´ 1 2 B M ε

Questions § Is the classical system well-behaved, i.e., is the Cauchy problem well-posed? § Can one quantize the system? If yes, what is the interplay between bulk and boundary fields?

Outline The wave equation Quantization Conclusion

Variation of S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M yields the equations of motion ´ l g � ` µ 2 � “ 0 in M , (1) ´ l h � ` µ 2 � “ ´ c ´ 1 B K � in B M . (2) Using (1), one may write (2) alternatively as B 2 K � “ ´ c ´ 1 B K � in B M . (3) Such boundary conditions are known in the mathematical literature as generalized Wentzell, Wentzell-Feller type, kinematic, or dynamical boundary conditions.

Variation of S “ ´ 1 ª d d ` 1 x ´ c ª ´ g µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ ´ h µ ⌫ B µ � B ⌫ � ` µ 2 � 2 ¯ d d x 2 2 B M M yields the equations of motion ´ l g � ` µ 2 � “ 0 in M , (1) ´ l h � ` µ 2 � “ ´ c ´ 1 B K � in B M . (2) Using (1), one may write (2) alternatively as B 2 K � “ ´ c ´ 1 B K � in B M . (3) Such boundary conditions are known in the mathematical literature as generalized Wentzell, Wentzell-Feller type, kinematic, or dynamical boundary conditions. Di ff erent interpretations possible: § (3) as boundary condition for wave equation (1). § (1), (2) as wave equations for the bulk and the boundary field, coupled by § The bulk field providing a source for the boundary field; § The boundary field providing the boundary value of the bulk field.

Strategy § Write full system as ´B 2 t Φ “ ∆Φ with ∆ a self-adjoint operator on some Hilbert space H . § Using ∆ , rewrite the full system as a first order equation on suitable energy Hilbert spaces for the Cauchy data. This yields well-posedness for smooth initial data with suitable fall-o ff and global energy estimates. § Derive causal propagation by local energy estimates. § By glueing, this yields global well-posedness for smooth initial data.

Strategy § Write full system as ´B 2 t Φ “ ∆Φ with ∆ a self-adjoint operator on some Hilbert space H .

§ The following symplectic form is conserved: ª ª σ pp φ , 9 φ q , p ψ , 9 φ 9 ψ ´ 9 φ 9 ψ ´ 9 ψ qq “ φψ ` c φψ . Σ B Σ

§ The following symplectic form is conserved: ª ª � pp � , 9 � q , p , 9 � 9 ´ 9 � 9 ´ 9 qq “ � ` c � . Σ B Σ § It is thus natural to consider the Hilbert space H “ L 2 p Σ q ‘ cL 2 pB Σ q with scalar product xp � bk , � bd q , p bk , bd qy “ x � bk , bk y L 2 p Σ q ` c x � bd , bd y L 2 pB Σ q so that � pp � , 9 � q , p , 9 qq “ xp ¯ � , ¯ � | B Σ q , p 9 , 9 | B Σ qy ´ xp ¯ , ¯ | B Σ q , p 9 � , 9 � | B Σ qy .

§ The following symplectic form is conserved: ª ª � pp � , 9 � q , p , 9 � 9 ´ 9 � 9 ´ 9 qq “ � ` c � . Σ B Σ § It is thus natural to consider the Hilbert space H “ L 2 p Σ q ‘ cL 2 pB Σ q with scalar product xp � bk , � bd q , p bk , bd qy “ x � bk , bk y L 2 p Σ q ` c x � bd , bd y L 2 pB Σ q so that � pp � , 9 � q , p , 9 qq “ xp ¯ � , ¯ � | B Σ q , p 9 , 9 | B Σ qy ´ xp ¯ , ¯ | B Σ q , p 9 � , 9 � | B Σ qy . § We may write the wave equation as ´ ∆ Σ ` µ 2 ˆ 0 ˙ ˆ ˙ � bk ´B 2 t Φ “ ∆Φ “ , c ´ 1 B K ¨ | B Σ ´ ∆ B Σ ` µ 2 � bd where the boundary condition � bk | B Σ “ � bd is encoded in the domain ! ) p � bk , � bd q P H | � bk P H 2 p Σ q , � bd P H 2 pB Σ q , � bk | B Σ “ � bd dom p ∆ q “ .

Proposition ∆ is self-adjoint with spectrum contained in r µ 2 , 8q . Proof. For Φ P dom p ∆ q , we compute (with µ “ 0): ª ª ¯ � bd B K � bk | ´ c ¯ ¯ x Φ , ∆Φ y “ ´ � bk ∆ Σ � bk ` � bd ∆ B Σ � bd Σ B Σ ª ª B i ¯ B j ¯ “ � bk B i � bk ` c � bd B j � bd • 0 . B Σ Σ This entails the bound on the spectrum. The claim on self-adjointness follows similarly by integration by parts: One shows that also on dom p ∆ ˚ q the boundary condition � bk | B Σ “ � bd has to be satisfied.

Strategy § Write full system as ´B 2 t Φ “ ∆Φ with ∆ a self-adjoint operator on some Hilbert space H . § Using ∆ , rewrite the full system as a first order equation on suitable energy Hilbert spaces for the Cauchy data. This yields well-posedness for smooth initial data with suitable fall-o ff and global energy estimates.