Pauli matrices

File:Wolfgang Pauli.jpg

Wolfgang Pauli (1900–1958), c. 1924. Pauli received the Nobel Prize in Physics in 1945, nominated by Albert Einstein, for the Pauli exclusion principle.

In mathematical physics and mathematics, the Pauli matrices are a set of three Template:Math complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (Template:Mvar), they are occasionally denoted by tau (Template:Mvar) when used in connection with isospin symmetries. <math display="block"> \begin{align}

 \sigma_1 = \sigma_x &=
   \begin{pmatrix}
     0&1\\
     1&0
   \end{pmatrix}, \\
 \sigma_2 = \sigma_y &=
   \begin{pmatrix}
     0& -i \\
     i&0
   \end{pmatrix}, \\
 \sigma_3 = \sigma_z &=
   \begin{pmatrix}
     1&0\\
     0&-1
   \end{pmatrix}. \\

\end{align} </math>

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 Template:Mvar (sometimes considered as the zeroth Pauli matrix Template:Math), the Pauli matrices form a basis of the vector space of Template:Math Hermitian matrices over the real numbers, under addition. This means that any Template:Math Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

The Pauli matrices satisfy the useful product relation:<math display="block">\begin{align}

\sigma_i \sigma_j = \delta_{ij}+i\epsilon_{ijk}\sigma_k.

\end{align}</math>

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, Template:Mvar represents the observable corresponding to spin along the Template:Mvarth coordinate axis in three-dimensional Euclidean space <math>\mathbb{R}^3 .</math>

The Pauli matrices (after multiplication by Template:Mvar to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices Template:Math form a basis for the real Lie algebra <math>\mathfrak{su}(2)</math>, which exponentiates to the special unitary group SU(2).Template:Efn The algebra generated by the three matrices Template:Math is isomorphic to the Clifford algebra of <math> \mathbb{R}^3,</math><ref> Template:Cite journal </ref> and the (unital) associative algebra generated by Template:Math functions identically (is isomorphic) to that of quaternions (<math>\mathbb{H}</math>).

Algebraic properties

Cayley table; the entry shows the value of the row times the column.

×	<math>\sigma_x</math>	<math>\sigma_y</math>	<math>\sigma_z</math>
<math>\sigma_x</math>	<math>I</math>	<math>i \sigma_z</math>	<math>-i \sigma_y</math>
<math>\sigma_y</math>	<math>-i \sigma_z</math>	<math>I</math>	<math>i \sigma_x</math>
<math>\sigma_z</math>	<math>i \sigma_y</math>	<math>-i \sigma_x</math>	<math>I</math>

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

<math>

 \sigma_j = 
   \begin{pmatrix}
     \delta_{j3}                   &  \delta_{j1} - i\,\delta_{j2}\\
     \delta_{j1} + i\,\delta_{j2}  & -\delta_{j3}
   \end{pmatrix},

</math> where Template:Mvar is the Kronecker delta, which equals Template:Math if Template:Math and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of Template:Math in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are involutory:

<math>\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,</math>

where Template:Mvar is the identity matrix.

The determinants and traces of the Pauli matrices are

<math>\begin{align}

              \det \sigma_j &= -1, \\
 \operatorname{tr} \sigma_j &= 0,

\end{align}</math> from which we can deduce that each matrix Template:Mvar has eigenvalues +1 and −1.

With the inclusion of the identity matrix Template:Mvar (sometimes denoted Template:Math), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space <math>\mathcal{H}_2</math> of Template:Math Hermitian matrices over <math>\mathbb{R}</math>, and the Hilbert space <math>\mathcal{M}_{2,2}(\mathbb{C})</math> of all complex Template:Math matrices over <math>\mathbb{C}</math>.

Commutation and anti-commutation relations

Commutation relations

The Pauli matrices obey the following commutation relations:

<math>[\sigma_j, \sigma_k] = 2 i \varepsilon_{j k l}\,\sigma_l, </math>

where the Levi-Civita symbol Template:Math is used.

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra <math>(\mathbb{R}^3, \times) \cong \mathfrak{su}(2) \cong \mathfrak{so}(3) .</math>

Anticommutation relations

They also satisfy the anticommutation relations:

<math>\{\sigma_j, \sigma_k\} = 2 \delta_{j k}\,I,</math>

where <math>\{\sigma_j, \sigma_k\}</math> is defined as <math>\sigma_j \sigma_k + \sigma_k \sigma_j,</math> and Template:Math is the Kronecker delta. Template:Mvar denotes the Template:Math identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for <math>\mathbb{R}^3,</math> denoted <math>\mathrm{Cl}_3(\mathbb{R}) .</math>

The usual construction of generators <math>\sigma_{jk} = \tfrac{1}{4} [\sigma_j, \sigma_k]</math> of <math>\mathfrak{so}(3)</math> 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
<math>\begin{align} \left[\sigma_1, \sigma_1\right] &= 0 \\ \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 \end{align}</math>Template:Quad	<math>\begin{align} \left\{\sigma_1, \sigma_1\right\} &= 2I \\ \left\{\sigma_1, \sigma_2\right\} &= 0 \\ \left\{\sigma_2, \sigma_3\right\} &= 0 \\ \left\{\sigma_3, \sigma_1\right\} &= 0 \end{align}</math>

