Pauli matrices

File:Wolfgang Pauli.jpg

Wolfgang Pauli (1900–1958), c. 1924. Pauli received the Nobel Prize in Physics in 1945, nominated by Albert Einstein, for the Pauli exclusion principle.

In mathematical physics and mathematics, the Pauli matrices are a set of three $2\times 2$ complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter $\sigma$ (sigma), and occasionally by $\tau$ (tau) when used in connection with isospin symmetries.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\\\end{aligned}}}

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is Hermitian, and together with the identity matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{0}} ), the Pauli matrices form a basis of the vector space of $2\times 2$ Hermitian matrices over the real numbers, under addition. This means that any $2\times 2$ Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

The Pauli matrices satisfy the useful product relation:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}}

where $\delta _{ij}$ is the Kronecker delta, which equals $+1$ if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i=j} otherwise $0$, and the Levi-Civita symbol Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varepsilon _{ijk}} is used.

Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{k}} represents the observable corresponding to spin along the $k$th coordinate axis in three-dimensional Euclidean space $\mathbb {R} ^{3}$.

The Pauli matrices (after multiplication by $i$ to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: The matrices Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{1}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{2}} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{3}} form a basis for the real Lie algebra Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special unitary group SU(2). The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ \mathbb {R} ^{3}} and the (unital) associative algebra generated by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{1}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{2}} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{3}} functions identically (is isomorphic) to that of quaternions ($\mathbb {H}$).

Algebraic properties

Cayley table; the entry shows the value of the row times the column.

×	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{x}}	$\sigma _{y}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{z}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{x}}	$I$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\ \sigma _{z}}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -i\ \sigma _{y}}
$\sigma _{y}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -i\ \sigma _{z}}	$I$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\ \sigma _{x}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{z}}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\sigma _{y}}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -i\ \sigma _{x}}	$I$

All three of the Pauli matrices can be compacted into a single expression:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{j}={\begin{pmatrix}\delta _{j3}&\delta _{j1}-i\ \delta _{j2}\\\delta _{j1}+i\ \delta _{j2}&-\delta _{j3}\end{pmatrix}}~.}

This expression is useful for "selecting" any one of the matrices numerically by substituting values of $j\in \{1,2,3\}$ in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are involutory:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\ \sigma _{1}\ \sigma _{2}\ \sigma _{3}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\mathbb {I} ,}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {I} } is the identity matrix.

The determinants and traces of the Pauli matrices are

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\det \sigma _{j}&=-1\ ,\\\operatorname {tr} \sigma _{j}&=0\ ,\end{aligned}}}

from which we can deduce that each matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{j}} has eigenvalues $\pm 1$.

With the inclusion of the identity matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {I} } (sometimes denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sigma _{0}} ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal{H}_2\ } of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2} Hermitian matrices over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} and the Hilbert space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M}_{2,2}(\mathbb{C})} of all complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2} matrices over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} .

Commutation and anti-commutation relations

Commutation relations

The Pauli matrices obey the following commutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\sigma_j, \sigma_k] = 2\ i\ \varepsilon_{j k l}\ \sigma_l ~. }

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}^3, \times)\ \cong\ \mathfrak{su}(2)\ \cong\ \mathfrak{so}(3) ~.}

Anticommutation relations

They also satisfy the anticommutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\sigma_j, \sigma_k\} = 2\ \delta_{j k}\ I\ ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\sigma_j, \sigma_k\}} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma_j\ \sigma_k + \sigma_k\ \sigma_j\ ,} and δ_jk is the Kronecker delta. I denotes the 2 × 2 identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbb{R}^3\ ,} denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{Cl}_3(\mathbb{R}) ~.}

The usual construction of generators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma_{jk} = \tfrac{1}{4} [\sigma_j, \sigma_k]\ } of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathfrak{so}(3)\ } using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

A few explicit commutators and anti-commutators are given below as examples:

Commutators

Anticommutators

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \bigl[\ \sigma_1, \sigma_1\ \bigr] &= ~~~ 0 \\ \bigl[\ \sigma_1, \sigma_2\ \bigr] &= 2\ i\ \sigma_3 \\ \bigl[\ \sigma_2, \sigma_3\ \bigr] &= 2\ i\ \sigma_1 \\ \bigl[\ \sigma_3, \sigma_1\ \bigr] &= 2\ i\ \sigma_2 \end{align}} Template:Quad

