  The Dirac equation with chiral symmetry is derived using the irreducible
representations of the Poincar\'{e} group, the Lagrangian formalism, and a
novel method of projection operators that takes as its starting point the
minimal assumption of four linearly independent physical states. We thereby
demonstrate the fundamental nature of this form of the Dirac equation. The
resulting equation is then examined within the context of spacetime and CPT
symmetries with a discussion of the implications for the general formulation of
physical theories.
