Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary.^[1] Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.

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.

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ₀), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices. This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex 2-dimensional Hilbert space. In the context of Pauli's work, σ_k represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}.}$

The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ₁, iσ₂, iσ₃ form a basis for the real Lie algebra ${\displaystyle {\mathfrak {su}}(2)}$, which exponentiates to the special unitary group SU(2).^[a] The algebra generated by the three matrices σ₁, σ₂, σ₃ is isomorphic to the Clifford algebra of ${\displaystyle \mathbb {R} ^{3}}$^{[citation needed]}, and the (unital associative) algebra generated by iσ₁, iσ₂, iσ₃ is effectively identical (isomorphic) to that of quaternions (${\displaystyle \mathbb {H} }$).

Algebraic properties[]

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

${\displaystyle \sigma _{j}={\begin{pmatrix}\delta _{j3}&\delta _{j1}-i\,\delta _{j2}\\\delta _{j1}+i\,\delta _{j2}&-\delta _{j3}\end{pmatrix}}}$

where i = √−1   is the "imaginary unit", and δ_jk is the Kronecker delta, which equals +1 if j = k and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of 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:

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\,\sigma _{1}\sigma _{2}\sigma _{3}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I}$

where I is the identity matrix.

The determinants and traces of the Pauli matrices are:

${\displaystyle {\begin{aligned}\det \sigma _{j}&~=\,-1\,,\\\operatorname {tr} \sigma _{j}&~=~~~\;0~.\end{aligned}}}$

From which, we can deduce that each matrix   σ_jk   has eigenvalues   +1 and −1 .

With the inclusion of the identity matrix, I (sometimes denoted σ₀ ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space of real 2 × 2 Hermitian matrices, ${\displaystyle {\mathcal {H}}_{2}(\mathbb {C} )}$, and the Hilbert space of all complex 2 × 2 matrices, ${\displaystyle {\mathcal {M}}_{2,2}(\mathbb {C} )}$.

Eigenvectors and eigenvalues[]

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

${\displaystyle {\begin{aligned}\psi _{x+}&={\frac {1}{\sqrt {2\,}}}{\begin{bmatrix}1\\1\end{bmatrix}}\;,&\psi _{x-}&={\frac {1}{\sqrt {2\,}}}{\begin{bmatrix}1\\-1\end{bmatrix}}\;,\\\psi _{y+}&={\frac {1}{\sqrt {2\,}}}{\begin{bmatrix}1\\i\end{bmatrix}}\;,&\psi _{y-}&={\frac {1}{\sqrt {2\,}}}{\begin{bmatrix}1\\-i\end{bmatrix}}\;,\\\psi _{z+}&={\begin{bmatrix}1\\0\end{bmatrix}}\;,&\psi _{z-}&={\begin{bmatrix}0\\1\end{bmatrix}}~.\end{aligned}}}$

Pauli vector[]

The Pauli vector is defined by^[b]

${\displaystyle {\vec {\sigma }}=\sigma _{1}{\hat {x}}_{1}+\sigma _{2}{\hat {x}}_{2}+\sigma _{3}{\hat {x}}_{3}~,}$

where ${\displaystyle \;{\hat {x}}_{1},{\hat {x}}_{2},\,{\text{ and }}\,{\hat {x}}_{3}\;}$ are an equivalent notation for the more familiar ${\displaystyle \;{\hat {x}},{\hat {y}},\,{\text{ and }}\,{\hat {z}}\,;\;}$ the subscripted notation ${\displaystyle \,{\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3}\,}$ is more compact than the old ${\displaystyle \,{\hat {x}},{\hat {y}},{\hat {z}}\,}$ form.

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis^[2] as follows,

${\displaystyle {\begin{aligned}{\vec {a}}\cdot {\vec {\sigma }}&=\left(a_{k}{\hat {x}}_{k}\right)\cdot \left(\sigma _{\ell }{\hat {x}}_{\ell }\right)=a_{k}\sigma _{\ell }{\hat {x}}_{k}\cdot {\hat {x}}_{\ell }\\\\&=a_{k}\sigma _{\ell }\delta _{k\ell }=a_{k}\sigma _{k}\\\\&=~a_{1}\;{\begin{pmatrix}0&1\\1&0\end{pmatrix}}~+~a_{2}\;i{\begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix}}~+~a_{3}\;{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}~=~{\begin{pmatrix}a_{3}&a_{1}-ia_{2}\\a_{1}+ia_{2}&-a_{3}\end{pmatrix}}\end{aligned}}}$

using Einstein's summation convention. Further,

${\displaystyle \det {\bigl (}{\vec {a}}\cdot {\vec {\sigma }}{\bigr )}=-{\vec {a}}\cdot {\vec {a}}=-\left|{\vec {a}}\right|^{2},}$

its eigenvalues being ${\displaystyle \pm |{\vec {a}}|}$, and moreover (see § completeness relation, below)

