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Question on Fierz identity

by Einj
Tags: fierz, identity
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Einj
#1
Mar27-14, 12:17 PM
P: 305
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?
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Bill_K
#2
Mar27-14, 12:26 PM
Sci Advisor
Thanks
Bill_K's Avatar
P: 4,160
Quote Quote by Einj View Post
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 Fij including the ψ's, with the opposite sign.
http://hep-www.px.tsukuba.ac.jp/~yuj...fierzTrans.pdf
http://onlinelibrary.wiley.com/doi/1...48887.app5/pdf
Einj
#3
Mar27-14, 12:30 PM
P: 305
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.


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