Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infrared physics of QED and gravity from representation theory

Laura Donnay, Yannick Herfray

Abstract

The infrared structure of QED and gravity is known to be governed by an infinite-dimensional symmetry group which extends the Poincaré group to include, respectively, large $U(1)$ transformations and BMS supertranslations. We describe how the unitary irreducible representations (UIRs) of these asymptotic symmetry groups encode universal infrared features of a scattering process. Motivated by the goal of defining an infrared-finite $S$-matrix based on these UIRs, we also study supermomentum eigenstates and contrast our construction with the dressed-state approach for infrared-safe amplitudes.

Infrared physics of QED and gravity from representation theory

Abstract

The infrared structure of QED and gravity is known to be governed by an infinite-dimensional symmetry group which extends the Poincaré group to include, respectively, large transformations and BMS supertranslations. We describe how the unitary irreducible representations (UIRs) of these asymptotic symmetry groups encode universal infrared features of a scattering process. Motivated by the goal of defining an infrared-finite -matrix based on these UIRs, we also study supermomentum eigenstates and contrast our construction with the dressed-state approach for infrared-safe amplitudes.
Paper Structure (84 sections, 3 theorems, 271 equations, 3 tables)

This paper contains 84 sections, 3 theorems, 271 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Part I. QED
  3. Hard representations of the QED asymptotic symmetry group
  4. Scalar fields
  5. Poincaré UIRs
  6. Hard representations of the asymptotic symmetry group
  7. Hard representations as scattering data
  8. Spinning fields
  9. Why hard representations are not enough
  10. Generic representations
  11. QED supermomentum
  12. Asymptotic little group
  13. Hard representations
  14. Soft representations
  15. Hard/soft decomposition of supermomenta
  16. ...and 69 more sections

Key Result

Theorem B.1

All UIRs of the group $SO(3,1) \ltimes \mathbb{R}^4$ can be constructed in the following way: The corresponding UIR of $ISO(3,1)$ with mass square $m^2$ and Poincaré spin $\rho$ is then given by the vector space of square-integrable sections of the homogeneous vector bundle $E\,:=\,SL(2,\mathbb{C}) \times_{\rho} V$ over the coset space $\frac{SL(2,\mathbb{C})}{\ell_{{p}}}$ or, equivalently,

Theorems & Definitions (3)

  • Theorem B.1: Induced representations of Poincaré group; Wigner 1939
  • Theorem B.2: Induced representations of group of the form $SO(3,1) \ltimes A$; Mackey 1968
  • Theorem B.3: Piard 1977