Linearized gravity and Hadamard states

Transcription

1 Linearized gravity and Hadamard states Simone Murro Department of Mathematics University of Regensburg Séminaires Math-Physique Dijon, 10th of May 2017 Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

2 General Framework Is there a quantum theory of General Relativity? 1) String theory, loop quantum gravity. 2) Algebraic approach to quantum gravity. [Brunetti, Fewster, Fredenhagen, Giesel, Majid, Rejzner, Tiemann,...] Algebraic approach rigorous approach to QFT [Araki, Bär, Brunetti, Buchholz, Dappiaggi, Dimock, Finster, Fredenhagen, Gérard, Ginoux, Haag, Kay, Moretti, Pinamonti, Strohmaier, Verch, Wald,...] Why is the construction of a Hadamard state important? 1) Evaluation of the influence of gravitational field on physical observables. 2) Understand in the algebraic approach structural problems of quantum gravity. [Brunetti, Fredenhagen, Rejzner: Quantum gravity from the point of view of locally covariant quantum field theory ] Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

3 Outline Outline On the algebraic approach to quantum field theory Linearized gravity on globally hyperbolic spacetimes Hadamard states for linearized gravity on asymptotically flat spacetimes Based on: M. Benini, C. Dappiaggi and S.M. - J. Math. Phys (2014) Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

4 Algebraic Quantum Field Theory PART I: On the algebraic approach to quantum field theory Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

5 Algebraic Quantum Field Theory AQFT I - Scalar field [C. Bär, N. Ginoux, F. Pfäffle: Wave Equations on Lorentzian Manifolds and Quantization - European Mathematical Society (2007)] Goal: Outline AQFT via a good example! M R Σ is 4-dim is a globally hyperbolic spacetime ds 2 = β 2 dt 2 h t; β C (M; R + ) and h t Riem(Σ); t R ϕ : M R is a conformally real scalar field Pϕ = ( + 16 ) R ϕ = 0 P : C (M) C (M) is normally hyperbolic then G ± : C c (M) C sc (M) (i) P G ± f = f (ii) G ± P f = f (iii) supp(g ± f ) J ± (supp(f )) All dynamical configurations of a real scalar field are Sol(M) = {ϕ C sc (M) f C c (M) and ϕ = Gf } Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

6 Algebraic Quantum Field Theory AQFT II - Classical Observables [C. Dappiaggi, G. Nosari, N. Pinamonti: The Casimir effect from the point of view of algebraic quantum field theory - Math. Phys. Anal. Geom. 19, 12 (2016)] A classical observable is an assignment of a real number to each dynamical configuration ˆ ϕ C (M) there exists α Cc (M) O α(ϕ) := dµ g ϕ(x)α(x) Space of classical observables { E(M) := O [α] : Sol(M) R ϕ [α] C c ˆ (M) P[Cc (M)] O [α](ϕ) := We have identified classical observables as the vector space E(M) C c (M) P[Cc (M)] Why do we believe it is the right choice? Paradigm is: M M } dµ g ϕ(x)α(x) E(M) is separating: ϕ, ϕ Sol(M), [α] E(M) s. t. O [α] (ϕ) O [α] (ϕ ) E(M) is optimal: [α], [α ] E(M), ϕ Sol(M) s. t. O [α] (ϕ) O [α ](ϕ) Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

7 Algebraic Quantum Field Theory AQFT III - Algebra of Observables [J. Dimock: Algebras of Local Observables on a Manifold - Commun. Math. Phys. 77, 219 (1980)] Gaol: From E(M; C) = E(M) C build the algebra of fields (1) Construct the unital Borchers-Uhlmann -algebra: A = E(M; C) n where E(M; C) 0 = C and the -operation is complex conjugation (2) Construct the ideal I (M) A (M) generated by elements of the form n=0 [α] [α ] [α ] [α] ı G([α], [α ])1, where 1 is the unit in A (M) and G([α], [α ]) =. ( α Gα ) ˆ = (3) Define the Algebra of Fields F (M). = A (M) I (M) M dµ g α(x)g(α )(x) Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

