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?
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
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.