Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Index theorems for hyperbolic operators and particle creation Alexander Strohmaier University of Leeds York, LQP meeting 06. April 2017
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory The Gauß-Bonnet Theorem Theorem (C.F . Gauß, P .O. Bonnet, 1827–1848) � 1 K ( x ) dx = χ ( S ) 2 π S
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory The Gauß-Bonnet Theorem Theorem (C.F . Gauß, P .O. Bonnet, 1827–1848) � 1 K ( x ) dx = χ ( S ) 2 π S Here χ ( S ) is the Euler-number of S : χ ( S ) = # triangles − # edges +# vertices = − 2 g + 2 http://commons.wikimedia.org/wiki/File:Tri-brezel.png#/media/File:Tri-brezel.png
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory The Gauß-Bonnet-Chern Theorem For curved spaces M of higher dimension n (Riemannian manifolds). Theorem (C.F . Gauß, P .O. Bonnet, S.-S. Chern, 1945) � ( 2 π ) − n / 2 Pf (Ω) = χ ( M ) M
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Hodge-deRham-Theorie exterior differential d : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k + 1 ) codifferential: δ : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k − 1 ) together the form the Euler-operator: d + δ : C ∞ ( M , Λ even ) → C ∞ ( M , Λ odd )
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Hodge-deRham-Theorie exterior differential d : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k + 1 ) codifferential: δ : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k − 1 ) together the form the Euler-operator: d + δ : C ∞ ( M , Λ even ) → C ∞ ( M , Λ odd ) Hodge-deRham-Theory: χ ( M ) = ind ( d + δ ) := dim ker ( d + δ ) − dim coker ( d + δ )
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Hodge-deRham-Theorie exterior differential d : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k + 1 ) codifferential: δ : C ∞ ( M , Λ k ) → C ∞ ( M , Λ k − 1 ) together the form the Euler-operator: d + δ : C ∞ ( M , Λ even ) → C ∞ ( M , Λ odd ) Hodge-deRham-Theory: χ ( M ) = ind ( d + δ ) := dim ker ( d + δ ) − dim coker ( d + δ ) Therefore: � ( 2 π ) − n / 2 Pf (Ω) = ind ( d + δ ) M
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Singer Index Theorem • M closed Riemannian manifold, • D : C ∞ ( M , E ) → C ∞ ( M , F ) elliptic first order operator
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Singer Index Theorem • M closed Riemannian manifold, • D : C ∞ ( M , E ) → C ∞ ( M , F ) elliptic first order operator basic example (if a spin structure is given): Dirac-Operator D : C ∞ ( M , S + M ) → C ∞ ( M , S − M ) n � D = γ ( e j ) ∇ e j j = 1
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Singer Index Theorem Theorem (M. Atiyah, I. Singer, 1968) � ind ( D ) = algebr . expression M
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Singer Index Theorem Theorem (M. Atiyah, I. Singer, 1968) � ind ( D ) = algebr . expression M Example Dirac-Operator � � ind ( D ) = A (Ω) M
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Singer Index Theorem Theorem (M. Atiyah, I. Singer, 1968) � ind ( D ) = algebr . expression M Example Dirac-Operator � � ind ( D ) = A (Ω) ∧ ch ( E ) M E is a twist bundle.
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Dimension 4 The Atiyah-Singer integrand in dimension 4 (twisted case): � � 1 tr ( F 2 ) + 1 ˆ 48 tr ( R 2 ) A ∧ ch ( E )( x ) = . ( 2 π ) 2 tr ( F 2 ) = 1 4 ǫ abcd tr ( F ab F cd ) , similarly for R .
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Spectral Boundary Conditions If ∂ M � = ∅ we need boundary conditions.
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Spectral Boundary Conditions If ∂ M � = ∅ we need boundary conditions. Choose a boundary defining function „Fermi coordinates“ r : M → R and write � ∂ � D = γ ∂ r + A r A 0 is an elliptic self-adjoint operator on ∂ M . P + = χ [ 0 , ∞ ) ( A 0 ) = spectral projection
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Spectral Boundary Conditions If ∂ M � = ∅ we need boundary conditions. Choose a boundary defining function „Fermi coordinates“ r : M → R and write � ∂ � D = γ ∂ r + A r A 0 is an elliptic self-adjoint operator on ∂ M . P + = χ [ 0 , ∞ ) ( A 0 ) = spectral projection APS-boundary conditions: P + ( f | ∂ M ) = 0
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Atiyah-Patodi-Singer Index Theorem Theorem (M. Atiyah, V. Patodi, I. Singer, 1975) imposing APS boundary conditions we have � ind ( D ) = alg . expr . (Ω) M � alg . expr (Ω , 2 . FF ) − h ( A 0 ) + η ( A 0 ) + 2 ∂ M where • h ( A ) = dim ker ( A ) � sign ( λ ) · | λ | − s • η ( A ) = η A ( 0 ) , and η A ( s ) = λ ∈ spec ( A ) λ � = 0
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Lorentzian Manifolds Replace „space “ by „space-times “, that is Riemannian by Lorentzian manifolds.
