Pauli matrices

The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. They are:

$\sigma _{1}={\begin{pmatrix}0&&1\\1&&0\end{pmatrix}}$

$\sigma _{2}={\begin{pmatrix}0&&-i\\i&&0\end{pmatrix}}$

$\sigma _{3}={\begin{pmatrix}1&&0\\0&&-1\end{pmatrix}}$

The determinants and traces of the Pauli matrices are:

${\begin{matrix}{\hbox{det}}(\sigma _{i})&=&-1&\\{\hbox{Tr}}(\sigma _{i})&=&0&\quad {\hbox{for}}\ i=1,2,3\end{matrix}}$

The Pauli matrices obey the following commutation and anticommutation relations:

${\begin{matrix}[\sigma _{i},\sigma _{j}]&=&2i\,\epsilon _{ijk}\,\sigma _{k}\\\{\sigma _{i},\sigma _{j}\}&=&2\delta _{ij}\end{matrix}}$

where ε_ijk is the Levi-Civita symbol and δ_ij is the Kronecker delta.

The above commutation relation defines the Lie algebra su(2), and the Pauli matrices generate the corresponding Lie group SU(2). In fact, su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, the Pauli matrices are the lowest-dimensional realization of infinitesimal rotations in three-dimensional space.

In quantum mechanics, the Pauli matrices represent the generators of rotation acting on non-relativistic particles with spin 1/2. The state of the particles are represented as two-component spinors, which is the fundamental representation of SU(2). An interesting property of spin 1/2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration. This is due to the fact that SU(2) and SO(3) are not globally isomorphic, even though their generators su(2) and so(3) are. SU(2) is actually a "double cover" of SO(3), meaning that each element of SO(3) actually corresponds to two elements in SU(2).

Together with the identity matrix I (which is sometimes written as &sigma₀), the Pauli matrices form a basis for the set of 2 × 2 complex Hermitian matrices.

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See also:

• Angular momentum
• Lorentz-Poincare group