  It is of general theoretical interest to investigate the properties of
superluminal matter wave equations for spin one-half particles. One can either
enforce superluminal propagation by an explicit substitution of the real mass
term for an imaginary mass, or one can use a matrix representation of the
imaginary unit that multiplies the mass term. The latter leads to the tachyonic
Dirac equation, while the equation obtained by the substitution m->i*m in the
Dirac equation is naturally referred to as the imaginary-mass Dirac equation.
Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commute
with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also
possess additional modified pseudo-Hermitian properties, leading to constraints
on the resonance eigenvalues. Here, by an explicit calculation, we show that
specific sum rules over the spectrum hold for the wave functions corresponding
to the well-defined real energy eigenvalues and complex resonance and
anti-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 a
Gupta--Bleuler type condition.