${\displaystyle {\frac {1}{2}}\operatorname {tr} {\Bigl (}{\bigl (}{\vec {a}}\cdot {\vec {\sigma }}{\bigr )}{\vec {\sigma }}{\Bigr )}={\vec {a}}~.}$

Its normalized eigenvectors are

${\displaystyle \psi _{+}={\frac {1}{\sqrt {2|{\vec {a}}|(a_{3}+|{\vec {a}}|)}}}{\begin{bmatrix}a_{3}+|{\vec {a}}|\\a_{1}+ia_{2}\end{bmatrix}};\qquad \psi _{-}={\frac {1}{\sqrt {2|{\vec {a}}|(a_{3}+|{\vec {a}}|)}}}{\begin{bmatrix}ia_{2}-a_{1}\\a_{3}+|{\vec {a}}|\end{bmatrix}}~.}$

Commutation relations[]

The Pauli matrices obey the following commutation relations:

${\displaystyle [\sigma _{j},\sigma _{k}]=2i\varepsilon _{jk\ell }\,\sigma _{\ell }~,}$

and anticommutation relations:

${\displaystyle \{\sigma _{j},\sigma _{k}\}=2\delta _{jk}\,I~.}$

where the structure constant ε_abc is the Levi-Civita symbol, Einstein summation notation is used, δ_jk is the Kronecker delta, and I is the 2 × 2 identity matrix.

For example,

commutators	anticommutators
${\displaystyle {\begin{aligned}\left[\sigma _{1},\sigma _{2}\right]&=2i\sigma _{3}\\\left[\sigma _{2},\sigma _{3}\right]&=2i\sigma _{1}\\\left[\sigma _{3},\sigma _{1}\right]&=2i\sigma _{2}\\\left[\sigma _{1},\sigma _{1}\right]&=0\end{aligned}}}$	${\displaystyle {\begin{aligned}\left\{\sigma _{1},\sigma _{1}\right\}&=2I\\\left\{\sigma _{1},\sigma _{2}\right\}&=0\\\left\{\sigma _{1},\sigma _{3}\right\}&=0\\\left\{\sigma _{2},\sigma _{2}\right\}&=2I.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\left[\sigma _{j},\sigma _{k}\right]+\{\sigma _{j},\sigma _{k}\}&=(\sigma _{j}\sigma _{k}-\sigma _{k}\sigma _{j})+(\sigma _{j}\sigma _{k}+\sigma _{k}\sigma _{j})\\2i\varepsilon _{jk\ell }\,\sigma _{\ell }+2\delta _{jk}I&=2\sigma _{j}\sigma _{k}\end{aligned}}}$

so that,

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

${\displaystyle ~~{\begin{aligned}a_{j}b_{k}\sigma _{\ell }\sigma _{k}&=a_{j}b_{k}\left(i\varepsilon _{jk\ell }\,\sigma _{\ell }+\delta _{jk}I\right)\\a_{j}\sigma _{j}b_{k}\sigma _{k}&=i\varepsilon _{jk\ell }\,a_{j}b_{k}\sigma _{\ell }+a_{j}b_{k}\delta _{jk}I\end{aligned}}~.~}$

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

${\displaystyle ~~\left({\vec {a}}\cdot {\vec {\sigma }}\right)\left({\vec {b}}\cdot {\vec {\sigma }}\right)=\left({\vec {a}}\cdot {\vec {b}}\right)\,I+i\left({\vec {a}}\times {\vec {b}}\right)\cdot {\vec {\sigma }}~~}$

(1)

If i is identified with the pseudoscalar σ_xσ_yσ_z then the right hand side becomes ${\displaystyle a\cdot b+a\wedge b}$ which is also the definition for the product of two vectors in geometric algebra.

