  We develop the auxiliary field formalism corresponding to a dilute system of
spin-1/2 fermions. This theory represents the Fermi counterpart of the BEC
theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402
(2010)] to describe a dilute gas of Bose particles. Assuming tunable
interactions, this formalism is appropriate for the study of the crossover from
the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of
Bose-Einstein condensation (BEC) in ultracold fermionic atom gases. We show
that when applied to the Fermi case at zero temperature, the leading-order
auxiliary field (LOAF) approximation gives the same equations as those obtained
in the standard BCS variational picture. At finite temperature, LOAF leads to
the theory discussed by by Sa de Melo, Randeria, and Engelbrecht [Phys. Rev.
Lett. 71, 3202(1993); Phys. Rev. B 55, 15153(1997)]. As such, LOAF provides a
unified framework to study the interacting Fermi gas. The mean-field results
discussed here can be systematically improved upon by calculating the
one-particle irreducible (1-PI) action corrections, order by order.
