In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter σ {\displaystyle \sigma } (sigma), and occasionally by τ {\displaystyle \tau } (tau) when used in connection with isospin symmetries. σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 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 physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the identity matrix I {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix σ 0 {\displaystyle \sigma _{0}} ), the Pauli matrices form a basis of the vector space of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices over the real numbers, under addition. This means that any 2 × 2 {\displaystyle 2\times 2} Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. The Pauli matrices satisfy the useful product relation:
σ i σ j = δ i j I + i ε i j k σ k , {\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}}
where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, which equals + 1 {\displaystyle +1} if i = j {\displaystyle i=j} otherwise 0 {\displaystyle 0} , and the Levi-Civita symbol ε i j k {\displaystyle \varepsilon _{ijk}} is used. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, σ k {\displaystyle \sigma _{k}} represents the observable corresponding to spin along the k {\displaystyle k} th coordinate axis in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} . The Pauli matrices (after multiplication by i {\displaystyle i} to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: The matrices i σ 1 {\displaystyle i\sigma _{1}} , i σ 2 {\displaystyle i\sigma _{2}} , and i σ 3 {\displaystyle i\sigma _{3}} form a basis for the real Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special unitary group SU(2). The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of R 3 {\displaystyle \ \mathbb {R} ^{3}} and the (unital) associative algebra generated by i σ 1 {\displaystyle i\sigma _{1}} , i σ 2 {\displaystyle i\sigma _{2}} , and i σ 3 {\displaystyle i\sigma _{3}} functions identically (is isomorphic) to that of quaternions ( H {\displaystyle \mathbb {H} } ).
Algebraic properties
All three of the Pauli matrices can be compacted into a single expression:
σ j = ( δ j 3 δ j 1 − i δ j 2 δ j 1 + i δ j 2 − δ j 3 ) . {\displaystyle \sigma _{j}={\begin{pmatrix}\delta _{j3}&\delta _{j1}-i\ \delta _{j2}\\\delta _{j1}+i\ \delta _{j2}&-\delta _{j3}\end{pmatrix}}~.}
This expression is useful for "selecting" any one of the matrices numerically by substituting values of j ∈ { 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 are involutory:
σ 1 2 = σ 2 2 = σ 3 2 = − i σ 1 σ 2 σ 3 = ( 1 0 0 1 ) = 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} ,}
where I {\displaystyle \mathbb {I} } is the identity matrix. The determinants and traces of the Pauli matrices are
det σ j = − 1 , tr σ j = 0 , {\displaystyle {\begin{aligned}\det \sigma _{j}&=-1\ ,\\\operatorname {tr} \sigma _{j}&=0\ ,\end{aligned}}}
from which we can deduce that each matrix σ j {\displaystyle \sigma _{j}} has eigenvalues ± 1 {\displaystyle \pm 1} . With the inclusion of the identity matrix I {\displaystyle \mathbb {I} } (sometimes denoted σ 0 {\displaystyle \sigma _{0}} ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space H 2 {\displaystyle \ {\mathcal {H}}_{2}\ } of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices over R {\displaystyle \mathbb {R} } and the Hilbert space M 2 , 2 ( C ) {\displaystyle {\mathcal {M}}_{2,2}(\mathbb {C} )} of all complex 2 × 2 {\displaystyle 2\times 2} matrices over C {\displaystyle \mathbb {C} } . That is, any 2 × 2 {\displaystyle 2\times 2} matrix A {\displaystyle A} can be written ∑ i = 0 3 a i σ i {\displaystyle \sum _{i=0}^{3}a_{i}\sigma _{i}} , and the Hilbert-Schmidt inner product with another such matrix B = ∑ i = 0 3 b i σ i {\displaystyle B=\sum _{i=0}^{3}b_{i}\sigma _{i}} is then ⟨ A , B ⟩ = 2 ∑ i = 0 3 a i b i ∗ {\displaystyle \langle A,B\rangle =2\sum _{i=0}^{3}a_{i}b_{i}^{*}} since ⟨ σ i , σ j ⟩ = 2 δ i , j {\displaystyle \langle \sigma _{i},\sigma _{j}\rangle =2\delta _{i,j}} . The components of A {\displaystyle A} in this basis can be obtained directly via a i = ⟨ A , σ i ⟩ / | | σ i | | 2 = 1 2 ⟨ A , σ i ⟩ {\displaystyle a_{i}=\langle A,\sigma _{i}\rangle /||\sigma _{i}||^{2}={\frac {1}{2}}\langle A,\sigma _{i}\rangle } . Likewise, the matrices σ i / 2 {\displaystyle \sigma _{i}/{\sqrt {2}}} (including σ 0 = I {\displaystyle \sigma _{0}=\mathbb {I} } ) form a complete orthonormal basis for 2 × 2 {\displaystyle 2\times 2} complex matrices.
