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!