Introduction GR JT gravity Conclusions Covariant Phase Space with Boundaries Jie-qiang Wu (MIT) Based on work in progress with Daniel Harlow Tsinghua Sanya International Mathematics Forum Jan 8th, 2019 Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions In general relativity, covariant phase space method was developed by Iyer, Lee, Wald, and Zoupas to study Hamiltonian and black hole first law without breaking covariance In this work, we study the covariant phase space with more careful treatment of the boundary terms With this formalism, we give an explicit algorithm to calculate the Hamiltonian (without dealing with the B term � � δ ∂ ξ · B = ∂ ξ · Θ( φ, δφ )) To understand the covariant phase space method, we study the phase space and the symplectic form for JT gravity With this symplectic form, we give an explanation for the traversable wormhole In this work, we only focus on classical mechanics Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Outline Introduction 1 GR 2 JT gravity 3 Conclusions 4 Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Classical mechanics In classical mechanics, the Hamiltonian formalism is defined by phase space, Hamiltonian, and the Possion bracket or Dirac bracket (symplectic form) The phase space and symplectic form include everything in classical mechanics In statistical mechanics, the microscopic state is the volume of the phase space In quantum mechanics, the classical phase space is the first step of canonical quantization Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Hamiltonian vs general relativity Hamiltonian formalism is not convenient to describe general relativity Hamiltonian formalism: a special coordinate and time direction General relativity: diffeomorphism symmetry Covariant phase space method Lee, Wald J.Math.Phys31 725(1990) Iyer, Wald gr-qc/9403028 gr-qc/9503052 Wald Zoupas gr-qc/9911095 Phase space: (up to gauge equivalence,) every solution for the equation of motion corresponds to one point in the phase space Symplectic form: derived from the action Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Simple example: point particle � t f t i dt 1 x 2 Point particle S = 2 ˙ Taking a variation � t f δ S = t i dt ( − ¨ x ) δ x ( t ) + ˙ x δ x | f − ˙ x δ x | i The pre-symplectic potential is defined as the initial (or final) surface term under variation of the action; 1-form in configuration space The pre-symplectic form is the derivative of symplectic potential in configuration space; 2-form in configuration space Symplectic potential: θ [ x , δ x ] = ˙ x ( t ) δ x ( t ) = p δ x Symplectic form ω [ x , δ 1 x , δ 2 x ] = δ 1 θ [ x , δ 2 x ] − δ 2 θ [ x , δ 1 x ] = δ 1 p δ 2 x − δ 2 p δ 1 x x ] − L = 1 2 p 2 Hamiltonian H = θ [ x , ˙ Hamiltonian equation δ H = ω [ x , δ x , δ 2 x = ˙ x ] or ( δ H ) A = ω AB ξ B , where ξ is a flow in phase space Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions GR For more complicated theories in higher dimension even with gauge symmetry, the prescription still works L ( n ) + � � γ l ( n − 1) Action: S = Diffeomorphism symmetry: δφ = L ξ φ , δ L ( n ) = L ξ L ( n ) δ l ( n − 1) = L ξ l ( n − 1) ξ is parallel to the boundary φ denote all of the fields including the matter fields and metric Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Taking a variation for the action δ L ( n ) = E ( n ) ( φ ) δφ + d Θ ( n − 1) ( φ, δφ ) − Θ ( n − 1) ( φ, δφ ) + δ l ( n − 1) = F ( n − 1) ( φ ) δφ + dC ( n − 2) ( φ, δφ ) � � E ( n ) ( φ ) δφ + F ( n − 1) ( φ ) δφ δ S = ∂ � � Θ ( n − 1) ( φ, δφ ) + C ( n − 2) ( φ, δφ )) +( Σ ( n − 1) , f ∂ Σ ( n − 2) , f � � Θ ( n − 1) ( φ, δφ ) + C ( n − 2) ( φ, δφ )) − ( Σ ( n − 1) , i ∂ Σ ( n − 2) , i Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Pre-symplectic potential Σ ( n − 1) Θ ( n − 1) ( φ, δφ ) + ∂ Σ ( n − 2) C ( n − 2 ) ( φ, δφ ) � � Θtot[ φ, δφ ] = Pre-symplectic form Ωtot = δ 1 Θtot( φ, δ 2 φ ) − δ 2 Θtot( φ, δ 1 φ ) � Σ ( n − 1) ω ( n − 1) ( φ, δ 1 φ, δ 2 φ ) = ∂ Σ ( n − 2) ( δ 1 