View source for Pauli matrices

{{Short description|Matrices important in quantum mechanics and the study of spin}}
{{Use American English|date=March 2019}}
{{MOS|article|date=July 2025| [[MOS:FORMULA]] - avoid mixing {{tag|math}} and {{tl|math}} in the same expression}}
[[File:Wolfgang Pauli.jpg|thumb|right|[[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 {{math|2 × 2}} [[complex number|complex]] [[matrix (mathematics)|matrices]] that are [[traceless]], [[Hermitian matrix|Hermitian]], [[Involutory matrix|involutory]] and [[Unitary matrix|unitary]]. Usually indicated by the [[Greek (alphabet)|Greek]] letter [[sigma]] ({{mvar|σ}}), they are occasionally denoted by [[tau]] ({{mvar|τ}}) when used in connection with [[isospin]] symmetries.
<math display="block">
\begin{align}
  \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{align}
</math>

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 (physics)|spin]] of a particle with an external [[electromagnetic field]]. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45&nbsp;degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is [[Hermitian matrix|Hermitian]], and together with the identity matrix {{mvar|I}} (sometimes considered as the zeroth Pauli matrix {{math|''σ''{{sub|0}} }}), the Pauli matrices form a [[Basis (linear algebra)|basis]] of the [[vector space]] of {{math|2 × 2}} Hermitian matrices over the [[real number|real numbers]], under addition. This means that any {{math|2 × 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:
:<math display="block">\begin{align}
 \sigma_i\ \sigma_j = \delta_{ij}\ I + i\ \varepsilon_{ijk}\ \sigma_k\ ,
\end{align}</math>
where {{mvar|δ{{sub|ij}} }} is the [[Kronecker delta]], which equals {{math|+1}} if {{nobr| {{math|''i'' {{=}} ''j''}} ,}} otherwise {{math|0}}, and the [[Levi-Civita symbol]] {{math|''ε{{sub|ijk}}''}} is used.

[[Hermitian operator]]s represent [[observable]]s in quantum mechanics, so the Pauli matrices span the space of observables of the [[complex number|complex]] two-dimensional [[Hilbert space]]. In the context of Pauli's work, {{mvar|σ{{sub|k}}}} represents the observable corresponding to spin along the {{mvar|k}}th coordinate axis in three-dimensional [[Euclidean space]] <math>\mathbb{R}^3 ~.</math>

The Pauli matrices (after multiplication by {{mvar|i}} to make them [[skew-Hermitian|anti-Hermitian]]) also generate transformations in the sense of [[Lie algebra]]s: The matrices {{math|{{nobr|''i σ''{{sub|1}},}} {{nobr|''i σ''{{sub|2}},}} {{nobr|''i σ''{{sub|3}}}} }} form a basis for the real Lie algebra <math>\ \mathfrak{su}(2)\ ,</math> which [[Exponential map (Lie theory)|exponentiates]] to the special unitary group [[SU(2)#n = 2|SU(2)]].{{efn|
This conforms to the convention ''in mathematics'' for the [[matrix exponential]], {{nobr|{{math|''i σ'' ⟼ exp(''i σ'')}} .}} In the convention ''in physics'', {{nobr| {{math|''σ'' ⟼ exp(−''i σ'')}} ,}} hence in it no pre-multiplication by {{mvar|i}} is necessary to land in {{nobr|{{math|SU(2)}} .}}
}} The [[Algebra over a field|algebra]] generated by the three matrices {{math|''σ''{{sub|1}}, ''σ''{{sub|2}}, ''σ''{{sub|3}} }} is [[isomorphic]] to the [[Clifford algebra]] of <math>\ \mathbb{R}^3\ ,</math><ref>{{cite journal |last1=Gull |first1=S.F. |last2=Lasenby |first2=A.N. |last3=Doran |first3=C.J.L. |date=January 1993 |title=Imaginary numbers are not Real – the geometric algebra of spacetime |journal=[[Foundations of Physics]] |volume=23 |issue=9 |pages=1175–1201 |doi=10.1007/BF01883676 |bibcode=1993FoPh...23.1175G |s2cid=14670523 |url=http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf |via=geometry.mrao.cam.ac.uk |url-status=dead |access-date=2023-05-05 |df=dmy-all |archive-url=https://web.archive.org/web/20231009095333/http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf |archive-date=2023-10-09 }}</ref> and the (unital) [[associative algebra]] generated by {{math|''iσ''{{sub|1}}, ''iσ''{{sub|2}}, ''iσ''{{sub|3}} }} functions identically ([[isomorphism|is isomorphic]]) to that of [[quaternion]]s (<math>\mathbb{H}</math>).

