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Pauli matrices

From Wikipedia, the free encyclopedia
Matrices important in quantum mechanics and the study of spin

Wolfgang Pauli (1900–1958), c. 1924. Pauli received theNobel Prize in Physics in 1945, nominated byAlbert Einstein, for thePauli exclusion principle.

Inmathematical physics andmathematics, thePauli matrices are a set of three2×2{\displaystyle 2\times 2}complexmatrices that aretraceless,Hermitian,involutory andunitary. They are usually denoted by theGreek letterσ{\displaystyle \sigma } (sigma), and occasionally byτ{\displaystyle \tau } (tau) when used in connection withisospin symmetries.σ1=σx=(0110),σ2=σy=(0−ii0),σ3=σz=(100−1).{\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 physicistWolfgang Pauli. Inquantum mechanics, they occur in thePauli equation, which takes into account the interaction of thespin of a particle with an externalelectromagnetic 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 isHermitian, and together with the identity matrixI{\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrixσ0{\displaystyle \sigma _{0}}), the Pauli matrices form abasis of thevector space of2×2{\displaystyle 2\times 2} Hermitian matrices over thereal numbers, under addition. This means that any2×2{\displaystyle 2\times 2}Hermitian matrix can be written in a unique way as alinear combination of Pauli matrices, with all coefficients being real numbers.

The Pauli matrices satisfy the useful product relation:

σi σj=δij I+i εijk σk ,{\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}}

whereδij{\displaystyle \delta _{ij}} is theKronecker delta, which equals+1{\displaystyle +1} ifi=j{\displaystyle i=j} otherwise0{\displaystyle 0}, and theLevi-Civita symbolεijk{\displaystyle \varepsilon _{ijk}}is used.

Hermitian operators representobservables in quantum mechanics, so the Pauli matrices span the space of observables of thecomplex two-dimensionalHilbert space. In the context of Pauli's work,σk{\displaystyle \sigma _{k}} represents the observable corresponding to spin along thek{\displaystyle k}th coordinate axis in three-dimensionalEuclidean spaceR3{\displaystyle \mathbb {R} ^{3}}.

The Pauli matrices (after multiplication byi{\displaystyle i} to make themanti-Hermitian) also generate transformations in the sense ofLie algebras: The matricesiσ1{\displaystyle i\sigma _{1}},iσ2{\displaystyle i\sigma _{2}}, andiσ3{\displaystyle i\sigma _{3}} form a basis for the real Lie algebrasu(2){\displaystyle {\mathfrak {su}}(2)}, whichexponentiates to the special unitary groupSU(2).[a] Thealgebra generated by the three Pauli matrices isisomorphic to theClifford algebra of R3{\displaystyle \ \mathbb {R} ^{3}}[1] and the (unital)associative algebra generated byiσ1{\displaystyle i\sigma _{1}},iσ2{\displaystyle i\sigma _{2}}, andiσ3{\displaystyle i\sigma _{3}} functions identically (is isomorphic) to that ofquaternions (H{\displaystyle \mathbb {H} }).

Algebraic properties

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Cayley table; the entry shows the value of the row times the column.
×σx{\displaystyle \sigma _{x}}σy{\displaystyle \sigma _{y}}σz{\displaystyle \sigma _{z}}
σx{\displaystyle \sigma _{x}}I{\displaystyle I}i σz{\displaystyle i\ \sigma _{z}}−i σy{\displaystyle -i\ \sigma _{y}}
σy{\displaystyle \sigma _{y}}−i σz{\displaystyle -i\ \sigma _{z}}I{\displaystyle I}i σx{\displaystyle i\ \sigma _{x}}
σz{\displaystyle \sigma _{z}}iσy{\displaystyle i\sigma _{y}}−i σx{\displaystyle -i\ \sigma _{x}}I{\displaystyle I}

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

σj=(δj3δj1−i δj2δj1+i δj2−δj3) .{\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 ofj∈{1,2,3}{\displaystyle 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 areinvolutory:

σ12=σ22=σ32=−i σ1 σ2 σ3=(1001)=I,{\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} ,}

whereI{\displaystyle \mathbb {I} } is theidentity matrix.

Thedeterminants andtraces of the Pauli matrices are

detσj=−1 ,tr⁡σj=0 ,{\displaystyle {\begin{aligned}\det \sigma _{j}&=-1\ ,\\\operatorname {tr} \sigma _{j}&=0\ ,\end{aligned}}}

from which we can deduce that each matrixσj{\displaystyle \sigma _{j}} haseigenvalues±1{\displaystyle \pm 1}.

With the inclusion of the identity matrixI{\displaystyle \mathbb {I} } (sometimes denotedσ0{\displaystyle \sigma _{0}}), the Pauli matrices form an orthogonal basis (in the sense ofHilbert–Schmidt) of theHilbert space H2 {\displaystyle \ {\mathcal {H}}_{2}\ } of2×2{\displaystyle 2\times 2} Hermitian matrices overR{\displaystyle \mathbb {R} } and the Hilbert spaceM2,2(C){\displaystyle {\mathcal {M}}_{2,2}(\mathbb {C} )} of allcomplex2×2{\displaystyle 2\times 2} matrices overC{\displaystyle \mathbb {C} }.

