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. They are
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) form a basis for the vector space of 2 × 2 Hermitian matrices.
Hermitian operators represent 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} represents 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 , which exponentiates to the special unitary group SU(2).^{[nb 1]} The algebra generated by the three matrices σ_{1}, σ_{2}, σ_{3} is isomorphic to the Clifford algebra of ℝ^{3}.
Algebraic properties
All three of the Pauli matrices can be compacted into a single expression:
where i = √−1 is the imaginary unit, and δ_{ab} is the Kronecker delta, which equals +1 if a = b and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of a = 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:
where I is the identity matrix.
 The determinants and traces of the Pauli matrices are:
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:
Pauli vector
The Pauli vector is defined by^{[nb 2]}
and provides a mapping mechanism from a vector basis to a Pauli matrix basis^{[2]} as follows,
using the summation convention. Further,
its eigenvalues being , and moreover (see completeness, below)
Its (unnormalized) eigenvectors are
Commutation relations
The Pauli matrices obey the following commutation relations:
and anticommutation relations:
where the structure constant 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,
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
so that,
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
Finally, translating the index notation for the dot product and cross product results in


(1 ) 
If is identified with the pseudoscalar then the right hand side becomes which is also the definition for the product of two vectors in geometric algebra.
Some trace relations
Following traces can be derived using the commutation and anticommutation relations.
Exponential of a Pauli vector
For
one has, for even powers,
which can be shown first for the case using the anticommutation relations. For convenience, the case is taken to be by convention.
For odd powers,
Matrix exponentiating, and using the Taylor series for sine and cosine,
 .
In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,


(2 ) 
which is analogous to Euler's formula, extended to quaternions.
Note that
 ,
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]}
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).^{[nb 3]} One may directly solve for c in
which specifies the generic group multiplication, where, manifestly,
the spherical law of cosines. Given c, then,
Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to^{[4]}
(Of course, when n̂ is parallel to m̂, so is k̂, 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,
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
 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
 where c is a complex number, and a is a 3component complex vector. It is straightforward to show, using the properties listed above, that
 where "tr" denotes the trace, and hence that
 and
 which can be rewritten in terms of matrix indices as
 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 alternatively be expressed as
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.
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},
This operator can also be written more explicitly as Dirac's spin exchange operator,
Its eigenvalues are therefore^{[5]} 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 is the 3dimensional real algebra spanned by the set {iσ_{j}}. In compact notation,
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
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 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 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 twotoone group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).
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):
Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,^{[6]}
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.
Multiplying any two Pauli matrices always yield the matrix of a quaternion unit.
Physics
Classical mechanics
In classical mechanics, Pauli matrices are useful in the context of the CayleyKlein parameters.^{[7]} The matrix P corresponding to the position of a point in space is defined in terms of the above Pauli vector matrix,
Consequently, the transformation matrix for rotations about the xaxis through an angle θ may be written in terms of Pauli matrices and the unit matrix as ^{[7]}
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 ½ 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 nonrelativistic particles with spin ½. The states of the particles are represented as twocomponent spinors. In the same way, the Pauli matrices are related to the isospin operator.
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. 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.^{[8]}
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.
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
It follows from this definition that matrices have the same algebraic properties as matrices.
However, relativistic angular momentum is not a threevector, but a second order fourtensor. Hence needs to be replaced by , the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the are also antisymmetric. Hence there are only six independent matrices.
The first three are the The remaining three, , are the Dirac matrices defined as
The relativistic spin matrices are written in compact form in terms of commutator of gamma matrices as
 .
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
 Spinors in three dimensions
 Gamma matrices
 Angular momentum
 GellMann matrices
 Poincaré group
 Generalizations of Pauli matrices
 Bloch sphere
 Euler's foursquare identity
 For higher spin generalizations of the Pauli matrices, see spin (physics) § Higher spins
Remarks
 ↑ This conforms to the mathematics convention for the matrix exponential, iσ ↦ exp(iσ). In the physics convention, σ ↦ exp(−iσ), hence in it no premultiplication by i is necessary to land in SU(2).
 ↑ The Pauli vector is a formal device. It may be thought of as an element of M_{2}(ℂ) ⊗ ℝ^{3}, where the tensor product space is endowed with a mapping ⋅: ℝ^{3} × M_{2}(ℂ) ⊗ ℝ^{3} → M_{2}(ℂ) induced by the dot product on ℝ^{3}.
 ↑ N.B. The relation among a, b, c, n, m, k derived here in the 2 × 2 representation holds for all representations of SU(2), being a group identity. Note that, by virtue of the standard normalization of that group's generators as half the Pauli matrices, the parameters a,b,c correspond to half the rotation angles of the rotation group.
Notes
 ↑ "Pauli matrices". Planetmath website. 28 March 2008. Retrieved 28 May 2013.
 ↑ See the spinor map.
 ↑ Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. ISBN 9780521632355. OCLC 43641333.
 ↑ cf. J W Gibbs (1884). Elements of Vector Analysis, New Haven, 1884, p. 67. In fact, however, the formula goes back to Olinde Rodrigues, 1840, replete with halfangle: "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", J. Math. Pures Appl. 5 (1840), 380–440;
 ↑ Explicitly, in the convention of "rightspace matrices into elements of leftspace matrices", it is
 ↑ Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN 9780750306065 , pp. xxii.
 1 2 Goldstein, Herbert (1959). Classical Mechanics. AddisonWesley. pp. 109–118.
 ↑ 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.
References
 Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.
 Schiff, Leonard I. (1968). Quantum Mechanics. McGrawHill. ISBN 9780070552876.
 Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 0521145058.