== Algebraic properties ==

{| class="wikitable floatright" style="text-align: center;"
|+ [[Cayley table]]; the entry shows the value of the row times the column.
! ×
! style=width:3em | <math>\sigma_x</math>
! style=width:3em | <math>\sigma_y</math>
! style=width:3em | <math>\sigma_z</math>
|-
! <math>\sigma_x</math>
| <math>I</math> || <math>i\ \sigma_z</math> || <math>-i\ \sigma_y</math>
|-
! <math>\sigma_y</math>
| <math>-i\ \sigma_z</math> || <math>I</math> || <math>i\ \sigma_x</math>
|-
! <math>\sigma_z</math>
| <math>i \sigma_y</math> || <math>-i\ \sigma_x</math> || <math>I</math>
|}

All three of the Pauli matrices can be compacted into a single expression:
:<math>
  \sigma_j = 
    \begin{pmatrix}
      \delta_{j3}                   &  \delta_{j1} - i\ \delta_{j2}\\
      \delta_{j1} + i\ \delta_{j2}  & -\delta_{j3}
    \end{pmatrix} ~.
</math>
This expression is useful for "selecting" any one of the matrices numerically by substituting values of {{nobr| {{math|''j'' {{=}} 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 matrix|''involutory'']]:
:<math>\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\ \sigma_1\ \sigma_2\ \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I\ ,</math>
where {{mvar|I}} is the [[identity matrix]].

The [[determinant]]s and [[trace of a matrix|trace]]s of the Pauli matrices are
:<math>\begin{align}
               \det \sigma_j &= -1\ , \\
  \operatorname{tr} \sigma_j &= 0\ ,
\end{align}</math>
from which we can deduce that each matrix {{mvar|σ{{sub|j}} }} has [[eigenvalues]] {{math|+1}} and {{nobr|{{math|−1}} .}}

With the inclusion of the identity matrix {{mvar|I}} (sometimes denoted {{math|''σ''{{sub|0}}}}), the Pauli matrices form an orthogonal basis (in the sense of [[Hilbert–Schmidt operator|Hilbert–Schmidt]]) of the [[Hilbert space]] <math>\ \mathcal{H}_2\ </math> of {{math|2 × 2}} Hermitian matrices over <math>\ \mathbb{R}\ ,</math> and the Hilbert space <math>\mathcal{M}_{2,2}(\mathbb{C})</math> of all [[complex number|complex]] {{math|2 × 2}} matrices over <math>\ \mathbb{C} ~.</math>

=== Commutation and anti-commutation relations ===
==== Commutation relations====
The Pauli matrices obey the following [[commutator|commutation]] relations:
:<math>[\sigma_j, \sigma_k] = 2\ i\ \varepsilon_{j k l}\ \sigma_l ~. </math>

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra <math>(\mathbb{R}^3, \times)\ \cong\ \mathfrak{su}(2)\ \cong\ \mathfrak{so}(3) ~.</math>

==== Anticommutation relations====
They also satisfy the [[anticommutator|anticommutation]] relations:
:<math>\{\sigma_j, \sigma_k\} = 2\ \delta_{j k}\ I\ ,</math>

where <math>\{\sigma_j, \sigma_k\}</math> is defined as <math>\ \sigma_j\ \sigma_k + \sigma_k\ \sigma_j\ ,</math> and {{math|''δ{{sub|jk}}''}} is the [[Kronecker delta]]. {{mvar|I}} denotes the {{math|2 × 2}} identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the [[Clifford algebra]] for <math>\ \mathbb{R}^3\ ,</math> denoted <math>\ \mathrm{Cl}_3(\mathbb{R}) ~.</math>

The usual construction of generators <math>\ \sigma_{jk} = \tfrac{1}{4} [\sigma_j, \sigma_k]\ </math> of <math>\ \mathfrak{so}(3)\ </math> 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:
{| style="text-align:left;"
! Commutators
! Anticommutators
|-
| <math>\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}</math>{{quad}}
| <math>\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}</math>
|}

=== Eigenvectors and eigenvalues ===
Each of the ([[Hermitian matrix|Hermitian]]) Pauli matrices has two [[eigenvalues]]: {{math|+1}} and {{math|−1}}. The corresponding [[Normalisable wavefunction|normalized]] [[eigenvectors]] are
:<math>\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}</math>

== Pauli vectors ==
The Pauli vector is defined by{{efn|
The Pauli vector is a formal device. It may be thought of as an element of <math>\ \mathcal M_2(\Complex) \otimes \R^3\ ,</math> where the [[Tensor product|tensor product space]] is endowed with a mapping <math>\ \cdot : \mathbb{R}^3 \times (\mathcal M_2(\Complex) \otimes \R^3) \to \mathcal M_2(\Complex)\ </math> induced by the [[dot product]] on <math>\ \mathbb{R}^3 ~.</math>
}}
<math display="block">
\vec{\sigma} = \sigma_1 \hat{x}_1 + \sigma_2 \hat{x}_2 + \sigma_3 \hat{x}_3\ ,
</math>
where <math>\ \hat{x}_1\ ,</math> <math>\ \hat{x}_2\ ,</math> and <math>\ \hat{x}_3\ </math> are an equivalent notation for the more familiar <math>\ \hat{x}\ ,</math> <math>\ \hat{y}\ ,</math> and <math>\ \hat{z} ~.</math>

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis<ref>See the [[Lorentz group#Relation to the Möbius group|spinor map]].</ref> as follows:
<math display="block">\begin{align}
     \vec{a} \cdot \vec{\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} </math>

