  We develop relativistic wave equations in the framework of the new
non-hermitian ${\cal PT}$ quantum mechanics. The familiar Hermitian Dirac
equation emerges as an exact result of imposing the Dirac algebra, the criteria
of ${\cal PT}$-symmetric quantum mechanics, and relativistic invariance.
However, relaxing the constraint that in particular the mass matrix be
Hermitian also allows for models that have no counterpart in conventional
quantum mechanics. For example it is well-known that a quartet of Weyl spinors
coupled by a Hermitian mass matrix reduces to two independent Dirac fermions;
here we show that the same quartet of Weyl spinors, when coupled by a
non-Hermitian but $\cal{PT}$ symmetric mass matrix, describes a single
relativistic particle that can have massless dispersion relation even though
the mass matrix is non-zero.The ${\cal PT}$-generalized Dirac equation is also
Lorentz invariant, unitary in time, and CPT respecting, even though as a
non-interacting theory it violates ${\cal P}$ and ${\cal T}$ individually. The
relativistic wave equations are reformulated as canonical fermionic field
theories to facilitate the study of interactions, and are shown to maintain
many of the canonical structures from Hermitian field theory, but with new and
interesting new possibilities permitted by the non-hermiticity parameter $m_2$.

High Energy Physics - Theory;Quantum Physics