Fierz Identity Question: Understanding the Transformation and Matrices

[Einj]

Hi everyone, I have a doubt on Fierz identities. If we define the following quantities: $S=1,\; V=\gamma_\mu,\; T=\sigma_{\mu\nu},\; A=\gamma_\mu\gamma_5,\;P=\gamma_5$, then we have the identity:

$$
(\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},
$$
where $\Gamma_i$ 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 $VV+AA=-VV-AA$ 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?

 

[Bill_K]

Hi everyone, I have a doubt on Fierz identities. If we define the following quantities: $S=1,\; V=\gamma_\mu,\; T=\sigma_{\mu\nu},\; A=\gamma_\mu\gamma_5,\;P=\gamma_5$, then we have the identity:

$$
(\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},
$$
where $\Gamma_i$ 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 $VV+AA=-VV-AA$ 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?

Because the ψ's anticommute? I think it matters whether you just give the relation between matrices, as Wikipedia does, or include the ψ's. Both of these references give the table for F_ij including the ψ's, with the opposite sign.
http://hep-www.px.tsukuba.ac.jp/~yuji/mdoc/fierzTrans.pdf
http://onlinelibrary.wiley.com/doi/10.1002/9783527648887.app5/pdf

 

[Einj]

I think you are right. Once we write the identity for the matrices then we need to switch the two field and this should give an extra minus sign.