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. They are

\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}
which may be compacted into a single expression using the Kronecker delta,

\sigma_j = \begin{pmatrix} \delta_{j3} & \delta_{j1}  i\delta_{j2}\\ \delta_{j1} + i\delta_{j2} & \delta_{j3} \end{pmatrix}.
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 σ_{0}), the Pauli matrices (multiplied by real coefficients) span the full vector space of 2 × 2 Hermitian matrices.
In the language of quantum mechanics, Hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2dimensional complex Hilbert space. In the context of Pauli's work, σ_{k} is the observable corresponding to spin along the kth coordinate axis in threedimensional Euclidean space ℝ^{3}.
The Pauli matrices (after multiplication by i to make them antiHermitian), also generate transformations in the sense of Lie algebras: the matrices iσ_{1}, iσ_{2}, iσ_{3} form a basis for \mathfrak{su}_2, which exponentiates to the spin group SU(2), and for the identical Lie algebra \mathfrak{so}_3, which exponentiates to the Lie group SO(3) of rotations of 3dimensional space. The algebra generated by the three matrices σ_{1}, σ_{2}, σ_{3} is isomorphic to the Clifford algebra of ℝ^{3}, called the algebra of physical space.
Contents

Algebraic properties 1

Eigenvectors and eigenvalues 1.1

Pauli vector 1.2

Commutation relations 1.3

Relation to dot and cross product 1.4

Exponential of a Pauli vector 1.5

Completeness relation 1.6

Relation with the permutation operator 1.7

SU(2) 2

SO(3) 2.1

Quaternions 2.2

Physics 3

Quantum mechanics 3.1

Quantum information 3.2

See also 4

Notes 5

References 6
Algebraic properties
The matrices are involutory:

\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.

\begin{align} \det (\sigma_i) &= 1, \\ \operatorname{Tr} (\sigma_i) &= 0 . \end{align}
From above we can deduce that the eigenvalues of each σ_{i} are ±1.

Together with the 2 × 2 identity matrix I (sometimes written as σ_{0}), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.
Eigenvectors and eigenvalues
Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

\begin{array}{lclc} \psi_{x+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{1}\end{pmatrix}, & \psi_{x}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{1}\end{pmatrix}, \\ \psi_{y+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{i}\end{pmatrix}, & \psi_{y}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{i}\end{pmatrix}, \\ \psi_{z+}= & \begin{pmatrix}{1}\\{0}\end{pmatrix}, & \psi_{z}= & \begin{pmatrix}{0}\\{1}\end{pmatrix}. \end{array}
Pauli vector
The Pauli vector is defined by

\vec{\sigma} = \sigma_1 \hat{x} + \sigma_2 \hat{y} + \sigma_3 \hat{z} \,
and provides a mapping mechanism from a vector basis to a Pauli matrix basis^{[2]} as follows,

\begin{align} \vec{a} \cdot \vec{\sigma} &= (a_i \hat{x}_i) \cdot (\sigma_j \hat{x}_j ) \\ &= a_i \sigma_j \hat{x}_i \cdot \hat{x}_j \\ &= a_i \sigma_j \delta_{ij} \\ &= a_i \sigma_i =\begin{pmatrix} a_3&a_1ia_2\\a_1+ia_2&a_3\end{pmatrix} \end{align}
using the summation convention. Further,

\det \vec{a} \cdot \vec{\sigma} =  \vec{a} \cdot \vec{a}= \vec{a}^2.
Commutation relations
The Pauli matrices obey the following commutation relations:

[\sigma_a, \sigma_b] = 2 i \varepsilon_{a b c}\,\sigma_c \, ,
and anticommutation relations:

