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

TL;DR

This paper reveals a geometric interpretation of the Dirac equation in projective space, introducing a non-Hermitian but PT-symmetric Hamiltonian with real spectra under unbroken PT symmetry.

Contribution

It establishes a novel connection between projective geometry and PT-symmetric quantum mechanics in the context of the Dirac equation.

Findings

The Dirac equation can be expressed in terms of projective geometry.

The Hamiltonian with a gamma_5 mass term is PT-symmetric but non-Hermitian.

Unbroken PT symmetry leads to real energy spectra with shifted mass.

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.