More formally, this defines a map from <math>\mathbb{R}^3</math> to the vector space of traceless Hermitian <math>2\times 2</math> matrices. This map encodes structures of <math>\mathbb{R}^3</math> 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 <math>\ 2 \times 2\ </math> Hermitian traceless matrix-valued dual vector, that is, an element of <math>\ \mathrm{Mat}_{2\times 2}(\mathbb{C}) \otimes (\mathbb{R}^3)^*\ </math> that maps <math>\ \vec a \mapsto \vec a \cdot \vec \sigma ~.</math>

=== Completeness relation ===

Each component of <math>\vec a</math> can be recovered from the matrix (see [[#completeness anchor|completeness relation]] below) 
<math display="block">
 \frac{1}{2} \operatorname{tr} \Bigl(\ \bigl(\ \vec{a} \cdot \vec{\sigma}\ \bigr)\ \vec{\sigma}\ \Bigr) = \vec{a} ~.
</math>
This constitutes an inverse to the map <math>\ \vec a\ \mapsto\ \vec a \cdot \vec \sigma\ ,</math> making it manifest that the map is a bijection.

=== 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}\ =\ -\left|\ \vec{a}\ \right|^2 ~.
</math>
Then, considering the conjugation action of an <math>\ \mathrm{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>\ \mathrm{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>

=== Cross-product ===

The cross-product is given by the matrix commutator (up to a factor of <math>\ 2\ i\ </math>)
<math display="block">
 \left[\ \vec a \cdot \vec \sigma,\ \vec b \cdot \vec \sigma\ \right] = 2\ i\ \left( \vec a \times \vec b \right) \cdot \vec \sigma ~.
</math>
In fact, the existence of a norm follows from the fact that <math>\ \mathbb{R}^3\ </math> 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 <math>\ \vec a \cdot \vec \sigma\ </math> are <math>\ \pm |\vec{a}| ~.</math> 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 <math>\ (\vec a \cdot \vec \sigma)^2 - |\vec a|^2 = 0\ ,</math> since this can be factorised into <math>\ (\vec a \cdot \vec \sigma - |\vec a|)(\vec a \cdot \vec \sigma + |\vec a|)= 0 ~.</math> A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is [[Diagonalizable_matrix|diagonalizable]]) means this implies <math>\ \vec a \cdot \vec \sigma\ </math> is diagonalizable with possible eigenvalues <math>\ \pm |\vec a| ~.</math> The tracelessness of <math>\ \vec a \cdot \vec \sigma\ </math> means it has exactly one of each eigenvalue.

Its normalized eigenvectors are 
<math display="block">
  \psi_+ = \frac{1}{\sqrt{2 \left|\vec{a} \right|\ (a_3+\left|\vec{a}\right|)\ }\ } \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} ~.
</math>
These expressions become singular for <math>\ a_3 \to -\left|\ \vec{a}\ \right| ~.</math> They can be rescued by letting <math>\vec{a} = \left|\ \vec{a}\ \right| \left( \epsilon,\ 0,\ -\left( 1 - \tfrac{\epsilon^2}{2} \right) \right)\ </math> and taking the limit <math>\ \epsilon \to 0\ ,</math> which yields the correct eigenvectors {{nobr| {{math|(0,1) }} }} and {{nobr|{{math| (1,0) }} }} of <math>\ \sigma_z ~.</math>

Alternatively, one may use spherical coordinates <math>\ \vec{a} = a\ \bigl(\ \sin \vartheta\ \cos \varphi,\ \sin \vartheta\ \sin \varphi,\ \cos\vartheta\ \bigr)\ </math> to obtain the eigenvectors <math>\ \psi_{+} = \left(\ \cos \tfrac{\vartheta}{2}, \; \sin \tfrac{\vartheta}{2}\ e^{+i\varphi}\ \right)\ </math> and <math>\ \psi_{-} = \left(\ -\sin \tfrac{\vartheta}{2}\ e^{-i\varphi}, \; \cos \tfrac{\vartheta}{2}\ \right) ~.</math>

=== Pauli 4-vector ===

The Pauli 4-vector, used in spinor theory, is written <math>\ \sigma^\mu\ </math> with components
:<math>\ \sigma^\mu = \bigl(\ I,\ \vec\sigma\ \bigr) ~.</math>
This defines a map from <math>\ \mathbb{R}^{1,3}\ </math> to the vector space of Hermitian matrices,
:<math>\ x_\mu \mapsto x_\mu\sigma^\mu\ ,</math>
which also encodes the [[Minkowski metric]] (with ''mostly minus'' convention) in its determinant:
:<math>\ \det\bigl(\ x_\mu\sigma^\mu\ \bigr) = \eta(x,x) ~.</math>

This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector
:<math>\ \bar\sigma^\mu = \bigl(\ I, -\vec\sigma\ \bigr) ~.</math>
and allow raising and lowering using the Minkowski metric tensor. The relation can then be written
<math display="block">\ x_\nu = \tfrac{1}{2} \operatorname{tr}\!\Bigl(\ \bar\sigma_\nu\bigl( x_\mu \sigma^\mu \bigr)\   \Bigr) ~.</math>

Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on <math>\ \mathbb{R}^{1,3}\ ;</math> in this case the matrix group is <math>\ \mathrm{SL}( 2, \mathbb{C} )\ ,</math> and this shows <math>\ \mathrm{SL}(2,\mathbb{C})\ \cong\ \mathrm{Spin}(1,3) ~.</math> Similarly to above, this can be explicitly realized for <math>\ S \in \mathrm{SL}(2,\mathbb{C})\ </math> with components
:<math>\ \Lambda(S)^\mu{}_\nu = \tfrac{1}{2}\operatorname{tr}\!\left(\ \bar\sigma_\nu\ S\ \sigma^\mu\ S^{\dagger}\ \right) ~.</math>

In fact, the determinant property follows abstractly from trace properties of the <math>\ \sigma^\mu ~.</math> For <math>\ 2\times 2\ </math> matrices, the following identity holds:
:<math>\ \det(\ A + B\ )\ =\ \det(A)\ +\ \det(B)\ +\ \operatorname{tr}(A)\ \operatorname{tr}(B)\ -\ \operatorname{tr}(\ A\ B\ ) ~.</math>
That is, the 'cross-terms' can be written as traces. When <math>\ A,B\ </math> are chosen to be different <math>\ \sigma^\mu\ ,</math> the cross-terms vanish. It then follows, now showing summation explicitly,
<math display="inline">\det\left(\sum_\mu x_\mu \sigma^\mu\right) = \sum_\mu \det\left(x_\mu\sigma^\mu\right).</math> Since the matrices are <math>\ 2 \times 2\ ,</math> this is equal to <math display="inline">\ \sum_\mu x_\mu^2 \det(\sigma^\mu) = \eta(x,x) ~.</math>

===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
:<math>
\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}
</math>
so that,  
{{Equation box 1
 |indent =:
 |equation =  <math>~~ \sigma_j \sigma_k = \delta_{j k}I + i\varepsilon_{j k \ell}\,\sigma_\ell ~ .~</math>
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

[[tensor contraction|Contracting]] each side of the equation with components of two {{math|3}}-vectors {{math|''a{{sub|p}}''}} and {{math|''b{{sub|q}}''}} (which commute with the Pauli matrices, i.e., {{math|1=''a{{sub|p}}σ{{sub|q}}'' = ''σ{{sub|q}}a{{sub|p}}'')}} for each matrix {{math|''σ{{sub|q}}''}} and vector component {{math|''a{{sub|p}}''}} (and likewise with {{math|''b{{sub|q}}''}}) yields
:<math>~~ \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} ~.</math>

Finally, translating the index notation for the [[dot product]] and [[cross product#Index notation for tensors|cross product]] results in 
{{NumBlk||{{Equation box 1
  |indent =:
  |equation = <math>~~\Bigl(\vec{a} \cdot \vec{\sigma}\Bigr)\Bigl(\vec{b} \cdot \vec{\sigma}\Bigr) = \Bigl(\vec{a} \cdot \vec{b}\Bigr) \, I + i \Bigl(\vec{a} \times \vec{b}\Bigr) \cdot \vec{\sigma}~~</math>
  |cellpadding= 6
  |border
  |border colour = #0073CF
  |bgcolor=#F9FFF7
 }}
 |{{EquationRef|1}}
}}

If {{mvar|i}} is identified with the pseudoscalar {{nobr| {{math|''σ{{sub|x}} σ{{sub|y}} σ{{sub|z}}''}} }} then the right hand side becomes <math>\ a \cdot b + a \wedge b\ ,</math> which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as {{nobr| {{math| '''''J''''' {{=}} {{sfrac|''ħ''|2}}'''''σ'''''}} ,}} then {{math|1='''''J'''''}} satisfies the commutation relation:<math display="block">\ \mathbf{J} \times \mathbf{J} = i\ \hbar \mathbf{J}\ </math> Or equivalently, the Pauli vector satisfies:<math display="block">\ \frac{\vec{\sigma}}{2} \times \frac{\vec{\sigma}}{2} = i\ \frac{\vec{\sigma}}{2} ~.</math>

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

:<math>\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} ~.</math>

If the matrix {{math| ''σ''{{sub|0}} {{=}} ''I'' }} is also considered, these relationships become

<math display=block>\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} ~.</math>

where Greek indices {{math|''α'', ''β'', ''γ''}} and {{mvar|μ}} assume values from {{math|{0, ''x'', ''y'', ''z''}<nowiki/>}} and the notation <math display="inline">\sum_{(\alpha \ldots)}</math> is used to denote the sum over the [[cyclic permutation]] of the included indices.

