Now that we have some basic understanding of FeynCalc objects, let us
do something with them. Contractions of Lorentz indices are one of the
most essential operations in symbolic QFT calculations. In FeynCalc the
corresponding function is called Contract
FV[p, \[Mu]] MT[\[Mu], \[Nu]]
Contract[%]\overline{p}^{\mu } \bar{g}^{\mu \nu }
\overline{p}^{\nu }
FV[p, \[Alpha]] FV[q, \[Alpha]]
Contract[%]\overline{p}^{\alpha } \overline{q}^{\alpha }
\overline{p}\cdot \overline{q}
Notice that when we enter noncommutative objects, such as Dirac
matrices, we use Dot (.) and not
Times (*)
FV[p, \[Alpha]] MT[\[Beta], \[Gamma]] GA[\[Alpha]] . GA[\[Beta]] . GA[\[Gamma]]
Contract[%]\overline{p}^{\alpha } \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma } \bar{g}^{\beta \gamma }
\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\gamma }.\bar{\gamma }^{\gamma }
This is because Times is commutative, so writing
something like
GA[\[Delta]] GA[\[Beta]] GA[\[Alpha]]\bar{\gamma }^{\alpha } \bar{\gamma }^{\beta } \bar{\gamma }^{\delta }
will give you completely wrong results. It is also a very common beginner’s mistake!