  The relativistic Dirac equation in four-dimensional spacetime reveals a
coherent relation between the dimensions of spacetime and the degrees of
freedom of fermionic spinors. A massless Dirac fermion generates new symmetries
corresponding to chirality spin and charge spin as well as conformal scaling
transformations. With the introduction of intrinsic W-parity, a massless Dirac
fermion can be treated as a Majorana-type or Weyl-type spinor in a
six-dimensional spacetime that reflects the intrinsic quantum numbers of
chirality spin. A generalized Dirac equation is obtained in the six-dimensional
spacetime with a maximal symmetry. Based on the framework of gravitational
quantum field theory proposed in Ref. \cite{YLWU} with the postulate of gauge
invariance and coordinate independence, we arrive at a maximally symmetric
gravitational gauge field theory for the massless Dirac fermion in
six-dimensional spacetime. Such a theory is governed by the local spin gauge
symmetry SP(1,5) and the global Poincar\'e symmetry P(1,5)= SO(1,5)$\ltimes
P^{1,5} $ as well as the charge spin gauge symmetry SU(2). The theory leads to
the prediction of doubly electrically charged bosons. A scalar field and
conformal scaling gauge field are introduced to maintain both global and local
conformal scaling symmetries. A generalized gravitational Dirac equation for
the massless Dirac fermion is derived in the six-dimensional spacetime. The
equations of motion for gauge fields are obtained with conserved currents in
the presence of gravitational effects. The dynamics of the gauge-type
gravifield as a Goldstone-like boson is shown to be governed by a conserved
energy-momentum tensor, and its symmetric part provides a generalized Einstein
equation of gravity. An alternative geometrical symmetry breaking mechanism for
the mass generation of Dirac fermions is demonstrated.