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \bigl\{\ \sigma_1, \sigma_1\ \bigr\} &= 2\ I \\ \bigl\{\ \sigma_1, \sigma_2\ \bigr\} &= ~ 0 \\ \bigl\{\ \sigma_2, \sigma_3\ \bigr\} &= ~ 0 \\ \bigl\{\ \sigma_3, \sigma_1\ \bigr\} &= ~ 0 \end{align}}

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm 1} . The corresponding normalized eigenvectors are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \psi_{x+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}, & \psi_{x-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \\ \psi_{y+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ i \end{bmatrix}, & \psi_{y-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -i \end{bmatrix}, \\ \psi_{z+} &= \begin{bmatrix} 1 \\ 0 \end{bmatrix}, & \psi_{z-} &= \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \end{align}}

Pauli vectors

The Pauli vector is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\sigma} = \sigma_1 \boldsymbol{\hat{x}}_1 + \sigma_2 \boldsymbol{\hat{x}}_2 + \sigma_3 \boldsymbol{\hat{x}}_3, } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{x}}_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{x}}_2} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{x}}_3} are an equivalent notation for the more familiar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{x}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{y}}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\hat{z}}} .

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \boldsymbol{a} \cdot \boldsymbol{\sigma} &= \sum_{k,l} a_k\, \sigma_\ell\, \hat{x}_k \cdot \hat{x}_\ell \\ &= \sum_k a_k\, \sigma_k \\ &= \begin{pmatrix} a_3 & a_1 - i a_2 \\ a_1 + i a_2 & -a_3 \end{pmatrix} ~. \end{align} }

More formally, this defines a map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} to the vector space of traceless Hermitian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2} matrices. This map encodes structures of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory.

Another way to view the Pauli vector is as a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2 \times 2\ } Hermitian traceless matrix-valued dual vector, that is, an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{Mat}_{2\times 2}(\mathbb{C}) \otimes (\mathbb{R}^3)^*\ } that maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{a} \mapsto \boldsymbol{a} \cdot \boldsymbol{\sigma}}

Completeness relation

Each component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{a}} can be recovered from the matrix (see completeness relation below) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} \operatorname{tr} \Bigl[ \bigl(\ \boldsymbol{a} \cdot \boldsymbol{\sigma}\ \bigr)\ \boldsymbol{\sigma}\ \Bigr] = \boldsymbol{a} } This constitutes an inverse to the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{a} \mapsto \boldsymbol{a} \cdot \boldsymbol{\sigma}} , making it manifest that the map is a bijection.

Determinant

The norm is given by the determinant (up to a minus sign) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\!\bigl(\ \vec{a} \cdot \vec{\sigma}\ \bigr)\ =\ -\vec{a} \cdot \vec{a}\ =\ -\left|\ \vec{a}\ \right|^2 ~. } Then, considering the conjugation action of an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SU}(2)\ } matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} on this space of matrices,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U * \vec a \cdot \vec \sigma\ :=\ U\ \vec a \cdot \vec \sigma\ U^{-1}\ ,}

we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \det(U * \vec a \cdot \vec\sigma)\ =\ \det(\vec a \cdot \vec \sigma)\ ,} and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U * \vec a \cdot \vec \sigma\ } is Hermitian and traceless. It then makes sense to define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U * \vec a \cdot \vec\sigma\ =\ \vec a' \cdot \vec\sigma\ ,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vec a'\ } has the same norm as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec a,} and therefore interpret Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} as a rotation of three-dimensional space. In fact, it turns out that the special restriction on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} implies that the rotation is orientation preserving. This allows the definition of a map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ R: \mathrm{SU}(2) \to \mathrm{SO}(3)\ } given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U * \vec a \cdot \vec \sigma\ =\ \vec a' \cdot \vec \sigma\ =:\ (R(U)\ \vec a) \cdot \vec \sigma\ ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ R(U)\ \in\ \mathrm{SO}(3) ~.} This map is the concrete realization of the double cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SO}(3)\ } by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SU}(2)\ ,} and therefore shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SU}(2)\ \cong\ \mathrm{Spin}(3) ~.} The components of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(U)} can be recovered using the tracing process above:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ R(U)_{ij} = \frac{1}{2}\ \operatorname{tr}\!\left(\ \sigma_i U \sigma_j U^{-1}\ \right) ~.}

Cross-product

The cross-product is given by the matrix commutator (up to a factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2\ i\ } ) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\ \vec a \cdot \vec \sigma,\ \vec b \cdot \vec \sigma\ \right] = 2\ i\ \left( \vec a \times \vec b \right) \cdot \vec \sigma ~. } In fact, the existence of a norm follows from the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbb{R}^3\ } is a Lie algebra (see Killing form).

