  Microscopically conserving reduced models of many-body systems have a long,
highly successful history. Established theories of this type are the
random-phase approximation for Coulomb fluids and the particle-particle ladder
model for nuclear matter. There are also more physically comprehensive
approximations such as the induced-interaction and parquet theories.
Notwithstanding their explanatory power, some theories have lacked an explicit
Hamiltonian from which all significant system properties, static and dynamic,
emerge canonically. This absence can complicate evaluation of the conserving
sum rules, essential consistency checks on the validity of any model. In a
series of papers Kraichnan introduced a stochastic embedding procedure to
generate explicit Hamiltonians for common approximations for the full many-body
problem. Existence of a Hamiltonian greatly eases the task of securing
fundamental identities in such models. I revisit Kraichnan's method to apply it
to correlation theories for which such a canonical framework has not been
available. I exhibit Hamiltonians for more elaborate correlated models
incorporating both long-range screening and short-range scattering phenomena.
These are relevant to the study of strongly interacting electrons and condensed
quantum systems broadly.