\{\sigma_a, \sigma_b\} = 2 \delta_{a b}\,I.
where ε_{abc} is the LeviCivita symbol, Einstein summation notation is used, δ_{ab} is the Kronecker delta, and I is the 2 × 2 identity matrix.
For example,

\begin{align} \left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \,,\\ \left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \,,\\ \left[\sigma_2, \sigma_1\right] &= 2i\sigma_3 \,,\\ \left[\sigma_1, \sigma_1\right] &= 0\,,\\ \left\{\sigma_1, \sigma_1\right\} &= 2I\,,\\ \left\{\sigma_1, \sigma_2\right\} &= 0\,.\\ \end{align}
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

\begin{align} \left[\sigma_a, \sigma_b\right] + \{\sigma_a, \sigma_b\} &= (\sigma_a \sigma_b  \sigma_b \sigma_a ) + (\sigma_a \sigma_b + \sigma_b \sigma_a) \\ 2i\sum_c\varepsilon_{a b c}\,\sigma_c + 2 \delta_{a b}I &= 2\sigma_a \sigma_b \end{align}
so that, cancelling the factors of 2,

\sigma_a \sigma_b = i\sum_c\varepsilon_{a b c}\,\sigma_c + \delta_{a b}I~ .

Contracting each side of the equation with components of two 3vectors a_{p} and b_{q} (which commute with the Pauli matrices, i.e., a_{p}σ_{q} = σ_{q}a_{p}) for each matrix σ_{q} and vector component a_{p} (and likewise with b_{q}), and relabeling indices a, b, c → p, q, r, to prevent notational conflicts, yields

\begin{align} a_p b_q \sigma_p \sigma_q & = a_p b_q \left(i\sum_r\varepsilon_{pqr}\,\sigma_r + \delta_{pq}I\right) \\ a_p \sigma_p b_q \sigma_q & = i\sum_r\varepsilon_{pqr}\,a_p b_q \sigma_r + a_p b_q \delta_{pq}I ~. \end{align}
Finally, translating the index notation for the dot product and cross product results in

(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma}


(1)

Exponential of a Pauli vector
For a→ = an̂ with n̂ = 1, one has, for even powers,

(\hat{n} \cdot \vec{\sigma})^{2n} = I \,
which can be shown first for the n = 1 case using the anticommutation relations.
Thus, for odd powers,

(\hat{n} \cdot \vec{\sigma})^{2n+1} = \hat{n} \cdot \vec{\sigma} \, .
Matrix exponentiating, and using the Taylor series for sine and cosine,

\begin{align} e^{i a(\hat{n} \cdot \vec{\sigma})} &= \sum_{n=0}^\infty{\frac{i^n \left[a (\hat{n} \cdot \vec{\sigma})\right]^n}{n!}} \\ &= \sum_{n=0}^\infty{\frac{(1)^n (a\hat{n}\cdot \vec{\sigma})^{2n}}{(2n)!}} + i\sum_{n=0}^\infty{\frac{(1)^n (a\hat{n}\cdot \vec{\sigma})^{2n + 1}}{(2n + 1)!}} \\ &= I\sum_{n=0}^\infty{\frac{(1)^n a^{2n}}{(2n)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{n=0}^\infty{\frac{(1)^n a^{2n+1}}{(2n + 1)!}}\\ \end{align}
and, in the last line, the first sum is the cosine, while the second sum is the sine; so, finally,


e^{i a(\hat{n} \cdot \vec{\sigma})} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \,



(2)

which is analogous to Euler's formula. Note

\det[i a(\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 this formula, (2), for a general 2×2 matrix can be found in the article on matrix exponentials.
A straightforward application of this formula provides a parameterization of the group composition law of SU(2): One may directly solve for c in

e^{i a(\hat{n} \cdot \vec{\sigma})} e^{i b(\hat{n}' \cdot \vec{\sigma})} = I\cos{c} + i (\hat{n}'' \cdot \vec{\sigma}) \sin{c} = e^{i c \left(\hat{n}'' \cdot \vec{\sigma}\right)},
which specifies the generic group multiplication, where

\cos c = \cos a \cos b  \hat{n} \cdot\hat{n}' \sin a \sin b~,
the spherical law of cosines. Given c, then,

