Pauli spin matrices

The Pauli spin matrices (named after physicist Wolfgang Ernst Pauli) are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-{\mathit {i}}\\{\mathit {i}}&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

Algebraic properties

$\sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=I$

For i = 1, 2, 3:

${\mbox{det}}(\sigma _{i})=-1\,$

${\mbox{Tr}}(\sigma _{i})=0\,$

${\mbox{eigenvalues}}=\pm 1\,$

Commutation relations

$\sigma _{1}\sigma _{2}=i\sigma _{3}\,\!$

$\sigma _{3}\sigma _{1}=i\sigma _{2}\,\!$

$\sigma _{2}\sigma _{3}=i\sigma _{1}\,\!$

$\sigma _{i}\sigma _{j}=-\sigma _{j}\sigma _{i}{\mbox{ for }}i\neq j\,\!$

The Pauli matrices obey the following commutation and anticommutation relations:

${\begin{matrix}[\sigma _{i},\sigma _{j}]&=&2i\,\varepsilon _{ijk}\,\sigma _{k}\\[1ex]\{\sigma _{i},\sigma _{j}\}&=&2\delta _{ij}\cdot I\end{matrix}}$

where $\varepsilon _{ijk}$ is the Levi-Civita symbol, $\delta _{ij}$ is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

$\sigma _{i}\sigma _{j}=\delta _{ij}\cdot I+i\varepsilon _{ijk}\sigma _{k}.\,$

Rotations

The commutation relations for the Pauli spin matrices can be rearranged as:

${\frac {1}{2}}\sigma _{\alpha }{\frac {1}{2}}\sigma _{\beta }-{\frac {1}{2}}\sigma _{\beta }{\frac {1}{2}}\sigma _{\alpha }=i\ \varepsilon _{\alpha \beta \gamma }{\frac {1}{2}}\sigma _{\gamma }\ ,$

with αβγ any combination of xyz.

These commutation relations are the same as those satisfied by the generators of infinitesimal rotations in three-dimensional space. If the Pauli matrices are considered to act on a two-dimensional "spin" space, finite rotations in this space can be connected to rotations in three-dimensional space. These spin-space rotations are generated by the Pauli spin matrices, with a finite rotation in three-space of angle θ about the axis aligned with unit vector û becoming in spin-space the rotation:

$R_{\hat {\mathbf {u} }}(\theta )=e^{i\theta {\hat {\mathbf {u} }}\mathbf {\cdot {\boldsymbol {\sigma }}} /2}.$

If û is in the z direction, for example:

$R_{z}(\theta )={\begin{pmatrix}e^{i\theta /2}&0\\0&e^{-i\theta /2}\end{pmatrix}}\ ,$

as can be verified using the Taylor series expansion:

$e^{i\theta \sigma _{z}/2}=1+i\theta {\sigma _{z}}/2+{\frac {1}{2}}\left(i\theta {\sigma _{z}}/2\right)^{2}...$

Given a set of Euler angles α, β, γ describing orientation of an object in ordinary three-dimensional space, the general spin-space rotation corresponding to these angles is described as:

${\begin{pmatrix}e^{i\gamma /2}&0\\0&e^{-i\gamma /2}\end{pmatrix}}{\begin{pmatrix}\cos(\beta /2)&\sin(\beta /2)\\-\sin(\beta /2)&\cos(\beta /2)\end{pmatrix}}{\begin{pmatrix}e^{i\alpha /2}&0\\0&e^{-i\alpha /2}\end{pmatrix}}$

$={\begin{pmatrix}e^{i(\alpha +\gamma )/2}\cos(\beta /2)&e^{-i(\alpha -\gamma )/2}\sin(\beta /2)\\-e^{i(\alpha -\gamma )/2}\sin(\beta /2)&e^{-i(\alpha +\gamma )/2}\cos(\beta /2)\end{pmatrix}}\ .$

The two-dimensional matrices describing all rotations in "spin space" form a representation of the special unitary group of transformations of two complex variables, usually denoted as SU(2), formally described as the the group of transformations of two-dimensional complex vectors leaving their inner product fixed, and hence also the norm of a complex vector.

Extensions

These commutation relations can be viewed as applying in general, and the question opened as to what general mathematical objects might satisfy these rules. A set of symbols with a defined sum and a product taken as a commutator of the symbols is called a Lie algebra.

One can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called representation of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way. The Pauli matrices are the basis for an irreducible two-dimensional representation of this Lie algebra.