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Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

Max Lewandowski

Abstract

According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of Hadamard form iff they are given by a linear combination of distinguished parametrices $\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F} + \widetilde{G}_A - \widetilde{G}_R\big)$ in the sense of Duistermaat-Hörmander. Inspired by the construction of the corresponding advanced and retarded Green operator $G_A,G_R$ as done in Bär, Ginoux, Pfäffle 2007, we construct the remaining two Green operators $G_F, G_{aF}$ locally in terms of Hadamard series. Afterwards, we provide the global construction of $\frac{i}{2}\big(\widetilde{G}_{aF} - \widetilde{G}_{F}\big)$, which relies on new techniques like a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e. the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.

Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes

Abstract

According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of Hadamard form iff they are given by a linear combination of distinguished parametrices in the sense of Duistermaat-Hörmander. Inspired by the construction of the corresponding advanced and retarded Green operator as done in Bär, Ginoux, Pfäffle 2007, we construct the remaining two Green operators locally in terms of Hadamard series. Afterwards, we provide the global construction of , which relies on new techniques like a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e. the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.

Paper Structure

This paper contains 18 sections, 39 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Quantum field theory on curved space times
  3. States, quasifree states and Hadamard states
  4. This work
  5. Preliminaries
  6. Cauchy problems
  7. The prototype
  8. Local Construction
  9. Families of Riesz-like distributions
  10. The Hadamard series
  11. Local parametrices and Hadamard bidistributions
  12. Local fundamental solutions and Hadamard bisolutions
  13. Global Hadamard two-point-functions
  14. Global construction of symmetric bisolutions
  15. Positivity
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $M,N$ be globally hyperbolic Lorentzian manifolds with Cauchy hypersurfaces $\Sigma,\Xi$ and unit normal fields $\mu,\nu$. Furthermore, let $P,Q$ denote linear differential operators of second order acting on smooth sections in vector bundles $E,F$ over $M,N$, which admit well-posed Cauchy probl Let $Z:=(\oplus^2C^\infty(M\times N,E\boxtimes F))\oplus(\oplus^4C^\infty(\Sigma\times\Xi,E\boxtime

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5: Théorème 2 of M1954
  • Theorem 2.6: Théorème 1 of M1954
  • Proposition 2.7
  • ...and 68 more