Commutation and anti-commutation relations

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Commutation relations

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The Pauli matrices obey the followingcommutation relations:

[σj,σk]=2 i εjkl σl .{\displaystyle [\sigma _{j},\sigma _{k}]=2\ i\ \varepsilon _{jkl}\ \sigma _{l}~.}

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra(R3,×) ≅ su(2) ≅ so(3) .{\displaystyle (\mathbb {R} ^{3},\times )\ \cong \ {\mathfrak {su}}(2)\ \cong \ {\mathfrak {so}}(3)~.}

Anticommutation relations

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They also satisfy theanticommutation relations:

{σj,σk}=2 δjk I ,{\displaystyle \{\sigma _{j},\sigma _{k}\}=2\ \delta _{jk}\ I\ ,}

where{σj,σk}{\displaystyle \{\sigma _{j},\sigma _{k}\}} is defined as σj σk+σk σj ,{\displaystyle \ \sigma _{j}\ \sigma _{k}+\sigma _{k}\ \sigma _{j}\ ,} andδjk is theKronecker delta.I denotes the2 × 2 identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of theClifford algebra for R3 ,{\displaystyle \ \mathbb {R} ^{3}\ ,} denoted Cl3(R) .{\displaystyle \ \mathrm {Cl} _{3}(\mathbb {R} )~.}

The usual construction of generators σjk=14[σj,σk] {\displaystyle \ \sigma _{jk}={\tfrac {1}{4}}[\sigma _{j},\sigma _{k}]\ } of so(3) {\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:

CommutatorsAnticommutators
[ σ1,σ1 ]=   0[ σ1,σ2 ]=2 i σ3[ σ2,σ3 ]=2 i σ1[ σ3,σ1 ]=2 i σ2{\displaystyle {\begin{aligned}{\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{aligned}}}    { σ1,σ1 }=2 I{ σ1,σ2 }= 0{ σ2,σ3 }= 0{ σ3,σ1 }= 0{\displaystyle {\begin{aligned}{\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{aligned}}}

Eigenvectors and eigenvalues

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Each of the (Hermitian) Pauli matrices has twoeigenvalues:±1{\displaystyle \pm 1}. The correspondingnormalizedeigenvectors are

ψx+=12[11],ψx−=12[1−1],ψy+=12[1i],ψy−=12[1−i],ψz+=[10],ψz−=[01].{\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 vectors

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The Pauli vector is defined by[b]σ=σ1x^1+σ2x^2+σ3x^3,{\displaystyle {\boldsymbol {\sigma }}=\sigma _{1}{\boldsymbol {\hat {x}}}_{1}+\sigma _{2}{\boldsymbol {\hat {x}}}_{2}+\sigma _{3}{\boldsymbol {\hat {x}}}_{3},}wherex^1{\displaystyle {\boldsymbol {\hat {x}}}_{1}},x^2{\displaystyle {\boldsymbol {\hat {x}}}_{2}}, andx^3{\displaystyle {\boldsymbol {\hat {x}}}_{3}} are an equivalent notation for the more familiarx^{\displaystyle {\boldsymbol {\hat {x}}}},y^{\displaystyle {\boldsymbol {\hat {y}}}}, andz^{\displaystyle {\boldsymbol {\hat {z}}}}.

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis[2] as follows:a⋅σ=∑k,lakσℓx^k⋅x^ℓ=∑kakσk=(a3a1−ia2a1+ia2−a3) .{\displaystyle {\begin{aligned}{\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}-ia_{2}\\a_{1}+ia_{2}&-a_{3}\end{pmatrix}}~.\end{aligned}}}

More formally, this defines a map fromR3{\displaystyle \mathbb {R} ^{3}} to the vector space of traceless Hermitian2×2{\displaystyle 2\times 2} matrices. This map encodes structures ofR3{\displaystyle \mathbb {R} ^{3}} as a normed vector space and as a Lie algebra (with thecross-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 2×2 {\displaystyle \ 2\times 2\ } Hermitian traceless matrix-valued dual vector, that is, an element of Mat2×2(C)⊗(R3)∗ {\displaystyle \ \mathrm {Mat} _{2\times 2}(\mathbb {C} )\otimes (\mathbb {R} ^{3})^{*}\ } that mapsa↦a⋅σ{\displaystyle {\boldsymbol {a}}\mapsto {\boldsymbol {a}}\cdot {\boldsymbol {\sigma }}}

Completeness relation

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Each component ofa{\displaystyle {\boldsymbol {a}}} can be recovered from the matrix (seecompleteness relation below)12tr⁡[( a⋅σ ) σ ]=a{\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 mapa↦a⋅σ{\displaystyle {\boldsymbol {a}}\mapsto {\boldsymbol {a}}\cdot {\boldsymbol {\sigma }}}, making it manifest that the map is a bijection.

