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Relativistic Non-Hermitian Quantum Mechanics

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 $m_2$.