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues: Template:Math and Template:Math. The corresponding normalized eigenvectors are

<math>\begin{align}

 \psi_{x+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\  1 \end{bmatrix}, &
 \psi_{x-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \\
 \psi_{y+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\  i \end{bmatrix}, &
 \psi_{y-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -i \end{bmatrix}, \\
 \psi_{z+} &=                  \begin{bmatrix} 1 \\  0 \end{bmatrix}, &
 \psi_{z-} &=                  \begin{bmatrix} 0 \\  1 \end{bmatrix}.

\end{align}</math>

Pauli vectors

The Pauli vector is defined byTemplate:Efn <math display="block"> \vec{\sigma} = \sigma_1 \hat{x}_1 + \sigma_2 \hat{x}_2 + \sigma_3 \hat{x}_3, </math> where <math>\hat{x}_1</math>, <math>\hat{x}_2</math>, and <math>\hat{x}_3</math> are an equivalent notation for the more familiar <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math>.

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis<ref>See the spinor map.</ref> as follows: <math display="block">\begin{align}

    \vec{a} \cdot \vec{\sigma} 
 &= \sum_{k,l} a_k\, \sigma_\ell\, \hat{x}_k \cdot \hat{x}_\ell \\
 &= \sum_k a_k\, \sigma_k \\
 &= \begin{pmatrix}
                   a_3 &  a_1 - i a_2 \\
           a_1 + i a_2 & -a_3
       \end{pmatrix}.

\end{align} </math>

More formally, this defines a map from <math>\mathbb{R}^3</math> to the vector space of traceless Hermitian <math>2\times 2</math> matrices. This map encodes structures of <math>\mathbb{R}^3</math> 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 <math>2\times 2</math> Hermitian traceless matrix-valued dual vector, that is, an element of <math>\text{Mat}_{2\times 2}(\mathbb{C}) \otimes (\mathbb{R}^3)^*</math> that maps <math>\vec a \mapsto \vec a \cdot \vec \sigma.</math>

Completeness relation

Each component of <math>\vec a</math> can be recovered from the matrix (see completeness relation below) <math display="block">

\frac{1}{2} \operatorname{tr} \Bigl( \bigl( \vec{a} \cdot \vec{\sigma} \bigr) \vec{\sigma} \Bigr) = \vec{a}.

</math> This constitutes an inverse to the map <math>\vec a \mapsto \vec a \cdot \vec \sigma</math>, making it manifest that the map is a bijection.

Determinant

The norm is given by the determinant (up to a minus sign) <math display="block"> \det \bigl( \vec{a} \cdot \vec{\sigma} \bigr) = -\vec{a} \cdot \vec{a} = -|\vec{a}|^2. </math> Then, considering the conjugation action of an <math>\text{SU}(2)</math> matrix <math>U</math> on this space of matrices,

<math>U * \vec a \cdot \vec \sigma := U \, \vec a \cdot \vec \sigma \, U^{-1},</math>

we find <math>\det(U * \vec a \cdot \vec\sigma) = \det(\vec a \cdot \vec \sigma),</math> and that <math>U * \vec a \cdot \vec \sigma</math> is Hermitian and traceless. It then makes sense to define <math>U * \vec a \cdot \vec\sigma = \vec a' \cdot \vec\sigma,</math> where <math>\vec a'</math> has the same norm as <math>\vec a,</math> and therefore interpret <math>U</math> as a rotation of three-dimensional space. In fact, it turns out that the special restriction on <math>U</math> implies that the rotation is orientation preserving. This allows the definition of a map <math>R: \mathrm{SU}(2) \to \mathrm{SO}(3)</math> given by

<math>U * \vec a \cdot \vec \sigma = \vec a' \cdot \vec \sigma =: (R(U)\ \vec a) \cdot \vec \sigma,</math>

where <math>R(U) \in \mathrm{SO}(3).</math> This map is the concrete realization of the double cover of <math>\mathrm{SO}(3)</math> by <math>\mathrm{SU}(2),</math> and therefore shows that <math>\text{SU}(2) \cong \mathrm{Spin}(3).</math> The components of <math>R(U)</math> can be recovered using the tracing process above:

<math>R(U)_{ij} = \frac{1}{2} \operatorname{tr} \left( \sigma_i U \sigma_j U^{-1} \right).</math>

Cross-product

The cross-product is given by the matrix commutator (up to a factor of <math>2i</math>) <math display="block">

[\vec a \cdot \vec \sigma, \vec b \cdot \vec \sigma] = 2i\,(\vec a \times \vec b) \cdot \vec \sigma.

</math> In fact, the existence of a norm follows from the fact that <math>\mathbb{R}^3</math> is a Lie algebra (see Killing form).

