Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. 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. 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 (sometimes considered as the zeroth Pauli matrix σ_{0} ), 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 threedimensional Euclidean space
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 the real Lie algebra , which exponentiates to the special unitary group SU(2).^{[a]} The algebra generated by the three matrices σ_{1}, σ_{2}, σ_{3} is isomorphic to the Clifford algebra of ^{[1]} and the (unital associative) algebra generated by iσ_{1}, iσ_{2}, iσ_{3} functions identically (is isomorphic) to that of quaternions ().
Algebraic properties[edit]
×  

All three of the Pauli matrices can be compacted into a single expression:
where the solution to i^{2} = −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:
where I is the identity matrix.
The determinants and traces of the Pauli matrices are:
From which, we can deduce that each matrix σ_{j} has eigenvalues +1 and −1.
With the inclusion of the identity matrix, I (sometimes denoted σ_{0} ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space of 2 × 2 Hermitian matrices, , over , and the Hilbert space of all complex 2 × 2 matrices, , over .
Commutation and anticommutation relations[edit]
The Pauli matrices obey the following commutation relations:
where the structure constant ε_{ijk} is the LeviCivita symbol and Einstein summation notation is used.
These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra
They also satisfy the anticommutation relations:
where is defined as and δ_{ij} is the Kronecker delta. I denotes the 2 × 2 identity matrix.
These anticommutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for denoted
The usual construction of generators of using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.
A few explicit commutators and anticommutators are given below as examples:
Commutators  Anticommutators 

Eigenvectors and eigenvalues[edit]
Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:
Pauli vectors[edit]
The Pauli vector is defined by^{[b]}
The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis^{[2]} as follows,
More formally, this defines a map from to the vector space of traceless Hermitian matrices. This map encodes structures of as a normed vector space and as a Lie algebra (with the crossproduct 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 Hermitian traceless matrixvalued dual vector, that is, an element of which maps
Completeness relation[edit]
Each component of can be recovered from the matrix (see completeness relation below)
This constitutes an inverse to the map , making it manifest that the map is a bijection.
Determinant[edit]
The norm is given by the determinant (up to a minus sign)
Then considering the conjugation action of an matrix on this space of matrices,
we find and that is Hermitian and traceless. It then makes sense to define where has the same norm as and therefore interpret as a rotation of 3dimensional space. In fact, it turns out that the special restriction on implies that the rotation is orientation preserving. This allows the definition of a map given by
where This map is the concrete realization of the double cover of by and therefore shows that The components of can be recovered using the tracing process above:
Crossproduct[edit]
The crossproduct is given by the matrix commutator (up to a factor of )
In fact, the existence of a norm follows from the fact that is a Lie algebra: see Killing form.
This crossproduct can be used to prove the orientationpreserving property of the map above.
Eigenvalues and eigenvectors[edit]
The eigenvalues of are 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 since this can be factorised into A standard result in linear algebra (a linear map which satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies is diagonal with possible eigenvalues The tracelessness of means it has exactly one of each eigenvalue.
Its normalized eigenvectors are
Aternatively, one may use spherical coordinates to obtain the eigenvectors and .
Pauli 4vector[edit]
The Pauli 4vector, used in spinor theory, is written with components
This defines a map from to the vector space of Hermitian matrices,
which also encodes the Minkowski metric (with mostly minus convention) in its determinant:
This 4vector also has a completeness relation. It is convenient to define a second Pauli 4vector
and allow raising and lowering using the Minkowski metric tensor. The relation can then be written
Similarly to the Pauli 3vector case, we can find a matrix group which acts as isometries on in this case the matrix group is and this shows Similarly to above, this can be explicitly realized for with components
In fact, the determinant property follows abstractly from trace properties of the For matrices, the following identity holds:
That is, the 'crossterms' can be written as traces. When are chosen to be different the crossterms vanish. It then follows, now showing summation explicitly, Since the matrices are this is equal to
Relation to dot and cross product[edit]
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}) yields
Finally, translating the index notation for the dot product and cross product results in


