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Index theorems for hyperbolic operators and particle creation - - PowerPoint PPT Presentation

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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

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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
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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 2π

  • S

K(x) dx = χ(S)

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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 2π

  • S

K(x) dx = χ(S) Here χ(S) is the Euler-number of S: χ(S) = #triangles−#edges+#vertices = −2g+2

http://commons.wikimedia.org/wiki/File:Tri-brezel.png#/media/File:Tri-brezel.png

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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)

(2π)−n/2

  • M

Pf(Ω) = χ(M)

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Hodge-deRham-Theorie

exterior differential d : C∞(M, Λk) → C∞(M, Λk+1) codifferential: δ : C∞(M, Λk) → C∞(M, Λk−1) together the form the Euler-operator: d + δ : C∞(M, Λeven) → C∞(M, Λodd)

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Hodge-deRham-Theorie

exterior differential d : C∞(M, Λk) → C∞(M, Λk+1) codifferential: δ : C∞(M, Λk) → C∞(M, Λk−1) together the form the Euler-operator: d + δ : C∞(M, Λeven) → C∞(M, Λodd) Hodge-deRham-Theory: χ(M) = ind(d + δ) := dim ker(d + δ) − dim coker(d + δ)

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Hodge-deRham-Theorie

exterior differential d : C∞(M, Λk) → C∞(M, Λk+1) codifferential: δ : C∞(M, Λk) → C∞(M, Λk−1) together the form the Euler-operator: d + δ : C∞(M, Λeven) → C∞(M, Λodd) Hodge-deRham-Theory: χ(M) = ind(d + δ) := dim ker(d + δ) − dim coker(d + δ) Therefore: (2π)−n/2

  • M

Pf(Ω) = ind(d + δ)

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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
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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) D =

n

  • j=1

γ(ej)∇ej

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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) =

  • M
  • algebr. expression
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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) =

  • M
  • algebr. expression

Example Dirac-Operator

ind(D) =

  • M
  • A(Ω)
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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) =

  • M
  • algebr. expression

Example Dirac-Operator

ind(D) =

  • M
  • A(Ω)∧ch(E)

E is a twist bundle.

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Dimension 4

The Atiyah-Singer integrand in dimension 4 (twisted case): ˆ A ∧ ch(E)(x) = 1 (2π)2

  • tr(F 2) + 1

48tr(R2)

  • .

tr(F 2) = 1 4ǫabcdtr(FabFcd), similarly for R.

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Spectral Boundary Conditions

If ∂M = ∅ we need boundary conditions.

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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 + Ar

  • A0 is an elliptic self-adjoint operator on ∂M.

P+ = χ[0,∞)(A0) = spectral projection

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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 + Ar

  • A0 is an elliptic self-adjoint operator on ∂M.

P+ = χ[0,∞)(A0) = spectral projection APS-boundary conditions: P+(f|∂M) = 0

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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) =

  • M
  • alg. expr.(Ω)

+

  • ∂M
  • alg. expr(Ω, 2. FF)−h(A0) + η(A0)

2 where

  • h(A) = dim ker(A)
  • η(A) = ηA(0), and ηA(s) =
  • λ∈spec(A)

λ=0

sign(λ) · |λ|−s

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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.

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Lorentzian Manifolds

Replace „space “ by „space-times “, that is Riemannian by Lorentzian manifolds. model signature natural op. Dirac-op. Euclid positive definite Laplace

n

  • j=1

∂2

j

elliptic Minkowski (n − 1, 1) d’Alembert −∂2

1 + n

  • j=2

∂2

j

hyperbolic

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Closedness?

Problem 1: compact Lorentzian manifolds (without boundary) violate causality conditions ⇒ not suitable for models in GR

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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.

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The Lorentzian Index Theorem

Let M be a compact globally hyperbolic space-time with boundary ∂M = Σ1 ⊔ Σ2 Σj smooth spacelike Cauchy surfaces D Dirac-Operator

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The Lorentzian Index Theorem

Let M be a compact globally hyperbolic space-time with boundary ∂M = Σ1 ⊔ Σ2 Σj 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

ind(D) =

  • M
  • A(Ω) +
  • ∂M

T A(Ω, 2. FF) −h(A1) + h(A2) + η(A1) − η(A2) 2

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Physical Interpretation: One particle picture

Wave Evolution Operator Q : C∞(Σ1) → C∞(Σ2): For ϕ ∈ C∞(Σ1) solve DΦ = 0 with initial condition Φ|Σ1 = ϕ. Then Qϕ = Φ|Σ2. Q extends to a unitary operator L2(Σ1) → L2(Σ2).

