Quantum backflow of the relativistic spin-$1/2$ free particle

It is shown that the free Dirac particle in a superposition of positive momenta can in certain cases bear a negative probability current --- i.e., the so-called quantum backflow --- as is in analogy with that in the nonrelativistic limit. Further, more variations are brought about and investigated when the negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise even with a well-defined momentum but different signs of energies, or more remarkably with different signs of both momenta and energies, in the superposition. Neither of these cases has counterpart in nonrelativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.