FCCanonicalizeDummyIndices[expr] canonicalizes dummy indices in the expression.
Following index types are supported: LorentzIndex, CartesianIndex, SUNIndex, SUNFIndex, DiracIndex, PauliIndex
In the case of Lorentz indices the option Momentum provides a possibility to limit the canonicalization only to particular Momenta. The option LorentzIndexNames can be used to assign specific names to the canonicalized indices, to have say \mu, \nu, \rho etc. instead of some random names.
For other index types the corresponding options are called CartesianIndexNames, SUNIndexNames, SUNFIndexNames, DiracIndexNames and PauliIndexNames.
Overview, FCRenameDummyIndices.
Canonicalization of Lorentz indices
FVD[q, mu] FVD[p, mu] + FVD[q, nu] FVD[p, nu] + FVD[q, si] FVD[r, si]
FCCanonicalizeDummyIndices[%] // Factor2p^{\text{mu}} q^{\text{mu}}+p^{\text{nu}} q^{\text{nu}}+q^{\text{si}} r^{\text{si}}
q^{\text{FCGV}(\text{li191})} \left(2 p^{\text{FCGV}(\text{li191})}+r^{\text{FCGV}(\text{li191})}\right)
Uncontract[SPD[q, p]^2, q, p, Pair -> All]
FCCanonicalizeDummyIndices[%, LorentzIndexNames -> {\[Mu], \[Nu]}]p^{\text{\$AL}(\text{\$28})} p^{\text{\$AL}(\text{\$29})} q^{\text{\$AL}(\text{\$28})} q^{\text{\$AL}(\text{\$29})}
p^{\mu } p^{\nu } q^{\mu } q^{\nu }
Canonicalization of Cartesian indices
CVD[p, i] CVD[q, i] + CVD[p, j] CVD[r, j]
FCCanonicalizeDummyIndices[%] // Factor2p^i q^i+p^j r^j
p^{\text{FCGV}(\text{ci391})} \left(q^{\text{FCGV}(\text{ci391})}+r^{\text{FCGV}(\text{ci391})}\right)
CVD[p, i] CVD[q, i] + CVD[p, j] CVD[r, j]
FCCanonicalizeDummyIndices[%, CartesianIndexNames -> {a}] // Factor2p^i q^i+p^j r^j
p^a \left(q^a+r^a\right)
Canonicalization of color indices
SUNT[a, b, a] + SUNT[c, b, c]
FCCanonicalizeDummyIndices[%]T^a.T^b.T^a+T^c.T^b.T^c
2 T^{\text{FCGV}(\text{sun601})}.T^b.T^{\text{FCGV}(\text{sun601})}
SUNT[a, b, a] + SUNT[c, b, c]
FCCanonicalizeDummyIndices[%, SUNIndexNames -> {u}]T^a.T^b.T^a+T^c.T^b.T^c
2 T^u.T^b.T^u
Canonicalization of Dirac indices
DCHN[GA[mu], i, j] DCHN[GA[nu], j, k]
FCCanonicalizeDummyIndices[%]\left(\bar{\gamma }^{\text{mu}}\right){}_{ij} \left(\bar{\gamma }^{\text{nu}}\right){}_{jk}
\left(\bar{\gamma }^{\text{mu}}\right){}_{i\text{FCGV}(\text{di771})} \left(\bar{\gamma }^{\text{nu}}\right){}_{\text{FCGV}(\text{di771})k}
DCHN[GA[mu], i, j] DCHN[GA[nu], j, k]
FCCanonicalizeDummyIndices[%, DiracIndexNames -> {a}]\left(\bar{\gamma }^{\text{mu}}\right){}_{ij} \left(\bar{\gamma }^{\text{nu}}\right){}_{jk}
\left(\bar{\gamma }^{\text{mu}}\right){}_{ia} \left(\bar{\gamma }^{\text{nu}}\right){}_{ak}
Canonicalization of Pauli indices
PCHN[CSI[a], i, j] PCHN[CSI[b], j, k]
FCCanonicalizeDummyIndices[%]\left(\overline{\sigma }^a\right){}_{ij} \left(\overline{\sigma }^b\right){}_{jk}
\left(\overline{\sigma }^a\right){}_{i\text{FCGV}(\text{pi921})} \left(\overline{\sigma }^b\right){}_{\text{FCGV}(\text{pi921})k}
PCHN[CSI[a], i, j] PCHN[CSI[b], j, k]
FCCanonicalizeDummyIndices[%, PauliIndexNames -> {l}]\left(\overline{\sigma }^a\right){}_{ij} \left(\overline{\sigma }^b\right){}_{jk}
\left(\overline{\sigma }^a\right){}_{il} \left(\overline{\sigma }^b\right){}_{lk}
Using the option Head one can specify which index heads should be canonicalized, while the rest will be ignored.
(QuantumField[Superscript[\[Phi], "+"], PauliIndex[k1], PauliIndex[k2],
R, r] . QuantumField[FCPartialD[{CartesianIndex[i], r}],
FCPartialD[{CartesianIndex[i], r}], \[Phi], PauliIndex[k2], PauliIndex[k1], R, r])
FCCanonicalizeDummyIndices[%, CartesianIndexNames -> {j}, Head -> {CartesianIndex}]
\phi ^{+\text{k1}\;\text{k2}Rr}.\left(\partial _{\{i,r\}}\partial _{\{i,r\}}\phi ^{\text{k2}\;\text{k1}Rr}\right)
\phi ^{+\text{k1}\;\text{k2}Rr}.\left(\partial _{\{j,r\}}\partial _{\{j,r\}}\phi ^{\text{k2}\;\text{k1}Rr}\right)