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  • Open Access

Dirac brackets for classical radiative observables

Francesco Alessio*

Riccardo Gonzo†

Canxin Shi‡

  • INFN, Laboratori Nazionali di Frascati, 00044 Frascati (RM), Italy

  • *Contact author: Francesco.Alessio@lnf.infn.it
  • †Contact author: rgonzo@ed.ac.uk
  • ‡Contact author: shicanxin@itp.ac.cn
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Phys. Rev. D 112, 104060 – Published 21 November, 2025

DOI: https://doi.org/10.1103/ykgq-jqd5

Abstract

We introduce a new coherent state expansion of the exponential representation of the S-matrix for the classical gravitational two-body problem. By combining the Kosower-Maybee-O’Connell (KMOC) formalism with the Dirac bracket structure emerging in the classical limit, we derive compact and gauge-invariant expressions for scattering observables in the presence of radiation. This causal formulation bypasses the calculation of KMOC cuts and provides a direct link between observables and a minimal set of classical matrix elements extracted from amplitudes. We illustrate our method with several examples, including the impulse, spin kick, angular momentum, waveform and the related radiative fluxes. Finally, using our formalism we evaluate for the first time the spin kick and the change in angular momentum of each particle up to O(G2s1j1s2j2) with j1+j2≤11.

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Phys. Rev. D 112, 104060– Published 21 November, 2025
  • Received 2 July 2025
  • Accepted 30 October 2025
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