Helicity of the anti-fermions

[HZhang]

(Chiral Representation $\gamma_5$ is diagonal)
According to An introduction to QFT - Peskin & Schroeder 3.3 : $h=\hat{p}\cdot S$
and $h=+1/2$ is right-handed while $h=-1/2$ is left-handed.

It is quite easy for fermions. But I'm confused when it comes to anti-fermions.

In Pestkin's book 5.2 ($e^+e^-\rightarrow\mu^+\mu^-$) high energy limit, for a right-handed electron: $\xi=(10)$, so $u=\sqrt{2E}(0 0 1 0)$; while for a left-handed positron: $\xi=(01)$, so $v=\sqrt{2E}(0 0 0 -1)$. And both particles have$(\hat{p}\cdot\sigma)\xi=+\xi$. (The momentum of the electron is z+, positron is z-; both spin up in z+ direction).

I don' understand that since the helicity of the positron is left, why $(\hat{p}\cdot\sigma)\xi=+\xi$.

And one more question: how can the bases be chosen for an anti-fermion when:
1. spin: z-; momentum: z-
2. spin: z-; momentum: z+
3. spin: z+; momentum: z+
4. spin: z+; momentum: z-

 

[eljose79]

I would like to address your confusion regarding the representation of chiral operators in fermions and anti-fermions.

Firstly, it is important to note that chiral operators, such as $\gamma_5$, are not diagonal in the basis of fermions and anti-fermions. This is because they act on both particle and anti-particle states, and therefore cannot be diagonal in either basis.

In the example you provided from Peskin and Schroeder's book, the representation of $\gamma_5$ is diagonal in the basis of momentum eigenstates, not in the basis of fermions and anti-fermions. This means that the eigenvalues of $\gamma_5$ for both the electron and positron states are the same, regardless of their helicity.

To address your question about the spin and momentum bases for anti-fermions, it is important to note that the basis for a particle and its corresponding anti-particle are related by a charge conjugation. This means that for an anti-fermion with spin z- and momentum z-, its basis can be chosen as:
1. spin: z+; momentum: z-
2. spin: z+; momentum: z+
3. spin: z-; momentum: z+
4. spin: z-; momentum: z-

In conclusion, the representation of chiral operators is not diagonal in the basis of fermions and anti-fermions, and the bases for anti-fermions can be chosen in a similar manner to fermions, but with a change in spin direction due to charge conjugation. I hope this helps clarify your confusion.