  We obtain a Lorentz covariant wave equation whose complex wave function
transforms under a Lorentz boost according to the following rule,
$\Psi(x)\rightarrow e^{\frac{i}{\hbar}f(x)}\Psi(x)$. We show that the spacetime
dependent phase $f(x)$ is the most natural relativistic extension of the phase
associated with the transformation rule for the non-relativistic Schroedinger
wave function when it is subjected to a Galilean transformation. We then
generalize the previous analysis by postulating that $\Psi(x)$ transforms
according to the above rule under proper Lorentz transformations (boosts or
spatial rotations). This is the most general transformation rule compatible
with a Lorentz invariant physical theory whose observables are bilinear
functions of the field $\Psi(x)$. We use the previous wave equations to
describe several physical systems. In particular, we solve the bound state and
scattering problems of two particles which interact both electromagnetically
and gravitationally (static electromagnetic and gravitational fields). The
former interaction is modeled via the minimal coupling prescription while the
latter enters via an external potential. We also formulate logically consistent
classical and quantum field theories associated with these Lorentz covariant
wave equations. We show that it is possible to make those theories equivalent
to the Klein-Gordon theory whenever we have self-interacting terms that do not
break their Lorentz invariance or if we introduce electromagnetic interactions
via the minimal coupling prescription. For interactions that break Lorentz
invariance, we show that the present theories imply that particles and
antiparticles behave differently at decaying processes, with the latter being
more unstable. This suggests a possible connection between Lorentz
invariance-breaking interactions and the matter-antimatter asymmetry problem.