This cross-product can be used to prove the orientation-preserving property of the map above.

Eigenvalues and eigenvectors

The eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vec a \cdot \vec \sigma\ } are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \pm |\vec{a}| ~.} This follows immediately from tracelessness and explicitly computing the determinant.

More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (\vec a \cdot \vec \sigma)^2 - |\vec a|^2 = 0\ ,} since this can be factorised into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (\vec a \cdot \vec \sigma - |\vec a|)(\vec a \cdot \vec \sigma + |\vec a|)= 0 ~.} A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonalizable) means this implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vec a \cdot \vec \sigma\ } is diagonalizable with possible eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \pm |\vec a| ~.} The tracelessness of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vec a \cdot \vec \sigma\ } means it has exactly one of each eigenvalue.

Its normalized eigenvectors are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_+ = \frac{1}{\sqrt{2|\vec{a}|(a_3+|\vec{a}|)}} \begin{bmatrix} a_3 + \left|\vec{a}\right| \\ a_1 + ia_2 \end{bmatrix}\ ; \qquad \psi_- = \frac{1}{\sqrt{2|\vec{a}|(a_3+|\vec{a}|)}} \begin{bmatrix} ia_2 - a_1 \\ a_3 + |\vec{a}| \end{bmatrix} ~. } These expressions become singular for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_3 \to -\left|\ \vec{a}\ \right| ~.} They can be rescued by letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a} = \left|\ \vec{a}\ \right| \left( \epsilon,\ 0,\ -\left( 1 - \tfrac{\epsilon^2}{2} \right) \right)\ } and taking the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \epsilon \to 0\ ,} which yields the correct eigenvectors (0,1) and (1,0) of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma_z ~.}

Alternatively, one may use spherical coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vec{a} = a\ \bigl(\ \sin \vartheta\ \cos \varphi,\ \sin \vartheta\ \sin \varphi,\ \cos\vartheta\ \bigr)\ } to obtain the eigenvectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \psi_{+} = \left(\ \cos \tfrac{\vartheta}{2}, \; \sin \tfrac{\vartheta}{2}\ e^{+i\varphi}\ \right)\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \psi_{-} = \left(\ -\sin \tfrac{\vartheta}{2}\ e^{-i\varphi}, \; \cos \tfrac{\vartheta}{2}\ \right) ~.}

Pauli 4-vector

The Pauli 4-vector, used in spinor theory, is written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma^\mu\ } with components

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma^\mu = \bigl(\ I,\ \vec\sigma\ \bigr) ~.}

This defines a map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbb{R}^{1,3}\ } to the vector space of Hermitian matrices,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x_\mu \mapsto x_\mu\sigma^\mu\ ,}

which also encodes the Minkowski metric (with mostly minus convention) in its determinant:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \det\bigl(\ x_\mu\sigma^\mu\ \bigr) = \eta(x,x) ~.}

This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \bar\sigma^\mu = \bigl(\ I, -\vec\sigma\ \bigr) ~.}

and allow raising and lowering using the Minkowski metric tensor. The relation can then be written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x_\nu = \tfrac{1}{2} \operatorname{tr}\!\Bigl(\ \bar\sigma_\nu\bigl( x_\mu \sigma^\mu \bigr)\ \Bigr) ~.}

Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbb{R}^{1,3}\ ;} in this case the matrix group is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SL}( 2, \mathbb{C} )\ ,} and this shows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{SL}(2,\mathbb{C})\ \cong\ \mathrm{Spin}(1,3) ~.} Similarly to above, this can be explicitly realized for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ S \in \mathrm{SL}(2,\mathbb{C})\ } with components

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \Lambda(S)^\mu{}_\nu = \tfrac{1}{2}\operatorname{tr}\!\left(\ \bar\sigma_\nu\ S\ \sigma^\mu\ S^{\dagger}\ \right) ~.}

