  This dissertation focuses on a theoretical study of interacting electrons in
one dimension. The research elucidates the ground state (zero temperature)
electronic phase diagram of an aluminum arsenide quantum wire which is an
example of an interacting one dimensional electron liquid. Using one
dimensional field theoretic methods involving abelian bosonization and the
renormalization group we show the existence of a spin gapped quantum wire with
electronic ground states such as charge density wave and singlet
superconductivity. The superconducting state arises due to the unique umklapp
interaction present in the aluminum arsenide quantum wire bandstructure
discussed in this dissertation. It is characterized by Cooper pairs carrying a
finite pairing momentum. This is a realization of the Fulde-Ferrell-Larkin-
Ovchinnikov state which is known to lead to inhomogeneous superconductivity.
The dissertation also presents a theoretical analysis of the finite temperature
single hole spectral function of the one dimensional electron liquid with
gapless spin and charge modes (Luttinger liquid). The hole spectral function is
measured in angle resolved photoemission spectroscopy experiments. The results
predict a kink in the effective electronic dispersion of the Luttinger liquid.
A systematic study of the temperature and interaction dependence of the kink
provides an alternative way to detect spincharge separation in one dimensional
systems where the peak due to the spin part of the spectral function is
suppressed.
