Fermionic Projectors and Hadamard States Simone Murro Fakultät für Mathematik Universität Regensburg Foundations and Constructive Aspects of QFT Göttingen, 16th of January 2016 To the memory of Rudolf Haag, one of the founding fathers of the algebraic quantum field theory Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 1 / 19
Outline Outline Based on: The algebraic approach to QFT Quasi-free states for CAR algebras Fermionic Projector and the Hadamard condition: FP states in a strip of spacetime FP states and the mass oscillation property FP states via the Møller operator Conclusions F. Finster, S. M., C. Röken: The Fermionic Projector in a time-dependent external potential: mass oscillation property and Hadamard states N. Drago, S. M.: A new class of Fermionic Projectors: Møller operators and mass oscillation properties." (work in progress) Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 2 / 19
Algebraic Quantum Field Theory Algebraic Quantum Field Theory Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 3 / 19
Algebraic Quantum Field Theory AQFT - I: Dirac field Spinor bundle SM M C 4 and cospinor bundle S M M (C 4 ), with M R Σ 4-dim globally hyperbolic spacetime : ds 2 = β 2 dt 2 h t; β C (M; R + ) and h t Riem(Σ); t R. Spinors ψ C sc (M, C 4 ) and cospinors ϕ C sc (M, (C 4 ) ). Dirac conjugation map: A : Csc (M, C 4 ) Csc (M, (C 4 ) ). ( ) Spin scalar product: x : Csc (M, C 4 ) Csc (M, C 4 ) C ψ ψ x := ( (Aψ) ψ ) (x). spacetime inner product: < > : Csc (M, C 4 ) Cc (M, C 4 ) C ˆ <ψ ϑ> = ψ ϑ x dµ g. M Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 4 / 19
Algebraic Quantum Field Theory AQFT - II: Dynamics Dirac operator on SM and its dual on S M: Dψ m. = (iγ µ µ + B m)ψ m = 0, D ϕ m = ( iγ µ µ + B m)ϕ m = 0. Causal propagators: G ( ) : C c (M, (C 4 ) ( ) ) C sc (M, (C 4 ) ( ) ) Hilbert spaces: ( Hm c := ( Hm s := D ( ) G ( ) = 0 = G ( ) D ( ) C c (M,(C 4 ) ( ) supp(g ( ) (f )) J(supp(f )), f C c (M, (C 4 ) ( ) ) Sol(D) C c (M, C 4 ) DCc (M, C 4 ), ( )s m Sol(D )) C c (M, (C 4 ) ) D Cc (M, C 4 ) ), ( )c m ˆ ). = /ν x dσ Σ ˆ ). = A 1 /ν A 1 x dσ Σ Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 5 / 19
Algebraic Quantum Field Theory AQFT - III: CAR Algebra unital Borchers-Uhlmann -algebra: A = - 1 = {1, 0, 0,...}, Φ(ψ m) = - -operation: { 0, { ( ) ( ) ψm 0, 0, ϕ m ψm,... ϕ m k=0 ( ) } ψm 0, 0,... } { = ( ) k Sol(D m) Sol(Dm) { ( ) } 0 0, ϕ m, 0,... ( ) ( ) } A 1 ϕ m A ψ A 1 ϕ m,... m Aψ m, Ψ(ϕ m) = 0, 0, We encode the CARs in the -ideal I A generated by: - Φ(ψ m) Φ( ψ m) + Φ( ψ m) Φ(ψ m) - Ψ(ϕ m) Ψ( ϕ m) + Ψ( ϕ m) Ψ(ϕ m) - Ψ(ϕ m) Φ(ψ m) + Φ(ψ m) Ψ(ϕ m) ( ) A 1 s ϕ m ψ m 1 m Algebra of fields: F. = A I Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 6 / 19
Algebraic Quantum Field Theory AQFT - IV: States Algebraic state ω : F C such that: ω(1) = 1, ω(h h) 0, h F. N.B.: Choosing a state ω is equivalent to assigning ω n(h 1,..., h n) n N and h i F. Quasi-free states: ω 2n+1(h 1,..., h 2n+1) = 0 ω 2n(h 1,..., h 2n) = ( 1) sign(σ) n σ S 2n i=1 ω 2 ( hσ(2i 1), h σ(2i) ). Question: Are all states physically acceptable? Of course not! Minimal physical requirements are: i) covariant construction of Wick polynomials to deal with interactions, ii) same UV behaviour of the Minkowski vacuum, iii) finite quantum fluctuations of all observables. Answer: Hadamard States Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 7 / 19
