PauliSigma[x, dim] is the internal representation of a Pauli matrix with a Lorentz or Cartesian index or a contraction of a Pauli matrix and a Lorentz or Cartesian vector.
PauliSigma[x,3] simplifies to PauliSigma[x].
PauliSigma[LorentzIndex[\[Alpha]]]\bar{\sigma }^{\alpha }
PauliSigma[CartesianIndex[i]]\overline{\sigma }^i
A Pauli matrix contracted with a Lorentz or Cartesian vector is displayed as \sigma \cdot p
PauliSigma[Momentum[p]]\bar{\sigma }\cdot \overline{p}
PauliSigma[CartesianMomentum[p]]\overline{\sigma }\cdot \overline{p}
PauliSigma[Momentum[q]] . PauliSigma[Momentum[p - q]]
% // PauliSigmaExpand\left(\bar{\sigma }\cdot \overline{q}\right).\left(\bar{\sigma }\cdot \left(\overline{p}-\overline{q}\right)\right)
\left(\bar{\sigma }\cdot \overline{q}\right).\left(\bar{\sigma }\cdot \overline{p}-\bar{\sigma }\cdot \overline{q}\right)
PauliSigma[CartesianMomentum[q]] . PauliSigma[CartesianMomentum[p - q]]
% // PauliSigmaExpand
\left(\overline{\sigma }\cdot \overline{q}\right).\left(\overline{\sigma }\cdot \left(\overline{p}-\overline{q}\right)\right)
\left(\overline{\sigma }\cdot \overline{q}\right).\left(\overline{\sigma }\cdot \overline{p}-\overline{\sigma }\cdot \overline{q}\right)