Slower-Than-Light Spin-$\frac{1}{2}$ Particles Endowed with Negative Mass Squared

Extending in a straightforward way the standard Dirac theory, we study a quantum mechanical wave equation describing free spinning particles — which we propose to call Pseudotachyons (PT's) — which behave like tachyons in the momentum space (p2 = -m2), but like subluminal particles (v < c) in the ordinary space. This is allowed since, as it happens in every quantum theory for spin-[Formula: see text] particles, the momentum operator, -i ∇, (that is conserved), and the velocity operator α, (that is not), are independent operators, which refer to independent quantities: [Formula: see text]. As a consequence, at variance with ordinary Dirac particles, for PT's the average velocity [Formula: see text] is not equal to the classical velocity v cl = p/ε, but actually to the velocity "dual" of v cl : εp/p2. Being reciprocal of |v cl |, the speed of PT's is therefore smaller than the light speed. Since a lot of experimental data seems to involve a negative mass squared for neutrinos, we suggest that these particles might be PT's, travelling, because of their very small mass, at subluminal speeds very close to the light one. The present theory is shown to be separately invariant under the C, P, T transformations; the covariance under Lorentz transformations is also proven. Furthermore, we derive the kinematical constraints linking 4-impulse, 4-velocity and 4-polarization of free PT's.