- #1
Einj
- 470
- 59
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:
$$
(\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},
$$
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?
$$
(\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j F_{ij}(\Gamma_j)_{\alpha\xi}(\Gamma_j)_{\gamma\beta},
$$
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?