Editing Pauli matrices (section)

=== Relativistic quantum mechanics ===
In [[relativistic quantum mechanics]], the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as
:<math>\mathsf{\Sigma}_k = \begin{pmatrix} \mathsf{\sigma}_k & 0 \\ 0 & \mathsf{\sigma}_k \end{pmatrix} .</math>

It follows from this definition that the <math>\ \mathsf{ \Sigma }_k \ </math> matrices have the same algebraic properties as the {{mvar| σ{{sub|k}} }} matrices.

However, [[relativistic angular momentum]] is not a three-vector, but a second order [[four-tensor]]. Hence <math>\ \mathsf{\Sigma}_k\ </math> needs to be replaced by {{mvar|Σ{{sub|μν}} }}, the generator of [[representation theory of the Lorentz group#The (1/2, 0) ⊕ (0, 1/2) spin representation|Lorentz transformations on spinors]]. By the antisymmetry of angular momentum, the {{math|Σ''{{sub|μν}}''}} are also antisymmetric. Hence there are only six independent matrices.

The first three are the <math>\ \Sigma_{k\ell} \equiv \epsilon_{jk\ell}\mathsf{\Sigma}_j .</math> The remaining three, <math>\ -i\ \Sigma_{0k} \equiv \mathsf{\alpha}_k\ ,</math> where the [[Dirac equation|Dirac {{math|''α{{sub|k}}''}} matrices]] are defined as

:<math>
\mathsf{\alpha}_k =
\begin{pmatrix}
 0   &   \mathsf{\sigma}_k \\
 \mathsf{\sigma}_k   &   0
\end{pmatrix} .
</math>

The relativistic spin matrices {{math|Σ''{{sub|μν}}''}} are written in compact form in terms of commutator of [[gamma matrices]] as
:<math>\Sigma_{\mu\nu} = \frac{i}{2} \bigl[ \gamma_\mu, \gamma_\nu \bigr] .</math>