===Exponential of a Pauli vector===
For 
:<math>\vec{a} = a\ \hat{n}, \quad \left|\ \hat{n}\ \right| = 1\ ,</math>

one has, for even powers, {{math| 2 ''p'', ''p'' {{=}} 0, 1, 2, 3, ...}}
:<math>\ (\hat{n} \cdot \vec{\sigma})^{2p} = I\ ,</math>

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

For odd powers, {{math| 2 ''q'' + 1, ''q'' {{=}} 0, 1, 2, 3, ...}}
:<math>\ \left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} ~.</math>

[[Matrix exponential|Matrix exponentiating]], and using the [[Taylor series#List of Maclaurin series of some common functions|Taylor series for sine and cosine]],
:<math>\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} ~.</math>

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,
{{NumBlk||{{Equation box 1
  |indent =:
  |equation = <math>~~e^{i\ a\left(\hat{n} \cdot \vec{\sigma}\right)} = I\ \cos{a} + i\ (\hat{n} \cdot \vec{\sigma}) \sin{a} ~~</math> 
  |cellpadding= 6
  |border
  |border colour = #0073CF
  |bgcolor=#F9FFF7
 }}
 |{{EquationRef|2}}
}}

which is [[quaternions and spatial rotation#Using quaternion as rotations|analogous]] to [[Euler's formula]], extended to [[quaternions]]. In particular,

<math>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} ~.</math>

Note that
:<math>\det\!\left[\ i\ a\ \left(\hat{n} \cdot \vec{\sigma} \right)\ \right] = a^2\ ,</math>

while the determinant of the exponential itself is just {{math|1}}, which makes it the '''generic group element of [[SU(2)]]'''.

A more abstract version of formula {{EquationNote|(2)}} for a general {{math|2 × 2}} matrix can be found in the article on [[Matrix exponential#Evaluation by Laurent series|matrix exponentials]]. A general version of {{EquationNote|(2)}} for an analytic (at {{mvar|a}} and {{mvar|−a}}) function  is provided by application of  [[Sylvester's formula]],<ref>
{{cite book |title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A. |author-link1=Michael Nielsen |last2=Chuang |first2=Isaac L. |author-link2=Isaac Chuang |year=2000 |publisher=Cambridge University Press |location=Cambridge, UK |isbn=978-0-521-63235-5 |oclc=43641333}}
</ref>
:<math>\ 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} ~.</math>

====The group composition law of {{math|SU(2)}}====
A straightforward application of  formula {{EquationNote|(2)}}  provides a parameterization of the composition law of  the group {{math|SU(2)}}.{{efn|
The relation among {{math|''a, b, c,'' ''' n, m, k '''}} derived here in the {{math|2 × 2}} representation holds for ''all representations'' of {{math|SU(2)}}, being a ''group identity''. Note that, by virtue of the standard normalization of that group's generators as ''half'' the Pauli matrices, the parameters {{math|''a'',''b'',''c''}} correspond to ''half'' the rotation angles of the rotation group. That is, the Gibbs formula linked amounts to <math>\ \hat k \tan \tfrac{c}{2} =  (\hat n\ \tan \tfrac{a}{2} + \hat m\ \tan \tfrac{b}{2} - \hat m\ \times \hat n\ \tan \tfrac{a}{2} ~ \tan \tfrac{b}{2} )/(1 - \hat m\cdot \hat n\ \tan \tfrac{a}{2} ~ \tan \tfrac{b}{2} ) ~.</math>
}} One may directly solve for {{mvar|c}} in 
<math display=block>\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}</math>

which specifies the generic group multiplication, where, manifestly,  
<math display=block>\ \cos c = \cos a\ \cos b\ -\ \hat{n} \cdot \hat{m}\ \sin a\ \sin b\ ,</math>
the [[spherical law of cosines]].  Given {{mvar|c}}, then, 
<math display=block>\ \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) ~.</math>

Consequently, the composite rotation parameters in this group element (a closed form of the respective [[Baker–Campbell–Hausdorff formula|BCH expansion]] in this case) simply amount to<ref>{{cite book |first=J.W. |last=Gibbs |year=1884 |title=Elements of Vector Analysis |place=New Haven, CT |page=67 |author-link=J. W. Gibbs |chapter=4. Concerning the differential and integral calculus of vectors |chapter-url={{GBurl|VurzAAAAMAAJ|p=67}} |publisher=Tuttle, Moorehouse & Taylor }} In fact, however, the formula goes back to [[Olinde Rodrigues]] (1840), replete with half-angle: {{cite journal |first=Olinde |last=Rodrigues |author-link=Olinde Rodrigues |year=1840 |title=Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire |journal=[[J. Math. Pures Appl.]] |volume=5 |pages=380–440 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A39_0.pdf }}</ref>

<math display=block>\ 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) ~.</math>

(Of course, when <math>\ \hat{n}\ </math> is parallel to <math>\ \hat{m}\ ,</math> so are <math>\ \hat{k}\ </math> and {{nobr| {{math| ''c'' {{=}} ''a + b''}} .)}}
{{see also|Rotation formalisms in three dimensions#Rodrigues vector|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 <math>a</math> along any axis <math>\hat n</math>:
<math display=block>  
  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)) ~ .
</math>

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 <math display="inline">\ 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 ~.</math>

{{see also|Rodrigues' rotation formula}}

=== Completeness relation<span class="anchor" id="completeness_anchor"></span> ===
An alternative notation that is commonly used for the Pauli matrices is to write the vector index {{mvar|k}} in the superscript, and the matrix indices as subscripts, so that the element in row {{mvar|α}} and column {{mvar|β}} of the {{mvar|k}}-th Pauli matrix is {{nobr|{{math| ''σ {{sup|k}} {{sub|αβ}}''}} .}}

In this notation, the ''completeness relation'' for the Pauli matrices can be written
:<math> \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} ~.</math>