In fact, the determinant property follows abstractly from trace properties of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma^\mu ~.} For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2\times 2\ } matrices, the following identity holds:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \det(\ A + B\ )\ =\ \det(A)\ +\ \det(B)\ +\ \operatorname{tr}(A)\ \operatorname{tr}(B)\ -\ \operatorname{tr}(\ A\ B\ ) ~.}

That is, the 'cross-terms' can be written as traces. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A,B\ } are chosen to be different Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sigma^\mu\ ,} the cross-terms vanish. It then follows, now showing summation explicitly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \det\left(\sum_\mu x_\mu \sigma^\mu\right) = \sum_\mu \det\left(x_\mu\sigma^\mu\right).} Since the matrices are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2 \times 2\ ,} this is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \ \sum_\mu x_\mu^2 \det(\sigma^\mu) = \eta(x,x) ~.}

Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[ \sigma_j, \sigma_k\right] + \{\sigma_j, \sigma_k\} &= (\sigma_j \sigma_k - \sigma_k \sigma_j ) + (\sigma_j \sigma_k + \sigma_k \sigma_j) \\ 2i\varepsilon_{j k \ell}\,\sigma_\ell + 2 \delta_{j k}I &= 2\sigma_j \sigma_k \end{align} }

so that, Template:Equation box 1

Contracting each side of the equation with components of two 3-vectors a_p and b_q (which commute with the Pauli matrices, i.e., a_pσ_q = σ_qa_p) for each matrix σ_q and vector component a_p (and likewise with b_q) yields

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~~ \begin{align} a_j b_k \sigma_j \sigma_k & = a_j b_k \left(i\varepsilon_{jk\ell}\,\sigma_\ell + \delta_{jk}I\right) \\ a_j \sigma_j b_k \sigma_k & = i\varepsilon_{jk\ell}\,a_j b_k \sigma_\ell + a_j b_k \delta_{jk}I \end{align} ~.}

Finally, translating the index notation for the dot product and cross product results in

Template:Equation box 1

(1)

If i is identified with the pseudoscalar σ_x σ_y σ_z then the right hand side becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a \cdot b + a \wedge b\ ,} which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as J = ħ/2σ , then J satisfies the commutation relation:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbf{J} \times \mathbf{J} = i\ \hbar \mathbf{J}\ } Or equivalently, the Pauli vector satisfies:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \frac{\vec{\sigma}}{2} \times \frac{\vec{\sigma}}{2} = i\ \frac{\vec{\sigma}}{2} ~.}

Some trace relations

The following traces can be derived using the commutation and anticommutation relations.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \operatorname{tr}\left(\sigma_j \right) &= 0 \\ \operatorname{tr}\left(\sigma_j \, \sigma_k \right) &= 2\delta_{jk} \\ \operatorname{tr}\left(\sigma_j \, \sigma_k \, \sigma_\ell \right) &= 2i\varepsilon_{jk\ell} \\ \operatorname{tr}\left(\sigma_j \, \sigma_k \, \sigma_\ell \, \sigma_m \right) &= 2\left(\delta_{jk}\, \delta_{\ell m} - \delta_{j\ell} \, \delta_{km} + \delta_{jm}\, \delta_{k\ell}\right) \end{align} ~.}

If the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_0 = \mathbb{I}} is also considered, these relationships become

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \operatorname{tr}\left(\sigma_\alpha \right) &= 2\delta_{0 \alpha} \\ \operatorname{tr}\left(\sigma_\alpha \sigma_\beta \right) &= 2\delta_{\alpha \beta} \\ \operatorname{tr}\left(\sigma_\alpha \sigma_\beta \sigma_\gamma \right) &= 2 \sum_{(\alpha \beta \gamma)} \delta_{\alpha \beta} \delta_{0 \gamma} - 4 \delta_{0 \alpha} \delta_{0 \beta} \delta_{0 \gamma} + 2i\varepsilon_{0 \alpha \beta \gamma} \\ \operatorname{tr}\left(\sigma_\alpha \sigma_\beta \sigma_\gamma \sigma_\mu \right) &= 2\left(\delta_{\alpha \beta}\delta_{\gamma \mu} - \delta_{\alpha \gamma}\delta_{\beta \mu} + \delta_{\alpha \mu}\delta_{\beta \gamma}\right) + 4\left(\delta_{\alpha \gamma} \delta_{0 \beta} \delta_{0 \mu} + \delta_{\beta \mu} \delta_{0 \alpha} \delta_{0 \gamma}\right) - 8 \delta_{0 \alpha} \delta_{0 \beta} \delta_{0 \gamma} \delta_{0 \mu} + 2 i \sum_{(\alpha \beta \gamma \mu)} \varepsilon_{0 \alpha \beta \gamma} \delta_{0 \mu} \end{align} ~.}

where Greek indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha, \beta, \gamma} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} assume values from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0, x, y, z\}} and the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum_{(\alpha \ldots)}} is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a} = a\ \hat{n}, \quad \left|\ \hat{n}\ \right| = 1\ ,}

one has, for even powers, 2 p, p = 0, 1, 2, 3, ...

