FeynCalc manual (development version)

Pair

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. gμνg^{\mu \nu}) is understood.

If x and y have head Momentum, a scalar product (e.g. pqp \cdot q) is meant.

If one of x and y has head LorentzIndex and the other Momentum, a Lorentz vector (e.g. pμp^{\mu }) is implied.

See also

Overview, FV, FVD, MT, MTD, ScalarProduct, SP, SPD.

Examples

This represents a 44-dimensional metric tensor

Pair[LorentzIndex[\[Alpha]], LorentzIndex[\[Beta]]]

gˉαβ\bar{g}^{\alpha \beta }

This is a D-dimensional metric tensor

Pair[LorentzIndex[\[Alpha], D], LorentzIndex[\[Beta], D]]

gαβg^{\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]]]

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A 44-dimensional Lorentz vector

Pair[LorentzIndex[\[Alpha]], Momentum[p]]

pα\overline{p}^{\alpha }

A DD-dimensional Lorentz vector

Pair[LorentzIndex[\[Alpha], D], Momentum[p, D]]

pαp^{\alpha }

44-dimensional scalar products of Lorentz vectors

Pair[Momentum[q], Momentum[p]]

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

Pair[Momentum[p], Momentum[p]]

p2\overline{p}^2

Pair[Momentum[p - q], Momentum[p]]

p(pq)\overline{p}\cdot (\overline{p}-\overline{q})

Pair[Momentum[p], Momentum[p]]^2

p4\overline{p}^4

Pair[Momentum[p], Momentum[p]]^3

p6\overline{p}^6

ExpandScalarProduct[Pair[Momentum[p - q], Momentum[p]]]

p2pq\overline{p}^2-\overline{p}\cdot \overline{q}

Pair[Momentum[-q], Momentum[p]] + Pair[Momentum[q], Momentum[p]]

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