{{math proof | proof = The fact that the Pauli matrices, along with the identity matrix {{mvar|I}}, form an orthogonal basis for the Hilbert space of all {{math| 2 × 2 }} [[complex number|complex]] matrices <math>\ \mathcal{M}_{2,2}(\mathbb{C})\ </math> over <math>\ \mathbb{C}\ ,</math> means that we can express any {{math| 2 × 2 }} complex matrix {{mvar|M}} as
<math display="block"> M = c\ I + \sum_k a_k\ \sigma^k </math>
where {{mvar|c}} is a complex number, and {{mvar|a}} is a 3-component, complex vector. It is straightforward to show, using the properties listed above, that
<math display="block"> \operatorname{tr}\left( \sigma^j\,\sigma^k \right) = 2\ \delta_{jk} </math>
where "{{math|tr}}" denotes the [[trace (linear algebra)|trace]], and hence that 
<math display="block">\begin{align}
                   c &={} \tfrac{1}{2}\ \operatorname{tr}\, M\ ,\begin{align}&& a_k &= \tfrac{1}{2}\  \operatorname{tr}\ \sigma^k\ M \end{align} ~.\\[3pt]
  \therefore ~~ 2\,M &= I\,\operatorname{tr}\, M + \sum_k \sigma^k\,\operatorname{tr}\, \sigma^k M\ ,
\end{align}</math>
which can be rewritten in terms of matrix indices as
<math display="block">2\ M_{\alpha\beta} = \delta_{\alpha\beta}\ M_{\gamma\gamma} + \sum_k \sigma^k_{\alpha\beta}\  \sigma^k_{\gamma\delta}\ M_{\delta\gamma}\ ,</math>
where [[Einstein notation|summation over the repeated indices is implied]] {{mvar|γ}} and {{mvar|δ}}. Since this is true for any choice of the matrix {{mvar|M}}, the completeness relation follows as stated above. [[Q.E.D.]]
}}

As noted above, it is common to denote the 2 × 2 unit matrix by {{nobr| {{math|''σ''{{sub|0}}}} ,}} so {{nobr| {{math| ''σ{{sup|0}}{{sub|αβ}}'' {{=}} ''δ{{sub|αβ}}'' .}} }} The completeness relation can alternatively be expressed as
<math display="block">\ \sum_{k=0}^3 \sigma^k_{\alpha\beta}\ \sigma^k_{\gamma\delta} = 2\ \delta_{\alpha\delta}\ \delta_{\beta\gamma} ~.</math>

The fact that any Hermitian [[complex number|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 state (physics)|mixed state]]s’ density matrix, ([[Positive semidefinite 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 {{math|{''σ''{{sub|0}}, ''σ''{{sub|1}}, ''σ''{{sub|2}}, ''σ''{{sub|3}}<nowiki>}</nowiki>}} as above, and then imposing the positive-semidefinite and [[Trace (linear algebra)|trace]] {{math|1}} conditions.

For a pure state, in polar coordinates, <math display="block">\vec{a} = \begin{pmatrix}\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end{pmatrix},</math> the [[idempotent]] density matrix
<math display="block">
  \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}
</math>

acts on the state eigenvector <math>\ \begin{pmatrix}\cos\left(\frac{\ \theta\ }{2}\right) & e^{+i\phi}\ \sin\left(\frac{\ \theta\ }{2}\right) \end{pmatrix}\ </math> with eigenvalue +1, hence it acts like a [[projection (linear algebra)|projection operator]].

=== Relation with the permutation operator ===
Let {{math|''P{{sub|jk}}''}} be the [[transposition (mathematics)|transposition]] (also known as a permutation) between two spins {{math|''σ{{sub|j}}''}} and {{math|''σ{{sub|k}}''}} living in the [[tensor product]] space {{nobr| {{tmath|\Complex^2 \otimes \Complex^2}} ,}}
:<math>P_{jk} \left| \sigma_j \sigma_k \right\rangle =  \left| \sigma_k \sigma_j \right\rangle .</math>

This operator can also be written more explicitly as [[Exchange interaction#Inclusion of spin|Dirac's spin exchange operator]],
:<math>\ P_{jk} = \frac{1}{2}\ \left(\vec{\sigma}_j \cdot \vec{\sigma}_k + 1 \right) ~.</math>

Its eigenvalues are therefore{{efn|
Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is <math>\left(\begin{smallmatrix}
  1 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 \\  
  0 & 1 & 0 & 0 \\  
  0 & 0 & 0 & 1
\end{smallmatrix}\right) ~ .</math>
}} 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 matrix|unitary]] {{math|2 × 2}} matrices with unit determinant; its [[Lie algebra]] is the set of all {{math|2 × 2}} anti-Hermitian matrices  with trace 0.  Direct calculation, as above, shows that the [[Lie algebra]] <math>\mathfrak{su}_2</math> is the three-dimensional real algebra [[linear span|spanned]] by the set {{math|{''iσ{{sub|k}}''}<nowiki/>}}. In compact notation,
:<math> \mathfrak{su}(2) = \operatorname{span} \{\; i\,\sigma_1\, ,\; i\,\sigma_2\, , \; i\,\sigma_3 \;\}.</math>

