Theory of Strongly Interacting Quantum Liquids. I. Formulation

We make use of our understanding of the ground state ${\psi }_0$ and elementary excitations ${\psi }_q$ of the strongly interacting Bose gas to construct a theory of strongly interacting mixed and Fermi gases. We consider the overcomplete basis set ${\phi }^k{\psi }_q$, where ${\phi }^k$ is a determinant of plane waves, and interpret the off-diagonal part of the hamiltonian as a three-point vertex, coupling the Fermi quasiparticles to the collective modes. This vertex dresses each fermion with its backflow and induces a two-fermion scattering via the interaction of the backflows. There is no residual (screened) two-body interaction in the problem. Finally we discuss the expansion parameter for strong interactions.