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Correlators are simpler than wavefunctions

Nima Arkani-Hamed, Ross Glew, Francisco Vazão

TL;DR

This work demonstrates that equal-time correlators are structurally simpler than their wavefunction counterparts because they arise from integrating Feynman propagators over the full spacetime, unlike the half-space domain of the wavefunction. The correlator exhibits a reduced and more tractable pole structure, with a Laurent expansion around each pole whose first subleading term always vanishes, and a total-energy pole that can be generated by a compact differential operator acting on the amplitude. These features extend to the full correlator in Tr $\phi^3$ theory and can be understood via soft limits, factorization, and a momentum-polygon framework, offering a promising bootstrap-like approach to correlators. The results illuminate why correlators are more directly connected to flat-space scattering amplitudes and motivate generalizations to cosmological correlators and more complex interactions while highlighting limitations for non-commuting operators.

Abstract

Several recent works have revealed a simplicity in equal-time correlators that is absent in their wavefunction counterparts. In this letter, we show that this arises from the simple fact that the correlator is obtained by integrating Feynman propagators over the full spacetime, as opposed to the half-space for the wavefunction. Several striking new properties of correlators for any graph are made obvious from this picture. Certain patterns of poles that appear in the wavefunction do not appear in the correlator. The correlator also enjoys several remarkable factorization properties in various limits. Most strikingly, the correlator admits a systematic Laurent expansion in the neighborhood of every pole, with the first subleading term vanishing for every pole. There is an especially simple understanding of the expansion around the total energy pole up to second order, given by a differential operator acting on the amplitude. Finally, we show how these results extend beyond single graphs to the full correlator in Tr $φ^3$ theory.

Correlators are simpler than wavefunctions

TL;DR

This work demonstrates that equal-time correlators are structurally simpler than their wavefunction counterparts because they arise from integrating Feynman propagators over the full spacetime, unlike the half-space domain of the wavefunction. The correlator exhibits a reduced and more tractable pole structure, with a Laurent expansion around each pole whose first subleading term always vanishes, and a total-energy pole that can be generated by a compact differential operator acting on the amplitude. These features extend to the full correlator in Tr theory and can be understood via soft limits, factorization, and a momentum-polygon framework, offering a promising bootstrap-like approach to correlators. The results illuminate why correlators are more directly connected to flat-space scattering amplitudes and motivate generalizations to cosmological correlators and more complex interactions while highlighting limitations for non-commuting operators.

Abstract

Several recent works have revealed a simplicity in equal-time correlators that is absent in their wavefunction counterparts. In this letter, we show that this arises from the simple fact that the correlator is obtained by integrating Feynman propagators over the full spacetime, as opposed to the half-space for the wavefunction. Several striking new properties of correlators for any graph are made obvious from this picture. Certain patterns of poles that appear in the wavefunction do not appear in the correlator. The correlator also enjoys several remarkable factorization properties in various limits. Most strikingly, the correlator admits a systematic Laurent expansion in the neighborhood of every pole, with the first subleading term vanishing for every pole. There is an especially simple understanding of the expansion around the total energy pole up to second order, given by a differential operator acting on the amplitude. Finally, we show how these results extend beyond single graphs to the full correlator in Tr theory.
Paper Structure (7 sections, 51 equations, 1 figure)

This paper contains 7 sections, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Wavefunction vs correlator
  3. Soft Limits
  4. Expanding the correlator
  5. $\text{Tr}(\phi^3)$ correlator
  6. Outlook
  7. Some Derivations

Figures (1)

  • Figure 1: Example graph contributions to the wavefunction coefficients at tree-level and one-loop.