PauliIndexDelta[PauliIndex[i], PauliIndex[j]] is the Kronecker-delta in the Pauli space with two explicit Pauli indices i and j.
Overview, PauliChain, PCHN, PauliIndex, DIDelta, PauliChainJoin, PauliChainCombine, PauliChainExpand, PauliChainFactor.
PauliIndexDelta[PauliIndex[i], PauliIndex[j]]\delta _{ij}
PauliIndexDelta[PauliIndex[i], PauliIndex[j]]^2
PauliChainJoin[%]
PauliChainJoin[%%, TraceOfOne -> D]\delta _{ij}^2
4
D
PauliIndexDelta[PauliIndex[i], PauliIndex[j]] PauliIndexDelta[PauliIndex[j], PauliIndex[k]]
PauliChainJoin[%]\delta _{ij} \delta _{jk}
\delta _{ik}
PauliChain[PauliEta[-I], PauliIndex[i0]] PIDelta[i0, i1] // FCI // PauliChainJoin\left(\eta ^{\dagger }\right){}_{\text{i1}}
PauliIndexDelta[PauliIndex[i2], PauliIndex[i3]] PauliIndexDelta[PauliIndex[i4], PauliIndex[i5]] PauliChain[PauliIndex[i7], PauliXi[I]] PauliChain[PauliEta[-I], PauliIndex[i0]] PauliChain[PauliSigma[CartesianIndex[a]], PauliIndex[i1], PauliIndex[i2]] PauliChain[PauliSigma[CartesianIndex[b]], PauliIndex[i5], PauliIndex[i6]] PauliChain[m + PauliSigma[CartesianMomentum[p]], PauliIndex[i3], PauliIndex[i4]]
PauliChainJoin[%](\xi )_{\text{i7}} \left(\eta ^{\dagger }\right){}_{\text{i0}} \delta _{\text{i2}\;\text{i3}} \delta _{\text{i4}\;\text{i5}} \left(\overline{\sigma }^a\right){}_{\text{i1}\;\text{i2}} \left(\overline{\sigma }^b\right){}_{\text{i5}\;\text{i6}} \left(\overline{\sigma }\cdot \overline{p}+m\right)_{\text{i3}\;\text{i4}}
(\xi )_{\text{i7}} \left(\eta ^{\dagger }\right){}_{\text{i0}} \left(\overline{\sigma }^a.\left(\overline{\sigma }\cdot \overline{p}+m\right).\overline{\sigma }^b\right){}_{\text{i1}\;\text{i6}}
PauliChainJoin[% PIDelta[i0, i1]](\xi )_{\text{i7}} \left(\eta ^{\dagger }.\overline{\sigma }^a.\left(\overline{\sigma }\cdot \overline{p}+m\right).\overline{\sigma }^b\right){}_{\text{i6}}
PauliChainJoin[% PIDelta[i7, i6]]\eta ^{\dagger }.\overline{\sigma }^a.\left(\overline{\sigma }\cdot \overline{p}+m\right).\overline{\sigma }^b.\xi