8 Algebraic Quantum Field Theory AQFT IV - Hadamard States [M. J. Radzikowski: Micro-local approach to the Hadamard condition in QFT on curved space-time - Commun. Math. Phys. 179, 529 (1996)] An algebraic state ω : F (M) C is a linear functional such that: ω(1) = 1 (normalized) ω(a a) 0 (positive) Notice that choosing a state ω : F (M) C is equivalent to assigning ω n(α 1,..., α n) n N and α i C c (M) Which criteria to choose a physical state on a curved spacetime? (1) Quasifree (ω 2n+1 0 and ω 2n is determined by ω 2), (2) Hadamard: WF (ω 2) = { (x, y, ξ x, ξ y ) T M 2 \ 0 (x, ξx) (y, ξ y ), ξ x 0 } same ultraviolet behaviour as the vacuum state, quantum fluctuations of all observables are finite, covariant construction of Wick polynomials to deal with interactions. Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

9 Linearized gravity PART II: Linearized gravity on globally hyperbolic spacetimes Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

10 Linearized gravity Linearized Einstein equations I On a globally hyperbolic spacetime (M, g) such that Ric(g) = 0, (L h) µν = 1 2 ( hµν gµν hα α)+r α β µν h βα + (µ α h ν)α 1 2 µ νhα α 1 2 gµν α β h αβ = 0. (1) This system exhibits a gauge symmetry: h h χ Γ(T M) such that h = h + sχ where ( sχ) µν = 1 ( µχν + νχµ) 2 Gauge equivalence class of solutions Sol(M)/G where Sol(M). = {h Γ sc( 2 s T M) (L h) µν = 0}. G(M) = {L ξ g Γ sc( 2 s T M) ξ Γ sc(t M)} Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

11 Linearized gravity Linearized Einstein equations II - de Donder gauge Lemma For every [h] Sol(M)/G, there exists a representative h such that { Ph = ( 2Riem) I h = 0 div I h = 0 where (div (h)) µ = ν h µν and (I h) µν = h µν 1 2 gµνha a Notice that: P = 2Riem is normally hyperbolic causal propagator: P G P = G P P = 0 the trace reversal I does not spoil hyperbolicity, let G P = G P I G P P = P G P = 0 Since div I G ± P = G ± div, the fixing is not complete: [h] there exist h, h such that h h = sχ where χ = 0 Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

12 Linearized gravity Linearized Einstein equations III - de Donder gauge Theorem There exists a 1:1 correspondence Sol(M) G Kerc(div) Im c(l ) Ker c(div). = { ε Γ c( 2 s T M) div(ε) = 0 } Im c(l ). = { ε Γ c( 2 s T M) ε = L γ with γ Γ c( 2 s T M) } The isomorphism is realized by the map Ker c(div) Im c(l ) [ε] [G P(ε)] Sol(M) G Important: There is a topological obstruction in implementing the TT gauge: whenever the Cauchy surface is compact! [ C. J. Fewster, D. S. Hunt : Quantization of linearized gravity in cosmological vacuum spacetimes - Rev. Math. Phys. 25, (2013) ] Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

13 Linearized gravity Classical observables and the algebra of fields Mimicking the case of the scalar field, the classical observables are E(M) = E inv Im c(l ), with E inv =. { ε Γ c( 2 s T M) div(ε) = 0 } The Borchers-Uhlmann algebra A(M) is defined as follow: A(M) = E(M; C) n n=0 Take a quotient of A(M) by the ideal I generated by ˆ [ε] [ε ] [ε ] [ε] ıg([ε], [ε ])1, G([ε], [ε ]) = The resulting algebra of fields: F(M). = A(M)/I. M dµ g α(x)g(α )(x) Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

14 Hadamard states PART III: Hadamard states for linearized gravity on asymptotically flat spacetimes Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

