Pauli matrices

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

<math>

\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} </math>

<math>

\sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} </math>

<math>

\sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} </math>

1 Properties
- 1.1 Identities
- 1.2 Additional properties
2 Physics
3 See also
4 References

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Properties

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Identities

<math>

\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I</math> where I is the identity matrix.

<math>\sigma_1\sigma_2 = i\sigma_3\,\!</math>

<math>\sigma_3\sigma_1 = i\sigma_2\,\!</math>

<math>\sigma_2\sigma_3 = i\sigma_1\,\!</math>

<math>\sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i\ne j\,\!</math>

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Additional properties

The determinants and traces of the Pauli matrices are:

<math>\begin{matrix}

\det (\sigma_i) &=& -1 & \\[1ex] \operatorname{Tr} (\sigma_i) &=& 0 & \quad \hbox{for}\ i = 1, 2, 3 \end{matrix}</math>

and as a result, the eigenvalues of each matrix are ±1.

The Pauli matrices obey the following commutation and anticommutation relations:

<math>\begin{matrix}

[\sigma_i, \sigma_j] &=& 2 i\,\varepsilon_{i j k}\,\sigma_k \\[1ex] \{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I \end{matrix}</math>

where <math>\varepsilon_{ijk}</math> is the Levi-Civita symbol, <math>\delta_{ij}</math> is the Kronecker delta, and I is the identity matrix. The above relations can be verified using

<math>\sigma_i \sigma_j = i \varepsilon_{ijk} \sigma_k + \delta_{ij} \cdot I</math>.

The above commutation relations are similar to those of the Lie algebra su(2), and indeed su(2) may be identified with the Lie algebra of all real linear combinations of i times the Pauli matrices <math>i \sigma_j</math>, i.e. with the anti-Hermitian 2×2 matrices with trace 0. In this sense, the Pauli matrices generate su(2). As a result, <math>i \sigma_j</math> can be seen as infinitesimal generators of the corresponding Lie group SU(2).

The Lie algebra 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, <math>i \sigma_j</math> are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space.

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Physics

In quantum mechanics, <math>i \sigma_j</math> represent the generators of rotation acting on non-relativistic particles with spin ½. The state of the particles are represented as two-component spinors, which is the fundamental representation of SU(2). An interesting property of spin ½ particles is that they must be rotated by an angle of 4<math>\pi</math> 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 infinitesimal generators su(2) and so(3) are isomorphic. 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). Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli Matrices.

Together with the identity matrix I (which is sometimes written as σ₀), the Pauli matrices form a basis for the real vector space of 2 × 2 complex Hermitian matrices. This basis is equivalent to the quaternions, and when used as the basis for the spin-½ rotation operator is the same as the corresponding quaternion rotation representation.

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