FeynCalc offers further useful functions for the manipulations of Lorentz tensors and Dirac matrices. To expand scalar products
ex1 = SP[p + q + r, s + t](\overline{p}+\overline{q}+\overline{r})\cdot (\overline{s}+\overline{t})
or expressions like
ex2 = FV[p + q + r, \[Mu]]\left(\overline{p}+\overline{q}+\overline{r}\right)^{\mu }
one can use
ExpandScalarProduct[ex1]\overline{p}\cdot \overline{s}+\overline{p}\cdot \overline{t}+\overline{q}\cdot \overline{s}+\overline{q}\cdot \overline{t}+\overline{r}\cdot \overline{s}+\overline{r}\cdot \overline{t}
ExpandScalarProduct[ex2]\overline{p}^{\mu }+\overline{q}^{\mu }+\overline{r}^{\mu }
For the expansion of Eps tensors, we use
LC[][p1 + p2, q, r, s]
EpsEvaluate[%]\bar{\epsilon }^{\overline{\text{p1}}+\overline{\text{p2}}\overline{q}\overline{r}\overline{s}}
\bar{\epsilon }^{\overline{\text{p1}}\overline{q}\overline{r}\overline{s}}+\bar{\epsilon }^{\overline{\text{p2}}\overline{q}\overline{r}\overline{s}}
EpsEvaluate also reorders the arguments of Eps according to its antisymmetric properties
LC[\[Mu], \[Sigma], \[Rho], \[Nu]]
EpsEvaluate[%]\bar{\epsilon }^{\mu \sigma \rho \nu }
-\bar{\epsilon }^{\mu \nu \rho \sigma }
The inverse of ExpandScalarProduct is called MomentumCombine
3 FV[p, \[Mu]] + 4 FV[q, \[Mu]]
MomentumCombine[%]3 \overline{p}^{\mu }+4 \overline{q}^{\mu }
\left(3 \overline{p}+4 \overline{q}\right)^{\mu }
For Dirac matrices the corresponding functions are DiracGammaExpand and DiracGammaCombine
GA[\[Mu]] . GS[p + q] . GA[\[Nu]] . GS[r + s]
DiracGammaExpand[%]
DiracGammaCombine[%]\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{p}+\overline{q}\right)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{r}+\overline{s}\right)\right)
\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}+\bar{\gamma }\cdot \overline{q}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{r}+\bar{\gamma }\cdot \overline{s}\right)
\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{p}+\overline{q}\right)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{r}+\overline{s}\right)\right)
Notice the DiracGammaExpand does not expand the whole noncommutative product. If you need that, use DotSimplify
GA[\[Mu]] . GS[p + q] . GA[\[Nu]] . GS[r + s]
% // DiracGammaExpand // DotSimplify\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{p}+\overline{q}\right)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{r}+\overline{s}\right)\right)
\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{r}\right)+\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{s}\right)+\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{q}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{r}\right)+\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{q}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{s}\right)