My question is based on the pdf in this link

On page 8, the authors consider an example of massless fermions. I follow through most of the steps in here except for equation 41, which I believe should be:

$\overline{\mathbb{\psi}} \to \overline{\mathbb{\psi}} e^{+i\frac{\vec{\tau}}{2}\vec{\Theta}}$

Then when we reach the part right after equation 41, I am confused about why

$i\overline{\mathbb{\psi}}\not\partial \psi \to i\overline{\mathbb{\psi}}\not\partial \psi - i\vec{\Theta}(\bar{\psi}i\not\partial\frac{\vec{\tau}}{2}\psi - \bar{\psi}\frac{\vec{\tau}}{2}i\not\partial\psi) $ = $i\overline{\mathbb{\psi}}\not\partial\psi$?

Why are the terms inside the parenthesis canceling each other?

If I understand it correctly, $\vec{\tau}$ is a vector of pauli matrices and then $\not\partial = \gamma \cdot \partial$ which means we have Pauli matrices and $\gamma$ matrices, and they do not commute with each other. So how can we obtain equation 42?

P.S. I really did not find a way to write the slashed partial derivative properly, so if anyone knows a better way and is willing to edit it, thank you in advance.

  • 3
    $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. $\endgroup$ – Qmechanic Nov 7 '20 at 4:20
  • $\begingroup$ @Qmechanic, I will note that for future. $\endgroup$ – time12 Nov 7 '20 at 16:19

The Pauli matrices are operating on the implied isospin index that distinguishes the $u$ and $d$ components of $\psi$. The Dirac gamma matrices are operating on the implied Lorentz spinor index that distinguishes the four components of $u$ and the four components of $d$. Thus they commute.

  • $\begingroup$ @newUser You beat me to the answer. I did not see your answer until I finished mine. I’ve upvoted yours. $\endgroup$ – G. Smith Nov 7 '20 at 3:25
  • $\begingroup$ Don't worry! ;) $\endgroup$ – apt45 Nov 7 '20 at 3:26
  • $\begingroup$ Ah, thanks a lot. Since I am new to QFT, I still confuse this idea a lot. $\endgroup$ – time12 Nov 7 '20 at 16:19

I haven't opened the link, but I guess that the field transformation you are writing is a global symmetry (SU(2)) action, which means that Pauli matrices act on a different space than the gamma matrices.

In other words, gamma matrices act on the spinorial indexes, while Pauli matrices act on the internal symmetry indexes - so, they commute.

In particular, the spinor $\Psi$ have the following indexes structure $\Psi^{\,\alpha}_b$ and in the following expression

$$ (\vec{\tau})^{ab}\gamma^\mu_{\alpha\beta}\partial_\mu \Psi^{\,\beta}_b $$

it's clear that the Gamma matrices and Pauli matrices commute - they are matrices on different spaces.


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