  I derive a loop representation for the canonical and grand-canonical
partition functions for an interacting four-component Fermi gas in one spatial
dimension and an arbitrary external potential. The representation is free of
the "sign problem" irrespective of population imbalance, mass imbalance, and to
a degree, sign of the interaction strength. This property is in sharp contrast
with the analogous three-dimensional two-component interacting Fermi gas, which
exhibits a sign problem in the case of unequal masses, chemical potentials, and
repulsive interactions. The one-dimensional system is believed to exhibit many
phenomena in common with its three-dimensional counterpart, including an analog
of the BCS-BEC crossover, and nonperturbative universal few- and many-body
physics at scattering lengths much larger than the range of interaction, making
the theory an interesting candidate for numerical study. Positivity of the
probability measure for the partition function allows for a mean-field
treatment of the model; here, I present such an analysis for the interacting
Fermi gas in the SU(4) (unpolarized, mass-symmetric) limit, and demonstrate
that there exists a phase in which a continuum limit may be defined.