C ( n − 2 ) ( φ, δ 2 φ ) − δ 2 C ( n − 2 ) ( φ, δ 1 φ )) � + Compared with Wald, we have an extra boundary term related to C ( n − 2) Iyer, Wald gr-qc/9403028 gr-qc/9503052 In Einstein-Hilbert action C ∼ δ g ab n a τ b In Einstein-Hilbert action, JT gravity, f ( R ) gravity, Lovelock gravity, the C term vanish if we choose the gauge that the foliation is orthogonal to the boundary � ∇ a R bc ∇ a R bc Non-zero C term: S = It is convenient to keep the gauge redundancy and non-zero C term in calculation Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Hamiltonian Noether current: j ( n − 1) = Θ ( n − 1) [ φ, L ξ φ ] − ξ · L ( n ) ξ Noether charge: dj ( n − 1) j ( n − 1) = dQ ( n − 2) = 0 ξ ξ ξ (under on-shell condition) Relation with symplectic form current: δ j ( n − 1) = ω ( n − 1) ( φ, δφ, L ξ φ ) + d ( ξ · Θ ( n − 1) ) ξ � � Σ ω ( φ, δφ, L ξ φ ) = ∂ Σ ( δ Q ξ − ξ · Θ( φ, δφ )) Boundary action variation − Θ ( n − 1) ( φ, δφ ) + δ l ( n − 1) = F ( n − 1) ( φ ) δφ + dC ( n − 2) ( φ, δφ ) Hamiltonian Σ ω ( n − 1) ( φ, δφ, L ξ φ )+ ∂ Σ δ C ( n − 2) ( φ, L ξ φ ) −L ξ C ( n − 2) ( φ, δφ ) � � ∂ Σ δ ( Q ( n − 2) � + C ( n − 2) ( φ, L ξ φ ) − ξ · l ( n − 1) ) = ξ Hamiltonian equation Ωtot[ φ, δφ, L ξ φ ] = δ H ξ � ∂ Σ ( Q ξ + C ( n − 2) ( φ, L ξ φ ) − ξ · l ( n − 1) ) H ξ = Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Hamiltonian ( Q ( n − 2) = dj ( n − 1) ) ∂ Σ ( Q ξ + C ( n − 2) ( φ, L ξ φ ) − ξ · l ( n − 1) ) � H ξ = Σ Θ ( n − 1) ( φ, δφ, L ξ φ ) + ∂ Σ C ( n − 2) ( φ, δφ, L ξ φ ) � � = Σ ξ · L ( n ) ( φ ) − ∂ Σ ξ · l ( n − 1) ( φ ) � � − Classical mechanics: H = p ˙ q − L Hawking, Horowitz gr-qc/9501014 Ambiguity I: L ( n ) + Γ l ( n − 1) ⇒ L → L + dX � � S = l → l + X Ambiguity II: δ L ( n ) = E ( n ) ( φ ) δφ + d Θ ( n − 1) ( φ, δφ ) − Θ ( n − 1) ( φ, δφ ) + δ l ( n − 1) = F ( n − 1) ( φ ) δφ + dC ( n − 2) ( φ, δφ ) ⇒ Θ → Θ + dY C → C − Y The Hamiltonian have no ambiguities Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Relation with Brown York tensor In Einstein-Hilbert action, JT gravity, H ξ in our algorithm matches with the Brown York tensor’s calculation A general proof: Taking a variation δφ = L ξ φ ξ | ∂ � = 0 √− γ ∇ a ξ b T ab � δ S = Θtot , f [ φ, L ξ φ ] − Θtot , i [ φ, L ξ φ ] + γ ∂ Σ f dx n − 2 √ � h λ a ξ b T ab ) = (Θtot , f [ φ, L ξ φ ] − ∂ Σ i dx n − 2 √ � h λ a ξ b T ab ) − (Θtot , i [ φ, L ξ φ ] − Diffeomorphism symmetry δ L = L ξ L δ l = L ξ l L ξ L ( n ) + γ L ξ l ( n − 1) � � δ S = Σ f ξ · L ( n ) + Σ i ξ · L ( n ) + � � ∂ Σ f ξ · l ( n − 1) ) − ( � � ∂ Σ i ξ · l ( n − 1) ) = ( Compare the two equations � � H ξ = Θtot[ φ, L ξ φ ] − ( Σ ξ · L + ∂ Σ ξ · l ) ∂ Σ f dx n − 2 √ � h λ a ξ b T ab = Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Black hole first law The C term don’t change black hole first law � � δ H ξ = Σ ω ( φ, δφ, L ξ φ ) + ∂ Σ ( δ C ( φ, L ξ φ ) − L ξ C ( φ, δφ )) � � = ∂ Σ ( δ Q ξ − ξ · Θ( φ, δφ ))+ ∂ Σ ( δ C ( φ, L ξ φ ) −L ξ C ( φ, δφ )) Under stationary black hole background L ξ φ = 0, the C related term vanish � ∂ Σ ( δ C ( φ, L ξ φ ) − L ξ C ( φ, δφ )) = 0 The first law goes back to Wald’s derivation Iyer, Wald gr-qc/9403028 Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Gauge invariance When ξ | ∂ = 0 , H ξ = 0 H ξ only depend on ξ | ∂ so is gauge invariant � ∂ Σ ( Q ξ + C ( n − 2) ( φ, L ξ φ ) − ξ · l ( n − 1) ) H ξ = Criteria of gauge invariance of Θtot and Ωtot: Θtot[ φ, L ξ φ ] = 0 Ωtot[ φ, δφ, L ξ φ ] = 0 ( ξ | ∂ = 0) Ωtot[ φ, δφ, L ξ φ ] = δ H ξ = 0 Ωtot is gauge invariant � � Θtot[ φ, L ξ φ ] − ξ · L − ξ · l = H ξ = 0 Θtot is gauge invariant if and only if the bulk Lagrangian density vanish under on-shell condition Jie-qiang Wu Covariant Phase Space with Boundaries
Introduction GR JT gravity Conclusions Symplectic form To have a better understanding for covariant phase space, we explicitly build the phase space and calculate the symplectic form in JT gravity Jie-qiang Wu Covariant Phase Space with Boundaries
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