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Linearized gravity and Hadamard states
Simone Murro
Department of MathematicsUniversity of Regensburg
Séminaires Math-Physique
Dijon, 10th of May 2017
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 1 / 20
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 locallycovariant quantum field theory ]
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 2 / 20
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:
I M. Benini, C. Dappiaggi and S.M. - J. Math. Phys. 55 082301 (2014)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 3 / 20
Algebraic Quantum Field Theory
PART I:
On the algebraic approachto quantum field theory
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 4 / 20
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
ds2 = β2dt2 − ht ; β ∈ C∞(M;R+) and ht ∈ Riem(Σ);∀t ∈ R
ϕ : M → R is a conformally real scalar field
Pϕ =
(−+
16R
)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, 2017 5 / 20
Algebraic Quantum Field Theory
AQFT II - Classical Observables[C. Dappiaggi, G. Nosari, N. Pinamonti: The Casimir effect from the point of view of algebraicquantum field theory - Math. Phys. Anal. Geom. 19, 12 (2016)]
A classical observable is an assignment of a real number to each dynamical configuration
∀ϕ ∈ C∞(M) there exists α ∈ C∞c (M) 7→ Oα(ϕ) :=
ˆM
dµgϕ(x)α(x)
Space of classical observables
E(M) :=
O[α] : Sol(M)→ R
∣∣∣ ∀ϕ ∃ [α] ∈ C∞c (M)
P[C∞c (M)]7→ O[α](ϕ) :=
ˆM
dµgϕ(x)α(x)
We have identified classical observables as the vector space E(M) ' C∞c (M)
P[C∞c (M)]
Why do we believe it is the right choice? Paradigm is:
E(M) is separating: ∀ ϕ,ϕ′ ∈ Sol(M), ∃ [α] ∈ E(M) s. t. O[α](ϕ) 6= O[α](ϕ′)
E(M) is optimal: ∀ [α], [α′] ∈ E(M), ∃ϕ ∈ Sol(M) s. t. O[α](ϕ) 6= O[α′](ϕ)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 6 / 20
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 =∞⊕n=0
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
[α]⊗ [α′]− [α′]⊗ [α]− ıG([α], [α′])11 ,
where 11 is the unit in A (M) and
G([α], [α′]).
=(α | Gα′
)=
ˆM
dµgα(x)G(α′)(x)
(3) Define the Algebra of Fields F (M).
= A (M)I (M)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 7 / 20
Algebraic Quantum Field Theory
AQFT IV - Hadamard States[M. J. Radzikowski: Micro-local approach to the Hadamard condition in QFT on curvedspace-time - Commun. Math. Phys. 179, 529 (1996)]
An algebraic state ω : F (M)→ C is a linear functional such that:
ω(11) = 1 (normalized) ω(a∗a) ≥ 0 (positive)
Notice that choosing a state ω : F (M)→ C is equivalent to assigning
ωn(α1, . . . , αn) ∀n ∈ N and ∀αi ∈ C∞c (M)
Which criteria to choose a physical state on a curved spacetime?
(1) Quasifree (ω2n+1 ≡ 0 and ω2n is determined by ω2),
(2) Hadamard: WF (ω2) =
(x , y , ξx , ξy ) ∈ T ∗M⊗2 \ 0∣∣ (x , ξx) ∼ (y ,−ξy ), ξx . 0
I same ultraviolet behaviour as the vacuum state,
I quantum fluctuations of all observables are finite,
I covariant construction of Wick polynomials to deal with interactions.
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 8 / 20
Linearized gravity
PART II:
Linearized gravity onglobally hyperbolic spacetimes
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 9 / 20
Linearized gravity
Linearized Einstein equations I
On a globally hyperbolic spacetime (M, g) such that Ric(g) = 0,
(L h)µν =−12
(2hµν − gµν2hαα)+Rα βµν hβα
+∇(µ∇αhν)α −12∇µ∇νhαα −
12gµν∇α∇βhαβ = 0 .
(1)
This system exhibits a gauge symmetry:
h ∼ h′ ⇐⇒ ∃χ ∈ Γ(T ∗M) such that h′ = h +∇sχ
where (∇sχ)µν = 12 (∇µχν +∇νχµ)
Gauge equivalence class of solutions Sol(M)/G where
Sol(M).
= h ∈ Γsc(⊗2sT∗M) | (L h)µν = 0.
G(M) = Lξg ∈ Γsc(⊗2sT∗M) | ξ ∈ Γsc(T ∗M)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 10 / 20
Linearized gravity
Linearized Einstein equations II - de Donder gauge
Lemma
For every [h] ∈ Sol(M)/G, there exists a representative h such thatPh = (− 2Riem) I h = 0div I h = 0
where (div (h))µ = ∇νhµν and (I h)µν = hµν − 12gµνh
aa
Notice that:
P = − 2Riem is normally hyperbolic ⇒ causal propagator: P GP = GP P = 0
the trace reversal I does not spoil hyperbolicity, let GP = GP I
GP P = P GP = 0
Since div IG±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, 2017 11 / 20
Linearized gravity
Linearized Einstein equations III - de Donder gauge
Theorem
There exists a 1:1 correspondenceSol(M)
G ←→ Kerc(div)Imc(L )
Kerc(div).
