Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two-point Functions and Quantum Fields in de Sitter Universe

J. Bros, U. Moschella

TL;DR

This work develops a comprehensive analytic framework for quantum fields on de Sitter space by replacing the Minkowski spectral condition with a geodesic spectral condition anchored in maximal analyticity and perikernel structures on the complex quadric ${X^{(c)}_d}$. It introduces de Sitter plane waves, a double analytic structure, and a Laplace/Fourier calculus that yield Källén-Lehmann-type representations for general two-point functions and a conjugate Euclidean (Schwinger) perspective, including a thermal interpretation via geodesic observers. The paper establishes a rigorous free-field theory on ${X_d}$, derives the Reeh-Schlieder property, and demonstrates how Minkowski QFTs emerge as contraction limits, thereby linking curved-space QFT to the flat-space paradigm. It then outlines an axiomatic, Wightman-style approach for interacting de Sitter QFTs grounded in tuboid analyticity, offering a pathway toward a full constructive/axiomatic theory in curved spacetime with potential implications for cosmology and inflationary physics.

Abstract

We present a theory of general two-point functions and of generalized free fields in d-dimensional de Sitter space-time which closely parallels the corresponding minkowskian theory. The usual spectral condition is now replaced by a certain geodesic spectral condition, equivalent to a precise thermal characterization of the corresponding ``vacuum''states. Our method is based on the geometry of the complex de Sitter space-time and on the introduction of a class of holomorphic functions on this manifold, called perikernels, which reproduce mutatis mutandis the structural properties of the two-point correlation functions of the minkowskian quantum field theory. The theory contains as basic elementary case the linear massive field models in their ``preferred'' representation. The latter are described by the introduction of de Sitter plane waves in their tube domains which lead to a new integral representation of the two-point functions and to a Fourier-Laplace type transformation on the hyperboloid. The Hilbert space structure of these theories is then analysed by using this transformation. In particular we show the Reeh-Schlieder property. For general two-point functions, a substitute to the Wick rotation is defined both in complex space-time and in the complex mass variable, and substantial results concerning the derivation of Kallen-Lehmann type representation are obtained.

Two-point Functions and Quantum Fields in de Sitter Universe

TL;DR

This work develops a comprehensive analytic framework for quantum fields on de Sitter space by replacing the Minkowski spectral condition with a geodesic spectral condition anchored in maximal analyticity and perikernel structures on the complex quadric . It introduces de Sitter plane waves, a double analytic structure, and a Laplace/Fourier calculus that yield Källén-Lehmann-type representations for general two-point functions and a conjugate Euclidean (Schwinger) perspective, including a thermal interpretation via geodesic observers. The paper establishes a rigorous free-field theory on , derives the Reeh-Schlieder property, and demonstrates how Minkowski QFTs emerge as contraction limits, thereby linking curved-space QFT to the flat-space paradigm. It then outlines an axiomatic, Wightman-style approach for interacting de Sitter QFTs grounded in tuboid analyticity, offering a pathway toward a full constructive/axiomatic theory in curved spacetime with potential implications for cosmology and inflationary physics.

Abstract

We present a theory of general two-point functions and of generalized free fields in d-dimensional de Sitter space-time which closely parallels the corresponding minkowskian theory. The usual spectral condition is now replaced by a certain geodesic spectral condition, equivalent to a precise thermal characterization of the corresponding ``vacuum''states. Our method is based on the geometry of the complex de Sitter space-time and on the introduction of a class of holomorphic functions on this manifold, called perikernels, which reproduce mutatis mutandis the structural properties of the two-point correlation functions of the minkowskian quantum field theory. The theory contains as basic elementary case the linear massive field models in their ``preferred'' representation. The latter are described by the introduction of de Sitter plane waves in their tube domains which lead to a new integral representation of the two-point functions and to a Fourier-Laplace type transformation on the hyperboloid. The Hilbert space structure of these theories is then analysed by using this transformation. In particular we show the Reeh-Schlieder property. For general two-point functions, a substitute to the Wick rotation is defined both in complex space-time and in the complex mass variable, and substantial results concerning the derivation of Kallen-Lehmann type representation are obtained.

Paper Structure

This paper contains 25 sections, 32 theorems, 196 equations, 2 figures.

Table of Contents

  1. Introduction and description of the results
  2. Two-point functions and generalized free fields in the maximal analyticity framework
  3. Notations
  4. Local scalar fields in de Sitter space-time; generalized free fields
  5. The property of maximal analyticity of two-point functions; perikernels on $X^{(c)}_d$
  6. The thermal interpretation and the geodesic spectral condition for generalized free fields
  7. Feynman-type algebra, double analytic structure, and complex mass representation of the de Sitter propagators
  8. De Sitterian propagators: Feynman-type algebra and a substitute to the "Wick-rotation"
  9. The Laplace transform of the retarded propagators
  10. The euclidean propagators in the mass variable. The FG property
  11. Inversion of the Laplace transformation and Källen-Lehmann-type representations
  12. Quantum theory of free fields
  13. The de Sitter-Klein-Gordon equation and its plane-wave type solutions
  14. Plane-wave analysis of the free fields
  15. Fourier transform of test functions on $X_d$
  16. ...and 10 more sections

Key Result

Proposition 2.1

The "tubes" ${\cal T}^{+}$ and ${\cal T}^{-}$ are tuboids above $X_{d}$ in $X^{(c)}_{d}$, whose profiles are respectively $\Lambda^+ = \bigcup_{x\in X_{d}} (x,\Lambda^+_x)$ and $\Lambda^- = \bigcup_{x\in X_{d}} (x,\Lambda^-_x)$, where: $T_x X_{d}$ denoting the tangent space to $X_{d}$ at the point x, identified here to a hyperplane of the vector space $T_x{{\Bbb R}}^{d+1}\approx {{\Bbb R}}^{d+1}$

Figures (2)

  • Figure 1: Causal domains associated with the point $x_0$
  • Figure 2: Self-energy diagrams.

Theorems & Definitions (57)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.6
  • ...and 47 more