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.

  • $\begingroup$ recently i saw a work on graphene ;Nature Physics 2, 620 - 625 (2006) Published online: 20 August 2006 | doi:10.1038/nphys384 M. I. Katsnelson,K. S. Novoselov,A. K. Geim Chiral tunnelling and the Klein paradox in graphene does this help in clarifying the picture of "klein Paradox" M. I. Katsnelson1, K. S. Novoselov2 & A. K. Geim2 $\endgroup$
    – drvrm
    Mar 2, 2016 at 17:41

3 Answers 3


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.

enter image description here

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.

  • 4
    $\begingroup$ Welcome to Physics! It's always great when authors join the Q&A on their own papers. $\endgroup$ Sep 9, 2021 at 15:48
  • $\begingroup$ @AlexH How is your answer consistent with the book that I quote in my question? It is straightforward to see that the apparent breaking of unitarity in the 1-particle problem arises due to solutions that carry a current with the wrong sign. If one solves the one-particle scattering problem by evolving a wave packet numerically, there won't be any violation of unitarity. At least, this is how the problem is dealt with in condensed-matter physics. Here, the negative-energy solutions correspond to the Fermi sea, and there is no need to invoke quantum field theory in the absence of interactions. $\endgroup$
    – Praan
    Sep 16, 2021 at 17:34
  • $\begingroup$ I see no inconsistencies between our work and the quote from the Thaller book. Perhaps the following talk given by Fin Ravndal in 2011 on the Klein paradox will make our treatment clearer: link. $\endgroup$ Sep 27, 2021 at 13:53
  • $\begingroup$ As for the condensed-matter version of the Klein paradox, it is not the same problem. There are a number characteristics of this problem that are similar to the relativistic Klein problem, but this is not enough to guarantee that the solution of the relativistic Klein paradox carries over to the other one. I have not studied the latter, so I do not know whether the solution carries over or not. $\endgroup$ Sep 27, 2021 at 14:10

There are pro-forma classical solutions available mixing on one side of potential step particles with, on the other, antiparticles. Such situation implies the need for understanding of what is the physics problem one is solving. This situation was addressed and studied in depth in early-mid seventies and detailed studies were published as a part of a Physics Reports https://doi.org/10.1016/0370-1573(78)90116-3 and there is another presentation in this book https://www.amazon.com/Quantum-Electrodynamics-Strong-Fields-Introduction/dp/3642822746 . Alex Hansen and Finn Ravndal paper as I recall was an outsider view point. Another insight: Oscar Klein's "paradox" paper was sandwiched in-between the computation he was doing of the Klein-Nishina formula for Compton scattering BEFORE it was understood how to interpret E<-m solutions of the Dirac equation. Klein needed to clarify the role of these states as they were finding that all of the scattering was due to E<-m solutions. Thus before positrons were postulated, and before QED was formulated the formal solutions with coexistent outgoing (pair) waves were a (Klein's) paradox. I would say there is little paradoxical in this today. Giving paper clip answer how to deal with specific physics problems involving strong fields is not possible in these pages.


Really interesting Question. To me, this Seems like Quantum-mechanical-hydraulic jump. As it's mostly studied on fluid flows, it can't be very obvious why I parallel with this Klein-paradox; Hydraulic jump happens when Froude number is above one. (If $Fr=\sqrt3$ this jump is dissipative.)

When Froude Number is one. The inertia is in balance with external field when $Fr=1=\frac{v}{\sqrt{gl}}$,
which I prefere to write in from $1=\frac{v^2}{{gl}}$

If we change these values to Energy (or metric); we have another (metric)ratio; $1/2$
as Kinetic energy is $E_k=1/2mv^2$ and Potential energy is $E_p=mgl$ when $(l=h)$.

So what do we have in Klein paradox? $V_0$ is the total Potential step of height. $E$ is the kinetic energy of the particle and $mc^2$ is the mass energy of Particle, and as these have a rule
$V_0 > E +mc^2> 2 mc^2$, which is the paradox, as it's violating the conservation of the energy.

Let's take a closer look; as
$E +mc^2> 2 mc^2$ thus also
$E >mc^2$

So if the Froude Number would be one, we would have
$V_0 = E + 2 mc^2 = 3 mc^2$ thus we can conclude that
$V_0 > E +mc^2> 2 mc^2$ has obviously $Fr>1$,
more precisely it has $E /mc^2$ which means that the Metric ratio is $1/1$, which simply means $Fr=\sqrt2$


Is this the correct picture?

A: No.


Are the textbooks wrong on this one?

A: Yes.


Also, does pair creation at the potential violate energy conservation or is the energy supplied by the static potential?

A: Yes, it does violates the Energy conservation, or rather, as this can't be violated proves that the Current physical theories are wrong.

Here's a part of the page 120 of the book provided in Question;

Klein's Paradox

At the text there is word "kinetic Energy" and "External Field" underlined with red. This is done because I expect this answer to be deleted because many will think it "Doesn't answer the question". So please do give a minus vote, but don't delete simply because of the lack of understanding. -Thanks.


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