The Fierz reshuffle theorem

Prom the fact that the As are traceless matrices, using (A2.6.1 and 2) 

It sometimes happens, when dealing with the matrix element corresponding to a Feynman diagram involving spin particles, that it is convenient to rearrange the order of the spinors compared with the order they acquire directly from the Feynman diagram. An example of this occurred in Section 5.1 where it was helpful to go from the form (5.1.17 ) to (5.1.18). 

Let Pi stand for the above set of matrices with their Lorentz indices 

After some labour one can obtain the following relation 

Clearly, analogous relations can be worked out for any product of the F matrices. Results may be found in Section 2.2B of Marshak, Riazuddin and Ryan (1969). 

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For later convenience we rearrange (5.1.17) using the Fierz reshuffle theorem on direct products of 7-matrices (see Appendix 2.7). Then... 

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