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Lorentzian Manifolds Replace „space “ by „space-times “, that is Riemannian by Lorentzian manifolds. model signature natural op. Dirac-op. Laplace � n Euclid positive definite elliptic ∂ 2 j j = 1 d’Alembert � n Minkowski ( n − 1 , 1 ) hyperbolic − ∂ 2 ∂ 2 1 + j j = 2
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Closedness? Problem 1: compact Lorentzian manifolds (without boundary) violate causality conditions ⇒ not suitable for models in GR
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Closedness? Problem 1: compact Lorentzian manifolds (without boundary) violate causality conditions ⇒ not suitable for models in GR Problem 2: hyperbolic PDE-Theory does not work on such space-times ⇒ no Lorentzian analog of the Atiyah-Singer Theorem.
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory The Lorentzian Index Theorem Let M be a compact globally hyperbolic space-time with boundary ∂ M = Σ 1 ⊔ Σ 2 Σ j smooth spacelike Cauchy surfaces D Dirac-Operator
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory The Lorentzian Index Theorem Let M be a compact globally hyperbolic space-time with boundary ∂ M = Σ 1 ⊔ Σ 2 Σ j smooth spacelike Cauchy surfaces D Dirac-Operator Theorem (A. S. , C. Bär, 2015) With APS-boundary conditions D is a Fredholm opera- tor. Its kernel consists of smooth spinors and one has � � � T � ind ( D ) = A (Ω) + A (Ω , 2 . FF ) ∂ M M − h ( A 1 ) + h ( A 2 ) + η ( A 1 ) − η ( A 2 ) 2
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Physical Interpretation: One particle picture Wave Evolution Operator Q : C ∞ (Σ 1 ) → C ∞ (Σ 2 ) : For ϕ ∈ C ∞ (Σ 1 ) solve D Φ = 0 with initial condition Φ | Σ 1 = ϕ . Then Q ϕ = Φ | Σ 2 . Q extends to a unitary operator L 2 (Σ 1 ) → L 2 (Σ 2 ) .
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Physical Interpretation: One particle picture Wave Evolution Operator Q : C ∞ (Σ 1 ) → C ∞ (Σ 2 ) : For ϕ ∈ C ∞ (Σ 1 ) solve D Φ = 0 with initial condition Φ | Σ 1 = ϕ . Then Q ϕ = Φ | Σ 2 . Q extends to a unitary operator L 2 (Σ 1 ) → L 2 (Σ 2 ) . Decompose � Q ++ � Q + − Q = Q − + Q −− wr.t. the splitting L 2 (Σ 1 ) = L 2 [ 0 , ∞ ) (Σ 1 ) ⊕ L 2 ( −∞ , 0 ) (Σ 1 ) , L 2 (Σ 2 ) = L 2 ( 0 , ∞ ) (Σ 2 ) ⊕ L 2 ( −∞ , 0 ] (Σ 2 )
Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory Physical Interpretation: One particle picture Wave Evolution Operator Q : C ∞ (Σ 1 ) → C ∞ (Σ 2 ) : For ϕ ∈ C ∞ (Σ 1 ) solve D Φ = 0 with initial condition Φ | Σ 1 = ϕ . Then Q ϕ = Φ | Σ 2 . Q extends to a unitary operator L 2 (Σ 1 ) → L 2 (Σ 2 ) . Decompose � Q ++ � Q + − Q = Q − + Q −− wr.t. the splitting L 2 (Σ 1 ) = L 2 [ 0 , ∞ ) (Σ 1 ) ⊕ L 2 ( −∞ , 0 ) (Σ 1 ) , L 2 (Σ 2 ) = L 2 ( 0 , ∞ ) (Σ 2 ) ⊕ L 2 ( −∞ , 0 ] (Σ 2 ) Then we have ind ( D ) = dim ker ( Q −− ) − dim ker ( Q ++ )
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