  Exterior Differential Systems (EDS) and Cartan forms, set in the state space
of field variables taken together with four space-time variables, are
formulated for classical gauge theories of Maxwell and SU(2) Yang-Mills fields
minimally coupled to Dirac spinor multiplets. Cartan character tables are
calculated, showing whether the EDS, and so the Euler-Lagrange partial
differential equations, is well-posed. The first theory, with 22 dimensional
state space (10 Maxwell field and potential components and 8 components of a
Dirac field), anticipates QED. In the second, non-Abelian, case (30 Yang-Mills
field components and 16 Dirac), only if three additional "ghost" fields are
included (15 more scalar variables) is a well-posed EDS found. This classical
formulation anticipates the need for introduction of Fadeev-Popov ghost fields
in the quantum standard model.