Commutation and anti-commutation relations
Commutation relations The Pauli matrices obey the following commutation relations:
[ σ j , σ k ] = 2 i ε j k l σ l . {\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 ( R 3 , × ) ≅ s u ( 2 ) ≅ s o ( 3 ) . {\displaystyle (\mathbb {R} ^{3},\times )\ \cong \ {\mathfrak {su}}(2)\ \cong \ {\mathfrak {so}}(3)~.}
Anticommutation relations They also satisfy the anticommutation relations:
{ σ j , σ k } = 2 δ j k I , {\displaystyle \{\sigma _{j},\sigma _{k}\}=2\ \delta _{jk}\ I\ ,}
where { σ j , σ k } {\displaystyle \{\sigma _{j},\sigma _{k}\}} is defined as σ j σ k + σ k σ j , {\displaystyle \ \sigma _{j}\ \sigma _{k}+\sigma _{k}\ \sigma _{j}\ ,} and δjk is the Kronecker delta. I denotes the 2 × 2 identity matrix. These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for R 3 , {\displaystyle \ \mathbb {R} ^{3}\ ,} denoted C l 3 ( R ) . {\displaystyle \ \mathrm {Cl} _{3}(\mathbb {R} )~.}
The usual construction of generators σ j k = 1 4 [ σ j , σ k ] {\displaystyle \ \sigma _{jk}={\tfrac {1}{4}}[\sigma _{j},\sigma _{k}]\ } of s o ( 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:
Eigenvectors and eigenvalues Each of the (Hermitian) Pauli matrices has two eigenvalues: ± 1 {\displaystyle \pm 1} . The corresponding normalized eigenvectors are
ψ x + = 1 2 [ 1 1 ] , ψ x − = 1 2 [ 1 − 1 ] , ψ y + = 1 2 [ 1 i ] , ψ y − = 1 2 [ 1 − i ] , ψ z + = [ 1 0 ] , ψ z − = [ 0 1 ] . {\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 The Pauli vector is defined by
σ = σ 1 x ^ 1 + σ 2 x ^ 2 + σ 3 x ^ 3 , {\displaystyle {\boldsymbol {\sigma }}=\sigma _{1}{\boldsymbol {\hat {x}}}_{1}+\sigma _{2}{\boldsymbol {\hat {x}}}_{2}+\sigma _{3}{\boldsymbol {\hat {x}}}_{3},}
where x ^ 1 {\displaystyle {\boldsymbol {\hat {x}}}_{1}} , x ^ 2 {\displaystyle {\boldsymbol {\hat {x}}}_{2}} , and x ^ 3 {\displaystyle {\boldsymbol {\hat {x}}}_{3}} are an equivalent notation for the more familiar x ^ {\displaystyle {\boldsymbol {\hat {x}}}} , y ^ {\displaystyle {\boldsymbol {\hat {y}}}} , and z ^ {\displaystyle {\boldsymbol {\hat {z}}}} . The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows:
a ⋅ σ = ∑ k , l a k σ ℓ x ^ k ⋅ x ^ ℓ = ∑ k a k σ k = ( a 3 a 1 − i a 2 a 1 + i a 2 − a 3 ) . {\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 from R 3 {\displaystyle \mathbb {R} ^{3}} to the vector space of traceless Hermitian 2 × 2 {\displaystyle 2\times 2} matrices. This map encodes structures of R 3 {\displaystyle \mathbb {R} ^{3}} as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory. Another way to view the Pauli vector is as a 2 × 2 {\displaystyle \ 2\times 2\ } Hermitian traceless matrix-valued dual vector, that is, an element of M a t 2 × 2 ( C ) ⊗ ( R 3 ) ∗ {\displaystyle \ \mathrm {Mat} _{2\times 2}(\mathbb {C} )\otimes (\mathbb {R} ^{3})^{*}\ } that maps a ↦ a ⋅ σ {\displaystyle {\boldsymbol {a}}\mapsto {\boldsymbol {a}}\cdot {\boldsymbol {\sigma }}}
Completeness relation Each component of a {\displaystyle {\boldsymbol {a}}} can be recovered from the matrix (see completeness relation below)
1 2 tr [ ( 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 map a ↦ a ⋅ σ {\displaystyle {\boldsymbol {a}}\mapsto {\boldsymbol {a}}\cdot {\boldsymbol {\sigma }}} , making it manifest that the map is a bijection.
Determinant The norm is given by the determinant (up to a minus sign)
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 S U ( 2 ) {\displaystyle \ \mathrm {SU} (2)\ } matrix U {\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 → ⋅