  We build the fully relativistic quantum field theory related to the
asymmetric Dirac fields. These fields are solutions of the asymmetric Dirac
equation, a Lorentz covariant Dirac-like equation whose positive and "negative"
frequency plane wave solutions' dispersion relations are no longer degenerate.
At the second quantization level, we show that this implies that particles and
antiparticles sharing the same wave number have different energies and momenta.
In spite of that, we prove that by properly fixing the values of the
relativistic invariants that define the asymmetric Dirac free field Lagrangian
density, we can build a consistent, fully relativistic, and renormalizable
quantum electrodynamics (QED) that is empirically equivalent to the standard
QED. We discuss the reasons and implications of this non-trivial equivalence,
exploring qualitatively other scenarios in which the asymmetric Dirac fields
may lead to beyond the standard model predictions. We conjecture that this
non-degeneracy in the energies for particles and antiparticles may lead to a
fully relativistic understanding of the asymmetry between matter and antimatter
in the present day universe as well as to an alternative way of modeling the
gravitational interaction between a particle and an antiparticle. We give a
complete account of how the asymmetric Dirac fields and the corresponding
annihilation and creation operators transform under improper Lorentz
transformations (parity and time reversal operations) and under the charge
conjugation operation. We also prove that the present theory respects the CPT
theorem.
