Projective Geometry and $\mathcal{P}\mathcal{T}$-Symmetric Dirac Hamiltonian

The $(3+1)$-dimensional (generalized) Dirac equation is shown to have the same form as the equation expressing the condition that a given point lies on a given line in 3-dimensional projective space. The resulting Hamiltonian with a ${\gamma }_5$ mass term is not Hermitian, but is invariant under the combined transformation of parity reflection $\mathcal{P}$ and time reversal $\mathcal{T}$. When the $\mathcal{P}\mathcal{T}$ symmetry is unbroken, the energy spectrum of the free spin-$\frac {1}{2}$ theory is real, with an appropriately shifted mass.