Determinant

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The norm is given by the determinant (up to a minus sign)det( a→⋅σ→ ) = −a→⋅a→ = −| a→ |2 .{\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 SU(2) {\displaystyle \ \mathrm {SU} (2)\ } matrixU{\displaystyle U} on this space of matrices,

 U∗a→⋅σ→ := U a→⋅σ→ U−1 ,{\displaystyle \ U*{\vec {a}}\cdot {\vec {\sigma }}\ :=\ U\ {\vec {a}}\cdot {\vec {\sigma }}\ U^{-1}\ ,}

we find det(U∗a→⋅σ→) = det(a→⋅σ→) ,{\displaystyle \ \det(U*{\vec {a}}\cdot {\vec {\sigma }})\ =\ \det({\vec {a}}\cdot {\vec {\sigma }})\ ,} and that U∗a→⋅σ→ {\displaystyle \ U*{\vec {a}}\cdot {\vec {\sigma }}\ } is Hermitian and traceless. It then makes sense to define U∗a→⋅σ→ = a→′⋅σ→ ,{\displaystyle \ U*{\vec {a}}\cdot {\vec {\sigma }}\ =\ {\vec {a}}'\cdot {\vec {\sigma }}\ ,} where a→′ {\displaystyle \ {\vec {a}}'\ } has the same norm asa→,{\displaystyle {\vec {a}},} and therefore interpretU{\displaystyle U} as a rotation of three-dimensional space. In fact, it turns out that thespecial restriction onU{\displaystyle U} implies that the rotation is orientation preserving. This allows the definition of a map R:SU(2)→SO(3) {\displaystyle \ R:\mathrm {SU} (2)\to \mathrm {SO} (3)\ } given by

 U∗a→⋅σ→ = a→′⋅σ→ =: (R(U) a→)⋅σ→ ,{\displaystyle \ U*{\vec {a}}\cdot {\vec {\sigma }}\ =\ {\vec {a}}'\cdot {\vec {\sigma }}\ =:\ (R(U)\ {\vec {a}})\cdot {\vec {\sigma }}\ ,}

where R(U) ∈ SO(3) .{\displaystyle \ R(U)\ \in \ \mathrm {SO} (3)~.} This map is the concrete realization of the double cover of SO(3) {\displaystyle \ \mathrm {SO} (3)\ } by SU(2) ,{\displaystyle \ \mathrm {SU} (2)\ ,} and therefore shows that SU(2) ≅ Spin(3) .{\displaystyle \ \mathrm {SU} (2)\ \cong \ \mathrm {Spin} (3)~.} The components ofR(U){\displaystyle R(U)} can be recovered using the tracing process above:

 R(U)ij=12 tr( σiUσjU−1 ) .{\displaystyle \ R(U)_{ij}={\frac {1}{2}}\ \operatorname {tr} \!\left(\ \sigma _{i}U\sigma _{j}U^{-1}\ \right)~.}

Cross-product

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The cross-product is given by the matrix commutator (up to a factor of 2 i {\displaystyle \ 2\ i\ })[ a→⋅σ→, b→⋅σ→ ]=2 i (a→×b→)⋅σ→ .{\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 R3 {\displaystyle \ \mathbb {R} ^{3}\ } is a Lie algebra (seeKilling form).

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

Eigenvalues and eigenvectors

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The eigenvalues of a→⋅σ→ {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } are ±|a→| .{\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 (a→⋅σ→)2−|a→|2=0 ,{\displaystyle \ ({\vec {a}}\cdot {\vec {\sigma }})^{2}-|{\vec {a}}|^{2}=0\ ,} since this can be factorised into (a→⋅σ→−|a→|)(a→⋅σ→+|a→|)=0 .{\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 isdiagonalizable) means this implies a→⋅σ→ {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } is diagonalizable with possible eigenvalues ±|a→| .{\displaystyle \ \pm |{\vec {a}}|~.} The tracelessness of a→⋅σ→ {\displaystyle \ {\vec {a}}\cdot {\vec {\sigma }}\ } means it has exactly one of each eigenvalue.

Its normalized eigenvectors areψ+=12|a→|(a3+|a→|)[a3+|a→|a1+ia2] ;ψ−=12|a→|(a3+|a→|)[ia2−a1a3+|a→|] .{\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 a3→−| a→ | .{\displaystyle \ a_{3}\to -\left|\ {\vec {a}}\ \right|~.} They can be rescued by lettinga→=| a→ |(ϵ, 0, −(1−ϵ22)) {\displaystyle {\vec {a}}=\left|\ {\vec {a}}\ \right|\left(\epsilon ,\ 0,\ -\left(1-{\tfrac {\epsilon ^{2}}{2}}\right)\right)\ } and taking the limit ϵ→0 ,{\displaystyle \ \epsilon \to 0\ ,} which yields the correct eigenvectors(0,1) and (1,0) of σz .{\displaystyle \ \sigma _{z}~.}

