  Effective field theories have often been applied to systems with deeply
inelastic reactions that produce particles with large momenta outside the
domain of validity of the effective theory. The effects of the deeply inelastic
reactions have been taken into account in previous work by adding local
anti-Hermitian terms to the effective Hamiltonian. Here we show that when
multi-particle systems are considered, an additional modification is required
in equations governing the density matrix. We define an effective density
matrix by tracing over the states containing high-momentum particles, and show
that it satisfies a Lindblad equation, with local Lindblad operators determined
by the anti-Hermitian terms in the effective Hamiltonian density.