=ε ∈ Γc(⊗2
sT∗M) | div(ε) = 0
Imc(L )
.=ε ∈ Γc(⊗2
sT∗M) | ε = L γ with γ ∈ Γc(⊗2
sT∗M)
The isomorphism is realized by the map
Kerc(div)Imc(L )
3 [ε] 7→ [GP(ε)] ∈ Sol(M)
G
Important: There is a topological obstruction in implementing the TT gauge: wheneverthe Cauchy surface is compact!
[ C. J. Fewster, D. S. Hunt : Quantization of linearized gravity in cosmological vacuumspacetimes - Rev. Math. Phys. 25, 1330003 (2013) ]
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 12 / 20
Linearized gravity
Classical observables and the algebra of fields
Mimicking the case of the scalar field, the classical observables are
E(M) =E inv
Imc(L ), with E inv .
=ε ∈ Γc(⊗2
sT∗M) | div(ε) = 0
The Borchers-Uhlmann algebra A(M) is defined as follow:
A(M) =∞⊕n=0
E(M;C)⊗n
Take a quotient of A(M) by the ideal I generated by
[ε]⊗ [ε′]− [ε′]⊗ [ε]− ıG([ε], [ε′])11 , G([ε], [ε′]) =
ˆM
dµgα(x)G(α′)(x)
The resulting algebra of fields: F(M).
= A(M)/I.
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 13 / 20
Hadamard states
PART III:
Hadamard statesfor linearized gravity on
asymptotically flat spacetimes
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 14 / 20
Hadamard states
Algebraic holography[Dappiaggi, Pinamonti, Moretti: Rigorous steps towards holography in asymptotically flatspacetimes - Rev. Math. Phys. 18, 349 (2006) ]
1) Encode the information of a QFT defined on the manifold into a counterpart livingon the boundary.
2) Asymptotically flat spacetimes:
(i) (M, g) → (M, g = Ω2g) 7→ (M, ωg)
(ii) Ω has smooth extension on M andΩ|I+∪ı+ ≡ 0, dΩ|I+ 6= 0 and dΩ|ı+ = 0
(iii) I + is lightlike 3D submanifold of M(iv) nµ = ∇µΩ vector field tangent to I +
(v) (I +, q, n) ! universal structure
(vi) the BMS group ϕ : I + → I +
I + 7→ I +, q 7→ ω2q, n 7→ ω−1n
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 15 / 20
Hadamard states
The algebra of fields on I +
Inspired by Ashtekar & Magnon (1982) define
E(I +) = λ ∈ Γ(⊗2sT∗I +) |λµνnµ = 0, λµνqµν = 0,
(λ, λ) <∞, (∂µλ, ∂µλ) <∞
where nµ = ∇µΩ, qµν satisfies qµνqµαqνβ = qαβ and
(λ, λ) =
ˆI+
dµI+λµνλαβqµνqαβ
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− ıσI+(λ, λ′)11
σI+(γ1, γ2) =
ˆI+
((γ1)µν∂n(γ2)µν − (γ2)µν∂n(γ1)µν
)d` ∧ dS2(ϑ, ϕ)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 16 / 20
Hadamard states
Bulk to Boundary correspondence I
Goal: Project an element of E(M) to E(I +)
First difficulty : (M, g) 7→ (M,Ω2g) transform L into an operator with termsproportional to Ω−n . . . divergences on I + since Ω ≡ 0
Solution? If dimM > 4 the TT-gauge saves the day, while dimM =4 you needGeroch-Xanthopoulos gauge
Big Problem: Topological obstruction to implementing the G-X gauge
Theorem
• Let h′ = h +∇sχ, χ ∈ Γ(T ∗M). Then τµν = Ωh′µν is in the GX-gauge iff
∇µ∇[µχν] = −vν(h) vν(h) = ∇µhµν −∇νh (co-exact)
• Let h = GP(ε). Then vν(h) is co-exact iff Tr(ε) = gµνεµν is co-exact.
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 17 / 20
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 anyasymptotically flat, globally hyperbolic spacetime whose Cauchy surface has a S1
factor.
The map Γ : E rad(M)→ E(I +) defined by
G([ε], [ε′]) = σI (Γ[ε], Γ[ε′])
induces an injective ∗-homomorphism ι : F rad(M)→ F(I +)
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 18 / 20
Hadamard states
Bulk to Boundary correspondence III
Define a state on the boundary ωI : F(I +)→ C.
Pull ωI back to the bulk via ι to get a state ωM on F rad(M):
ωM.
= ι∗ωI = ωI 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 bycharacteristic Cauchy problem.]
invariant under the action of all isometries of the bulk. [Moretti : Uniquenesstheorem for BMS-invariant states of scalar QFT on the null boundary ofasymptotically flat spacetimes and bulk-boundary observable algebra correspondence]
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 19 / 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 anothergauge
Apply this method for specific more complicated black hole backgrounds
Simone Murro (Universität Regensburg) Algebraic Quantum Field Theory Dijon, 2017 20 / 20
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