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It is of general theoretical interest to investigate the properties ofsuperluminal matter wave equations for spin one-half particles. One can eitherenforce superluminal propagation by an explicit substitution of the real massterm for an imaginary mass, or one can use a matrix representation of theimaginary unit that multiplies the mass term. The latter leads to the tachyonicDirac equation, while the equation obtained by the substitution m->i*m in theDirac equation is naturally referred to as the imaginary-mass Dirac equation.Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commutewith the helicity operator. Both Hamiltonians are pseudo-Hermitian and alsopossess additional modified pseudo-Hermitian properties, leading to constraintson the resonance eigenvalues. Here, by an explicit calculation, we show thatspecific sum rules over the spectrum hold for the wave functions correspondingto the well-defined real energy eigenvalues and complex resonance andanti-resonance energies. In the quantized imaginary-mass Dirac field,one-particle states of right-handed helicity acquire a negative norm("indefinite metric") and can be excluded from the physical spectrum by aGupta--Bleuler type condition.

Dirac Hamiltonian with Imaginary Mass and Induced Helicity-Dependence by Indefinite Metric