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On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes

Nima Arkani-Hamed, Paolo Benincasa

TL;DR

The paper investigates how Lorentz invariance and unitarity emerge from the boundary wavefunction of the universe by studying the scattering facet of cosmological polytopes. It shows that the canonical form of the polytope encodes the wavefunction integrand and that facet factorization reproduces unitarity via cutting rules, while a contour-integral representation reveals Lorentz-invariant propagators with the correct iε prescription, once loop-time variables are integrated. This provides a geometric, ab-initio origin for these foundational principles, at both tree and loop levels, and suggests deep connections to associahedron-like structures in cosmology. The work opens avenues to extend geometric methods to squared amplitudes and other cosmological observables, potentially linking boundary geometry with bulk flat-space physics.

Abstract

The concepts of Lorentz invariance of local (flat space) physics, and unitarity of time evolution and the S-matrix, are famously rigid and robust, admitting no obvious consistent theoretical deformations, and confirmed to incredible accuracy by experiments. But neither of these notions seem to appear directly in describing the spatial correlation functions at future infinity characterizing the "boundary" observables in cosmology. How then can we see them emerge as {\it exact} concepts from a possible ab-initio theory for the late-time wavefunction of the universe? In this letter we examine this question in a simple but concrete setting, for the perturbative wavefunction in a class of scalar field models where an ab-initio description of the wavefunction has been given by "cosmological polytopes". Singularities of the wavefunction are associated with facets of the polytope. One of the singularities -- corresponding to the "total energy pole" -- is well known to be associated with the flat-space scattering amplitude. We show how the combinatorics and geometry of this {\it scattering facet} of the cosmological polytope straightforwardly leads to the emergence of Lorentz invariance and unitarity for the S-matrix. Unitarity follows from the way boundaries of the scattering facet factorize into products of lower-dimensional polytopes, while Lorentz invariance follows from a contour integral representation of the canonical form, which exists for any polytope, specialized to cosmological polytopes.

On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes

TL;DR

The paper investigates how Lorentz invariance and unitarity emerge from the boundary wavefunction of the universe by studying the scattering facet of cosmological polytopes. It shows that the canonical form of the polytope encodes the wavefunction integrand and that facet factorization reproduces unitarity via cutting rules, while a contour-integral representation reveals Lorentz-invariant propagators with the correct iε prescription, once loop-time variables are integrated. This provides a geometric, ab-initio origin for these foundational principles, at both tree and loop levels, and suggests deep connections to associahedron-like structures in cosmology. The work opens avenues to extend geometric methods to squared amplitudes and other cosmological observables, potentially linking boundary geometry with bulk flat-space physics.

Abstract

The concepts of Lorentz invariance of local (flat space) physics, and unitarity of time evolution and the S-matrix, are famously rigid and robust, admitting no obvious consistent theoretical deformations, and confirmed to incredible accuracy by experiments. But neither of these notions seem to appear directly in describing the spatial correlation functions at future infinity characterizing the "boundary" observables in cosmology. How then can we see them emerge as {\it exact} concepts from a possible ab-initio theory for the late-time wavefunction of the universe? In this letter we examine this question in a simple but concrete setting, for the perturbative wavefunction in a class of scalar field models where an ab-initio description of the wavefunction has been given by "cosmological polytopes". Singularities of the wavefunction are associated with facets of the polytope. One of the singularities -- corresponding to the "total energy pole" -- is well known to be associated with the flat-space scattering amplitude. We show how the combinatorics and geometry of this {\it scattering facet} of the cosmological polytope straightforwardly leads to the emergence of Lorentz invariance and unitarity for the S-matrix. Unitarity follows from the way boundaries of the scattering facet factorize into products of lower-dimensional polytopes, while Lorentz invariance follows from a contour integral representation of the canonical form, which exists for any polytope, specialized to cosmological polytopes.

Paper Structure

This paper contains 6 sections, 12 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Cosmological polytope
  3. Emergent unitarity
  4. Emergent Lorentz invariance
  5. Outlook
  6. Acknowledgments

Figures (6)

  • Figure 1: Feynman graph contribution to the wavefunction (left) and related reduced graph (right). The reduced graphs are obtained from the Feynman ones by suppressing the external edges.
  • Figure 2: Scattering facets for the one-loop (left) and two-loop (right) two-site graphs. They are identified by the vertices $\{{\bf 1},\,{\bf 2},\,{\bf 3},\,{\bf 4}\}\,\equiv\,\{{\bf x}_1+{\bf y}_a-{\bf x}_2,\,-{\bf x}_1+{\bf y}_a+{\bf x}_2\,{\bf x}_1+{\bf y}_b-{\bf x}_{2},\,-{\bf x}_1+{\bf y}_b+{\bf x}_2\}$ and $\{{\bf 1},\,{\bf 2},\,{\bf 3},\,{\bf 4},\,{\bf 5},\,{\bf 6}\}\,\equiv\,\{{\bf x}_1+{\bf y}_a-{\bf x}_2,\,-{\bf x}_1+{\bf y}_a+{\bf x}_2,\,{\bf x}_1+{\bf y}_b-{\bf x}_2,\,-{\bf x}_1+{\bf y}_b+{\bf x}_2,\,{\bf x}_1+{\bf y}_c-{\bf x}_2,\,-{\bf x}_1+{\bf y}_c+{\bf x}_2\}$ respectively.
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