PT symmetric fermionic particle oscillations in even dimensional representations

We describe a novel class of quantum mechanical particle oscillations in both relativistic and non-relativistic systems based on $PT$ symmetry and $T^2=-1$ (relevant for fermions), where $P$ is parity and $T$ is time reversal. The Hamiltonians are chosen at the outset to be self-adjoint with respect to a PT inner product. The quantum mechanical time evolution is based on a modified $CPT$ inner product constructed in terms of a suitable $C$ operator. The resulting quantum mechanical evolution is shown to be unitary and probability is conserved by the oscillations.