Hadamard States via Fermionic Projectors in a Time-Dependent External Potentials 1

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1 Hadamard States via Fermionic Projectors in a Time-Dependent External Potentials 1 Simone Murro Fakultät für Mathematik Universität Regensburg New Trends in Algebraic Quantum Field Theory Frascati, 13th of February Joint work with F. Finster and C. Röken Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

2 Motivations Motivation Our setting: Space-time: M = (R 4, η), with dimm = 4 and η = diag(+1, 1, 1, 1). Matter: ( ıγ µ µ + B m ) ψ m (x) = 0. Difficulties: Goals: No translational symmetries No canonical frequency splitting. No conformal covariance We cannot use the bulk to boundary corresponcence. Construct a Hadamard state; Test our construction in an external time-dependent potential as preparation for curved space-times. Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

3 Motivations Warning Our construction is considerably different from the (modified) S-J construction: [Brum-Fredenhagen, arxiv: Fewster-Verch, arxiv: ] There exists two constructions of the fermionic projector: The mass oscillation property is something completely new! Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

4 Outline Outline Mathematical Preliminaries Quasi-Free States and Fermionic Projectors Mass Oscillation Property Hadamard States Based on: F. Finster, S. M., C. Röken, arxiv: [math-ph] Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

5 Mathematical preliminaries An Old Story I: Dirac spinor Spinors ψ m C sc (M, SM) and cospinors σ C sc (M, S M). Dirac conjugation map. : C 0 (M, SM) ( ) C 0 (M, S M). Spin scalar product: x : C sc (M, SM) C sc (M, SM) C ψ m ϕ m x := ψ m (x) γ 0 ϕ m (x). Space-time inner product: < > : C sc (M, SM) C 0 <ψ m ϕ c > = R 4 ψ m ϕ c x d 4 x. Scalar product: ( ) m t : C sc (M, SM) C sc (M, SM) C (ϕ m ψ m ) m t := 2π R 3 ψ m γ 0 ϕ m (t, x) d 3 x. Hilbert space H m := ( C sc (M, SM), ( ) m ) (M, SM) C Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

6 Mathematical preliminaries An Old Story II: CAR Algebra and Quasi-Free States Space of pairs of spinorial test functions: D := C 0 (M, SM) C 0 (M, S M). Scalar product: (, ) D : D D R + defined as ( f g a b ) D := <f K m a> + < K m b g > Involution map: Γ : D D such that Γ 2 = 1 and ( Γh 1, Γh 2 )D = (h 2, h 1 ) D. The field algebra F: unital -algebra generated by the abstract elements B(h) with h D which satisfy: (i) Linearity: B(αf g + m βn) = αb(f g) + βb(m n) (ii) Hermiticity: B(f g) = B ( Γ(f g) ) (iii) Dynamics: B ( (D m)f (D m)g ) = 0 for all f g D (iv) CARs: { B(f g), B(m n) } = (f g m n) D 1 F. A quasi-free state ω : F C if ω 2n+1 (h 1,..., h 2n+1 ) = 0 and ω 2n (h 1,..., h 2n ) = n ( ) ( 1) sign(σ) ω 2 hσ(2i 1), h σ(2i). σ S 2n i=1 Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

7 Quasi-Free States and Fermionic Projectors Quasi-Free States and Projection Operators Lemma 3.3 (H. Araki: On quasifree states of CAR and Bogoliubov automorphisms. (1970/71).) Let R be a bounded symmetric operator on (H D, (..) D ) with the following properties (a) R + ΓRΓ = 1, (b) 0 R = R 1. Then there exists a unique quasi-free state ω on F such that ω ( B(h) B( h) ) = (h R h) D for all h, h H D. Our motto will be = Split the H ilbert space! (...but how?) Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

8 Quasi-Free States and Fermionic Projectors The Fermionic Projector in a Strip of Space-Time [F. Finster and M. Reintjes: A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I Space-times of finite lifetime, arxiv: [math-ph], to appear in Adv. Theor. Math. Phys. (2015).] Begin with a simple example, to explain the basic idea: Ω ( T, T ) Σ M. As before the scalar product: ( ) m : C sc (Ω, S Ω) C sc (Ω, S Ω) C (ϕ m ψ m ) m := 2π Σ ψ m γ 0 ϕ m (t, x) dµ Σ. H m := ( C sc (Ω, S Ω), ( ) m ) is an Hilbert space. Now the space-time inner product: < > : C sc (Ω, S Ω) C sc (Ω, S Ω) C <ψ m ϕ m > = Ω ψ m ϕ m x dµ Ω (well defined) <ϕ m ψ m > c ϕ m m ψ m m Via Riesz representation theorem <ϕ m ψ m > = (ϕ m S ψ m ) m. (bounded). Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

9 Quasi-Free States and Fermionic Projectors From the spectral theorem, we can construct an orthonormal projector ˆ χ ± ( S) = χ ± (λ) de λ. Finally we can obtain a quasi-free state in the all the space-time via: P := χ + ( S) K. Problems: For T = the space-time inner product is not well defined, P(x, y) is in general not Hadamard! [C. Fewster and B. Lang: Pure quasifree states of the Dirac field from the fermionic projector, arxiv: [math-ph].] σ Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