15 Hadamard states Algebraic holography [Dappiaggi, Pinamonti, Moretti: Rigorous steps towards holography in asymptotically flat spacetimes - Rev. Math. Phys. 18, 349 (2006) ] 1) Encode the information of a QFT defined on the manifold into a counterpart living on the boundary. 2) Asymptotically flat spacetimes: (i) (M, g) ( M, g = Ω 2 g) ( M, ω g) (ii) Ω has smooth extension on M and Ω I + ı + 0, dω I + 0 and dω ı + = 0 (iii) I + is lightlike 3D submanifold of M (iv) n µ = µ Ω vector field tangent to I + (v) (I +, q, n) universal structure (vi) the BMS group ϕ : I + I + I + I +, q ω 2 q, n ω 1 n Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

16 Hadamard states The algebra of fields on I + Inspired by Ashtekar & Magnon (1982) define E(I + ) = {λ Γ( 2 s T I + ) λ µνn µ = 0, λ µνq µν = 0, (λ, λ) <, ( µλ, µλ) < } where n µ = µ Ω, q µν satisfies q µν q µαq νβ = q αβ and ˆ (λ, λ) = dµ I +λ µνλ αβ q µν q αβ I + The algebra of fields F(I + ) is defined as follow: F(I + ) =. n=0 E(M; C) n I(I + ) where I is generated by λ λ λ λ ıσ I +(λ, λ )1 ˆ ) σ I +(γ 1, γ 2) = ((γ 1) µν n(γ 2) µν (γ 2) µν n(γ 1) µν dl ds 2 (ϑ, ϕ) I + Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

17 Hadamard states Bulk to Boundary correspondence I Goal: Project an element of E(M) to E(I + ) First difficulty: (M, g) ( M, Ω 2 g) transform L into an operator with terms proportional to Ω n... divergences on I + since Ω 0 Solution? If dim M > 4 the TT-gauge saves the day, while dim M =4 you need Geroch-Xanthopoulos gauge Big Problem: Topological obstruction to implementing the G-X gauge Theorem Let h = h + sχ, χ Γ(T M). Then τ µν = Ωh µν is in the GX-gauge iff µ [µ χ ν] = v ν(h) v ν(h) = µ h µν νh (co-exact) Let h = G P (ε). Then v ν(h) is co-exact iff Tr(ε) = g µν ε µν is co-exact. Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

18 Hadamard states Bulk to Boundary correspondence II Definition We say that [ε] E(M) is a radiative observable if there exists a representative ε [ε] whose trace is co-exact. The collection of all these observables is E rad (M) E(M) Big Question: Can E rad (M) = E(M)? Proposition E rad (M) = E(M) on Minkowski spacetime but E rad (M) E(M) on any asymptotically flat, globally hyperbolic spacetime whose Cauchy surface has a S 1 factor. The map Γ : E rad (M) E(I + ) defined by G([ε], [ε ]) = σ I (Γ[ε], Γ[ε ]) induces an injective -homomorphism ι : F rad (M) F(I + ) Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

19 Hadamard states Bulk to Boundary correspondence III Define a state on the boundary ω I : F(I + ) C. Pull ω I back to the bulk via ι to get a state ω M on F rad (M): ω M. = ι ω I = ω I ι. Distinguished choice: Invariance under the BMS group. Via pull-back, we get the state on the bulk. This state turns out to be: of Hadamard form, [C. Gérard, M. Wrochna: Construction of Hadamard states by characteristic Cauchy problem.] invariant under the action of all isometries of the bulk. [Moretti : Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence] Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20

20 Conclusions Conclusions: THANK YOU for your attention! Asymptotic analysis is a powerful tool to construct physically relevant states it works for all free fields with some problems for linearized gravity What comes next? Prove if the no-go result for the GX gauge cannot be circumvented with another gauge Apply this method for specific more complicated black hole backgrounds Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, / 20