Algebraic Quantum Field Theory AQFT - V: Hadamard States A (quasi-free) state ω satisfies the Hadamard condition if and only if WF (ω 2) = { (x, y, ξ x, ξ y ) T M 2 \ 0 (x, ξx) (y, ξ y ), ξ x 0 }. deformation arguments static spacetime Question: How many Hadamard states do we know? pseudodifferential calculus holographic techniques Question: Does there exist an explicit method that which does not make use of use symmetries? Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 8 / 19
Quasi-free states for CAR algebras Quasi-free states for CAR algebras Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 9 / 19
Quasi-free states for CAR algebras Araki s characterisation [H. Araki: On quasifree states of CAR and Bogoliubov automorphisms.] Lemma 3.2: For any quasi-free state ω on F there exists Q B(Hm) s satisfying ) (a) ω 2 (Ψ(ϕ m)φ(ψ m) = ( ) A 1 s ϕ m Qψ m m (b) Q + AQA = 1, (c) 0 Q = Q 1. Lemma 3.3: Let Q be a bounded symmetric operator on H s m with the following properties (a) Q + AQA = 1, (b) 0 Q = Q 1. Then there exists a unique quasi-free state ω on F such that ) ω 2 (Ψ(ϕ m)φ(ψ m) = ( A 1 ) s ϕ m Qψ m m Our motto will be = Split the H ilbert space! (...but how?) Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 10 / 19
Quasi-free states for CAR algebras The Fermionic Signature Operator As before H s m := ( Sol(D), ( ) s m) and let N(, ) : H s m H s m C be a symmetric and densely defined sesquilinear form satisfying : a) N(, ) C( ). The Fermionic Signature Operator is S L(H s m) built out of the Riesz theorem N(, ) = ( S ) s m. Taking S 2 := S S, we construct the Fermionic Projector χ ± (S) := 1 2 S (S ± S ) : Hs m H s m. N.B.: If N(, ), then S B(Hm) s is essentially self-adjoint and ˆ χ ± (S) = χ(λ)de λ σ(s) Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 11 / 19
Fermionic Projectors and the Hadamard condition Fermionic Projectors and the Hadamard condition Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 12 / 19
Fermionic Projectors and the Hadamard condition FP states in a strip of spacetime Consider Ω ( T, T ) Σ M and H s m(ω) := ( Sol(D), ( ) s m). The spacetime inner product: < > : H s m(ω) H s m(ω) C <ψ m ϕ m> = Ω ψm ϕm x dµ Ω (well defined) <ϕ m ψ m> c ϕ m m ψ m m (bounded). Then the Fermionic Signature operator is essentially self-adjoint and ˆ χ ± (S) = χ ± Araki (λ) de λ == ω FP on F. Problems: For T the spacetime inner product is not well defined, ω 2(x, y) is in general not Hadamard! σ [C. Fewster and B. Lang: Pure quasifree states of the Dirac field from the fermionic projector.] Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 13 / 19
Fermionic Projectors and the Hadamard condition FP states and the mass oscillation property The strong mass oscillation property Hilbert space: H = ( Ψ := (ψ m) m I=(mL,m R ), (, ) = I ( )m dm ) Smearing operator: p : H C sc (M, SM), pψ = Symmetric sesquilinear form: <p p > : H H C Q: Is the new sesquilinear form bounded? The strong mass oscillation property: <pψ pφ> c ψm ϕ m <pt Ψ pφ> = <pψ pt Φ> where T Ψ = (mψ m) m I and T Φ = (mϕ m) m I. I ψm dm Family of linear operators (S m) m I with S m B(H s m) and sup m I S m < m (ψ m S m ϕ m) m is continuous ˆ <pψ pφ> = (ψ m S m ϕ m) m dm I Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 14 / 19
