  Recently Wang and Cheng proposed a self-consistent effective Hamiltonian
theory (SCEHT) for many-body fermionic systems (Wang & Cheng, 2019). This paper
attempts to provide a mathematical foundation to the formulation of the SCEHT
that enables further study of excited states of the system in a more systematic
and theoretical manner. Gauge fields are introduced and correct total energy
functional in relations to the coupling gauge field is given. We also provides
a Monte-Carlo numerical scheme for the search of the ground state that goes
beyond the SCEHT.
