When you square an expression with dummy indices, you must rename them first. People often do this by hand, e.g. as in
ex1 = (FV[p, \[Mu]] + FV[q, \[Mu]]) FV[r, \[Mu]] FV[r, \[Nu]]\overline{r}^{\mu } \overline{r}^{\nu } \left(\overline{p}^{\mu }+\overline{q}^{\mu }\right)
ex1 (ex1 /. \[Mu] -> \[Rho])
Contract[%]\overline{r}^{\mu } \left(\overline{r}^{\nu }\right)^2 \overline{r}^{\rho } \left(\overline{p}^{\mu }+\overline{q}^{\mu }\right) \left(\overline{p}^{\rho }+\overline{q}^{\rho }\right)
\overline{r}^2 \left(\overline{p}\cdot \overline{r}+\overline{q}\cdot \overline{r}\right)^2
However, FeynCalc offers a function for that
FCRenameDummyIndices[ex1]\overline{r}^{\nu } \overline{r}^{\text{\$AL}(\text{\$19})} \left(\overline{p}^{\text{\$AL}(\text{\$19})}+\overline{q}^{\text{\$AL}(\text{\$19})}\right)
ex1 FCRenameDummyIndices[ex1]
Contract[%]\overline{r}^{\mu } \overline{r}^{\nu } \overline{r}^{\nu } \left(\overline{p}^{\mu }+\overline{q}^{\mu }\right) \overline{r}^{\text{\$AL}(\text{\$20})} \left(\overline{p}^{\text{\$AL}(\text{\$20})}+\overline{q}^{\text{\$AL}(\text{\$20})}\right)
\overline{r}^2 \left(\overline{p}\cdot \overline{r}+\overline{q}\cdot \overline{r}\right)^2
Notice that FCRenameDummyIndices does not canonicalize
the indices
FV[p, \[Nu]] FV[q, \[Nu]] - FV[p, \[Mu]] FV[q, \[Mu]]
FCRenameDummyIndices[%]\overline{p}^{\nu } \overline{q}^{\nu }-\overline{p}^{\mu } \overline{q}^{\mu }
\overline{p}^{\text{\$AL}(\text{\$22})} \overline{q}^{\text{\$AL}(\text{\$22})}-\overline{p}^{\text{\$AL}(\text{\$21})} \overline{q}^{\text{\$AL}(\text{\$21})}
There is a function for that too
FV[p, \[Nu]] FV[q, \[Nu]] - FV[p, \[Mu]] FV[q, \[Mu]]
FCCanonicalizeDummyIndices[%]\overline{p}^{\nu } \overline{q}^{\nu }-\overline{p}^{\mu } \overline{q}^{\mu }
0
Often we also need to uncontract already contracted indices. This is
done by Uncontract. By default, it handles only
contractions with Dirac matrices and Levi-Civita tensors
LC[][p, q, r, s]
Uncontract[%, p]
Uncontract[%%, p, q]\bar{\epsilon }^{\overline{p}\overline{q}\overline{r}\overline{s}}
\overline{p}^{\text{\$AL}(\text{\$31})} \bar{\epsilon }^{\text{\$AL}(\text{\$31})\overline{q}\overline{r}\overline{s}}
\overline{p}^{\text{\$AL}(\text{\$33})} \overline{q}^{\text{\$AL}(\text{\$32})} \left(-\bar{\epsilon }^{\text{\$AL}(\text{\$32})\text{\$AL}(\text{\$33})\overline{r}\overline{s}}\right)
SP[p, q]
Uncontract[%, p]\overline{p}\cdot \overline{q}
\overline{p}\cdot \overline{q}
To uncontract scalar products as well, use the option
Pair->All
Uncontract[%, p, Pair -> All]\overline{p}^{\text{\$AL}(\text{\$34})} \overline{q}^{\text{\$AL}(\text{\$34})}
Sometimes one might want to define custom symbolic tensors that are not specified in terms of the 4-vectors, metric tensors and Levi-Civitas. This is possible in FeynCalc, but the handling of such objects is not as good as that of the built-in quantities
DeclareFCTensor[myTensor];myTensor[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]] FV[p, \[Nu]] FV[q, \[Mu]]
ex = Contract[%]\overline{p}^{\nu } \overline{q}^{\mu } \;\text{myTensor}(\mu ,\nu )
\text{myTensor}\left(\overline{q},\overline{p}\right)
Uncontract[ex, p, q, Pair -> All]\overline{p}^{\text{\$AL}(\text{\$36})} \overline{q}^{\text{\$AL}(\text{\$35})} \;\text{myTensor}(\text{\$AL}(\text{\$35}),\text{\$AL}(\text{\$36}))
(myTensor[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]] MT[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]] +
myTensor[LorentzIndex[\[Alpha]], LorentzIndex[\[Beta]]] MT[LorentzIndex[\[Alpha]], LorentzIndex[\[Beta]]])
FCCanonicalizeDummyIndices[%, LorentzIndexNames -> {i1, i2}]\bar{g}^{\alpha \beta } \;\text{myTensor}(\alpha ,\beta )+\bar{g}^{\mu \nu } \;\text{myTensor}(\mu ,\nu )
2 \bar{g}^{\text{i1}\;\text{i2}} \;\text{myTensor}(\text{i1},\text{i2})
To extract the list of free or dummy indices present in the
expression, one can use FCGetFreeIndices and
FCGetDummyIndices respectively
FCI[FV[p, \[Mu]] FV[q, \[Nu]]]
FCGetFreeIndices[%, {LorentzIndex}]\overline{p}^{\mu } \overline{q}^{\nu }
\{\mu ,\nu \}
```mathematica FCI[FV[p, [Mu]] FV[q, [Mu]]] FCGetDummyIndices[%, {LorentzIndex}]
```mathematica
\overline{p}^{\mu } \overline{q}^{\mu }
\{\mu \}