Projective Geometry and $\cal PT$-Symmetric Dirac Hamiltonian

Abstract

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 $\cal P$ and time reversal $\cal T$. When the $\cal PT$ symmetry is unbroken, the energy spectrum of the free spin-$\frac {1}{2}$ theory is real, with an appropriately shifted mass.