FeynCalc manual (development version)

Handling indices

See also

Overview.

Manipulations of tensorial quantities

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]]

r‾μr‾ν(p‾μ+q‾μ)\overline{r}^{\mu } \overline{r}^{\nu } \left(\overline{p}^{\mu }+\overline{q}^{\mu }\right)

ex1 (ex1 /. \[Mu] -> \[Rho])
Contract[%]

r‾μ(r‾ν)2r‾ρ(p‾μ+q‾μ)(p‾ρ+q‾ρ)\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)

r‾2(p‾⋅r‾+q‾⋅r‾)2\overline{r}^2 \left(\overline{p}\cdot \overline{r}+\overline{q}\cdot \overline{r}\right)^2

However, since FeynCalc 9 there is a function for that

FCRenameDummyIndices[ex1]

r‾νr‾$AL($19)(p‾$AL($19)+q‾$AL($19))\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[%]

r‾μr‾νr‾ν(p‾μ+q‾μ)r‾$AL($20)(p‾$AL($20)+q‾$AL($20))\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)

r‾2(p‾⋅r‾+q‾⋅r‾)2\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[%]

p‾νq‾ν−p‾μq‾μ\overline{p}^{\nu } \overline{q}^{\nu }-\overline{p}^{\mu } \overline{q}^{\mu }

p‾$AL($22)q‾$AL($22)−p‾$AL($21)q‾$AL($21)\overline{p}^{\text{\$AL}(\text{\$22})} \overline{q}^{\text{\$AL}(\text{\$22})}-\overline{p}^{\text{\$AL}(\text{\$21})} \overline{q}^{\text{\$AL}(\text{\$21})}

But since FeynCalc 9.1 there is a function for that too

FV[p, \[Nu]] FV[q, \[Nu]] - FV[p, \[Mu]] FV[q, \[Mu]]
FCCanonicalizeDummyIndices[%]

p‾νq‾ν−p‾μq‾μ\overline{p}^{\nu } \overline{q}^{\nu }-\overline{p}^{\mu } \overline{q}^{\mu }

00

Finally, 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]

ϵˉp‾q‾r‾s‾\bar{\epsilon }^{\overline{p}\overline{q}\overline{r}\overline{s}}

p‾$AL($31)ϵˉ$AL($31)q‾r‾s‾\overline{p}^{\text{\$AL}(\text{\$31})} \bar{\epsilon }^{\text{\$AL}(\text{\$31})\overline{q}\overline{r}\overline{s}}

p‾$AL($33)q‾$AL($32)(−ϵˉ$AL($32)$AL($33)r‾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]

p‾⋅q‾\overline{p}\cdot \overline{q}

p‾⋅q‾\overline{p}\cdot \overline{q}

To uncontract scalar products as well, use the option Pair->All

Uncontract[%, p, Pair -> All]

p‾$AL($34)q‾$AL($34)\overline{p}^{\text{\$AL}(\text{\$34})} \overline{q}^{\text{\$AL}(\text{\$34})}