Alternatively, one may use spherical coordinates a→=a ( sin⁡ϑ cos⁡φ, sin⁡ϑ sin⁡φ, cos⁡ϑ ) {\displaystyle \ {\vec {a}}=a\ {\bigl (}\ \sin \vartheta \ \cos \varphi ,\ \sin \vartheta \ \sin \varphi ,\ \cos \vartheta \ {\bigr )}\ } to obtain the eigenvectors ψ+=( cos⁡ϑ2,sin⁡ϑ2 e+iφ ) {\displaystyle \ \psi _{+}=\left(\ \cos {\tfrac {\vartheta }{2}},\;\sin {\tfrac {\vartheta }{2}}\ e^{+i\varphi }\ \right)\ } and ψ−=( −sin⁡ϑ2 e−iφ,cos⁡ϑ2 ) .{\displaystyle \ \psi _{-}=\left(\ -\sin {\tfrac {\vartheta }{2}}\ e^{-i\varphi },\;\cos {\tfrac {\vartheta }{2}}\ \right)~.}

Pauli 4-vector

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The Pauli 4-vector, used in spinor theory, is written σμ {\displaystyle \ \sigma ^{\mu }\ } with components

 σμ=( I, σ→ ) .{\displaystyle \ \sigma ^{\mu }={\bigl (}\ I,\ {\vec {\sigma }}\ {\bigr )}~.}

This defines a map from R1,3 {\displaystyle \ \mathbb {R} ^{1,3}\ } to the vector space of Hermitian matrices,

 xμ↦xμσμ ,{\displaystyle \ x_{\mu }\mapsto x_{\mu }\sigma ^{\mu }\ ,}

which also encodes theMinkowski metric (withmostly minus convention) in its determinant:

 det( xμσμ )=η(x,x) .{\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

 σ¯μ=( I,−σ→ ) .{\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 xν=12tr( σ¯ν(xμσμ) ) .{\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 R1,3 ;{\displaystyle \ \mathbb {R} ^{1,3}\ ;} in this case the matrix group is SL(2,C) ,{\displaystyle \ \mathrm {SL} (2,\mathbb {C} )\ ,} and this shows SL(2,C) ≅ Spin(1,3) .{\displaystyle \ \mathrm {SL} (2,\mathbb {C} )\ \cong \ \mathrm {Spin} (1,3)~.} Similarly to above, this can be explicitly realized for S∈SL(2,C) {\displaystyle \ S\in \mathrm {SL} (2,\mathbb {C} )\ } with components

 Λ(S)μν=12tr( σ¯ν S σμ S† ) .{\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 σμ .{\displaystyle \ \sigma ^{\mu }~.} For 2×2 {\displaystyle \ 2\times 2\ } matrices, the following identity holds:

 det( A+B ) = det(A) + det(B) + tr⁡(A) tr⁡(B) − tr⁡( A B ) .{\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 A,B {\displaystyle \ A,B\ } are chosen to be different σμ ,{\displaystyle \ \sigma ^{\mu }\ ,} the cross-terms vanish. It then follows, now showing summation explicitly,det(∑μxμσμ)=∑μdet(xμσμ).{\textstyle \det \left(\sum _{\mu }x_{\mu }\sigma ^{\mu }\right)=\sum _{\mu }\det \left(x_{\mu }\sigma ^{\mu }\right).} Since the matrices are 2×2 ,{\displaystyle \ 2\times 2\ ,} this is equal to ∑μxμ2det(σμ)=η(x,x) .{\textstyle \ \sum _{\mu }x_{\mu }^{2}\det(\sigma ^{\mu })=\eta (x,x)~.}

Relation to dot and cross product

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

[σj,σk]+{σj,σk}=(σjσk−σkσj)+(σjσk+σkσj)2iεjkℓσℓ+2δjkI=2σjσk{\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,

  σjσk=δjkI+iεjkℓσℓ . {\displaystyle ~~\sigma _{j}\sigma _{k}=\delta _{jk}I+i\varepsilon _{jk\ell }\,\sigma _{\ell }~.~}

Contracting each side of the equation with components of two3-vectorsap andbq (which commute with the Pauli matrices, i.e.,apσq =σqap) for each matrixσq and vector componentap (and likewise withbq) yields

  ajbkσjσk=ajbk(iεjkℓσℓ+δjkI)ajσjbkσk=iεjkℓajbkσℓ+ajbkδjkI .{\displaystyle ~~{\begin{aligned}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{aligned}}~.}

Finally, translating the index notation for thedot product andcross product results in

1

Ifi is identified with the pseudoscalarσx σy σz then the right hand side becomes a⋅b+a∧b ,{\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 asJ =⁠ħ/2⁠σ , thenJ satisfies the commutation relation: J×J=i ℏJ {\displaystyle \ \mathbf {J} \times \mathbf {J} =i\ \hbar \mathbf {J} \ } Or equivalently, the Pauli vector satisfies: σ→2×σ→2=i σ→2 .{\displaystyle \ {\frac {\vec {\sigma }}{2}}\times {\frac {\vec {\sigma }}{2}}=i\ {\frac {\vec {\sigma }}{2}}~.}

