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% 
% AUTHORS' MACROS END HERE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
	
	\title{
%		Field Equations from Particle Phase Spaces
		Double Copy and the Double Poisson Bracket
	}
	
	\author{Joon-Hwi Kim}
	\affiliation{Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA}
	
	\begin{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 $\dpb{H}{H}_\cov = 0$,
		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.
	\end{abstract}
	
	\preprint{CALT-TH 2025-005}
	
	% \date{\today}
	
	\bibliographystyle{utphys-modified}
	
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	\maketitle
	
	\input{body.tex}
	
	\medskip
	\noindent\textit{Acknowledgements.}|%
		The author is grateful to
			Clifford Cheung,
			Toby Saunders-A'Court,
		and
			Sonja Klisch
		for discussions.
	%	
		The author would like to thank
			Lionel Mason
		for insightful discussions
		during the
		conference ``The Mathematics behind Scattering Amplitudes'' held in August 2024;
	%	
		the author thanks the Galileo Galilei
		Institute for Theoretical Physics, Florence for hospitality. 
	%
		The author is grateful to
			Sebastian Mizera
		for encouraging comments
		and
			Julio Parra-Martinez
		for bringing the history of Feynman brackets to his attention.
	%
		The author thanks
		the attendees of California Amplitudes Meeting 2023 on March 18\textsuperscript{th} for 
		comments on the idea of approaching double copy via Feynman brackets,
		presented in the talk \cite{caamps23spt}.
	%
		J.-H.K. is supported by the Department of Energy (Grant {No.}~DE-SC0011632) and by the Walter Burke Institute for Theoretical Physics.
	% 	
	
	\newpage
	\appendix
	\onecolumngrid
	\input{app.tex}
	
%	\phantom{.}
%	\newpage
%	\newpage
%	\input{supp.tex}

	\twocolumngrid
	\bibliography{references.bib}
	
\end{document}
