# Pauli matrices

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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.

## Contents

{\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{\mathrm {x} }&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\\sigma _{2}=\sigma _{\mathrm {y} }&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\\sigma _{3}=\sigma _{\mathrm {z} }&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\\\end{aligned}}}

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 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 2-dimensional complex 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 R3.

The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices 1, 2, 3 form a basis for the real Lie algebra ${\displaystyle {\mathfrak {su}}(2)}$, which exponentiates to the special unitary group SU(2). [nb 1] The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of R3, and the (unital associative) algebra generated by 1, 2, 3 is isomorphic to that of quaternions.

## Algebraic properties

All three of the Pauli matrices can be compacted into a single expression:

${\displaystyle \sigma _{a}={\begin{pmatrix}\delta _{a3}&\delta _{a1}-i\delta _{a2}\\\delta _{a1}+i\delta _{a2}&-\delta _{a3}\end{pmatrix}}}$

where i = −1 is the imaginary unit, and δab 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:

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\sigma _{1}\sigma _{2}\sigma _{3}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I}$

where I is the identity matrix.

The determinants and traces of the Pauli matrices are:

{\displaystyle {\begin{aligned}\det \sigma _{i}&=-1,\\\operatorname {tr} \sigma _{i}&=0.\end{aligned}}}

From which, we can deduce that the eigenvalues of each σi are ±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 real Hilbert space of 2 × 2 complex Hermitian matrices, ${\displaystyle {\mathcal {H}}_{2}(\mathbb {C} )}$, and the complex Hilbert space of all 2 × 2 matrices, ${\displaystyle {\mathcal {M}}_{2,2}(\mathbb {C} )}$.

### Eigenvectors and eigenvalues

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

{\displaystyle {\begin{aligned}\psi _{x+}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}},&\psi _{x-}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-1\end{pmatrix}},\\\psi _{y+}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\i\end{pmatrix}},&\psi _{y-}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-i\end{pmatrix}},\\\psi _{z+}&={\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}&={\begin{pmatrix}0\\1\end{pmatrix}}.\end{aligned}}}

### Pauli vector

The Pauli vector is defined by [nb 2]

${\displaystyle {\vec {\sigma }}=\sigma _{1}{\hat {x}}+\sigma _{2}{\hat {y}}+\sigma _{3}{\hat {z}}}$

and provides a mapping mechanism from a vector basis to a Pauli matrix basis [2] as follows,

{\displaystyle {\begin{aligned}{\vec {a}}\cdot {\vec {\sigma }}&=\left(a_{i}{\hat {x}}_{i}\right)\cdot \left(\sigma _{j}{\hat {x}}_{j}\right)=a_{i}\sigma _{j}{\hat {x}}_{i}\cdot {\hat {x}}_{j}\\&=a_{i}\sigma _{j}\delta _{ij}=a_{i}\sigma _{i}={\begin{pmatrix}a_{3}&a_{1}-ia_{2}\\a_{1}+ia_{2}&-a_{3}\end{pmatrix}}\end{aligned}}}

using the summation convention. Further,

${\displaystyle \det {\vec {a}}\cdot {\vec {\sigma }}=-{\vec {a}}\cdot {\vec {a}}=-\left|{\vec {a}}\right|^{2},}$

its eigenvalues being ${\displaystyle \pm |{\vec {a}}|}$, and moreover (see completeness, below)

${\displaystyle {\frac {1}{2}}\operatorname {tr} \left(\left({\vec {a}}\cdot {\vec {\sigma }}\right){\vec {\sigma }}\right)={\vec {a}}~.}$

Its normalized eigenvectors are

${\displaystyle \psi _{+}={\frac {1}{\sqrt {2|{\vec {a}}|(a_{3}+|{\vec {a}}|)}}}{\begin{pmatrix}a_{3}+|{\vec {a}}|\\a_{1}+ia_{2}\end{pmatrix}};\qquad \psi _{-}={\frac {1}{\sqrt {2|{\vec {a}}|(a_{3}+|{\vec {a}}|)}}}{\begin{pmatrix}ia_{2}-a_{1}\\a_{3}+|{\vec {a}}|\end{pmatrix}}.}$

### Commutation relations

The Pauli matrices obey the following commutation relations:

${\displaystyle [\sigma _{a},\sigma _{b}]=2i\varepsilon _{abc}\,\sigma _{c}\,,}$

and anticommutation relations:

${\displaystyle \{\sigma _{a},\sigma _{b}\}=2\delta _{ab}\,I.}$

where the structure constant εabc is the Levi-Civita symbol, Einstein summation notation is used, δab is the Kronecker delta, and I is the 2 × 2 identity matrix.

