# Question on Fierz identity

by Einj
Tags: fierz, identity
 P: 324 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?
 Quote by 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?