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I am trying to understand some points about the paradox, what I am doing is solving the for the step potential $$ V = V_0 ~\theta(z) $$

I have two solutions $$ \Phi_I = e^{ik_1 z} + r e^{-ik_1 z}$$

for $z<0$ $$ \Phi_{II} = t e^{ik_2 z} $$ for $z>0$,

where

$$ k_1 = \sqrt{E^2-m^2} $$ $$ k_2 = \pm \sqrt{(E-V_0)^2-m^2} $$

The first question is, why there is no $ \pm $ from the square root for $k_1$, why is not possible for $k_1$ to be negative

Why is possible for $k_2$ to be negative for strong potentials (that is $V_0 > E^2+m^2 $) making $$ T=\frac{4k_1 k_2}{k_1+k_2}<0 $$ therefore showing the paradox with this nonsense but it won't happen for weak or intermediate potentials (Thats $V_0 < E^2-m^2 $ and $E-m < V_0 < E+m $) even when the $\pm$ it's always there.

They may be silly questions, but I just don't see it.

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For the first question, the answer is: in this context we just want to study the case where an electron wave is coming from $-\infty$ to the barrier, so we just consider a positive $k_1$. One can of course also consider an electron coming from $\infty$ to the barrier, then one just uses $k_1=-\sqrt{E^2-m^2}$.

For the second question, the answer is simple and complicated. It is simple because that is what one gets from solving Dirac equation. As you said, it is this strange solution that made people (such as Klein) confused about the properties of Dirac equation. It is complicated because that solution actually indicates creation of electron-positron pair in strong field, which is a specially case of Schwinger effect: the quantum vacuum is unstable under the influence of an external electric field (http://www.phys.uconn.edu/~dunne/dunne_schwinger.html).

My favorite introduction to this topic is Advanced Quantum Mechanics by J. J. Sakurai, I hope you will find the discussion there helpful.

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