Fermionic Projectors and the Hadamard condition FP states and the strong mass oscillation property [F. Finster, S. M., C. Röken: The Fermionic Projector in a time-dependent external potential: mass oscillation property and Hadamard states.] i tψ m = γ 0 (i γ + B(t, x) m)ψ m =: Hψ m (Dirac equation) ˆ t ψ m t = U t,t ( ) 0 m ψ 0 + i γ 0 B ψ τ m dτ (Lippmann-Schwinger equation) Um t,τ t 0 Setting B(t, x) = 0: the strong mass oscillation property holds and B(H s m) S m( ξ) := ξ 0 =±ω( ξ) /ξ + m 2 ξ 0 ( ξ) γ0 = χ +( ) S m = Θ(ξ 0 ) }{{} frequencies splitting Assuming B(t) C 2 c ( 1 + t 2+ε) 1 and B(t) C 0 dt < 2 1, B( H m) s S m χ ± ( S m) = χ ± (S) + 1 ( S m λ) 1 S ( S D λ) 1 dλ. 2πi B 12 (±1) }{{} integral operator with smooth kernel Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 15 / 19
Fermionic Projectors and the Hadamard condition FP states and the mass oscillation property F. Finster, S. M., C. Röken: The Fermionic Projector in a time-dependent external potential: mass oscillation property and Hadamard states B(x) smooth B(x) = η(x 0 )B(x) B vac (x) = 0 Ω 1 Σ 1 η Ω1 = 1 Ω 1 η Ω2 = 0 Ω 2 Σ 2 η Ω2 = 0 Ω 2 P = χ + (S) G m + (smooth) P = χ + (S) Ğm + (smooth) Pvac = χ + (S) G m ω vac 2 (x, y) is Hadamard in Ω 0 and then in M via the prop. of singularities. ω 2(x, y) ω vac 2 (x, y) C (R 4 ) for all x, y Ω 0, since Ğ m(x, y) G m(x, y). ω 2(x, y) is Hadamard in Ω 0 and then in M via the prop. of singularities. ω 2(x, y) ω 2(x, y) C (R 4 ) for all x, y Ω 1, since G m(x, y) Ğ m(x, y). ω 2(x, y) is Hadamard in Ω 1 and then in M via the prop. of singularities. Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 16 / 19
Fermionic Projectors and the Hadamard condition FP states via the Møller operator (... still work in progress) [N. Drago, S. M.: A new class of Fermionic Projectors: Møller operators and mass oscillation properties".] Consider P V := V and P V := V P V = P V + V V = P V ( Id + E ± V (V V ) ) if (V V ) is compact or future/past compact. Møller operator: R ± V,V := ( Id + E ± V (V V ) ) 1 : Sol(P V ) Sol(P V ). Fix a Cauchy surface Σ, define ϱ + = 1 on J + (Σ ε) \ Σ ε and ϱ + = 0 on J (Σ ε) \ Σ ε, ϱ = 1 ϱ + and introduce m (m, m ) := ϱ + m + mϱ R + m,m := Id G ± m (m m ) = Id G ± m ϱ+ (m m) R m,m := Id G ± m (m m ) = Id G ± m ϱ (m m ). Ultra Møller operator: R m,m := R m,m R + m,m : Hm H m new families of solutions: ψ m Rψ m := (R β,m ψ m) β I Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 17 / 19
Fermionic Projectors and the Hadamard condition FP states via the Møller operator (... still work in progress) [N. Drago, S. M.: A new class of Fermionic Projectors: Møller operators and mass oscillation properties".] As before H s m := ( Sol(D), ( ) s m) and let N(, ) = <pr pr > : H s m H s m C If the weak mass oscillation property holds N(, ) C( ), then Fermionic Signature Operator is S L(H s m) built out of N(, ) = ( S ) s m. As before, we define the Fermionic Projector as χ ± (S) := 1 (S ± S ). 2 S Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 18 / 19
Conclusions Conclusions What we know: Araki s lemmas: ω : F C Q B(H s m) ; Spectral calculus: χ ± (S)χ ± (S) = χ ± (S) χ ± (S)χ (S) = 0. Benefit: No use of symmetries; Distinguished states. Price to pay: Not always available a sesquilinear form = mass oscillation property. Future investigations: Hadamard condition. Simone Murro (Universität Regensburg) Fermionic Projectors and Hadamard States LQP 2016 19 / 19