FeynCalc manual (development version)

PIDelta

PIDelta[i,j] is the Kronecker-delta in the Pauli space. PIDelta[i,j] is transformed into PauliIndexDelta[PauliIndex[i],PauliIndex[j]] by FeynCalcInternal.

See also

Overview, PauliChain, PCHN, PauliIndex, PauliIndexDelta, PauliChainJoin, PauliChainExpand, PauliChainFactor.

Examples

PIDelta[i, j]

\delta _{ij}

PIDelta[i, i] 
 
PauliChainJoin[%]

\delta _{ii}

4

PIDelta[i, j]^2 
 
PauliChainJoin[%]

\delta _{ij}^2

4

PIDelta[i, j] PIDelta[j, k] 
 
PauliChainJoin[%]

\delta _{ij} \delta _{jk}

\delta _{ik}

ex = PIDelta[i2, i3] PIDelta[i4, i5]  PCHN[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]]

(\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}}

PauliChainJoin[ex]

(\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[ex 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