Some trace relations[]

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

${\displaystyle {\begin{aligned}\operatorname {tr} \left(\sigma _{j}\right)&=0\\\operatorname {tr} \left(\sigma _{j}\sigma _{k}\right)&=2\delta _{jk}\\\operatorname {tr} \left(\sigma _{j}\sigma _{k}\sigma _{\ell }\right)&=2i\varepsilon _{jk\ell }\\\operatorname {tr} \left(\sigma _{j}\sigma _{k}\sigma _{\ell }\sigma _{m}\right)&=2\left(\delta _{jk}\delta _{\ell m}-\delta _{j\ell }\delta _{km}+\delta _{jm}\delta _{k\ell }\right)\end{aligned}}}$

If the matrix σ₀ = I is also considered, these relationships become

${\displaystyle {\begin{aligned}\operatorname {tr} \left(\sigma _{\alpha }\right)&=2\delta _{0\alpha }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\right)&=2\delta _{\alpha \beta }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\sigma _{\gamma }\right)&=2\sum _{(\alpha \beta \gamma )}\delta _{\alpha \beta }\delta _{0\gamma }-4\delta _{0\alpha }\delta _{0\beta }\delta _{0\gamma }+2i\varepsilon _{0\alpha \beta \gamma }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\sigma _{\gamma }\sigma _{\mu }\right)&=2\left(\delta _{\alpha \beta }\delta _{\gamma \mu }-\delta _{\alpha \gamma }\delta _{\beta \mu }+\delta _{\alpha \mu }\delta _{\beta \gamma }\right)+4\left(\delta _{\alpha \gamma }\delta _{0\beta }\delta _{0\mu }+\delta _{\beta \mu }\delta _{0\alpha }\delta _{0\gamma }\right)-8\delta _{0\alpha }\delta _{0\beta }\delta _{0\gamma }\delta _{0\mu }+2i\sum _{(\alpha \beta \gamma \mu )}\varepsilon _{0\alpha \beta \gamma }\delta _{0\mu }\end{aligned}}}$

where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation ${\textstyle \sum _{(\alpha \ldots )}}$ is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector[]

For

${\displaystyle {\vec {a}}=a{\hat {n}},\quad |{\hat {n}}|=1,}$

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

${\displaystyle ({\hat {n}}\cdot {\vec {\sigma }})^{2p}=I}$

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, 2q + 1, q = 0, 1, 2, 3, ...

${\displaystyle \left({\hat {n}}\cdot {\vec {\sigma }}\right)^{2q+1}={\hat {n}}\cdot {\vec {\sigma }}\,.}$

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

${\displaystyle {\begin{aligned}e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}&=\sum _{k=0}^{\infty }{\frac {i^{k}\left[a\left({\hat {n}}\cdot {\vec {\sigma }}\right)\right]^{k}}{k!}}\\&=\sum _{p=0}^{\infty }{\frac {(-1)^{p}(a{\hat {n}}\cdot {\vec {\sigma }})^{2p}}{(2p)!}}+i\sum _{q=0}^{\infty }{\frac {(-1)^{q}(a{\hat {n}}\cdot {\vec {\sigma }})^{2q+1}}{(2q+1)!}}\\&=I\sum _{p=0}^{\infty }{\frac {(-1)^{p}a^{2p}}{(2p)!}}+i({\hat {n}}\cdot {\vec {\sigma }})\sum _{q=0}^{\infty }{\frac {(-1)^{q}a^{2q+1}}{(2q+1)!}}\\\end{aligned}}}$.

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

${\displaystyle ~~e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}=I\cos {a}+i({\hat {n}}\cdot {\vec {\sigma }})\sin {a}~~}$

(2)

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

Note that

${\displaystyle \det[ia({\hat {n}}\cdot {\vec {\sigma }})]=a^{2}}$,

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,^[3]

${\displaystyle f(a({\hat {n}}\cdot {\vec {\sigma }}))=I{\frac {f(a)+f(-a)}{2}}+{\hat {n}}\cdot {\vec {\sigma }}{\frac {f(a)-f(-a)}{2}}~.}$

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).^[c] One may directly solve for c in

