<PERSON> paradox for bosons and fermions
I am reading this paper about the <PERSON> paradox, i.e. transmission of relativistic particles incident on a potential step of height $V_0 > E + mc^2 > 2mc^2$ with $E$ the energy of the incident particle.
In most textbooks (e.g. <PERSON> and <PERSON>) you read that the paradox is resolved by considering the production of particle-antiparticle pairs at the potential step which can be naturally incorporated in quantum field theory.
For spinless bosons described by the Klein-Gordon equation, this picture seems satisfactory. An incident particle moving to the right towards the step annihilates with a left-moving antiparticle at the potential step and the partner particle is transmitted to the right.
However, for fermions, the paper seems to conclude that the incident particle is completely reflected and that no pair creation can occur because the reflected particle already occupies this mode. Also, naively, the single-particle calculation seems to violate unitarity.
This other paper seems to conclude the same, i.e. there is no transmission for fermions, just total reflection.
Question
Is this the correct picture? Are the textbooks wrong on this one? Also, does pair creation at the potential violate energy conservation or is the energy supplied by the static potential?
Edit
I have found a very interesting discussion on page 307 of the book B. Thaller, The Dirac equation.
In 1928, <PERSON> discovered oscillatory solutions inside a potential step where a nonrelativistic solution would decay exponentially. He determined the reflection and transmission coefficients for a rectangular step potential $V_0 \theta(x)$. Subsequently <PERSON> investigated <PERSON>'s paradox for a smooth potential, which gave the same qualitative result, but with a much smaller transmission coefficient. 346
<PERSON>'s paradox is also described in the book of <PERSON> and <PERSON>, but in their "plane wave treatment" of the problem is a serious error, as was pointed out first by <PERSON>, <PERSON> and <PERSON>. <PERSON> and <PERSON> considered a solution as "transmitted" which in fact corresponds to an incoming particle. Obviously, they neglected the fact that the velocity of the transmitted wave is opposite to its momentum (which is typical for negative energy solutions ... and was already noted by <PERSON>). They concluded that more is reflected than comes in, which is incorrect and contradicts, e.g. the unitarity of time evolution. Unfortunately, the same error is contained also in many papers which treat that subject on a formal level.
Neither of the above papers are completely correct in my opinion. The conclusion is that one needs to take the correct solution with positive probability current in the <PERSON> paradox regime. Note that there is still a <PERSON> paradox which is just transmission through a huge potential and that unitarity is preserved. 682