# FeynCalc does not simplify Spinor relations

Is there a reason why sometimes FeynCalc does not simplify expressions that looks very simple like $$\bar{u}^s(p,m).u^s(p,m).$$

This does not happen all the time, eg. if I simply input

SpinorUBar[a, b].SpinorU[a, b] // Calc


I get 2b which is good. But in a longer expression FeynCalc does not simplify. Also why is the Gordon identity not used to simplify expression of the form

SpinorUBar[a, b].GA[c].SpinorU[a, b] // Calc


when they appear in longer expressions? Is there a way to tell FeynCalc to evaluate and simplify these?

• Better late then never: Gordon identities, including their chiral versions, are now supported (see my answer below).
– vsht
Sep 15, 2020 at 8:44

Can you provide a minimal working example, where spinor expressions are not simplified?

As for the Gordon identity, this is not a transformation that always leads to simpler expressions, so it is not useful to apply it everywhere. It is of course handy in the evaluation of vertex functions, where you want to extract the different form factors, but in other calculations it might not make much sense to trade $$\gamma^\mu$$ for $$\sigma^{\mu \nu}$$. Furthermore, it is very easy to apply Gordon identity via a replacement rule, see how it is done in

https://github.com/FeynCalc/feynhelpers/blob/master/Examples/QCD/NRQCDVertexMatchingInTheTwoQuarkSectorOneLoop.m

and

https://github.com/FeynCalc/feynhelpers/blob/master/Examples/QED/QEDRenormalizationOneLoop.m

EDIT 2020: Gordon identities have been recently added to the development version via a new function GordonSimplify

Evaluating

SpinorUBar[a, b].GA[c].SpinorU[a, b] // GordonSimplify // FCE


returns

(Spinor[Momentum[a], b, 1].Spinor[Momentum[a], b, 1] FV[a, c])/b