Hi everyone, I have a doubt on Fierz identities. If we define the following quantities: [itex] S=1,\; V=\gamma_\mu,\; T=\sigma_{\mu\nu},\; A=\gamma_\mu\gamma_5,\;P=\gamma_5[/itex], then we have the identity:(adsbygoogle = window.adsbygoogle || []).push({});

$$

(\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},

$$

where [itex]\Gamma_i[/itex] are the matrices define before. Moreover:

$$

F_{ij}=\frac{1}{8}\left(\begin{array}{ccccc}

2 & 2 & 1 & -2 & -2 \\

8&-4&0&-4&-8 \\

24&0&-4&0&24 \\

-8&-4&0&-4&8 \\

2&-2&1&2&2

\end{array}\right)

$$

Therefore, if we take the VV+AA combination it turns out that [itex]VV+AA=-VV-AA[/itex] with exchanged indices.

However I usually read the Fierz transformation to be:

$$

(\psi_1\Gamma P_L\psi_2)(\psi_3\Gamma P_L\psi_4)=(\psi_1\Gamma P_L\psi_4)(\psi_3\Gamma P_L\psi_2).

$$

Without any minus sign. Does anyone knows why?

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# Question on Fierz identity

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