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.
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, 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.
Best Answer
It is already 40 years since my MSc thesis advisor Finn Ravndal and I wrote the paper discussed here.
The inspiration to the key idea in our paper came from one of Feynman's lesser known papers. Figure 1 in this paper (see below) shows the classical path of a particle moving through a box potential high enough for the Klein conditions to set in. As Feynman remarks and this figure shows, pair production - and the Klein paradox situation - also happens at the classical level - it is not just a quantum phenomenon.
Much of the confusion over the years concerning the Klein paradox comes from interpreting the sign of the momentum of the particles inside the strong potential. By comparing with the classical analog (Feynman's paper), it was suddenly easy to figure out the sign. This made us sure that we got the one-particle quantum mechanics right and then to incorporate it in a field-theoretic treatment.
There are then three levels to the Klein paradox. It appears already at the classical level (Feynman's paper), leading to problems with causality. This problem is solved in a one-particle quantum treatment of the problem, but then there is a problem with unitarity. Only at the field theoretic level are both problems solved, and there is no more any paradox. This is the contents of our paper in a nutshell.
The author of the original question asks whether energy conservation is violated. The answer is no. The potential is external and static. Hence, it acts as an infinite energy reservoir.
The author of the original question also notes that the transmission coefficient for fermions is zero. Yes, this is a consequence of the Pauli principle. In order to have a non-zero transmission coefficient in the Klein regime, the incoming particle would have to induce a pair creation event into the channel it is already occupying, which is forbidden.