As a result, each {{math|''iσ{{sub|j}}''}} can be seen as an [[Lie group#The Lie algebra associated with a Lie group|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 theory of SU(2)|representation of {{math|su(2)}}]], as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is {{nobr| {{math| ''λ'' {{=}} {{sfrac|1|2}} }} ,}} so that
:<math>\ \mathfrak{su}(2) = \operatorname{span} \left\{\frac{\ i\ \sigma_1\ }{2}, \frac{\ i\ \sigma_2\ }{2}, \frac{\ i\ \sigma_3\ }{2} \right\} ~.</math>

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

=== SO(3) ===
The Lie algebra <math>\ \mathfrak{su}(2)\ </math> is [[isomorphism|isomorphic]] to the Lie algebra <math> \mathfrak{so}(3)</math>, which corresponds to the Lie group [[Rotation group SO(3)|SO(3)]], the [[group (mathematics)|group]] of [[rotation]]s in three-dimensional space. In other words, one can say that the {{nobr| {{mvar| i σ{{sub|j}} }} }} are a realization (and, in fact, the lowest-dimensional realization) of ''infinitesimal'' rotations in three-dimensional space. However, even though <math>\ \mathfrak{su}(2)\ </math> and <math> \mathfrak{so}(3)</math> are isomorphic as Lie algebras, {{math|SU(2)}} and {{math|SO(3)}} are not isomorphic as Lie groups. {{math|SU(2)}} is actually a [[Double covering group|double cover]] of {{math|SO(3)}}, meaning that there is a two-to-one group homomorphism from {{nobr|{{math|SU(2)}} ↦ {{math|SO(3)}} ,}} see [[Rotation group SO(3)#Connection between SO(3) and SU(2)|relationship between SO(3) and SU(2)]].

=== Quaternions ===
{{main|Spinor#Three dimensions}}
The real linear span of {{math|{''I'', '' iσ''{{sub|1}}, ''i σ''{{sub|2}}, ''i σ''{{sub|3}}<nowiki>}</nowiki>}} is isomorphic to the real algebra of [[quaternions]], <math>\mathbb{H}</math>, represented by the span of the basis vectors <math>\ \left\{\; \mathbf{1},\ \mathbf{i},\ \mathbf{j},\ \mathbf{k} \;\right\} ~.</math> The isomorphism from <math>\ \mathbb{H}\ </math> to this set is given by the following map (notice the reversed signs for the Pauli matrices):
<math display=block>
  \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 ~.
</math>

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,<ref>
{{cite book
 |first=Mikio |last=Nakahara
 |year=2003
 |title=Geometry, Topology, and Physics
 |edition=2nd  |page=[https://books.google.com/books?id=cH-XQB0Ex5wC&pg=PR22 xxii]
 |publisher=CRC Press
 |isbn=978-0-7503-0606-5
 |url=https://books.google.com/books?id=cH-XQB0Ex5wC&q=%22Pauli+matrices%22+OR+%22Pauli+matrix%22
 |via=Google Books
}}
</ref>
:<math>
  \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 ~ .
</math>

As the set of [[versor]]s <math>U\sub\mathbb{H}</math> forms a group isomorphic to {{math|SU(2)}}, {{mvar|U}} gives yet another way of describing {{math|SU(2)}}. The two-to-one homomorphism from {{math|SU(2)}} to {{math|SO(3)}} may be given in terms of the Pauli matrices in this formulation.

== Physics ==

=== Classical mechanics ===
{{Main|Quaternions and spatial rotation}}

In [[classical mechanics]], Pauli matrices are useful in the context of the Cayley–Klein parameters.<ref name=Goldstein-1959>
{{cite book
 |last=Goldstein |first=Herbert
 |year=1959
 |title=Classical Mechanics
 |pages=109–118
 |publisher=Addison-Wesley
 |oclc=3175838
}}
</ref> The matrix {{mvar|P}} corresponding to the position <math>\vec{x}</math> of a point in space is defined in terms of the above Pauli vector matrix,
:<math>P = \vec{x} \cdot \vec{\sigma} = x\,\sigma_x + y\,\sigma_y + z\,\sigma_z .</math>

Consequently, the transformation matrix {{math|''Q{{sub|θ}}''}} for rotations about the {{mvar|x}}-axis through an angle {{mvar|θ}} may be written in terms of Pauli matrices and the unit matrix as<ref name=Goldstein-1959/>
:<math>\ Q_\theta = \boldsymbol{1}\,\cos\frac{\theta}{2} + i\ \sigma_x \sin\frac{\theta}{2} ~.</math>

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 (physics)|spin]] of a [[spin-1/2|spin {{1/2}}]] particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, {{math|''iσ{{sub|j}}''}} are the generators of a [[projective representation]] ('''spin representation''') of the [[rotation group SO(3)]] acting on [[theory of relativity|non-relativistic]] particles with spin {{1/2}}. The [[mathematical formulation of quantum mechanics|states]] of the particles are represented as two-component [[Spinors in three dimensions|spinors]]. In the same way, the Pauli matrices are related to the [[Isospin|isospin operator]].

