FeynCalc manual (development version)

CartesianPair

CartesianPair[a, b] is a special pairing used in the internal representation. a and b may have heads CartesianIndex or CartesianMomentum. If both a and b have head CartesianIndex, the Kronecker delta is understood. If a and b have head CartesianMomentum, a Cartesian scalar product is meant. If one of a and b has head CartesianIndex and the other CartesianMomentum, a Cartesian vector pip^i is understood.

See also

Overview, Pair, TemporalPair.

Examples

This represents a three-dimensional Kronecker delta

CartesianPair[CartesianIndex[i], CartesianIndex[j]]

δˉij\bar{\delta }^{ij}

This is a D1D-1-dimensional Kronecker delta

CartesianPair[CartesianIndex[i, D - 1], CartesianIndex[j, D - 1]]

δij\delta ^{ij}

If the Cartesian indices live in different dimensions, this gets resolved according to the t’Hoft-Veltman-Breitenlohner-Maison prescription

CartesianPair[CartesianIndex[i, D - 1], CartesianIndex[j]]

δˉij\bar{\delta }^{ij}

CartesianPair[CartesianIndex[i, D - 1], CartesianIndex[j, D - 4]]

δ^ij\hat{\delta }^{ij}

CartesianPair[CartesianIndex[i], CartesianIndex[j, D - 4]]

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A 33-dimensional Cartesian vector

CartesianPair[CartesianIndex[i], CartesianMomentum[p]]

pi\overline{p}^i

A D1D-1-dimensional Cartesian vector

CartesianPair[CartesianIndex[i, D - 1], CartesianMomentum[p, D - 1]]

pip^i

33-dimensional scalar products of Cartesian vectors

CartesianPair[CartesianMomentum[q], CartesianMomentum[p]]

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

CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]

p2\overline{p}^2

CartesianPair[CartesianMomentum[p - q], CartesianMomentum[p]]

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

CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]^2

p4\overline{p}^4

CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]^3

p6\overline{p}^6

ExpandScalarProduct[CartesianPair[CartesianMomentum[p - q], CartesianMomentum[p]]]

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

CartesianPair[CartesianMomentum[-q], CartesianMomentum[p]] + 
  CartesianPair[CartesianMomentum[q], CartesianMomentum[p]]

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