I am reading this paper about the Klein 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. Bjorken and Drell) 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.
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?
I have found a very interesting discussion on page 307 of the book B. Thaller, The Dirac equation.
In 1928, O. Klein 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 F. Sauter investigated Klein's paradox for a smooth potential, which gave the same qualitative result, but with a much smaller transmission coefficient. Klein's paradox is also described in the book of Björken and Drell, but in their "plane wave treatment" of the problem is a serious error, as was pointed out first by Dosch, Jensen and Muller. Björken and Drell 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 Klein). 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 Klein paradox regime. Note that there is still a Klein paradox which is just transmission through a huge potential and that unitarity is preserved.