Editing Pauli matrices (section)

=== Relation with the permutation operator ===
Let {{math|''P{{sub|jk}}''}} be the [[transposition (mathematics)|transposition]] (also known as a permutation) between two spins {{math|''σ{{sub|j}}''}} and {{math|''σ{{sub|k}}''}} living in the [[tensor product]] space {{nowrap|{{tmath|\Complex^2 \otimes \Complex^2}},}}
:<math>P_{jk} \left| \sigma_j \sigma_k \right\rangle =  \left| \sigma_k \sigma_j \right\rangle .</math>

This operator can also be written more explicitly as [[Exchange interaction#Inclusion of spin|Dirac's spin exchange operator]],
:<math>P_{jk} = \frac{1}{2}\,\left(\vec{\sigma}_j \cdot \vec{\sigma}_k + 1\right) ~ .</math>

Its eigenvalues are therefore{{efn|
Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is <math>\left(\begin{smallmatrix}
  1 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 \\  
  0 & 1 & 0 & 0 \\  
  0 & 0 & 0 & 1
\end{smallmatrix}\right) ~ .</math>
}} 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.