(1) 
If i is identified with the pseudoscalar σ_{x}σ_{y}σ_{z} then the right hand side becomes which is also the definition for the product of two vectors in geometric algebra.
If we define the spin operator as J = ħ/2σ, then J satisfies the commutation relation:
Some trace relations[edit]
The following traces can be derived using the commutation and anticommutation relations.
If the matrix σ_{0} = I is also considered, these relationships become
where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation is used to denote the sum over the cyclic permutation of the included indices.
Exponential of a Pauli vector[edit]
For
one has, for even powers, 2p, p = 0, 1, 2, 3, ...
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, ...
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)[edit]
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
which specifies the generic group multiplication, where, manifestly,
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 is parallel to , so is , and c = a + b.)
Adjoint action[edit]
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle along any axis :
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 .
Completeness relation[edit]
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 kth Pauli matrix is σ ^{k}_{αβ}.
In this notation, the completeness relation for the Pauli matrices can be written
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
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 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 {σ_{0}, σ_{1}, σ_{2}, σ_{3}} as above, and then imposing the positivesemidefinite and trace 1 conditions.
For a pure state, in polar coordinates,
acts on the state eigenvector with eigenvalue +1, hence it acts like a projection operator.
Relation with the permutation operator[edit]
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 ,
This operator can also be written more explicitly as Dirac's spin exchange operator,
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)[edit]
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_{k}}. 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)[edit]
The Lie algebra is isomorphic to the Lie algebra , 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 and 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[edit]
The real linear span of {I, iσ_{1}, iσ_{2}, iσ_{3}} is isomorphic to the real algebra of quaternions, , represented by the span of the basis vectors The isomorphism from 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,^{[5]}
As the set of versors U ⊂ forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The twotoone homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.
Physics[edit]
Classical mechanics[edit]
In classical mechanics, Pauli matrices are useful in the context of the CayleyKlein parameters.^{[6]} 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 Q_{θ} for rotations about the xaxis through an angle θ may be written in terms of Pauli matrices and the unit matrix as^{[6]}
Similar expressions follow for general Pauli vector rotations as detailed above.
Quantum mechanics[edit]
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 nonrelativistic particles with spin 1⁄2. 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 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 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 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 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 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[edit]
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 the matrices have the same algebraic properties as the σ_{k} matrices.
However, relativistic angular momentum is not a threevector, but a second order fourtensor. Hence 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 The remaining three, where the Dirac α_{k} matrices are defined as
The relativistic spin matrices Σ_{μν} are written in compact form in terms of commutator of gamma matrices as
Quantum information[edit]
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[edit]
 Algebra of physical space
 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
 Exchange matrix (the first Pauli matrix is an exchange matrix of order two)
 Splitquaternion
Remarks[edit]
 ^ This conforms to the convention in mathematics for the matrix exponential, iσ ⟼ exp(iσ). In the convention in physics, 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 , where the tensor product space is endowed with a mapping induced by the dot product on
 ^ 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. That is, the Gibbs formula linked amounts to .
 ^ Explicitly, in the convention of "rightspace matrices into elements of leftspace matrices", it is
Notes[edit]
 ^ Gull, S.F.; Lasenby, A.N.; Doran, C.J.L. (January 1993). "Imaginary numbers are not Real – the geometric algebra of spacetime" (PDF). Found. Phys. 23 (9): 1175–1201. Bibcode:1993FoPh...23.1175G. doi:10.1007/BF01883676. S2CID 14670523. Retrieved 5 May 2023 – via geometry.mrao.cam.ac.uk.
 ^ 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.
 ^ Gibbs, J.W. (1884). Elements of Vector Analysis. New Haven, CT. p. 67.
{{cite book}}
: CS1 maint: location missing publisher (link) In fact, however, the formula goes back to Olinde Rodrigues (1840), replete with halfangle: 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.  ^ Nakahara, Mikio (2003). Geometry, Topology, and Physics (2nd ed.). CRC Press. p. xxii. ISBN 9780750306065 – via Google Books.
 ^ ^{a} ^{b} 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. S2CID 18776942.
References[edit]
 The Pauli spin matrices  The Feynman Lectures on Physics
 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 9780521145053.