FeynCalc manual (development version)

PairContract

PairContract is like Pair, but with (local) contraction properties. The function fully supports the BMHV algebra and will expand momenta inside scalar products when it leads to simpler expressions.

PairContract is an auxiliary function used in higher level FeynCalc functions that require fast contractions between multiple expressions, where Contract would be too slow.

See also

Overview, Contract, PairContract2, PairContract3.

Examples

Pair[LorentzIndex[\[Mu]], Momentum[p]] Pair[LorentzIndex[\[Mu]], Momentum[q]] 
 
% /. Pair -> PairContract

pμqμ\overline{p}^{\mu } \overline{q}^{\mu }

pq\overline{p}\cdot \overline{q}

Pair[LorentzIndex[\[Mu]], Momentum[p]] Pair[LorentzIndex[\[Nu]], Momentum[q]] Pair[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]] 
 
% /. Pair -> PairContract

pμqνgˉμν\overline{p}^{\mu } \overline{q}^{\nu } \bar{g}^{\mu \nu }

pq\overline{p}\cdot \overline{q}

Pair[LorentzIndex[\[Mu]], Momentum[p + q]] Pair[LorentzIndex[\[Mu]], Momentum[r + s]] 
 
% /. Pair -> PairContract

(p+q)μ(r+s)μ\left(\overline{p}+\overline{q}\right)^{\mu } \left(\overline{r}+\overline{s}\right)^{\mu }

(p+q)(r+s)(\overline{p}+\overline{q})\cdot (\overline{r}+\overline{s})

FCClearScalarProducts[];
SP[p1, p2] = s2;
SP[p1, p3] = s3;
FCI[SP[p1, p2 + p3]] /. Pair -> PairContract

s2+s3\text{s2}+\text{s3}