Proof that a Dirac particle has spin 1/2?

[AlbertEi]

Hi,

I am having trouble following the Peskin and Schroeder and their derivations to show that a Dirac particle is a spin 1/2 particle (page 60 and 61). I understand how he gets the first (unnumbered) equation on page 61. However, I don't understand how he gets to the second equation:
\begin{equation}
J_z a^{s \dagger}_0|0 \rangle = \frac{1}{2m} \sum_r \left(u^{s \dagger}(0) \frac{\Sigma^3}{2} u^r(0) \right) a_0^{r \dagger} |0 \rangle
\end{equation}
In particular I do not understand why there is only one "summation" symbol. I would have thought that the equation would be:
\begin{equation}
J_z a^{s \dagger}_0|0 \rangle = \frac{1}{2m} \sum_r \sum_{s} \left(u^{s \dagger}(0) \frac{\Sigma^3}{2} u^r(0) \right) a_0^{r \dagger} |0 \rangle
\end{equation}
Does anybody know what they have done with the second "summation" symbol?

Any help would much appreciated.

 

[JK423]

Hi AlbertEi,

On the left hand side you have the free index 's', so on the right hand side this index shouldn't be summed. This is just a remark without knowing the actual calculation.

 

[VoxCaelum]

Well, for starters the second equation you wrote down doesn't make too much sense, since there is a sum over s on the right hand side, but there is still an s on the left hand side.
Now the actual problem, you should convince yourself that when taking the commutator
$$[J_{z},a_{0}^{s\dagger}]$$
the only term that does not (anti-)commute (anti-commutator or commutator does not matter, J_{z} will hit the vacuum and annihilate it regardless), is the term
$$ [a_{p}^{r\dagger}a_{p}^{r},a_{0}^{s\dagger}].$$
Next you should convince yourself that this term is exactly what Peskin tells you it is. Now check what happens to the exponentials in the first line of the unnumbered equation when you integrate it against the δ(p-p^{\prime}) over p^{\prime}, (as a hint, the position integral should now be easy, and doing the position integral should allow you to do the final momentum integral).

 

[AlbertEi]

Those indices represent the spin, i.e. they do are not tensor indices so I don't think we can use the Einstein summation convention in this sense.

 

[AlbertEi]

Hi VoxCaelum, my previous reply was not directed towards you. I will look into your answer and come back if I have any questions. Thanks for your reply.

 

[Bill_K]

In particular I do not understand why there is only one "summation" symbol. Does anybody know what they have done with the second "summation" symbol?

To begin with there were two summations over r and r'. The commutator contains a Kronecker delta, δ^r's, which collapses the r' summation to a single term s, so just the r summation remains.

 

[AlbertEi]

That makes sense; I feel a bit silly now. Thank Bill_K!