  It has been proposed several times in the past that one can obtain an
equivalent, but in many aspects simpler description of fermions by first
reformulating their first-order (Dirac) Lagrangian in terms of two-component
spinors, and then integrating out the spinors of one chirality ($e.g.$ primed
or dotted). The resulting new Lagrangian is second-order in derivatives, and
contains two-component spinors of only one chirality. The new second-order
formulation simplifies the fermion Feynman rules of the theory considerably,
$e.g.$ the propagator becomes a multiple of an identity matrix in the field
space. The aim of this thesis is to work out the details of this formulation
for theories such as Quantum Electrodynamics, and the Standard Model of
elementary particles. After having developed the tools necessary to establish
the second-order formalism as an equivalent approach to spinor field theories,
we proceed with some important consistency checks that the new formulation is
required to pass, namely the presence or absence of anomalies in their
perturbative and non-perturbative description, and the unitarity of the
S-Matrix derived from their Lagrangian. Another aspect which is studied is
unification, where we seek novel gauge-groups that can be used to embed all of
the Standard Model content: forces and fermionic representations. Finally, we
will explore the possibility to unify gravity and the Standard Model when the
former is seen as a diffeomorphism invariant gauge-theory.