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (\hat{n} \cdot \vec{\sigma})^{2p} = I\ ,}

which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention.

For odd powers, 2 q + 1, q = 0, 1, 2, 3, ...

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} ~.}

Matrix exponentiating, and using the Taylor series for sine and cosine,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} &= \sum_{k=0}^\infty{\frac{i^k \left[a \left(\hat{n} \cdot \vec{\sigma}\right)\right]^k}{k!}} \\ &= \sum_{p=0}^\infty{\frac{(-1)^p (a\hat{n}\cdot \vec{\sigma})^{2p}}{(2p)!}} + i\sum_{q=0}^\infty{\frac{(-1)^q (a\hat{n}\cdot \vec{\sigma})^{2q + 1}}{(2q + 1)!}} \\ &= I\sum_{p=0}^\infty{\frac{(-1)^p a^{2p}}{(2p)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{q=0}^\infty{\frac{(-1)^q a^{2q+1}}{(2q + 1)!}}\\ \end{align} ~.}

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

Template:Equation box 1

(2)

which is analogous to Euler's formula, extended to quaternions. In particular,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\ a\ \sigma_1} = \begin{pmatrix} \cos a & i\ \sin a\\ i\ \sin a & \cos a \end{pmatrix} \ , \quad e^{i\ a\ \sigma_2} = \begin{pmatrix} \cos a & \sin a \\ - \sin a & \cos a \end{pmatrix} \ , \quad e^{i\ a\ \sigma_3} = \begin{pmatrix} e^{i\ a} & 0 \\ 0 & e^{-i\ a} \end{pmatrix} ~.}

Note that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\!\left[\ i\ a\ \left(\hat{n} \cdot \vec{\sigma} \right)\ \right] = a^2\ ,}

while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).

A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f(\ a(\hat{n} \cdot \vec{\sigma})\ )\ =\ I\ \frac{\ f(+a) + f(-a)\ }{2}\ +\ \hat{n} \cdot \vec{\sigma}\ \frac{\ f(+a) - f(-a)\ }{2} ~.}

The group composition law of SU(2)

A straightforward application of formula (2) provides a parameterization of the composition law of the group SU(2). One may directly solve for c in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} e^{i\ a\left(\hat{n} \cdot \vec{\sigma}\right)}\ e^{i\ b\ \left( \hat{m} \cdot \vec{\sigma} \right)} &= I\ \left(\ \cos a\ \cos b\ -\ \hat{n} \cdot \hat{m}\ \sin a\ \sin b\ \right)\ +\ i\ \left(\ \hat{n}\ \sin a\ \cos b\ +\ \hat{m}\ \sin b\ \cos a\ -\ \hat{n} \times \hat{m} ~ \sin a\ \sin b\ \right) \cdot \vec{\sigma} \\ &= I\ \cos{c}\ +\ i\ \left( \hat{k} \cdot \vec{\sigma} \right)\ \sin c \\ &= e^{i\ c\ \left(\hat{k} \cdot \vec{\sigma} \right) }\ , \end{align}}

which specifies the generic group multiplication, where, manifestly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \cos c = \cos a\ \cos b\ -\ \hat{n} \cdot \hat{m}\ \sin a\ \sin b\ ,} the spherical law of cosines. Given c, then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \hat{k}\ =\ \frac{1}{\sin c}\ \left(\ \hat{n}\ \sin a\ \cos b\ +\ \hat{m}\ \sin b\ \cos a - \hat{n}\times\hat{m}\ \sin a\ \sin b\ \right) ~.}

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ e^{ic \hat{k} \cdot \vec{\sigma}} = \exp \left( i\frac{c}{\sin c} \left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b\right) \cdot \vec{\sigma}\right) ~.}

(Of course, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \hat{n}\ } is parallel to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \hat{m}\ ,} so are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \hat{k}\ } and c = a + b .)