Some trace relations

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

tr⁡(σj)=0tr⁡(σjσk)=2δjktr⁡(σjσkσℓ)=2iεjkℓtr⁡(σjσkσℓσm)=2(δjkδℓm−δjℓδkm+δjmδkℓ) .{\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 matrixσ0=I{\displaystyle \sigma _{0}=\mathbb {I} } is also considered, these relationships become

tr⁡(σα)=2δ0αtr⁡(σασβ)=2δαβtr⁡(σασβσγ)=2∑(αβγ)δαβδ0γ−4δ0αδ0βδ0γ+2iε0αβγtr⁡(σασβσγσμ)=2(δαβδγμ−δαγδβμ+δαμδβγ)+4(δαγδ0βδ0μ+δβμδ0αδ0γ)−8δ0αδ0βδ0γδ0μ+2i∑(αβγμ)ε0αβγδ0μ .{\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 indicesα,β,γ{\displaystyle \alpha ,\beta ,\gamma } andμ{\displaystyle \mu } assume values from{0,x,y,z}{\displaystyle \{0,x,y,z\}} and the notation∑(α…){\textstyle \sum _{(\alpha \ldots )}} is used to denote the sum over thecyclic permutation of the included indices.

Exponential of a Pauli vector

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For

a→=a n^,| n^ |=1 ,{\displaystyle {\vec {a}}=a\ {\hat {n}},\quad \left|\ {\hat {n}}\ \right|=1\ ,}

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

 (n^⋅σ→)2p=I ,{\displaystyle \ ({\hat {n}}\cdot {\vec {\sigma }})^{2p}=I\ ,}

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

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

 (n^⋅σ→)2q+1=n^⋅σ→ .{\displaystyle \ \left({\hat {n}}\cdot {\vec {\sigma }}\right)^{2q+1}={\hat {n}}\cdot {\vec {\sigma }}~.}

Matrix exponentiating, and using theTaylor series for sine and cosine,

eia(n^⋅σ→)=∑k=0∞ik[a(n^⋅σ→)]kk!=∑p=0∞(−1)p(an^⋅σ→)2p(2p)!+i∑q=0∞(−1)q(an^⋅σ→)2q+1(2q+1)!=I∑p=0∞(−1)pa2p(2p)!+i(n^⋅σ→)∑q=0∞(−1)qa2q+1(2q+1)! .{\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,

2

which isanalogous toEuler's formula, extended toquaternions. In particular,

ei a σ1=(cos⁡ai sin⁡ai sin⁡acos⁡a) ,ei a σ2=(cos⁡asin⁡a−sin⁡acos⁡a) ,ei a σ3=(ei a00e−i a) .{\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

det[ i a (n^⋅σ→) ]=a2 ,{\displaystyle \det \!\left[\ i\ a\ \left({\hat {n}}\cdot {\vec {\sigma }}\right)\ \right]=a^{2}\ ,}

while the determinant of the exponential itself is just1, which makes it thegeneric group element ofSU(2).

A more abstract version of formula(2) for a general2 × 2 matrix can be found in the article onmatrix exponentials. A general version of(2) for an analytic (ata and−a) function is provided by application ofSylvester's formula,[3]

 f( a(n^⋅σ→) ) = I  f(+a)+f(−a) 2 + n^⋅σ→  f(+a)−f(−a) 2 .{\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 ofSU(2)

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A straightforward application of formula(2) provides a parameterization of the composition law of the groupSU(2).[c] One may directly solve forc inei a(n^⋅σ→) ei b (m^⋅σ→)=I ( cos⁡a cos⁡b − n^⋅m^ sin⁡a sin⁡b ) + i ( n^ sin⁡a cos⁡b + m^ sin⁡b cos⁡a − n^×m^ sin⁡a sin⁡b )⋅σ→=I cos⁡c + i (k^⋅σ→) sin⁡c=ei c (k^⋅σ→) ,{\displaystyle {\begin{aligned}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{aligned}}}

which specifies the generic group multiplication, where, manifestly, cos⁡c=cos⁡a cos⁡b − n^⋅m^ sin⁡a sin⁡b ,{\displaystyle \ \cos c=\cos a\ \cos b\ -\ {\hat {n}}\cdot {\hat {m}}\ \sin a\ \sin b\ ,}thespherical law of cosines. Givenc, then, k^ = 1sin⁡c ( n^ sin⁡a cos⁡b + m^ sin⁡b cos⁡a−n^×m^ sin⁡a sin⁡b ) .{\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 respectiveBCH expansion in this case) simply amount to[4]

 eick^⋅σ→=exp⁡(icsin⁡c(n^sin⁡acos⁡b+m^sin⁡bcos⁡a−n^×m^ sin⁡asin⁡b)⋅σ→) .{\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 n^ {\displaystyle \ {\hat {n}}\ } is parallel to m^ ,{\displaystyle \ {\hat {m}}\ ,} so are k^ {\displaystyle \ {\hat {k}}\ } andc =a + b .)