${\displaystyle {\begin{aligned}e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}e^{ib\left({\hat {m}}\cdot {\vec {\sigma }}\right)}&=I\left(\cos a\cos b-{\hat {n}}\cdot {\hat {m}}\sin a\sin b\right)+i\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}~\sin a\sin b\right)\cdot {\vec {\sigma }}\\&=I\cos {c}+i\left({\hat {k}}\cdot {\vec {\sigma }}\right)\sin c\\&=e^{ic\left({\hat {k}}\cdot {\vec {\sigma }}\right)},\end{aligned}}}$

which specifies the generic group multiplication, where, manifestly,

${\displaystyle \cos c=\cos a\cos b-{\hat {n}}\cdot {\hat {m}}\sin a\sin b~,}$

the spherical law of cosines. Given c, then,

${\displaystyle {\hat {k}}={\frac {1}{\sin c}}\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}\sin a\sin b\right)~.}$

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to^[4]

${\displaystyle e^{ic{\hat {k}}\cdot {\vec {\sigma }}}=\exp \left(i{\frac {c}{\sin c}}\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}~\sin a\sin b\right)\cdot {\vec {\sigma }}\right)~.}$

(Of course, when ${\displaystyle {\hat {n}}}$ is parallel to ${\displaystyle {\hat {m}}}$, so is ${\displaystyle {\hat {k}}}$, and c = a + b.)

Adjoint action[]

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle a,

${\displaystyle e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}~{\vec {\sigma }}~e^{-ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}={\vec {\sigma }}\cos(2a)+{\hat {n}}\times {\vec {\sigma }}~\sin(2a)+{\hat {n}}~{\hat {n}}\cdot {\vec {\sigma }}~(1-\cos(2a))~.}$

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

${\displaystyle {\vec {\sigma }}_{\alpha \beta }\cdot {\vec {\sigma }}_{\gamma \delta }\equiv \sum _{k=1}^{3}\sigma _{\alpha \beta }^{k}\,\sigma _{\gamma \delta }^{k}=2\,\delta _{\alpha \delta }\,\delta _{\beta \gamma }-\delta _{\alpha \beta }\,\delta _{\gamma \delta }~.}$

Proof: The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices means that we can express any matrix M as

${\displaystyle M=c\,I+\sum _{k}a_{k}\,\sigma ^{k}~}$

where c is a complex number, and a is a 3 component, complex vector. It is straightforward to show, using the properties listed above, that

${\displaystyle \operatorname {tr} \left(\sigma ^{j}\,\sigma ^{k}\right)=2\,\delta _{jk}}$

where "tr" denotes the trace, and hence that

${\displaystyle {\begin{aligned}c&={\tfrac {1}{2}}\,\operatorname {tr} \,M\,,\quad \ a_{k}={\tfrac {1}{2}}\,\operatorname {tr} \,\sigma ^{k}\,M~.\\[3pt]\therefore ~~2\,M&=I\,\operatorname {tr} \,M+\sum _{k}\sigma ^{k}\,\operatorname {tr} \,\sigma ^{k}M~,\end{aligned}}}$

which can be rewritten in terms of matrix indices as

${\displaystyle 2\,M_{\alpha \beta }=\delta _{\alpha \beta }\,M_{\gamma \gamma }+\sum _{k}\sigma _{\alpha \beta }^{k}\,\sigma _{\gamma \delta }^{k}\,M_{\delta \gamma }~,}$

where summation over the repeated indices is implied γ and δ. Since this is true for any choice of the matrix M, the completeness relation follows as stated above.◻

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

${\displaystyle \sum _{k=0}^{3}\sigma _{\alpha \beta }^{k}\,\sigma _{\gamma \delta }^{k}=2\,\delta _{\alpha \delta }\,\delta _{\beta \gamma }~.}$

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,

${\displaystyle {\vec {a}}={\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \end{pmatrix}}}$, the idempotent density matrix

${\displaystyle {\tfrac {1}{2}}\,\left(\,\mathbf {1} +{\vec {a}}\cdot {\vec {\sigma }}\,\right)={\begin{pmatrix}\cos ^{2}\left({\frac {\,\theta \,}{2}}\right)&e^{-i\,\phi }\sin \left({\frac {\,\theta \,}{2}}\right)\cos \left({\frac {\,\theta \,}{2}}\right)\\e^{+i\,\phi }\sin \left({\frac {\,\theta \,}{2}}\right)\cos \left({\frac {\,\theta \,}{2}}\right)&\sin ^{2}\left({\frac {\,\theta \,}{2}}\right)\end{pmatrix}}}$

acts on the state eigenvector ${\displaystyle \,{\begin{pmatrix}\cos \left({\frac {\,\theta \,}{2}}\right)&e^{+i\phi }\,\sin \left({\frac {\,\theta \,}{2}}\right)\end{pmatrix}}\,}$ 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 ${\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}$,

