  Over the past few decades, it has been realised that gauge theory scattering
amplitudes have structures much simpler than the traditional Feynman graph
driven approach would suggest. In particular, Parke and Taylor found a
particularly simple expression for the tree-level amplitudes with two gluons of
different helicity than the others (the so-called MHV amplitudes). Cachazo,
Svrcek and Witten (CSW) devised rules for constructing tree-level amplitudes by
sewing lower-valence MHV amplitudes together with scalar propagators. It was
shown by Mansfield in 2005 that a canonical change of the field variables could
be constructed that resulted in a lagrangian whose vertices were proportional
to MHV amplitudes, continued off-shell by CSW's prescription, the so-called
Canonical MHV Lagrangian. We derive the explicit form of this transformation
and use this to show that the vertices are indeed the Parke--Taylor amplitudes
for up to five gluons. Noting that CSW's MHV rules cannot be used to construct
the tree-level (-++) or one-loop (++++) amplitudes, we extend our work to
augment the MHV rules with so-called completion vertices. These permit
construction of these missing amplitudes by means of evasion of the S-matrix
equivalence theorem. Indeed, together they reconstruct off-shell light-cone
Yang-Mills amplitudes algebraically. We also give a prescription for
dimensional regularisation of the Canonical MHV Lagrangian. Finally, we
construct a canonical MHV lagrangian with massless fermions in the fundamental
representation using a similar methodology.