For example,

commutatorsanticommutators
{\displaystyle {\begin{aligned}\left[\sigma _{1},\sigma _{2}\right]&=2i\sigma _{3}\\\left[\sigma _{2},\sigma _{3}\right]&=2i\sigma _{1}\\\left[\sigma _{3},\sigma _{1}\right]&=2i\sigma _{2}\\\left[\sigma _{1},\sigma _{1}\right]&=0\end{aligned}}}    {\displaystyle {\begin{aligned}\left\{\sigma _{1},\sigma _{1}\right\}&=2I\\\left\{\sigma _{1},\sigma _{2}\right\}&=0\\\left\{\sigma _{1},\sigma _{3}\right\}&=0\\\left\{\sigma _{2},\sigma _{2}\right\}&=2I.\end{aligned}}}

### 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

{\displaystyle {\begin{aligned}\left[\sigma _{a},\sigma _{b}\right]+\{\sigma _{a},\sigma _{b}\}&=(\sigma _{a}\sigma _{b}-\sigma _{b}\sigma _{a})+(\sigma _{a}\sigma _{b}+\sigma _{b}\sigma _{a})\\2i\varepsilon _{abc}\,\sigma _{c}+2\delta _{ab}I&=2\sigma _{a}\sigma _{b}\end{aligned}}}

so that,

${\displaystyle \sigma _{a}\sigma _{b}=\delta _{ab}I+i\varepsilon _{abc}\,\sigma _{c}~.}$

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), and relabeling indices a, b, cp, q, r, to prevent notational conflicts, yields

{\displaystyle {\begin{aligned}a_{p}b_{q}\sigma _{p}\sigma _{q}&=a_{p}b_{q}\left(i\varepsilon _{pqr}\,\sigma _{r}+\delta _{pq}I\right)\\a_{p}\sigma _{p}b_{q}\sigma _{q}&=i\varepsilon _{pqr}\,a_{p}b_{q}\sigma _{r}+a_{p}b_{q}\delta _{pq}I~.\end{aligned}}}

Finally, translating the index notation for the dot product and cross product results in

${\displaystyle \left({\vec {a}}\cdot {\vec {\sigma }}\right)\left({\vec {b}}\cdot {\vec {\sigma }}\right)=\left({\vec {a}}\cdot {\vec {b}}\right)\,I+i\left({\vec {a}}\times {\vec {b}}\right)\cdot {\vec {\sigma }}}$

(1)

If i is identified with the pseudoscalar σxσyσz then the right hand side becomes ${\displaystyle a\cdot b+a\wedge b}$ 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.

{\displaystyle {\begin{aligned}\operatorname {tr} \left(\sigma _{a}\right)&=0\\\operatorname {tr} \left(\sigma _{a}\sigma _{b}\right)&=2\delta _{ab}\\\operatorname {tr} \left(\sigma _{a}\sigma _{b}\sigma _{c}\right)&=2i\varepsilon _{abc}\\\operatorname {tr} \left(\sigma _{a}\sigma _{b}\sigma _{c}\sigma _{d}\right)&=2\left(\delta _{ab}\delta _{cd}-\delta _{ac}\delta _{bd}+\delta _{ad}\delta _{bc}\right)\end{aligned}}}

