  We analyze the Lagrangian density and canonical stress-energy tensor for the
Dirac equation, where the Dirac bispinor has been recast as a multivector set
of fields. For the massless Dirac field, the sign of the energy density is
determined by the relative phase of uncoupled even- and odd-grade field
components. These components become coupled in the massive Dirac equation, and
the sign of the energy is determined by their spatial parity. The corresponding
stress-energy tensors for the second-order equations also admit negative energy
states, with the sign of the energy density again dependent on field parity. We
apply the same multivector approach to electromagnetism, constructing new
Lagrangian and energy densities in which the vector potential and the
electromagnetic field are treated as independent field degrees of freedom.