\hat{n}'' = (\hat{n} \sin a \cos b + \hat{n}' \sin b \cos a  \hat{n}\times\hat{n}' \sin a \sin b)/ \sin c ~.
Consequently, the composite rotation parameters in the exponent of this group element (a closed form of the respective BCH expansion in this case) simply amount to

c \hat{n}'' = \frac{c}{\sin c} (\hat{n} \sin a \cos b + \hat{n}' \sin b \cos a  \hat{n}\times\hat{n}' ~ \sin a \sin b ) ~.
The fact that any 2 × 2 complex Hermitian 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, (2 × 2 positive semidefinite matrices with trace 1). This can be seen by simply first writing an arbitrary Hermitian matrix as a real linear combination of {σ_{0}, σ_{1}, σ_{2}, σ_{3}} as above, and then imposing the positivesemidefinite and trace 1 conditions.
Completeness relation
An alternative notation that is commonly used for the Pauli matrices is to write the vector index i in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the ith Pauli matrix is σ ^{i}_{αβ}.
In this notation, the completeness relation for the Pauli matrices can be written

\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{i=1}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 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 complex Hilbert space of all 2 × 2 matrices means that we can express any matrix M as

M = c I + \sum_i a_i \sigma^i
where c is a complex number, and a is a 3component complex vector. It is straightforward to show, using the properties listed above, that

\mathrm{tr}\, \sigma^i\sigma^j = 2\delta_{ij}
where "tr" denotes the trace, and hence that c=\frac{1}{2}\mathrm{tr}\,M and a_i = \frac{1}{2}\mathrm{tr}\,\sigma^i M.
This therefore gives

2M = I \mathrm{tr}\, M + \sum_i \sigma^i \mathrm{tr}\, \sigma^i M
which can be rewritten in terms of matrix indices as

2M_{\alpha\beta} = \delta_{\alpha\beta} M_{\gamma\gamma} + \sum_i \sigma^i_{\alpha\beta} \sigma^i_{\gamma\delta} M_{\delta\gamma}
where summation is implied over the repeated indices 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 σ_{0}, so σ^{0}_{αβ} = δ_{αβ}. The completeness relation can therefore alternatively be expressed as

\sum_{i=0}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma}\,.
Relation with the permutation operator
Let P_{ij} be the transposition (also known as a permutation) between two spins σ_{i} and σ_{j} living in the tensor product space ℂ^{2} ⊗ ℂ^{2},

P_{ij}\sigma_i \sigma_j\rangle = \sigma_j \sigma_i\rangle \,.
This operator can also be written more explicitly as Dirac's spin exchange operator,

P_{ij} = \tfrac{1}{2}(\vec{\sigma}_i\cdot\vec{\sigma}_j + 1)\,.
Its eigenvalues are therefore^{[3]} 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 antiHermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra \mathfrak{su}_2 is the 3dimensional real algebra spanned by the set { iσ_{j} }. In compact notation,

\mathfrak{su}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2 , i \sigma_3 \}.
As a result, each iσ_{j} can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2, so that

\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 \mathfrak{su}_2 is isomorphic to the Lie algebra \mathfrak{so}_3, which corresponds to the Lie group SO(3), the group of rotations in threedimensional space. In other words, one can say that the iσ_{j} are a realization (and, in fact, the lowestdimensional realization) of infinitesimal rotations in threedimensional space. However, even though \mathfrak{su}_2 and \mathfrak{so}_3 are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a twotoone group homomorphism from SU(2) to SO(3).
Quaternions
The real linear span of {I, iσ_{1}, iσ_{2}, iσ_{3}} is isomorphic to the real algebra of quaternions H. The isomorphism from H to this set is given by the following map (notice the reversed signs for the Pauli matrices):

1 \mapsto I, \quad i \mapsto  i \sigma_1, \quad j \mapsto  i \sigma_2, \quad k \mapsto  i \sigma_3.
Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,^{[4]}

