Minimal Doubling Fermion and Hermiticity

We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity, and ${\gamma }_5$-hermiticity. R-hermiticity is a condition for Hermite action and it is related to ${\gamma }_5$-hermiticity and PT symmetry. Assuming that a translation-invariant kinetic term with continuum and periodic function does not have PT symmetry, it can have R-hermiticity or ${\gamma }_5$-hermiticity. We prove that a kinetic term with continuum and periodic function that is PT symmetric does not reduce doublers. As a simple example, we analyze the two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions. The minimal doubling fermions break PT symmetry and R-hermiticity, hence complex or non-Hermite coupling constants are caused by quantum correction.