${\displaystyle P_{jk}\left|\sigma _{j}\sigma _{k}\right\rangle =\left|\sigma _{k}\sigma _{j}\right\rangle ~.}$

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

${\displaystyle P_{jk}={\frac {1}{2}}\,\left({\vec {\sigma }}_{j}\cdot {\vec {\sigma }}_{k}+1\right)~.}$

Its eigenvalues are therefore^[d] 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 ${\displaystyle {\mathfrak {su}}_{2}}$ is the 3-dimensional real algebra spanned by the set {iσ_k}. In compact notation,

${\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \{\;i\,\sigma _{1}\,,\;i\,\sigma _{2}\,,\;i\,\sigma _{3}\;\}~.}$

As a result, each i σ_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

${\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \left\{{\frac {\,i\,\sigma _{1}\,}{2}},{\frac {\,i\,\sigma _{2}\,}{2}},{\frac {\,i\,\sigma _{3}\,}{2}}\right\}~.}$

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

SO(3)[]

The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, 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 su(2) and 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) to SO(3), see relationship between SO(3) and SU(2).

Quaternions[]

The real linear span of {  I,   i σ₁, i σ₂,   i σ₃ } is isomorphic to the real algebra of quaternions ${\displaystyle \mathbb {H} }$, represented by the span of the basis vectors ${\displaystyle ~\left\{\;\mathbf {1} ,\,\mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} \;\right\}~.}$ The isomorphism from ${\displaystyle \mathbb {H} }$ to this set is given by the following map (notice the reversed signs for the Pauli matrices):

${\displaystyle \mathbf {1} \mapsto I,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3}=-i\,\sigma _{1},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1}=-i\,\sigma _{2},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2}=-i\,\sigma _{3}.}$

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,^[5]

${\displaystyle \mathbf {1} \mapsto I,\quad \mathbf {i} \mapsto i\,\sigma _{3}\,,\quad \mathbf {j} \mapsto i\,\sigma _{2}\,,\quad \mathbf {k} \mapsto i\,\sigma _{1}~.}$

As the set of versors U ⊂ ${\displaystyle \mathbb {H} }$ 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[]

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters.^[6] The matrix P corresponding to the position ${\displaystyle {\vec {x}}}$ of a point in space is defined in terms of the above Pauli vector matrix,

${\displaystyle P={\vec {x}}\cdot {\vec {\sigma }}=x\,\sigma _{x}+y\,\sigma _{y}+z\,\sigma _{z}~.}$

Consequently, the transformation matrix Q_θ for rotations about the x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as^[6]

${\displaystyle Q_{\theta }={\boldsymbol {1}}\,\cos {\frac {\theta }{2}}+i\,\sigma _{x}\sin {\frac {\theta }{2}}~.}$

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 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, iσ_j are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 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 1⁄2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration. This is due to the 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 S², 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 J = ħ/2σ, 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 algebra. 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.^[7]

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli matrices.

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

${\displaystyle {\mathsf {\Sigma }}_{k}={\begin{pmatrix}{\mathsf {\sigma }}_{k}&0\\0&{\mathsf {\sigma }}_{k}\end{pmatrix}}.}$

It follows from this definition that the ${\displaystyle \;{\mathsf {\Sigma }}_{k}\;}$ 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 ${\displaystyle {\mathsf {\Sigma }}_{k}}$ 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 ${\displaystyle \;\Sigma _{jk}\equiv \epsilon _{jk\ell }{\mathsf {\Sigma }}_{j}~.}$ The remaining three, ${\displaystyle \;-i\,\Sigma _{0k}\equiv {\mathsf {\alpha }}_{k}\;,}$ where the Dirac α_k matrices are defined as

${\displaystyle {\mathsf {\alpha }}_{k}={\begin{pmatrix}0&{\mathsf {\sigma }}_{k}\\{\mathsf {\sigma }}_{k}&0\end{pmatrix}}~.}$

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

${\displaystyle \Sigma _{\mu \nu }={\frac {i}{2}}\left[\gamma _{\mu },\gamma _{\nu }\right]~.}$

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[]

• 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 second Pauli matrix is an exchange matrix of order two)

Remarks[]

Notes[]

References[]

• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
• Schiff, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 978-0070552876.
• Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 978-0-521-14505-3.