Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Double copy and the double Poisson bracket

Joon-Hwi Kim

TL;DR

This work presents a unified ambitwistor-space framework in which YM theory and general relativity emerge from gauge-covariant constrained quantization of massless particle phase spaces. By incorporating covariantized, star-product–based OPEs on ambitwistor worldlines and one-loop diagrams, the authors derive both first-order (magnetic/electric) and second-order (Penrose-type) field equations, suggesting a covariant double Poisson bracket structure $\{\{H,H\}\}_{\nabla}=0$. A central theme is the Fedosov construction of gauge-covariant associative algebras, which reconciles covariance with associativity and underpins a potential ambitwistor-based realization of BCJ duality through BAS-type equations. The framework extends to $\mathcal{N}=1$ and $\mathcal{N}=2$ ambitwistor particles for YM and GR, respectively, and culminates in a covariant BAS formulation of gravity, including teleparallel/Born-Infeld viewpoints. Overall, the paper proposes that the kinematic (double copy) algebra may be realized as the Poisson/double Poisson structure of ambitwistor space in general dimensions, offering a new route to understanding gauge–gravity relations beyond self-duality limits.

Abstract

We derive first-order and second-order field equations from ambitwistor spaces as phase spaces of massless particles. In particular, the second-order field equations of Yang-Mills theory and general relativity are formulated in a unified form $\{\{H,H\}\}_\nabla = 0$, whose left-hand side describes a doubling of Poisson bracket in a covariant sense. This structure originates from a one-loop diagram encoded in gauge-covariant, associative operator products on the ambitwistor worldlines. A conjecture arises that the kinematic algebra might manifest as the Poisson algebra of ambitwistor space.

Double copy and the double Poisson bracket

TL;DR

This work presents a unified ambitwistor-space framework in which YM theory and general relativity emerge from gauge-covariant constrained quantization of massless particle phase spaces. By incorporating covariantized, star-product–based OPEs on ambitwistor worldlines and one-loop diagrams, the authors derive both first-order (magnetic/electric) and second-order (Penrose-type) field equations, suggesting a covariant double Poisson bracket structure . A central theme is the Fedosov construction of gauge-covariant associative algebras, which reconciles covariance with associativity and underpins a potential ambitwistor-based realization of BCJ duality through BAS-type equations. The framework extends to and ambitwistor particles for YM and GR, respectively, and culminates in a covariant BAS formulation of gravity, including teleparallel/Born-Infeld viewpoints. Overall, the paper proposes that the kinematic (double copy) algebra may be realized as the Poisson/double Poisson structure of ambitwistor space in general dimensions, offering a new route to understanding gauge–gravity relations beyond self-duality limits.

Abstract

We derive first-order and second-order field equations from ambitwistor spaces as phase spaces of massless particles. In particular, the second-order field equations of Yang-Mills theory and general relativity are formulated in a unified form , whose left-hand side describes a doubling of Poisson bracket in a covariant sense. This structure originates from a one-loop diagram encoded in gauge-covariant, associative operator products on the ambitwistor worldlines. A conjecture arises that the kinematic algebra might manifest as the Poisson algebra of ambitwistor space.
Paper Structure (19 sections, 68 equations, 2 figures)

This paper contains 19 sections, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical Mechanics in Phase Space
  3. Gauge Covariance Versus Associativity: Classical
  4. Historical Remarks
  5. Ambitwistor Space as a Constrained Phase Space
  6. YM Theory from $\mathcal{N}{\,=\,}1$ Ambitwistor String
  7. From Strings to Particles
  8. Worldline OPE as Moyal Star Product
  9. Covariantized Brackets
  10. The Master Equations
  11. YM Theory from $\mathcal{N}{\,=\,}1$ Ambitwistor Particle
  12. General Relativity from $\mathcal{N}{\,=\,}2$ Ambitwistor Particle
  13. Gauge Covariance Versus Associativity: Quantum
  14. BAS Formulation of Gravity
  15. Conclusions
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The Moyal star product from path integral.
  • Figure 2: The web of field theories exhibiting color-kinematics duality. The arrows pointing to the left replace $\mathfrak{su}(N)$ with $\mathfrak{sdiff}(\mathbb{R}^d)$. The arrows pointing to the right replace $\mathfrak{su}(N)$ with $\mathfrak{g}_{\text{YM}}$, a mystery infinite-dimensional Lie algebra.