Editing Pauli matrices (section)

=== Determinant ===

The norm is given by the determinant (up to a minus sign)
<math display="block">
\det \bigl( \vec{a} \cdot \vec{\sigma} \bigr) = -\vec{a} \cdot \vec{a} = -|\vec{a}|^2.
</math>
Then, considering the conjugation action of an <math>\text{SU}(2)</math> matrix <math>U</math> on this space of matrices,
: <math>U * \vec a \cdot \vec \sigma := U \, \vec a \cdot \vec \sigma \, U^{-1},</math>
we find <math>\det(U * \vec a \cdot \vec\sigma) = \det(\vec a \cdot \vec \sigma),</math> and that <math>U * \vec a \cdot \vec \sigma</math> is Hermitian and traceless. It then makes sense to define <math>U * \vec a \cdot \vec\sigma = \vec a' \cdot \vec\sigma,</math> where <math>\vec a'</math> has the same norm as <math>\vec a,</math> and therefore interpret <math>U</math> as a rotation of three-dimensional space. In fact, it turns out that the ''special'' restriction on <math>U</math> implies that the rotation is orientation preserving. This allows the definition of a map <math>R: \mathrm{SU}(2) \to \mathrm{SO}(3)</math> given by
: <math>U * \vec a \cdot \vec \sigma = \vec a' \cdot \vec \sigma =: (R(U)\ \vec a) \cdot \vec \sigma,</math>
where <math>R(U) \in \mathrm{SO}(3).</math> This map is the concrete realization of the double cover of <math>\mathrm{SO}(3)</math> by <math>\mathrm{SU}(2),</math> and therefore shows that <math>\text{SU}(2) \cong \mathrm{Spin}(3).</math> The components of <math>R(U)</math> can be recovered using the tracing process above:
: <math>R(U)_{ij} = \frac{1}{2} \operatorname{tr} \left( \sigma_i U \sigma_j U^{-1} \right).</math>