Dirac brackets for classical radiative observables

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">O</a:mi> <a:mo stretchy="false">(</a:mo> <a:msup> <a:mi>G</a:mi> <a:mn>2</a:mn> </a:msup> <a:msubsup> <a:mi>s</a:mi> <a:mn>1</a:mn> <a:msub> <a:mi>j</a:mi> <a:mn>1</a:mn> </a:msub> </a:msubsup> <a:msubsup> <a:mi>s</a:mi> <a:mn>2</a:mn> <a:msub> <a:mi>j</a:mi> <a:mn>2</a:mn> </a:msub> </a:msubsup> <a:mo stretchy="false">)</a:mo> </a:math> with <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>j</f:mi> <f:mn>1</f:mn> </f:msub> <f:mo>+</f:mo> <f:msub> <f:mi>j</f:mi> <f:mn>2</f:mn> </f:msub> <f:mo>≤</f:mo> <f:mn>11</f:mn> </f:math> .