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It is known that contracting over the vector indices of two Pauli matrices (the 3d γ-matrices) can be simplified to a bunch of δ functions. This is done via the Fierz formula $$\delta_{ab}\sigma^a_{ij}\sigma^b_{kl}=\delta_{ij}\delta_{kl}-2\epsilon_{ik}\epsilon_{jl}=2\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}.$$ Here, $a,b,c=1,2,3$ are vector indices, and $i,j,k,l=1,2$ are spinor indices. Repeated indices are summed over.

However, I would like to know whether there is a similar formula for $$\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}~?$$ If so, what is it?

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$$\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}= 2i \Big(\delta ^i_q \delta ^k_j\delta ^p_l - \delta ^i_l\delta ^k_q\delta ^p_j\Big ), $$ with all the right pairwise interchange symmetries. (I have raised, for convenience, not terribly meaningfully, indices in the Kronecker δs to churn these interchanges more easily.)

The normalization ensues by taking the trace, $\delta ^i_i\delta ^k_k\delta ^p_p -\delta ^i_i=6$, given the SU(2) Lie algebra normalization/identity, $$ {\mathrm Tr} ~ \sigma^a \sigma^b \sigma^c =2i~\epsilon_{abc} ~. $$

(Pauli matrices are essentially clebsches, cf Cvitanovic 1984, but I doubt connection to Clebsch coefficient identities would be constructive here... Still, you may think of three SU(2) "quarks" emitting an SU(2) "gluon" each, and all three pseudogluons merging at a cubic vertex: what can the outgoing pseudoquark pseudocolors be?)

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  • $\begingroup$ Hi Prof. Zachos, thank you very much for the answer. $\endgroup$ – user34104 Oct 1 '17 at 15:48

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