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IR-fixed Euclidean vacuum for linearized gravity on de Sitter space

Christian Gérard, Michał Wrochna

Abstract

We consider the Euclidean vacuum for linearized gravity on the global de Sitter space, obtained from the Euclidean Green's function on the 4-sphere. We use the notion of Calderón projectors to recover a quantum state for the Lorentzian theory on de Sitter space. We show that while the state is gauge invariant and Hadamard, it is not positive on the whole of the phase space. We show however that a suitable modification at low energies yields a well-defined Hadamard state on global de Sitter space.

IR-fixed Euclidean vacuum for linearized gravity on de Sitter space

Abstract

We consider the Euclidean vacuum for linearized gravity on the global de Sitter space, obtained from the Euclidean Green's function on the 4-sphere. We use the notion of Calderón projectors to recover a quantum state for the Lorentzian theory on de Sitter space. We show that while the state is gauge invariant and Hadamard, it is not positive on the whole of the phase space. We show however that a suitable modification at low energies yields a well-defined Hadamard state on global de Sitter space.
Paper Structure (108 sections, 43 theorems, 292 equations)

This paper contains 108 sections, 43 theorems, 292 equations.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Plan of the paper
  4. Notation
  5. Isomorphisms of vector spaces
  6. Sesquilinear forms
  7. Operators on quotient spaces
  8. Sesquilinear forms on quotients
  9. Differential operators
  10. A guide to the use of sub- and superscripts
  11. Linearized gravity
  12. Notation and background
  13. Convention for the Riemann tensor
  14. Hermitian forms on tensors
  15. Decomposition of tensors
  16. ...and 93 more sections

Key Result

Theorem 1.2

The Euclidean Green's function on $\mathbb{S}^4$ does not define a state for linearized gravity on $dS^4$ by Wick rotation. More precisely, $c_2^\pm$ satisfy properties 1)--3) and the Hadamard condition, but do not satisfy 4) on the whole of ${\mathcal{E}}_{{\rm TT}}$.

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2: cf. Thm. \ref{['thm5.0']}--Props.\ref{['prop5.3']}--\ref{['prop-pos1']}
  • Theorem 1.3: cf. Thm. \ref{['thm5.1']}
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 50 more