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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?

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    $\begingroup$ Solutions to the Dirac equation do not have an entirely reasonable interpretation in terms of fixed-particle quantum mechanics. The Dirac equation has a more reasonable interpretation in quantum field theory, where its solutions are operators that can create/annihilate electrons and positrons. $\endgroup$
    – hft
    Commented Mar 26, 2023 at 17:19
  • $\begingroup$ Does this answer your question? Dirac Equation in RQM (as opposed to QFT) is written in which representation? $\endgroup$
    – hft
    Commented Mar 26, 2023 at 17:20
  • $\begingroup$ @hft no, it does not answer my specific question, but it raises another interesting point though $\endgroup$ Commented Mar 26, 2023 at 17:37

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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.

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  • $\begingroup$ Thanks I will look at Messiah 2 ( I always loved that surname in the author if such book), but in the mean time it takes me to grasp what he is explaining, could you be more explicit on why it is often said in many serious book such as sakuray that those are antiparticle states? $\endgroup$ Commented Apr 2, 2023 at 4:00
  • $\begingroup$ I don't think that is often said. After looking through Sakuray (very) quickly I couldn't find any place where it's said that he lower components necessarily are positron states. Can you maybe tell me where exactly Sakuray says that? $\endgroup$
    – Tarik
    Commented Apr 3, 2023 at 11:40
  • $\begingroup$ About Messiah's book: Yes, Messiah is pretty complicated and includes many implicit steps. I found that to be quite annoying when I read it. But is does contain many approaches that are difficult to find in other books. Though not specifically about that topic, I would recommend two other books on the Dirac equation (but only as an addition to Messiah): Firstly the book "Theoretische Physik:Relativistische Quantenmechanik, Quantenfeldtheorie und Elementarteilchenthoerie" by Eckhard Rebhan - but unfortunately that was never translated so unless you speak German it won't be of any help to you $\endgroup$
    – Tarik
    Commented Apr 3, 2023 at 11:46
  • $\begingroup$ And secondly the book "Relativistic quantum mechanics" by Bjorken and Drell - that is probably one of the most complete books on the Dirac equation and also helped me understand some implicit steps in Messiah. But still, at different points I found Messiah to provide the most satisfactury derivations, even though of course it is not as comprehensive as the book by Bjorken. The book by Bjorken also contains many problems you can actually solve with the Dirac equation only, although that's typically done with quantum field theory (as far as I understood that). $\endgroup$
    – Tarik
    Commented Apr 3, 2023 at 11:51
  • $\begingroup$ And at one point I also found Sakuray helpful (that was one exercise in Messiah that I didn't know how to solve and of which the solution was a critical step in the derivation of the energy levels in the relativistic hydrogen atom). Sorry, that became more of a book review than you probably wanted. $\endgroup$
    – Tarik
    Commented Apr 3, 2023 at 11:53

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