Editing Pauli matrices (section)

=== Pauli 4-vector ===

The Pauli 4-vector, used in spinor theory, is written <math>\ \sigma^\mu\ </math> with components
:<math>\sigma^\mu = (I, \vec\sigma).</math>
This defines a map from <math>\mathbb{R}^{1,3}</math> to the vector space of Hermitian matrices,
:<math>x_\mu \mapsto x_\mu\sigma^\mu\ ,</math>
which also encodes the [[Minkowski metric]] (with ''mostly minus'' convention) in its determinant:
:<math>\det (x_\mu\sigma^\mu) = \eta(x,x).</math>

This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector
:<math>\bar\sigma^\mu = (I, -\vec\sigma).</math>
and allow raising and lowering using the Minkowski metric tensor. The relation can then be written
<math display="block">x_\nu = \tfrac{1}{2} \operatorname{tr} \Bigl( \bar\sigma_\nu\bigl( x_\mu \sigma^\mu \bigr) \Bigr) .</math>

Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on <math>\ \mathbb{R}^{1,3}\ ;</math> in this case the matrix group is <math>\ \mathrm{SL}(2,\mathbb{C})\ ,</math> and this shows <math>\ \mathrm{SL}(2,\mathbb{C}) \cong \mathrm{Spin}(1,3).</math> Similarly to above, this can be explicitly realized for <math>\ S \in \mathrm{SL}(2,\mathbb{C})\ </math> with components
:<math>\Lambda(S)^\mu{}_\nu = \tfrac{1}{2}\operatorname{tr} \left( \bar\sigma_\nu S \sigma^\mu S^{\dagger}\right).</math>

In fact, the determinant property follows abstractly from trace properties of the <math>\ \sigma^\mu.</math> For <math>\ 2\times 2\ </math> matrices, the following identity holds:
:<math>\det(A + B) = \det(A) + \det(B) + \operatorname{tr}(A)\operatorname{tr}(B) - \operatorname{tr}(AB).</math>
That is, the 'cross-terms' can be written as traces. When <math>\ A,B\ </math> are chosen to be different <math>\ \sigma^\mu\ ,</math> the cross-terms vanish. It then follows, now showing summation explicitly,
<math display="inline">\det\left(\sum_\mu x_\mu \sigma^\mu\right) = \sum_\mu \det\left(x_\mu\sigma^\mu\right).</math> Since the matrices are <math>\ 2 \times 2\ ,</math> this is equal to <math display="inline">\sum_\mu x_\mu^2 \det(\sigma^\mu) = \eta(x,x).</math>