In mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices that 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 two-dimensional Hilbert space. In the context of Pauli's work, σ_{k} represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space

The Pauli matrices (after multiplication by i to make them anti-Hermitian) 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 ().

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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 .

The Pauli matrices obey the following commutation relations:

where the structure constant *ε _{jkl}* is the Levi-Civita 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 _{jk}* 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 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 anti-commutators are given below as examples:

Commutators | Anticommutators |
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Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

The Pauli vector is defined by^{[b]}

where , , and are an equivalent notation for the more familiar , , and .

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis^{[2]} as follows,

using Einstein's summation convention.

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 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 Hermitian traceless matrix-valued dual vector, that is, an element of that maps

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.

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 three-dimensional 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:

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

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 that 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

These expressions become singular for . They can be rescued by letting and taking the limit , which yields the correct eigenvectors (0,1) and (1,0) of .

Alternatively, one may use spherical coordinates to obtain the eigenvectors and .

The Pauli 4-vector, 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 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector

and allow raising and lowering using the Minkowski metric tensor. The relation can then be written

Similarly to the Pauli 3-vector case, we can find a matrix group that 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 'cross-terms' can be written as traces. When are chosen to be different the cross-terms vanish. It then follows, now showing summation explicitly, Since the matrices are this is equal to

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 3-vectors *a _{p}* and

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* =

Or equivalently, the Pauli vector satisfies:

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.

For

one has, for even powers, 2*p*, *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, 2*q* + 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]}

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,

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 is parallel to , so is , and *c* = *a + b*.)

See also: Rotation formalisms in three dimensions § Rodrigues vector, and Spinor § Three dimensions |

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 .

See also: Rodrigues' rotation formula |

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 k-th 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 Hermitian matrices means that we can express any Hermitian matrix M as

where c is a complex number, and a is a 3-component, complex vector. It is straightforward to show, using the properties listed above, that

where "tr" denotes the trace, and hence that

which can be rewritten in terms of matrix indices as

where summation over the repeated indices is implied γ and δ. Since this is true for any choice of the matrix M, the completeness relation follows as stated above. Q.E.D.

As noted above, it is common to denote the 2 × 2 unit matrix by *σ*_{0}, so *σ ^{0}_{αβ}* =

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 positive-semidefinite and trace 1 conditions.

For a pure state, in polar coordinates,

the idempotent density matrix

acts on the state eigenvector with eigenvalue +1, hence it acts like a projection operator.

Let *P _{jk}* be the transposition (also known as a permutation) between two spins

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.

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 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra is the three-dimensional 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

As SU(2) is a compact group, its Cartan decomposition is trivial.

The Lie algebra is isomorphic to the Lie algebra , which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the *iσ _{j}* are a realization (and, in fact, the lowest-dimensional realization) of

Main article: Spinor § Three dimensions |

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 two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.

Main article: Quaternions and spatial rotation |

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein 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 x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as

Similar expressions follow for general Pauli vector rotations as detailed above.

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 (

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 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 the 2-sphere *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* =

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group *G _{n}* is defined to consist of all n-fold tensor products of Pauli matrices.

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 three-vector, but a second order four-tensor. 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

In quantum information, single-qubit quantum gates are 2 × 2 unitary 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–Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X–Y *decomposition of a single-qubit gate*.