  We derive a Dirac-like equation, the asymmetric Dirac equation, where
particles and antiparticles sharing the same wave number have different
energies and momenta. We show that this equation is Lorentz covariant under
proper Lorentz transformations (boosts and spatial rotations) and also
determine the corresponding transformation law for its wave function. We obtain
a formal connection between the asymmetric Dirac equation and the standard
Dirac equation and we show that by properly adjusting the free parameters of
the present wave equation we can make it reproduce the predictions of the usual
Dirac equation. We show that the rest mass of a particle in the theoretical
framework of the asymmetric Dirac equation is a function of a set of four
parameters, which are relativistic invariants under proper Lorentz
transformations. These four parameters are the analog to the mass that appears
in the standard Dirac equation. We prove that in order to guarantee the
covariance of the asymmetric Dirac equation under parity and time reversal
operations (improper Lorentz transformations) as well as under the charge
conjugation operation, these four parameters change sign in exactly the same
way as the four components of a four-vector. The mass, though, being a function
of the square of those parameters remains an invariant. We also extensively
study the free particle plane wave solutions to the asymmetric Dirac equation
and derive its energy, helicity, and spin projection operators as well as
several Gordon's identities. The hydrogen atom is solved in the present context
after applying the minimal coupling prescription to the asymmetric Dirac
equation, which also allows us to appropriately obtain its non-relativistic
limit.