If the matrix σ0 = I is also considered, these relationships become

{\displaystyle {\begin{aligned}\operatorname {tr} \left(\sigma _{\alpha }\right)&=2\delta _{0\alpha }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\right)&=2\delta _{\alpha \beta }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\sigma _{\gamma }\right)&=2\sum _{(\alpha \beta \gamma )}\delta _{\alpha \beta }\delta _{0\gamma }-4\delta _{0\alpha }\delta _{0\beta }\delta _{0\gamma }+2i\varepsilon _{0\alpha \beta \gamma }\\\operatorname {tr} \left(\sigma _{\alpha }\sigma _{\beta }\sigma _{\gamma }\sigma _{\mu }\right)&=2\left(\delta _{\alpha \beta }\delta _{\gamma \mu }-\delta _{\alpha \gamma }\delta _{\beta \mu }+\delta _{\alpha \mu }\delta _{\beta \gamma }\right)+4\left(\delta _{\alpha \gamma }\delta _{0\beta }\delta _{0\mu }+\delta _{\beta \mu }\delta _{0\alpha }\delta _{0\gamma }\right)-8\delta _{0\alpha }\delta _{0\beta }\delta _{0\gamma }\delta _{0\mu }+2i\sum _{(\alpha \beta \gamma \mu )}\varepsilon _{0\alpha \beta \gamma }\delta _{0\mu }\end{aligned}}}

where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation ${\textstyle \sum _{(\alpha \ldots )}}$ is used to denote the sum over the cyclic permutation of the included indices.

### Exponential of a Pauli vector

For

${\displaystyle {\vec {a}}=a{\hat {n}},\quad |{\hat {n}}|=1,}$

one has, for even powers, 2p, p = 0, 1, 2, 3, ...

${\displaystyle ({\hat {n}}\cdot {\vec {\sigma }})^{2p}=I}$

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, ...

${\displaystyle \left({\hat {n}}\cdot {\vec {\sigma }}\right)^{2q+1}={\hat {n}}\cdot {\vec {\sigma }}\,.}$

Matrix exponentiating, and using the Taylor series for sine and cosine,

{\displaystyle {\begin{aligned}e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}&=\sum _{k=0}^{\infty }{\frac {i^{k}\left[a\left({\hat {n}}\cdot {\vec {\sigma }}\right)\right]^{k}}{k!}}\\&=\sum _{p=0}^{\infty }{\frac {(-1)^{p}(a{\hat {n}}\cdot {\vec {\sigma }})^{2p}}{(2p)!}}+i\sum _{q=0}^{\infty }{\frac {(-1)^{q}(a{\hat {n}}\cdot {\vec {\sigma }})^{2q+1}}{(2q+1)!}}\\&=I\sum _{p=0}^{\infty }{\frac {(-1)^{p}a^{2p}}{(2p)!}}+i({\hat {n}}\cdot {\vec {\sigma }})\sum _{q=0}^{\infty }{\frac {(-1)^{q}a^{2q+1}}{(2q+1)!}}\\\end{aligned}}}.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

${\displaystyle e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}=I\cos {a}+i({\hat {n}}\cdot {\vec {\sigma }})\sin {a}}$

(2)

which is analogous to Euler's formula, extended to quaternions.

Note that

${\displaystyle \det[ia({\hat {n}}\cdot {\vec {\sigma }})]=a^{2}}$,

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]

${\displaystyle f(a({\hat {n}}\cdot {\vec {\sigma }}))=I{\frac {f(a)+f(-a)}{2}}+{\hat {n}}\cdot {\vec {\sigma }}{\frac {f(a)-f(-a)}{2}}~.}$

#### 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

{\displaystyle {\begin{aligned}e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}e^{ib\left({\hat {m}}\cdot {\vec {\sigma }}\right)}&=I\left(\cos a\cos b-{\hat {n}}\cdot {\hat {m}}\sin a\sin b\right)+i\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}~\sin a\sin b\right)\cdot {\vec {\sigma }}\\&=I\cos {c}+i\left({\hat {k}}\cdot {\vec {\sigma }}\right)\sin c\\&=e^{ic\left({\hat {k}}\cdot {\vec {\sigma }}\right)},\end{aligned}}}

which specifies the generic group multiplication, where, manifestly,

${\displaystyle \cos c=\cos a\cos b-{\hat {n}}\cdot {\hat {m}}\sin a\sin b~,}$

the spherical law of cosines. Given c, then,

${\displaystyle {\hat {k}}={\frac {1}{\sin c}}\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}\sin a\sin b\right)~.}$

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to [4]

${\displaystyle e^{ic{\hat {k}}\cdot {\vec {\sigma }}}=\exp \left(i{\frac {c}{\sin c}}\left({\hat {n}}\sin a\cos b+{\hat {m}}\sin b\cos a-{\hat {n}}\times {\hat {m}}~\sin a\sin b\right)\cdot {\vec {\sigma }}\right)~.}$

(Of course, when ̂n is parallel to ̂m, so is ̂k, and c = a + b.)