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Physical Interpretation: One particle picture

Wave Evolution Operator Q : C∞(Σ1) → C∞(Σ2): For ϕ ∈ C∞(Σ1) solve DΦ = 0 with initial condition Φ|Σ1 = ϕ. Then Qϕ = Φ|Σ2. Q extends to a unitary operator L2(Σ1) → L2(Σ2). Decompose Q = Q++ Q+− Q−+ Q−−

  • wr.t. the splitting

L2(Σ1) = L2

[0,∞)(Σ1) ⊕ L2 (−∞,0)(Σ1) ,

L2(Σ2) = L2

(0,∞)(Σ2) ⊕ L2 (−∞,0](Σ2)

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Physical Interpretation: One particle picture

Wave Evolution Operator Q : C∞(Σ1) → C∞(Σ2): For ϕ ∈ C∞(Σ1) solve DΦ = 0 with initial condition Φ|Σ1 = ϕ. Then Qϕ = Φ|Σ2. Q extends to a unitary operator L2(Σ1) → L2(Σ2). Decompose Q = Q++ Q+− Q−+ Q−−

  • wr.t. the splitting

L2(Σ1) = L2

[0,∞)(Σ1) ⊕ L2 (−∞,0)(Σ1) ,

L2(Σ2) = L2

(0,∞)(Σ2) ⊕ L2 (−∞,0](Σ2)

Then we have ind(D) = dim ker(Q−−) − dim ker(Q++)

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Physical Interpretation: After second quantization

Shale-Stinespring: Q is implementable if Q+− is Hilbert-Schmidt (the case in dim 4).

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Physical Interpretation: After second quantization

Shale-Stinespring: Q is implementable if Q+− is Hilbert-Schmidt (the case in dim 4). The sector of charge N is mapped to sector with charge N + ind(D).

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Physical Interpretation: After second quantization

Shale-Stinespring: Q is implementable if Q+− is Hilbert-Schmidt (the case in dim 4). The sector of charge N is mapped to sector with charge N + ind(D). Normal ordering in Fock space depends on the state (i.e. on the Cauchy surface).

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Previous work

  • Gibbons (’79), Gibbons and Richer (’80), Lohiya (’83) in

explicit models

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Previous work

  • Gibbons (’79), Gibbons and Richer (’80), Lohiya (’83) in

explicit models

  • Matsui (’90), Bunke and Hirschmann (’92), Σ1 = Σ2
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Previous work

  • Gibbons (’79), Gibbons and Richer (’80), Lohiya (’83) in

explicit models

  • Matsui (’90), Bunke and Hirschmann (’92), Σ1 = Σ2
  • Delbourgo and Salam (’72) , Dowker (’78), Zahn (’15),

non-concervation of axial current

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Methods

  • evolution operator Q is a Fourier Integral Operator
  • compute principal symbol using the calculus by

Duistermaat and Hörmander.

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Methods

  • evolution operator Q is a Fourier Integral Operator
  • compute principal symbol using the calculus by

Duistermaat and Hörmander.

  • define the appropriate function spaces using the theory of

hyperbolic PDEs and wavefront sets

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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Methods

  • evolution operator Q is a Fourier Integral Operator
  • compute principal symbol using the calculus by

Duistermaat and Hörmander.

  • define the appropriate function spaces using the theory of

hyperbolic PDEs and wavefront sets

  • use spectral flow and to transform to usual APS theorem
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Curvature and Index Atiyah(-Patodi)-Singer Index Theorem Lorentzian Manifolds Quantum Field Theory

Methods

  • evolution operator Q is a Fourier Integral Operator
  • compute principal symbol using the calculus by

Duistermaat and Hörmander.

  • define the appropriate function spaces using the theory of

hyperbolic PDEs and wavefront sets

  • use spectral flow and to transform to usual APS theorem
  • Feynman parametrix constructed using a gluing

contruction

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Thanks for your attention & Happy Birthday