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 σ0 ), 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, σk 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 iσ1, iσ2, iσ3 form a basis for the real Lie algebra
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All three of the Pauli matrices can be compacted into a single expression:
where the solution to i2 = −1 is the "imaginary unit", and δjk 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 σj has eigenvalues +1 and −1.
With the inclusion of the identity matrix, I (sometimes denoted σ0), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space of 2 × 2 Hermitian matrices,
The Pauli matrices obey the following commutation relations:
where the structure constant εijk 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
These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for
The usual construction of generators
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[lower-alpha 2]
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
Another way to view the Pauli vector is as a
Each component of
This constitutes an inverse to the map
The norm is given by the determinant (up to a minus sign)
Then considering the conjugation action of an
we find
where
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
This cross-product can be used to prove the orientation-preserving property of the map above.
The eigenvalues of
More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from
Its normalized eigenvectors are
Alternatively, one may use spherical coordinates
The Pauli 4-vector, used in spinor theory, is written
This defines a map from
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 fact, the determinant property follows abstractly from trace properties of the
That is, the 'cross-terms' can be written as traces. When
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 ap and bq (which commute with the Pauli matrices, i.e., apσq = σqap) for each matrix σq and vector component ap (and likewise with bq) yields
Finally, translating the index notation for the dot product and cross product results in
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If i is identified with the pseudoscalar σxσyσz then the right hand side becomes
If we define the spin operator as J = ħ/2σ, then J satisfies the commutation relation:
The following traces can be derived using the commutation and anticommutation relations.
If the matrix σ0 = 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
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,
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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).[lower-alpha 3] 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
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle
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
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 kαβ.
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 σ0, so σ0αβ = δαβ. 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 {σ0, σ1, σ2, σ3} as above, and then imposing the positive-semidefinite and trace 1 conditions.
For a pure state, in polar coordinates,
acts on the state eigenvector
Let Pjk be the transposition (also known as a permutation) between two spins σj and σk living in the tensor product space
This operator can also be written more explicitly as Dirac's spin exchange operator,
Its eigenvalues are therefore[lower-alpha 4] 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
As a result, each iσj 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.
The Lie algebra
The real linear span of {I, iσ1, iσ2, iσ3} is isomorphic to the real algebra of quaternions,
Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,[5]
As the set of versors U ⊂
In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters.[6] The matrix P corresponding to the position
Consequently, the transformation matrix Qθ for rotations about the x-axis 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.
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, iσj are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 1⁄2. 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 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 S2, 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 Gn 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
However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence
The first three are the
The relativistic spin matrices Σμν 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.
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