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle a,

${\displaystyle e^{ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}~{\vec {\sigma }}~e^{-ia\left({\hat {n}}\cdot {\vec {\sigma }}\right)}={\vec {\sigma }}\cos(2a)+{\hat {n}}\times {\vec {\sigma }}~\sin(2a)+{\hat {n}}~{\hat {n}}\cdot {\vec {\sigma }}~(1-\cos(2a))~.}$

### 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 i-th Pauli matrix is σiαβ.

In this notation, the completeness relation for the Pauli matrices can be written

${\displaystyle {\vec {\sigma }}_{\alpha \beta }\cdot {\vec {\sigma }}_{\gamma \delta }\equiv \sum _{i=1}^{3}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}=2\delta _{\alpha \delta }\delta _{\beta \gamma }-\delta _{\alpha \beta }\delta _{\gamma \delta }.}$
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
${\displaystyle M=cI+\sum _{i}a_{i}\sigma ^{i}}$
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
${\displaystyle \operatorname {tr} \,\sigma ^{i}\sigma ^{j}=2\delta _{ij}}$
where "tr" denotes the trace, and hence that
{\displaystyle {\begin{aligned}c&={\frac {1}{2}}\operatorname {tr} \,M,\ \ a_{i}={\frac {1}{2}}\operatorname {tr} \,\sigma ^{i}M~.\\[3pt]\therefore ~~2M&=I\operatorname {tr} \,M+\sum _{i}\sigma ^{i}\operatorname {tr} \,\sigma ^{i}M~,\end{aligned}}}
which can be rewritten in terms of matrix indices as
${\displaystyle 2M_{\alpha \beta }=\delta _{\alpha \beta }M_{\gamma \gamma }+\sum _{i}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}M_{\delta \gamma }~,}$
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 σ0, so σ0αβ = δαβ. The completeness relation can alternatively be expressed as

${\displaystyle \sum _{i=0}^{3}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}=2\delta _{\alpha \delta }\delta _{\beta \gamma }~.}$

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 {σ0, σ1, σ2, σ3} as above, and then imposing the positive-semidefinite and trace 1 conditions.

For a pure state, in polar coordinates, ${\displaystyle {\vec {a}}={\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \end{pmatrix}}}$, the idempotent density matrix

${\displaystyle {\frac {1}{2}}\left(1\!\!1+{\vec {a}}\cdot {\vec {\sigma }}\right)={\begin{pmatrix}\cos ^{2}\left({\frac {\theta }{2}}\right)&e^{-i\phi }\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right)\\e^{i\phi }\sin \left({\frac {\theta }{2}}\right)\cos \left({\frac {\theta }{2}}\right)&\sin ^{2}\left({\frac {\theta }{2}}\right)\end{pmatrix}}}$

acts on the state eigenvector ${\displaystyle {\begin{pmatrix}\cos \left({\frac {\theta }{2}}\right)&e^{i\phi }\sin \left({\frac {\theta }{2}}\right)\end{pmatrix}}}$ with eigenvalue 1, hence like a projection operator for it.

### Relation with the permutation operator

Let Pij be the transposition (also known as a permutation) between two spins σi and σj living in the tensor product space ${\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}$,

${\displaystyle P_{ij}|\sigma _{i}\sigma _{j}\rangle =|\sigma _{j}\sigma _{i}\rangle \,.}$

This operator can also be written more explicitly as Dirac's spin exchange operator,

${\displaystyle P_{ij}={\frac {1}{2}}\left({\vec {\sigma }}_{i}\cdot {\vec {\sigma }}_{j}+1\right)\,.}$

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 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra ${\displaystyle {\mathfrak {su}}_{2}}$ is the 3-dimensional real algebra spanned by the set {j}. In compact notation,

${\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \{i\sigma _{1},i\sigma _{2},i\sigma _{3}\}.}$

As a result, each 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

${\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \left\{{\frac {i\sigma _{1}}{2}},{\frac {i\sigma _{2}}{2}},{\frac {i\sigma _{3}}{2}}\right\}.}$

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 three-dimensional space. In other words, one can say that the j are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional 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 two-to-one group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).

