Pair[x, y] is the head of a special pairing used in the internal representation: x and y may have heads LorentzIndex or Momentum.
If both x and y have head LorentzIndex, the metric tensor (e.g. g^{\mu \nu}) is understood.
If x and y have head Momentum, a scalar product (e.g. p \cdot q) is meant.
If one of x and y has head LorentzIndex and the other Momentum, a Lorentz vector (e.g. p^{\mu }) is implied.
Overview, FV, FVD, MT, MTD, ScalarProduct, SP, SPD.
This represents a 4-dimensional metric tensor
Pair[LorentzIndex[\[Alpha]], LorentzIndex[\[Beta]]]\bar{g}^{\alpha \beta }
This is a D-dimensional metric tensor
Pair[LorentzIndex[\[Alpha], D], LorentzIndex[\[Beta], D]]g^{\alpha \beta }
If the Lorentz indices live in different dimensions, this gets resolved according to the t’Hooft-Veltman-Breitenlohner-Maison prescription
Pair[LorentzIndex[\[Alpha], n - 4], LorentzIndex[\[Beta]]]0
A 4-dimensional Lorentz vector
Pair[LorentzIndex[\[Alpha]], Momentum[p]]\overline{p}^{\alpha }
A D-dimensional Lorentz vector
Pair[LorentzIndex[\[Alpha], D], Momentum[p, D]]p^{\alpha }
4-dimensional scalar products of Lorentz vectors
Pair[Momentum[q], Momentum[p]]\overline{p}\cdot \overline{q}
Pair[Momentum[p], Momentum[p]]\overline{p}^2
Pair[Momentum[p - q], Momentum[p]]\overline{p}\cdot (\overline{p}-\overline{q})
Pair[Momentum[p], Momentum[p]]^2\overline{p}^4
Pair[Momentum[p], Momentum[p]]^3\overline{p}^6
ExpandScalarProduct[Pair[Momentum[p - q], Momentum[p]]]\overline{p}^2-\overline{p}\cdot \overline{q}
Pair[Momentum[-q], Momentum[p]] + Pair[Momentum[q], Momentum[p]]0