  This study investigates the intricate relationship between dissipative
processes of open quantum systems and the non-Hermitian quantum field theory of
relativistic fermionic systems. By examining the influence of dissipative
effects on Dirac fermions via Lindblad formalism, we elucidate the effects of
the coupling of relativistic Dirac particles with the environment. Employing
rigorous theoretical analysis, we explore the impact of dissipative
interactions and find the Lyapunov equation of the relativistic
dissipation-driven fermionic system. By use of a thermal ansatz, one finds the
solution to the Lyapunov equations in terms of a stationary Wigner
distribution. Our results describe a non-hermitian fermionic system and provide
valuable insights into dissipative quantum phenomena' fundamental mechanisms in
relativistic fermionic systems, advancing our understanding of their behavior
in non-equilibrium scenarios.
