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I am reading Sakurai and the free particle solutions to the Dirac equation. For the positive energy solution, one of them is:

$u_R(p)=\begin{pmatrix} 1 \\ 0 \\ \frac{p}{E_p+m} \\0\end{pmatrix} $

He then says that the 3rd component is small in the non relativistic limit, and then ignores it in what follows.

But my question is, what does this solution mean in the relativistic regime? arent the two lower components positron states rather than electron states? I am not sure how to interpret this, is this state a superposition of an electron and a positron? If so, does it mean that even if I have a free electron and try to measure it there is a chance to detect a positron instead?

Pato Galmarini
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1 Answers1

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In the "normal" Dirac picture, it is not necessarily true that the upper components are the ones with positive energy and the lower ones the ones with negative energy (positrons). In fact, you can see that directly in your example: That spinor has the energy $E_p > 0$. There are two ways (that I know of) that achieve that separation in upper and lower components: Firstly, it is generally true in the non-relativistic limit (in the Dirac-picture). And secondly, for free particles, you can achieve this seperation exactly with the so-called Foldy-Wouthuysen transformation, a unitary transformation (a bit like going to the Heisenberg-picture, which btw is also possible in the Dirac-theory). A propos: In the case of a particle in an electromagnetic field, that transformation is no longer exactly possible. Rather, you can get the separation up until $O((\frac{v}{c})^n)$ with an arbitrary n (although I've never seen any case of n>2). If you want to see the details worked out, I recommend Messiah qm 2 - there is a long chapter on the Dirac-equation in it which contains many approaches that aren't easy to be found elsewhere.

So, to sum up: No, the two lower components are not the positron states - at least not in the Dirac picture. However, in the non-relativistic limit as well as after a Foldy-Wouthuysen transformation, you can achieve that.

Tarik
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