See also: Rotation formalisms in three dimensions § Rodrigues vector, and Spinor § Three dimensions

Adjoint action

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} along any axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat n} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_n(-a) ~ \vec{\sigma} ~ R_n(a) = e^{i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma} ~ e^{-i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} = \vec{\sigma}\cos (a) + \hat{n} \times \vec{\sigma} ~ \sin(a) + \hat{n} ~ \hat{n} \cdot \vec{\sigma} ~ (1 - \cos(a)) ~ . }

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \ R_y\mathord\left(-\frac{\pi}{2}\right)\, \sigma_x\, R_y\mathord\left(\frac{\pi}{2}\right) = \hat{x} \cdot \left(\hat{y} \times \vec{\sigma}\right) = \sigma_z ~.}

See also: Rodrigues' rotation formula

Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ ^k _αβ .

In this notation, the completeness relation for the Pauli matrices can be written

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta} \equiv \sum_{k=1}^3 \sigma^k_{\alpha\beta}\ \sigma^k_{\gamma\delta} = 2\ \delta_{\alpha\delta}\ \delta_{\beta\gamma} - \delta_{\alpha\beta}\ \delta_{\gamma\delta} ~.}

As noted above, it is common to denote the 2 × 2 unit matrix by σ₀ , so σ⁰_αβ = δ_αβ . The completeness relation can alternatively be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \sum_{k=0}^3 \sigma^k_{\alpha\beta}\ \sigma^k_{\gamma\delta} = 2\ \delta_{\alpha\delta}\ \delta_{\beta\gamma} ~.}

The fact that any Hermitian complex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {σ₀, σ₁, σ₂, σ₃} as above, and then imposing the positive-semidefinite and trace 1 conditions.

For a pure state, in polar coordinates, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a} = \begin{pmatrix}\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end{pmatrix},} the idempotent density matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2} \left(\mathbf{1} + \vec{a} \cdot \vec{\sigma}\right) = \begin{pmatrix} \cos^2\left(\frac{\,\theta\,}{2}\right) & e^{-i\,\phi}\sin\left(\frac{\,\theta\,}{2}\right)\cos\left(\frac{\,\theta\,}{2}\right) \\ e^{+i\,\phi}\sin\left(\frac{\,\theta\,}{2}\right)\cos\left(\frac{\,\theta\,}{2}\right) & \sin^2\left(\frac{\,\theta\,}{2}\right) \end{pmatrix} }

acts on the state eigenvector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \begin{pmatrix}\cos\left(\frac{\ \theta\ }{2}\right) & e^{+i\phi}\ \sin\left(\frac{\ \theta\ }{2}\right) \end{pmatrix}\ } with eigenvalue +1, hence it acts like a projection operator.

Relation with the permutation operator

Let P_jk be the transposition (also known as a permutation) between two spins σ_j and σ_k living in the tensor product space Template:Tmath ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{jk} \left| \sigma_j \sigma_k \right\rangle = \left| \sigma_k \sigma_j \right\rangle .}

This operator can also be written more explicitly as Dirac's spin exchange operator,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ P_{jk} = \frac{1}{2}\ \left(\vec{\sigma}_j \cdot \vec{\sigma}_k + 1 \right) ~.}

Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

SU(2)

The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{su}_2} is the three-dimensional real algebra spanned by the set {iσ_k}. In compact notation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{su}(2) = \operatorname{span} \{\; i\,\sigma_1\, ,\; i\,\sigma_2\, , \; i\,\sigma_3 \;\}.}

As a result, each iσ_j can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2 , so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathfrak{su}(2) = \operatorname{span} \left\{\frac{\ i\ \sigma_1\ }{2}, \frac{\ i\ \sigma_2\ }{2}, \frac{\ i\ \sigma_3\ }{2} \right\} ~.}

As SU(2) is a compact group, its Cartan decomposition is trivial.

SO(3)

The Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathfrak{su}(2)\ } is isomorphic to the Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{so}(3)} , which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the i σ_j are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathfrak{su}(2)\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{so}(3)} are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) ↦ SO(3) , see relationship between SO(3) and SU(2).

