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BMS symmetry, soft particles and memory

Atreya Chatterjee, David A. Lowe

TL;DR

This paper addresses how to formulate a holographic description of asymptotically flat spacetimes using the Bondi-Metzner-Sachs (BMS) group by classifying its unitary irreducible representations labeled not only by 4-momentum but also by an infinite set of supermomenta. It develops the representation theory by examining the BMS group structure, its little groups (including SU(2), Δ, SL(2,R), and Γ), and the corresponding invariant norms, connecting these to the Bondi mass aspect and to the emergence of soft gravitational modes. Through tensor-product-inspired scattering analyses, it shows that memory effects arise from conservation of supermomenta, allowing information about initial states to be encoded in outgoing soft modes, even in processes with vanishing four-momentum soft states. The results illuminate how soft hair and memory could be integral to a consistent S-matrix for gravity and outline both opportunities and obstacles for building a flat-space holographic dual that properly accounts for these soft sectors.

Abstract

In this work, we revisit unitary irreducible representations of the Bondi-Metzner-Sachs (BMS) group discovered by McCarthy. Representations are labelled by an infinite number of super-momenta in addition to four-momentum. Tensor products of these irreducible representations lead to particle-like states dressed by soft gravitational modes. Conservation of 4-momentum and supermomentum in the scattering of such states leads to a memory effect encoded in the outgoing soft modes. We note there exist irreducible representations corresponding to soft states with strictly vanishing four-momentum, which may nevertheless be produced by scattering of particle-like states. This fact has interesting implications for the S-matrix in gravitational theories.

BMS symmetry, soft particles and memory

TL;DR

This paper addresses how to formulate a holographic description of asymptotically flat spacetimes using the Bondi-Metzner-Sachs (BMS) group by classifying its unitary irreducible representations labeled not only by 4-momentum but also by an infinite set of supermomenta. It develops the representation theory by examining the BMS group structure, its little groups (including SU(2), Δ, SL(2,R), and Γ), and the corresponding invariant norms, connecting these to the Bondi mass aspect and to the emergence of soft gravitational modes. Through tensor-product-inspired scattering analyses, it shows that memory effects arise from conservation of supermomenta, allowing information about initial states to be encoded in outgoing soft modes, even in processes with vanishing four-momentum soft states. The results illuminate how soft hair and memory could be integral to a consistent S-matrix for gravity and outline both opportunities and obstacles for building a flat-space holographic dual that properly accounts for these soft sectors.

Abstract

In this work, we revisit unitary irreducible representations of the Bondi-Metzner-Sachs (BMS) group discovered by McCarthy. Representations are labelled by an infinite number of super-momenta in addition to four-momentum. Tensor products of these irreducible representations lead to particle-like states dressed by soft gravitational modes. Conservation of 4-momentum and supermomentum in the scattering of such states leads to a memory effect encoded in the outgoing soft modes. We note there exist irreducible representations corresponding to soft states with strictly vanishing four-momentum, which may nevertheless be produced by scattering of particle-like states. This fact has interesting implications for the S-matrix in gravitational theories.

Paper Structure

This paper contains 19 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Representations of the BMS group
  3. BMS group
  4. Little groups
  5. $SU(2)$
  6. $\Delta$
  7. $SL(2,C)$
  8. $SL(2,R)$
  9. $\Gamma$
  10. Indecomposable
  11. Non-connected subgroups
  12. Holography
  13. $SU(2)$ and $\Delta$
  14. $SL(2,R)$
  15. $\Gamma$
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The figure depicts two particles of mass $M$ moving in opposite directions forming a bound state.
  • Figure 2: The left figure represent two particle of mass $m$ moving along $z$-axis. They collide and move out along $x$-axis. The figure on the right represents the final state. Subtracting blue patch from the red patch on the celestial sphere gives the soft mode in the final state.