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

Adjoint action

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It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any anglea{\displaystyle a} along any axisn^{\displaystyle {\hat {n}}}:Rn(−a) σ→ Rn(a)=eia2(n^⋅σ→) σ→ e−ia2(n^⋅σ→)=σ→cos⁡(a)+n^×σ→ sin⁡(a)+n^ n^⋅σ→ (1−cos⁡(a)) .{\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 Ry(−π2)σxRy(π2)=x^⋅(y^×σ→)=σz .{\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

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An alternative notation that is commonly used for the Pauli matrices is to write the vector indexk in the superscript, and the matrix indices as subscripts, so that the element in rowα and columnβ of thek-th Pauli matrix isσkαβ .

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

σ→αβ⋅σ→γδ≡∑k=13σαβk σγδk=2 δαδ δβγ−δαβ δγδ .{\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 matrixI, form an orthogonal basis for the Hilbert space of all 2 × 2complex matrices M2,2(C) {\displaystyle \ {\mathcal {M}}_{2,2}(\mathbb {C} )\ } over C ,{\displaystyle \ \mathbb {C} \ ,} means that we can express any 2 × 2 complex matrixM asM=c I+∑kak σk{\displaystyle M=c\ I+\sum _{k}a_{k}\ \sigma ^{k}}wherec is a complex number, anda is a 3-component, complex vector. It is straightforward to show, using the properties listed above, thattr⁡(σjσk)=2 δjk{\displaystyle \operatorname {tr} \left(\sigma ^{j}\,\sigma ^{k}\right)=2\ \delta _{jk}}where "tr" denotes thetrace, and hence thatc=12 trM ,ak=12 tr⁡ σk M .∴  2M=ItrM+∑kσktrσkM ,{\displaystyle {\begin{aligned}c&={}{\tfrac {1}{2}}\ \operatorname {tr} \,M\ ,{\begin{aligned}&&a_{k}&={\tfrac {1}{2}}\ \operatorname {tr} \ \sigma ^{k}\ M\end{aligned}}~.\\[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 as2 Mαβ=δαβ Mγγ+∑kσαβk σγδk Mδγ ,{\displaystyle 2\ M_{\alpha \beta }=\delta _{\alpha \beta }\ M_{\gamma \gamma }+\sum _{k}\sigma _{\alpha \beta }^{k}\ \sigma _{\gamma \delta }^{k}\ M_{\delta \gamma }\ ,}wheresummation over the repeated indices is impliedγ andδ. Since this is true for any choice of the matrixM, the completeness relation follows as stated above.Q.E.D.

As noted above, it is common to denote the 2 × 2 unit matrix byσ0 , soσ0αβ =δαβ . The completeness relation can alternatively be expressed as ∑k=03σαβk σγδk=2 δαδ δβγ .{\displaystyle \ \sum _{k=0}^{3}\sigma _{\alpha \beta }^{k}\ \sigma _{\gamma \delta }^{k}=2\ \delta _{\alpha \delta }\ \delta _{\beta \gamma }~.}

The fact that any Hermitiancomplex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to theBloch sphere representation of 2 × 2mixed 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 andtrace1 conditions.

For a pure state, in polar coordinates,a→=(sin⁡θcos⁡ϕsin⁡θsin⁡ϕcos⁡θ),{\displaystyle {\vec {a}}={\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \end{pmatrix}},} theidempotent density matrix12(1+a→⋅σ→)=(cos2⁡(θ2)e−iϕsin⁡(θ2)cos⁡(θ2)e+iϕsin⁡(θ2)cos⁡(θ2)sin2⁡(θ2)){\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 (cos⁡( θ 2)e+iϕ sin⁡( θ 2)) {\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 aprojection operator.

Relation with the permutation operator

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LetPjk be thetransposition (also known as a permutation) between two spinsσj andσk living in thetensor product space⁠C2⊗C2{\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}⁠ ,

Pjk|σjσk⟩=|σkσj⟩.{\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 asDirac's spin exchange operator,

 Pjk=12 (σ→j⋅σ→k+1) .{\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)

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The groupSU(2) is theLie group ofunitary2 × 2 matrices with unit determinant; itsLie algebra is the set of all2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that theLie algebrasu2{\displaystyle {\mathfrak {su}}_{2}} is the three-dimensional real algebraspanned by the set{iσk}. In compact notation,

su(2)=span⁡{iσ1,iσ2,iσ3}.{\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \{\;i\,\sigma _{1}\,,\;i\,\sigma _{2}\,,\;i\,\sigma _{3}\;\}.}

As a result, eachiσj can be seen as aninfinitesimal 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 properrepresentation ofsu(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization isλ =⁠1/2⁠ , so that

 su(2)=span⁡{ i σ1 2, i σ2 2, i σ3 2} .{\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, itsCartan decomposition is trivial.