Quaternions

Main article: Spinor § Three dimensions

The real linear span of {I, iσ₁, i σ₂, i σ₃} is isomorphic to the real algebra of quaternions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{H}} , represented by the span of the basis vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \left\{\; \mathbf{1},\ \mathbf{i},\ \mathbf{j},\ \mathbf{k} \;\right\} ~.} The isomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathbb{H}\ } to this set is given by the following map (notice the reversed signs for the Pauli matrices): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{1} \mapsto I, \quad \mathbf{i} \mapsto - \sigma_2\sigma_3 = - i\,\sigma_1, \quad \mathbf{j} \mapsto - \sigma_3\sigma_1 = - i\,\sigma_2, \quad \mathbf{k} \mapsto - \sigma_1\sigma_2 = - i\,\sigma_3 ~. }

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{1} \mapsto I, \quad \mathbf{i} \mapsto i\,\sigma_3 \, , \quad \mathbf{j} \mapsto i\,\sigma_2 \, , \quad \mathbf{k} \mapsto i\,\sigma_1 ~ . }

As the set of versors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\sub\mathbb{H}} forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.

Physics

Classical mechanics

Main article: Quaternions and spatial rotation

In classical mechanics, Pauli matrices are useful in the context of the Cayley–Klein parameters. The matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} corresponding to the position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}} of a point in space is defined in terms of the above Pauli vector matrix,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \boldsymbol{x} \cdot \boldsymbol{\sigma} = x\,\sigma_x + y\,\sigma_y + z\,\sigma_z.}

Consequently, the transformation matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_\theta} for rotations about the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis through an angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} may be written in terms of Pauli matrices and the unit matrix as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ Q_\theta = \mathbb{I}\,\cos\frac{\theta}{2} + i\ \sigma_x \sin\frac{\theta}{2}.}

Similar expressions follow for general Pauli vector rotations as detailed above.

Quantum mechanics

In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a [[spin-1/2|spin Template:1/2]] particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \sigma_j} are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin Template:1/2. The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator.

An interesting property of spin Template:1/2 particles is that they must be rotated by an angle of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\pi} in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north–south pole on the 2-sphere Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^2} they are actually represented by orthogonal vectors in the two-dimensional complex Hilbert space.

For a spin Template:1/2 particle, the spin operator is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{J} = \frac{\hslash}{2} \boldsymbol{\sigma}} , the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_n} is defined to consist of all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -fold tensor products of Pauli matrices.

Relativistic quantum mechanics

In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathsf{\Sigma}_k = \begin{pmatrix} \mathsf{\sigma}_k & 0 \\ 0 & \mathsf{\sigma}_k \end{pmatrix} ~.}

It follows from this definition that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathsf{ \Sigma }_k\ } matrices have the same algebraic properties as the σ_k matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathsf{\Sigma}_k\ } needs to be replaced by Σ_μν, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σ_μν are also antisymmetric. Hence there are only six independent matrices.

The first three are the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \Sigma_{k\ell} \equiv \epsilon_{jk\ell}\mathsf{\Sigma}_j ~.} The remaining three, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -i\ \Sigma_{0k} \equiv \mathsf{\alpha}_k\ ,} where the Dirac α_k matrices are defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathsf{\alpha}_k = \begin{pmatrix} 0 & \mathsf{\sigma}_k \\ \mathsf{\sigma}_k & 0 \end{pmatrix} ~. }

The relativistic spin matrices Σ_μν are written in compact form in terms of commutator of gamma matrices as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \Sigma_{\mu\nu} = \frac{i}{2} \bigl[ \gamma_\mu, \gamma_\nu \bigr] ~.}

Quantum information

In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the "Z–Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X–Y decomposition of a single-qubit gate ".

See also

• Algebra of physical space
• Spinors in three dimensions
• Gamma matrices
  • § Dirac basis
• Angular momentum
• Gell-Mann matrices
• Poincaré group
• Generalizations of Pauli matrices
• Bloch sphere
• Euler's four-square identity
• For higher spin generalizations of the Pauli matrices, see Spin (physics) § Higher spins
• Exchange matrix (the first Pauli matrix is an exchange matrix of order two)
• Split-quaternion

References

• "The Pauli spin matrices". The Feynman Lectures on Physics.
• Liboff, Richard L. (2002). Introductory Quantum Mechanics (4th ed.). Addison-Wesley. ISBN 0-8053-8714-5. OCLC 837947786.
• Schiff, Leonard I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. ISBN 978-0-07-055287-6. OCLC 643977885.
• Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. doi:10.1017/CBO9780511806117. ISBN 978-0-521-14505-3. OCLC 855534544.