  We compute the contribution to the Lagrangian from the leading order (2.5
post-Newtonian) radiation reaction and the quadrupolar gravitational waves
emitted from a binary system using the effective field theory (EFT) approach of
Goldberger and Rothstein. We use an initial value formulation of the underlying
(quantum) framework to implement retarded boundary conditions and describe
these real-time dissipative processes. We also demonstrate why the usual
scattering formalism of quantum field theory inadequately accounts for these.
The methods discussed here should be useful for deriving real-time quantities
(including radiation reaction forces and gravitational wave emission) and
hereditary terms in the post-Newtonian approximation (including memory, tail
and other causal, history-dependent integrals) within the EFT approach. We also
provide a consistent formulation of the radiation sector in the equivalent
effective field theory approach of Kol and Smolkin.