SO(3)

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The Lie algebra su(2) {\displaystyle \ {\mathfrak {su}}(2)\ } isisomorphic to the Lie algebraso(3){\displaystyle {\mathfrak {so}}(3)}, which corresponds to the Lie groupSO(3), thegroup ofrotations in three-dimensional space. In other words, one can say that the i σj are a realization (and, in fact, the lowest-dimensional realization) ofinfinitesimal rotations in three-dimensional space. However, even though su(2) {\displaystyle \ {\mathfrak {su}}(2)\ } andso(3){\displaystyle {\mathfrak {so}}(3)} are isomorphic as Lie algebras,SU(2) andSO(3) are not isomorphic as Lie groups.SU(2) is actually adouble cover ofSO(3), meaning that there is a two-to-one group homomorphism fromSU(2) ↦SO(3) , seerelationship between SO(3) and SU(2).

Quaternions

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Main article:Spinor § Three dimensions

The real linear span of{I, iσ1,i σ2,i σ3} is isomorphic to the real algebra ofquaternions,H{\displaystyle \mathbb {H} }, represented by the span of the basis vectors {1, i, j, k} .{\displaystyle \ \left\{\;\mathbf {1} ,\ \mathbf {i} ,\ \mathbf {j} ,\ \mathbf {k} \;\right\}~.} The isomorphism from H {\displaystyle \ \mathbb {H} \ } to this set is given by the following map (notice the reversed signs for the Pauli matrices):1↦I,i↦−σ2σ3=−iσ1,j↦−σ3σ1=−iσ2,k↦−σ1σ2=−iσ3 .{\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]

1↦I,i↦iσ3,j↦iσ2,k↦iσ1 .{\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 ofversorsU⊂H{\displaystyle U\subset \mathbb {H} } forms a group isomorphic toSU(2),U gives yet another way of describingSU(2). The two-to-one homomorphism fromSU(2) toSO(3) may be given in terms of the Pauli matrices in this formulation.

Physics

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Classical mechanics

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Main article:Quaternions and spatial rotation

Inclassical mechanics, Pauli matrices are useful in the context of the Cayley–Klein parameters.[6] The matrixP{\displaystyle P} corresponding to the positionx{\displaystyle {\boldsymbol {x}}} of a point in space is defined in terms of the above Pauli vector matrix,

P=x⋅σ=xσx+yσy+zσz.{\displaystyle P={\boldsymbol {x}}\cdot {\boldsymbol {\sigma }}=x\,\sigma _{x}+y\,\sigma _{y}+z\,\sigma _{z}.}

Consequently, the transformation matrixQθ{\displaystyle Q_{\theta }} for rotations about thex{\displaystyle x}-axis through an angleθ{\displaystyle \theta } may be written in terms of Pauli matrices and the unit matrix as[6]

 Qθ=Icos⁡θ2+i σxsin⁡θ2.{\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

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Inquantum mechanics, each Pauli matrix is related to anangular momentum operator that corresponds to anobservable describing thespin of aspin1⁄2 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above,iσj{\displaystyle i\sigma _{j}} are the generators of aprojective representation (spin representation) of therotation group SO(3) acting onnon-relativistic particles with spin1⁄2. Thestates of the particles are represented as two-componentspinors. In the same way, the Pauli matrices are related to theisospin operator.

An interesting property of spin1⁄2 particles is that they must be rotated by an angle of4π{\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 the2-sphereS2{\displaystyle S^{2}} they are actually represented byorthogonal vectors in the two-dimensional complexHilbert space.

For a spin1⁄2 particle, the spin operator is given byJ=ℏ2σ{\displaystyle {\textbf {J}}={\frac {\hslash }{2}}{\boldsymbol {\sigma }}}, thefundamental representation ofSU(2). By takingKronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resultingspin operators for higher spin systems in three spatial dimensions, for arbitrarily largej, can be calculated using thisspin operator andladder operators. They can be found inRotation 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.[7]

Also useful in thequantum mechanics of multiparticle systems, the generalPauli groupGn{\displaystyle G_{n}} is defined to consist of alln{\displaystyle n}-foldtensor products of Pauli matrices.

Relativistic quantum mechanics

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Inrelativistic quantum mechanics, the spinors in four dimensions are4 × 1 (or1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as

Σk=(σk00σk) .{\displaystyle {\mathsf {\Sigma }}_{k}={\begin{pmatrix}{\mathsf {\sigma }}_{k}&0\\0&{\mathsf {\sigma }}_{k}\end{pmatrix}}~.}

It follows from this definition that the Σk {\displaystyle \ {\mathsf {\Sigma }}_{k}\ } matrices have the same algebraic properties as the σk matrices.

However,relativistic angular momentum is not a three-vector, but a second orderfour-tensor. Hence Σk {\displaystyle \ {\mathsf {\Sigma }}_{k}\ } needs to be replaced byΣμν, the generator ofLorentz transformations on spinors. By the antisymmetry of angular momentum, theΣμν are also antisymmetric. Hence there are only six independent matrices.