An interesting property of spin {{1/2}} particles is that they must be rotated by an angle of 4{{mvar|π}} 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]] {{nobr| {{math|''S''{{sup|2}} }},}} they are actually represented by [[orthogonal]] vectors in the two-dimensional complex [[Hilbert space]].

For a spin {{1/2}} particle, the spin operator is given by {{nobr| {{math| '''''J''''' {{=}} {{sfrac|''ħ''|2}}'''''σ'''''}} ,}} the [[fundamental representation]] of [[representation theory of SU(2)|SU(2)]]. By taking [[Kronecker product]]s of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting [[spin operator]]s for higher spin systems in three spatial dimensions, for arbitrarily large ''j'', can be calculated using this [[spin operator]] and [[Ladder operator#Angular momentum|ladder operators]]. They can be found in {{section link|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.<ref>{{cite journal |last1=Curtright |first1=T.L. |author1-link=Thomas Curtright |last2=Fairlie |first2=D.B. |author2-link=David Fairlie |last3=Zachos |first3=C.K. |author3-link=Cosmas Zachos |year=2014 |title=A compact formula for rotations as spin matrix polynomials |journal=[[SIGMA (journal)|SIGMA]] |volume=10 |page=084 |arxiv=1402.3541 |bibcode=2014SIGMA..10..084C |s2cid=18776942 |doi=10.3842/SIGMA.2014.084 }}</ref>

Also useful in the [[quantum mechanics]] of multiparticle systems, the general [[Pauli group]] {{math|''G{{sub|n}}''}} is defined to consist of all {{mvar|n}}-fold [[tensor]] products of Pauli matrices.

=== Relativistic quantum mechanics ===
In [[relativistic quantum mechanics]], the spinors in four dimensions are {{math|4 × 1}} (or {{nobr| {{math|1 × 4}})}} matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be {{nobr| {{math|4 × 4}} matrices.}} They are defined in terms of {{nobr| {{math| 2 × 2 }} }} Pauli matrices as
:<math>\mathsf{\Sigma}_k = \begin{pmatrix} \mathsf{\sigma}_k & 0 \\ 0 & \mathsf{\sigma}_k \end{pmatrix} ~.</math>

It follows from this definition that the <math>\ \mathsf{ \Sigma }_k\ </math> matrices have the same algebraic properties as the {{mvar| σ{{sub|k}} }} matrices.

However, [[relativistic angular momentum]] is not a three-vector, but a second order [[four-tensor]]. Hence <math>\ \mathsf{\Sigma}_k\ </math> needs to be replaced by {{math|Σ''{{sub|μν}}''}}, the generator of [[representation theory of the Lorentz group#The (1/2, 0) ⊕ (0, 1/2) spin representation|Lorentz transformations on spinors]]. By the antisymmetry of angular momentum, the {{math|Σ''{{sub|μν}}''}} are also antisymmetric. Hence there are only six independent matrices.

The first three are the <math>\ \Sigma_{k\ell} \equiv \epsilon_{jk\ell}\mathsf{\Sigma}_j ~.</math> The remaining three, <math>\ -i\ \Sigma_{0k} \equiv \mathsf{\alpha}_k\ ,</math> where the [[Dirac equation|Dirac {{math|''α{{sub|k}}''}} matrices]] are defined as

:<math>\  
\mathsf{\alpha}_k =
\begin{pmatrix}
 0   &   \mathsf{\sigma}_k \\
 \mathsf{\sigma}_k   &   0
\end{pmatrix} ~.
</math>

The relativistic spin matrices {{math|Σ''{{sub|μν}}''}} are written in compact form in terms of commutator of [[gamma matrices]] as
:<math>\ \Sigma_{\mu\nu} = \frac{i}{2} \bigl[ \gamma_\mu, \gamma_\nu \bigr] ~.</math>

=== Quantum information ===
In [[quantum information]], single-[[qubit]] [[quantum gate]]s 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]]
** {{section link|Gamma matrices|Dirac basis|nopage=y}}
* [[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 {{section link|Spin (physics)|Higher spins}}
* [[Exchange matrix]] (the first Pauli matrix is an exchange matrix of order two)
* [[Split-quaternion]]

== Remarks ==
{{notelist}}

== Notes ==
{{reflist|25em}}

== References ==
{{refbegin}}
*{{cite book |chapter-url=https://feynmanlectures.caltech.edu/III_11.html#Ch11-S1 |chapter=The Pauli spin matrices |title=The Feynman Lectures on Physics}}
*{{cite book |last=Liboff |first=Richard L. | author-link=Liboff, Richard L. | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5 |oclc=837947786 |edition=4th}}
*{{cite book |last=Schiff |first=Leonard I. | title=Quantum Mechanics | url=https://archive.org/details/quantummechanics0000schi | url-access=registration | publisher=McGraw-Hill | year=1968 | isbn=978-0-07-055287-6 |oclc=643977885 |edition=3rd}}
*{{cite book |last=Leonhardt |first=Ulf | title=Essential Quantum Optics | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-14505-3 |doi=10.1017/CBO9780511806117 |oclc=855534544}}
{{refend}}

{{Matrix classes}}

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[[Category:Matrices (mathematics)]]
[[Category:Rotational symmetry]]
[[Category:Articles containing proofs]]
[[Category:Mathematical physics]]
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