Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () 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.
Each Pauli matrix is Hermitian, and together with the identity matrix (sometimes considered as the zeroth Pauli matrix ), the Pauli matrices form a basis for the real vector space of Hermitian matrices. 
This means that any 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 -dimensional complex Hilbert space. In the context of Pauli's work, represents the observable corresponding to spin along the th coordinate axis in three-dimensional Euclidean space .
The Pauli matrices (after multiplication by to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices form a basis for the real Lie algebra formula_2, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices is isomorphic to the Clifford algebra of , and the (unital associative) algebra generated by is isomorphic to that of quaternions.
Algebraic properties.
All three of the Pauli matrices can be compacted into a single expression:
where is the imaginary unit, and is the Kronecker delta, which equals +1 if and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of , in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.
The matrices are involutory:
where is the identity matrix.
The determinants and traces of the Pauli matrices are:
From which, we can deduce that the eigenvalues of each are .
With the inclusion of the identity matrix, (sometimes denoted ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the real Hilbert space of complex Hermitian matrices, formula_6, and the complex Hilbert space of all matrices, formula_7.
Eigenvectors and eigenvalues.
Each of the (Hermitian) Pauli matrices has two eigenvalues, and . Using a convention in which prior to normalization, the 1 is placed into the top and bottom positions of the + and – wavefunctions respectively, the corresponding normalized eigenvectors are:
An advantage of using this convention is that the + and – wavefunctions may be related to one another, using the Pauli matrices themselves, by , , and .
Pauli vector.
The Pauli vector is defined by
and provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows,
using the summation convention. Further,
its eigenvalues being formula_12, and moreover (see completeness, below) 
Its normalized eigenvectors are 
Commutation relations.
The Pauli matrices obey the following commutation relations:
and anticommutation relations:
where the structure constant is the Levi-Civita symbol, Einstein summation notation is used, is the Kronecker delta, and is the 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 -vectors and (which commute with the Pauli matrices, i.e., for each matrix and vector component (and likewise with ), and relabeling indices , to prevent notational conflicts, yields
Finally, translating the index notation for the dot product and cross product results in 
If is identified with the pseudoscalar then the right hand side becomes formula_20 which is also the definition for the product of two vectors in geometric algebra.
Some trace relations.
The following traces can be derived using the commutation and anticommutation relations.
If the matrix is also considered, these relationships become
where Greek indices and assume values from and the notation formula_23 is used to denote the sum over the cyclic permutation of the included indices.
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,
which is analogous to Euler's formula, extended to quaternions.
Note that
while the determinant of the exponential itself is just , which makes it the generic group element of SU(2).
A more abstract version of formula for a general matrix can be found in the article on matrix exponentials. A general version of for an analytic (at "a" and −"a") function is provided by application of Sylvester's formula,
The group composition law of.
A straightforward application of formula provides a parameterization of the composition law of the group . One may directly solve for in 
which specifies the generic group multiplication, where, manifestly, 
the spherical law of cosines. Given , then, 
Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to 
Adjoint action.
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle ,
Completeness relation.
An alternative notation that is commonly used for the Pauli matrices is to write the vector index in the superscript, and the matrix indices as subscripts, so that the element in row and column of the -th Pauli matrix is .
In this notation, the completeness relation for the Pauli matrices can be written
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 unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of as above, and then imposing the positive-semidefinite and trace conditions.
For a pure state, in polar coordinates, formula_41, the idempotent density matrix 
acts on the state eigenvector formula_43 with eigenvalue 1, hence like a projection operator for it.
Relation with the permutation operator.
Let be the transposition (also known as a permutation) between two spins and living in the tensor product space 
This operator can also be written more explicitly as Dirac's spin exchange operator,
Its eigenvalues are therefore 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 matrices with unit determinant; its Lie algebra is the set of all anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra formula_46 is the 3-dimensional real algebra spanned by the set }. In compact notation,
As a result, each 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, as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is , so that
As SU(2) is a compact group, its Cartan decomposition is trivial.
SO(3).
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 are a realization (and, in fact, the lowest-dimensional realization) of "infinitesimal" rotations in three-dimensional space. However, even though and are isomorphic as Lie algebras, and are not isomorphic as Lie groups. is actually a double cover of , meaning that there is a two-to-one group homomorphism from to , see relationship between SO(3) and SU(2).
Quaternions.
The real linear span of is isomorphic to the real algebra of quaternions . 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,
As the set of versors "U" ⊂ ℍ forms a group isomorphic to , "U" gives yet another way of describing . The two-to-one homomorphism from to may be given in terms of the Pauli matrices in this formulation.
Physics.
Classical mechanics.
In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix "P" corresponding to the position formula_51 of a point in space is defined in terms of the above Pauli vector matrix, 
Consequently, the transformation matrix 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.
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, are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin ½. The states of the particles are represented as two-component 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 4 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 , 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 , 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.
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group is defined to consist of all -fold 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 formula_55 matrices have the same algebraic properties as matrices.
However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence formula_55 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 formula_57 The remaining three, formula_58, where the Dirac matrices are defined as
The relativistic spin matrices are written in compact form in terms of commutator of gamma matrices as
Quantum information.
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".