10 Fermionic Projectors and the Mass Oscillation Property Mass Oscillation Property [F. Finster and M. Reintjes: A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds II Space-times of infinite lifetime, arxiv: [math-ph].] Families of solutions of families of Dirac equations: Ψ := (ψ m ) m I=(mL,m R ) H. New scalar product: (, ) : H H C ˆ ˆ ˆ (Ψ Φ) = (ψ m ϕ m ) m dm = 2π ψ m γ 0 ϕ m (t, x) d 3 x dm. I I R 3 Integration over mass as operator ˆ p : H Csc (M, SM), pψ = ψ m dm. p space-time inner product: <p p > : H H C I Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

11 Fermionic Projectors and the Mass Oscillation Property Q: Is the new space-time inner product bounded? The Dirac operator has the strong mass oscillation property in I= (m L, m R ) if there exists a constant c > 0 such that ˆ ˆ <pψ pφ> = pψ pφ x d 4 x c ψ m m ϕ m m dm R 4 for all families of solutions Ψ, Φ H. Then there exists a family of linear operators ( S m ) m I with S m L(H m ) which are uniformly bounded sup m I S m <, such that ˆ <pψ pφ> = (ψ m S m ϕ m ) m dm for all Ψ, Φ H. I The operator S m is uniquely determined for every m I by demanding that for all Ψ, Φ H, the functions (ψ m S m ϕ m ) m are continuous in m. I Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

12 Fermionic Projectors and the Mass Oscillation Property The Fermionic Projector in the Minkowski Vacuum ( B = 0 ) Family of solutions of the Dirac equations Ψ = (ψ m ) m I H in momentum space via the Fourier transform of a solution : ψ m (k) = 2π δ(k 2 m 2 ) ε(k 0 ) (/k + m) γ 0 ˆψ0 m ( k). After integration over m (pψ)(k) = 2π χ I (m) 1 2m ε(k0 ) (/k + m) γ 0 ˆψ0 m ( k) m= k 2 we compute the p space-time inner product ˆ d 4 k <pψ pϕ> = 4π 2 χ I (m) 1 2m γ0 ˆψ m( 0 k) (/k + m) γ 0 ˆϕ 0 m( k) m= k 2 Reparametrizing the k 0 -integral as an integral over m, we obtain <pψ pϕ> = 1 ˆ ˆ d 3 k 4π 2 dm 2 k 0 γ0 ˆψ0 m ( k) (/k+m) γ 0 ˆϕ 0 m( k) k 0 =± k 2 +m 2. I R 3. Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

13 Fermionic Projectors and the Mass Oscillation Property Using the Schwarz inequality and applying Plancherel s theorem <pψ pϕ> 1 ˆ ˆ ˆ 4π 2 dm ˆψ m( 0 k) ˆϕ 0 m( k) d 3 k 2π ψ m m ϕ m m dm. I R 3 I Then exists an operator S m uniquely determined for every m I : S m ( k) := Applying the spectral theorem: k 0 =±ω( k) χ +( S m ) = Θ(k 0 ). /k + m 2 ω( k) γ0. Then the fermonic projector in momentum space is P = χ +( ) S m Km = Θ(k 0 ) }{{} 2π δ(k2 m 2 ) ε(k 0 ) (/k + m) γ 0 Select the positive frequencies Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

14 Fermionic Projectors and the Mass Oscillation Property External Potential in Minkowski space-time If the external potential B satisfies the conditions c B(t) C t 2+ε then the strong mass oscillation property holds. For every Ψ, Φ H, ˆ <pψ pφ> = (ψ m S m ϕ m ) m dm, I where S m : H m H m are bounded linear operators which act on the wave functions at time t 0 by ˆ S m = S m + i ε(t t 0 ) [ S m Um t0,t 2 1 (ˆ ˆ ˆ t0 ˆ t0 ) 2 t 0 t 0 + γ 0 B(t) Ũ t,t0 m Ũ t0,t m Ũ t0,t m γ 0 ] B(t) S m Um t,t0 dt γ 0 B(t) S m U t,t m γ 0 B(t ) Ũ t,t 0 m dt dt Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

15 Fermionic Projectors and the Mass Oscillation Property Using PDE methods, we worked hard! (This is the technical core of the paper) We decompose S m with respect to the above frequency splitting, S m = S D + S, where S D := S S and S := S + + S +. Under the assumption ˆ B(t) C 0 dt < 2 1 the operators χ ± ( S m ) have the representations χ ± ( S m ) = χ ± (H) + 1 ( S m λ) 1 S ( S D λ) 1 dλ, 2πi B 12 (±1) }{{} integral operator with smooth kernel The fermionic projector is given by P = χ + ( S m ) K m = χ + (H) K m + smooth contribution. Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

16 Hadamard States Hadamard States: Three Different External Potentials 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 + (H) K m + (smooth) P = χ + (H) K m + (smooth) P vac = χ + (H) K m P vac (x, y) is Hadamard in Ω 0 and then in the whole spacetime. P(x, y) P vac (x, y) C (R 4 ) for all x, y Ω 0, since k m (x, y) k m (x, y). P(x, y) is Hadamard in Ω 0 and then in the whole spacetime. P(x, y) P(x, y) C (R 4 ) for all x, y Ω 1, since k m (x, y) k m (x, y). P(x, y) is Hadamard in Ω 1 and then in the whole spacetime. Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17

17 Conclusions Conclusions What we know: To every projector it is possible to associate an algebraic state. The construction holds also in the presence of an external potential. The state is Hadamard and when B = 0 is the Poincaré vacuum. Benefit: This technique works without symmetries. Allows to construct states for massive particles. Future investigations: in which space-time the Dirac operator satisfies the mass oscillation property, if the states constructed turn out to be Hadamard. Simone Murro (Universität Regensburg) Hadamard States via Fermionic Projector AQFT / 17