The first three are the Σkℓ≡ϵjkℓΣj .{\displaystyle \ \Sigma _{k\ell }\equiv \epsilon _{jk\ell }{\mathsf {\Sigma }}_{j}~.} The remaining three, −i Σ0k≡αk ,{\displaystyle \ -i\ \Sigma _{0k}\equiv {\mathsf {\alpha }}_{k}\ ,} where theDiracαk matrices are defined as

 αk=(0σkσk0) .{\displaystyle \ {\mathsf {\alpha }}_{k}={\begin{pmatrix}0&{\mathsf {\sigma }}_{k}\\{\mathsf {\sigma }}_{k}&0\end{pmatrix}}~.}

The relativistic spin matricesΣμν are written in compact form in terms of commutator ofgamma matrices as

 Σμν=i2[γμ,γν] .{\displaystyle \ \Sigma _{\mu \nu }={\frac {i}{2}}{\bigl [}\gamma _{\mu },\gamma _{\nu }{\bigr ]}~.}

Quantum information

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Inquantum information, single-qubitquantum gates are 2 × 2unitary 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–Ydecomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X–Ydecomposition of a single-qubit gate ".

See also

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Remarks

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  1. ^This conforms to the conventionin mathematics for thematrix exponential,i σ ⟼ exp(i σ) . In the conventionin physics,σ ⟼ exp(−i σ) , hence in it no pre-multiplication byi is necessary to land inSU(2) .
  2. ^The Pauli vector is a formal device. It may be thought of as an element of M2(C)⊗R3 ,{\displaystyle \ {\mathcal {M}}_{2}(\mathbb {C} )\otimes \mathbb {R} ^{3}\ ,} where thetensor product space is endowed with a mapping ⋅:R3×(M2(C)⊗R3)→M2(C) {\displaystyle \ \cdot :\mathbb {R} ^{3}\times ({\mathcal {M}}_{2}(\mathbb {C} )\otimes \mathbb {R} ^{3})\to {\mathcal {M}}_{2}(\mathbb {C} )\ } induced by thedot product on R3 .{\displaystyle \ \mathbb {R} ^{3}~.}
  3. ^The relation amonga, b, c, n, m, k derived here in the2 × 2 representation holds forall representations ofSU(2), being agroup identity. Note that, by virtue of the standard normalization of that group's generators ashalf the Pauli matrices, the parametersa,b,c correspond tohalf the rotation angles of the rotation group. That is, the Gibbs formula linked amounts to k^tan⁡c2=(n^ tan⁡a2+m^ tan⁡b2−m^ ×n^ tan⁡a2 tan⁡b2)/(1−m^⋅n^ tan⁡a2 tan⁡b2) .{\displaystyle \ {\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}})~.}
  4. ^Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is(1000001001000001) .{\displaystyle \left({\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{smallmatrix}}\right)~.}

Notes

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  1. ^Gull, S.F.; Lasenby, A.N.; Doran, C.J.L. (January 1993)."Imaginary numbers are not Real – the geometric algebra of spacetime"(PDF).Foundations of Physics.23 (9):1175–1201.Bibcode:1993FoPh...23.1175G.doi:10.1007/BF01883676.S2CID 14670523. Archived fromthe original(PDF) on 9 October 2023. Retrieved5 May 2023 – via geometry.mrao.cam.ac.uk.
  2. ^See thespinor map.
  3. ^Nielsen, Michael A.;Chuang, Isaac L. (2000).Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press.ISBN 978-0-521-63235-5.OCLC 43641333.
  4. ^Gibbs, J.W. (1884)."4. Concerning the differential and integral calculus of vectors".Elements of Vector Analysis. New Haven, CT: Tuttle, Moorehouse & Taylor. p. 67. In fact, however, the formula goes back toOlinde Rodrigues (1840), replete with half-angle:Rodrigues, Olinde (1840)."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"(PDF).J. Math. Pures Appl.5:380–440.
  5. ^Nakahara, Mikio (2003).Geometry, Topology, and Physics (2nd ed.). CRC Press. p. xxii.ISBN 978-0-7503-0606-5 – via Google Books.
  6. ^abGoldstein, Herbert (1959).Classical Mechanics. Addison-Wesley. pp. 109–118.OCLC 3175838.
  7. ^Curtright, T.L.;Fairlie, D.B.;Zachos, C.K. (2014). "A compact formula for rotations as spin matrix polynomials".SIGMA.10: 084.arXiv:1402.3541.Bibcode:2014SIGMA..10..084C.doi:10.3842/SIGMA.2014.084.S2CID 18776942.

References

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Matrix classes
Explicitly constrained entries
Constant
Conditions oneigenvalues or eigenvectors
Satisfying conditions onproducts orinverses
With specific applications
Used instatistics
Used ingraph theory
Used in science and engineering
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