1 \mapsto I, \quad i \mapsto i \sigma_3, \quad j \mapsto i \sigma_2, \quad k \mapsto i \sigma_1.
As the quaternions of unit norm is groupisomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices. The twotoone homomorphism from SU(2) to SO(3) can also be explicitly given in terms of the Pauli matrices in this formulation.
Quaternions form a division algebra—every nonzero element has an inverse—whereas Pauli matrices do not. For a quaternionic version of the algebra generated by Pauli matrices see biquaternions, which is a venerable algebra of eight real dimensions.
Physics
Quantum mechanics
In quantum mechanics, each Pauli matrix is related to an operator that corresponds to an observable describing the spin of a spin ½ particle, in each of the three spatial directions. Also, as an immediate consequence of the Cartan decomposition mentioned above, iσ_{j} are the generators of rotation acting on nonrelativistic particles with spin ½. The state of the particles are represented as twocomponent spinors.
An interesting property of spin ½ 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 twotoone 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 2sphere S ^{2}, they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.
For a spin ½ 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.
For example, the resulting spin matrices for spin 1 and spin 3/2 are:
For j = 1

\begin{align} J_x &= \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 &1 &0\\ 1 &0 &1\\ 0 &1 &0 \end{pmatrix} \\ J_y &= \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 &i &0\\ i &0 &i\\ 0 &i &0 \end{pmatrix} \\ J_z &= \hbar \begin{pmatrix} 1 &0 &0\\ 0 &0 &0\\ 0 &0 &1 \end{pmatrix} \end{align}
For j=\textstyle\frac{3}{2}:

\begin{align} J_x &= \frac\hbar2 \begin{pmatrix} 0 &\sqrt{3} &0 &0\\ \sqrt{3} &0 &2 &0\\ 0 &2 &0 &\sqrt{3}\\ 0 &0 &\sqrt{3} &0 \end{pmatrix} \\ J_y &= \frac\hbar2 \begin{pmatrix} 0 &i\sqrt{3} &0 &0\\ i\sqrt{3} &0 &2i &0\\ 0 &2i &0 &i\sqrt{3}\\ 0 &0 &i\sqrt{3} &0 \end{pmatrix} \\ J_z &= \frac\hbar2 \begin{pmatrix} 3 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &1 &0\\ 0 &0 &0 &3 \end{pmatrix}. \end{align}
For j = \textstyle\frac{5}{2}:

\begin{align} J_x &= \frac\hbar2 \begin{pmatrix} 0 &\sqrt{5} &0 &0 &0 &0 \\ \sqrt{5} &0 &2\sqrt{2} &0 &0 &0 \\ 0 &2\sqrt{2} &0 &3 &0 &0 \\ 0 &0 &3 &0 &2\sqrt{2} &0 \\ 0 &0 &0 &2\sqrt{2} &0 &\sqrt{5} \\ 0 &0 &0 &0 &\sqrt{5} &0 \end{pmatrix} \\ J_y &= \frac\hbar2 \begin{pmatrix} 0 &i\sqrt{5} &0 &0 &0 &0 \\ i\sqrt{5} &0 &2i\sqrt{2} &0 &0 &0 \\ 0 &2i\sqrt{2} &0 &3i &0 &0 \\ 0 &0 &3i &0 &2i\sqrt{2} &0 \\ 0 &0 &0 &2i\sqrt{2} &0 &i\sqrt{5} \\ 0 &0 &0 &0 &i\sqrt{5} &0 \end{pmatrix} \\ J_z &= \frac\hbar2 \begin{pmatrix} 5 &0 &0 &0 &0 &0 \\ 0 &3 &0 &0 &0 &0 \\ 0 &0 &1 &0 &0 &0 \\ 0 &0 &0 &1 &0 &0 \\ 0 &0 &0 &0 &3 &0 \\ 0 &0 &0 &0 &0 &5 \end{pmatrix}. \end{align}
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.^{[5]}
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_{n} is defined to consist of all nfold tensor products of Pauli matrices.
Quantum information

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

^ "Pauli matrices". Planetmath website. 28 March 2008. Retrieved 28 May 2013.

^ See the spinor map.

^ Explicitly, in the convention of "rightspace matrices into elements of leftspace matrices", it is \begin{pmatrix} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\end{pmatrix}~.

^ Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. , pp. xxii.

^
References


Schiff, Leonard I. (1968). Quantum Mechanics. McGrawHill.

Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press.
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