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 p^i is understood.
This represents a three-dimensional Kronecker delta
CartesianPair[CartesianIndex[i], CartesianIndex[j]]\bar{\delta }^{ij}
This is a D-1-dimensional Kronecker delta
CartesianPair[CartesianIndex[i, D - 1], CartesianIndex[j, D - 1]]\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]]\bar{\delta }^{ij}
CartesianPair[CartesianIndex[i, D - 1], CartesianIndex[j, D - 4]]\hat{\delta }^{ij}
CartesianPair[CartesianIndex[i], CartesianIndex[j, D - 4]]0
A 3-dimensional Cartesian vector
CartesianPair[CartesianIndex[i], CartesianMomentum[p]]\overline{p}^i
A D-1-dimensional Cartesian vector
CartesianPair[CartesianIndex[i, D - 1], CartesianMomentum[p, D - 1]]p^i
3-dimensional scalar products of Cartesian vectors
CartesianPair[CartesianMomentum[q], CartesianMomentum[p]]\overline{p}\cdot \overline{q}
CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]\overline{p}^2
CartesianPair[CartesianMomentum[p - q], CartesianMomentum[p]]\overline{p}\cdot (\overline{p}-\overline{q})
CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]^2\overline{p}^4
CartesianPair[CartesianMomentum[p], CartesianMomentum[p]]^3\overline{p}^6
ExpandScalarProduct[CartesianPair[CartesianMomentum[p - q], CartesianMomentum[p]]]\overline{p}^2-\overline{p}\cdot \overline{q}
CartesianPair[CartesianMomentum[-q], CartesianMomentum[p]] +
CartesianPair[CartesianMomentum[q], CartesianMomentum[p]]0