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Revisiting zero modes and cluster decomposition at the late-time boundary of de Sitter

Murat Onem, Gizem Sengor

TL;DR

This work investigates locality on de Sitter through the lens of the de Sitter isometry group representations, separating matter and gauge sectors via the principal and discrete series. Using late-time, equal-time two-point functions computed in both Euclidean $S^4$ and Lorentzian $dS_4$ (and via the wavefunction formalism), the authors analyze cluster decomposition in position space and distance distributions in field space, with careful treatment of zero modes. They find that principal-series fields (matter) exhibit standard clustering in position space, while discrete-series fields (gauge) show logarithmic growth and violate clustering, a nontrivial feature that persists even after removing zero modes. In field space, zero modes are problematic for all representations, necessitating a renormalized distance $\delta_{12}$; a cumulant-based analysis reveals representation-dependent finite width in the superhorizon limit, highlighting memory effects tied to the representation category. Overall, the results clarify locality constraints in de Sitter QFT and motivate extensions to higher-spin and fermionic fields to generalize these locality criteria in cosmological contexts.

Abstract

We revisit the literature on locality on de Sitter with the goal to organize the main results with respect to the representation theory of the isometry group of four dimensional de Sitter. We make use of the late-time behavior of two-point functions of principal and discrete series representation, both in physical and in field space and compare the role of the zero modes. Our overall conclusion is that when it comes to locality on de Sitter, analyzed in terms of cluster decomposition, the principal series representation that capture matter fields and discrete series representations that capture gauge fields show different behavior. Focusing on scalars as a first analysis, matter fields show explicit signs of respecting cluster decomposition while gauge fields do not.

Revisiting zero modes and cluster decomposition at the late-time boundary of de Sitter

TL;DR

This work investigates locality on de Sitter through the lens of the de Sitter isometry group representations, separating matter and gauge sectors via the principal and discrete series. Using late-time, equal-time two-point functions computed in both Euclidean and Lorentzian (and via the wavefunction formalism), the authors analyze cluster decomposition in position space and distance distributions in field space, with careful treatment of zero modes. They find that principal-series fields (matter) exhibit standard clustering in position space, while discrete-series fields (gauge) show logarithmic growth and violate clustering, a nontrivial feature that persists even after removing zero modes. In field space, zero modes are problematic for all representations, necessitating a renormalized distance ; a cumulant-based analysis reveals representation-dependent finite width in the superhorizon limit, highlighting memory effects tied to the representation category. Overall, the results clarify locality constraints in de Sitter QFT and motivate extensions to higher-spin and fermionic fields to generalize these locality criteria in cosmological contexts.

Abstract

We revisit the literature on locality on de Sitter with the goal to organize the main results with respect to the representation theory of the isometry group of four dimensional de Sitter. We make use of the late-time behavior of two-point functions of principal and discrete series representation, both in physical and in field space and compare the role of the zero modes. Our overall conclusion is that when it comes to locality on de Sitter, analyzed in terms of cluster decomposition, the principal series representation that capture matter fields and discrete series representations that capture gauge fields show different behavior. Focusing on scalars as a first analysis, matter fields show explicit signs of respecting cluster decomposition while gauge fields do not.

Paper Structure

This paper contains 16 sections, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of Scalar de Sitter representations
  3. Review of two-point functions on the $S^4$ and $dS_4$
  4. Two-point functions on $S^4$ and the zero modes
  5. The trouble with the zero-mode contribution
  6. The wavefunction and two-point functions on $dS_4$
  7. Cluster decomposition in position space
  8. Principal series two-point function in position space
  9. Discrete series two-point function in position space
  10. Removing the zero-mode
  11. Cluster decomposition in field space
  12. The problem with the zero-mode
  13. Renormalized distance in field space
  14. Revisiting the distance distribution and its width
  15. Conclusions and outlook
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Redefined distance in field space according to Roberts:2012jw.