### Quaternions

The real linear span of {I, 1, 2, 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):

${\displaystyle 1\mapsto I,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3}=-i\sigma _{1},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1}=-i\sigma _{2},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2}=-i\sigma _{3}.}$

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order, [6]

${\displaystyle 1\mapsto I,\quad \mathbf {i} \mapsto i\sigma _{3},\quad \mathbf {j} \mapsto i\sigma _{2},\quad \mathbf {k} \mapsto i\sigma _{1}.}$

As the set of versors U${\displaystyle \mathbb {H} }$ 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.

## Physics

### Classical mechanics

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. [7] The matrix P corresponding to the position ${\displaystyle {\vec {x}}}$ of a point in space is defined in terms of the above Pauli vector matrix,

${\displaystyle P={\vec {x}}\cdot {\vec {\sigma }}=x\sigma _{x}+y\sigma _{y}+z\sigma _{z}~.}$

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 [7]

${\displaystyle Q_{\theta }=1\!\!1\cos {\frac {\theta }{2}}+i\sigma _{x}\sin {\frac {\theta }{2}}~.}$

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 12 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, j are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 12. 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 12 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 12 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 Gn is defined to consist of all n-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

${\displaystyle {\mathsf {\Sigma }}_{i}={\begin{pmatrix}{\mathsf {\sigma }}_{i}&0\\0&{\mathsf {\sigma }}_{i}\end{pmatrix}}.}$

It follows from this definition that ${\displaystyle {\mathsf {\Sigma }}_{i}}$ matrices have the same algebraic properties as σi matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence ${\displaystyle {\mathsf {\Sigma }}_{i}}$ needs to be replaced by Σμν, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σμν are also antisymmetric. Hence there are only six independent matrices.

The first three are the ${\displaystyle \Sigma _{jk}\equiv \epsilon _{ijk}{\mathsf {\Sigma }}_{i}.}$ The remaining three, ${\displaystyle -i\Sigma _{0i}\equiv {\mathsf {\alpha }}_{i}}$, where the Dirac αi matrices are defined as

${\displaystyle {\mathsf {\alpha }}_{i}={\begin{pmatrix}0&{\mathsf {\sigma }}_{i}\\{\mathsf {\sigma }}_{i}&0\end{pmatrix}}.}$

The relativistic spin matrices Σμν are written in compact form in terms of commutator of gamma matrices as

${\displaystyle \Sigma _{\mu \nu }={\frac {i}{2}}\left[\gamma _{\mu },\gamma _{\nu }\right]}$.

### 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.

## Remarks

1. This conforms to the mathematics convention for the matrix exponential, ↦ exp(). In the physics convention, σ ↦ exp(−), hence in it no pre-multiplication by i is necessary to land in SU(2).
2. The Pauli vector is a formal device. It may be thought of as an element of M2(C) ⊗ R3, where the tensor product space is endowed with a mapping  : R3 × (M2(C) ⊗ R3) → M2(C) induced by the dot product on R3.
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

1. "Pauli matrices". Planetmath website. 28 March 2008. Retrieved 28 May 2013.
2. See the spinor map.
3. Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. ISBN   978-0-521-63235-5. OCLC   43641333.
4. 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 half-angle: "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; online
5. Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is ${\displaystyle \left({\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{smallmatrix}}\right)~.}$
6. Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN   978-0-7503-0606-5., pp. xxii.
7. Goldstein, Herbert (1959). Classical Mechanics. Addison-Wesley. pp. 109–118.
8. Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10: 084. arXiv:. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID   18776942.

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In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.

In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation. If A is a covariant vector,

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In mathematical physics, the gamma matrices, , also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles.

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any fermionic particle that is its own anti-particle.

The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.

In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations. In this basis, the spin is quantized along the axis in the direction of motion of the particle.

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

Electric dipole transition is the dominant effect of an interaction of an electron in an atom with the electromagnetic field.

The Maxwell–Bloch equations, also called the optical Bloch equations describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.