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

Eigenvalues and eigenvectors

The eigenvalues of <math>\ \vec a \cdot \vec \sigma\ </math> are <math>\ \pm |\vec{a}|.</math> 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 <math>\ (\vec a \cdot \vec \sigma)^2 - |\vec a|^2 = 0\ ,</math> since this can be factorised into <math>\ (\vec a \cdot \vec \sigma - |\vec a|)(\vec a \cdot \vec \sigma + |\vec a|)= 0.</math> A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies <math>\ \vec a \cdot \vec \sigma\ </math> is diagonal with possible eigenvalues <math>\ \pm |\vec a|.</math> The tracelessness of <math>\ \vec a \cdot \vec \sigma\ </math> means it has exactly one of each eigenvalue.

Its normalized eigenvectors are <math display="block">

 \psi_+ = \frac{1}{\sqrt{2 \left|\vec{a} \right|\ (a_3+\left|\vec{a}\right|)\ }\ } \begin{bmatrix} a_3 + \left|\vec{a}\right| \\ a_1 + ia_2 \end{bmatrix}; \qquad
 \psi_- = \frac{1}{\sqrt{2|\vec{a}|(a_3+|\vec{a}|)}} \begin{bmatrix} ia_2 - a_1 \\  a_3 + |\vec{a}| \end{bmatrix} ~ .

</math> These expressions become singular for <math>a_3\to

-\left|\vec{a} \right|</math>. They can be rescued by letting <math>\vec{a}=\left|\vec{a} \right|(\epsilon,0,-(1-\epsilon^2/2))</math> and taking the limit <math>\epsilon\to0</math>, which yields the correct eigenvectors (0,1) and (1,0) of <math>\sigma_z</math>.

Alternatively, one may use spherical coordinates <math>\vec{a}=a(\sin\vartheta\cos\varphi, \sin\vartheta\sin\varphi, \cos\vartheta)</math> to obtain the eigenvectors <math>\psi_+=(\cos(\vartheta/2), \sin(\vartheta/2)\exp(i\varphi))</math> and <math>\psi_-=(-\sin(\vartheta/2)\exp(-i\varphi), \cos(\vartheta/2))</math>.

Pauli 4-vector

The Pauli 4-vector, used in spinor theory, is written <math>\ \sigma^\mu\ </math> with components

<math>\sigma^\mu = (I, \vec\sigma).</math>

This defines a map from <math>\mathbb{R}^{1,3}</math> to the vector space of Hermitian matrices,

<math>x_\mu \mapsto x_\mu\sigma^\mu\ ,</math>

which also encodes the Minkowski metric (with mostly minus convention) in its determinant:

<math>\det (x_\mu\sigma^\mu) = \eta(x,x).</math>

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

<math>\bar\sigma^\mu = (I, -\vec\sigma).</math>

and allow raising and lowering using the Minkowski metric tensor. The relation can then be written <math display="block">x_\nu = \tfrac{1}{2} \operatorname{tr} \Bigl( \bar\sigma_\nu\bigl( x_\mu \sigma^\mu \bigr) \Bigr) .</math>

Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on <math>\ \mathbb{R}^{1,3}\ ;</math> in this case the matrix group is <math>\ \mathrm{SL}(2,\mathbb{C})\ ,</math> and this shows <math>\ \mathrm{SL}(2,\mathbb{C}) \cong \mathrm{Spin}(1,3).</math> Similarly to above, this can be explicitly realized for <math>\ S \in \mathrm{SL}(2,\mathbb{C})\ </math> with components

<math>\Lambda(S)^\mu{}_\nu = \tfrac{1}{2}\operatorname{tr} \left( \bar\sigma_\nu S \sigma^\mu S^{\dagger}\right).</math>

In fact, the determinant property follows abstractly from trace properties of the <math>\ \sigma^\mu.</math> For <math>\ 2\times 2\ </math> matrices, the following identity holds:

<math>\det(A + B) = \det(A) + \det(B) + \operatorname{tr}(A)\operatorname{tr}(B) - \operatorname{tr}(AB).</math>

That is, the 'cross-terms' can be written as traces. When <math>\ A,B\ </math> are chosen to be different <math>\ \sigma^\mu\ ,</math> the cross-terms vanish. It then follows, now showing summation explicitly, <math display="inline">\det\left(\sum_\mu x_\mu \sigma^\mu\right) = \sum_\mu \det\left(x_\mu\sigma^\mu\right).</math> Since the matrices are <math>\ 2 \times 2\ ,</math> this is equal to <math display="inline">\sum_\mu x_\mu^2 \det(\sigma^\mu) = \eta(x,x).</math>

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

<math>

\begin{align}

 \left[ \sigma_j, \sigma_k\right] + \{\sigma_j, \sigma_k\}  &= (\sigma_j \sigma_k - \sigma_k \sigma_j ) + (\sigma_j \sigma_k + \sigma_k \sigma_j) \\
   2i\varepsilon_{j k \ell}\,\sigma_\ell + 2 \delta_{j k}I &= 2\sigma_j \sigma_k 

\end{align} </math> so that, Template:Equation box 1

Contracting each side of the equation with components of two Template:Math-vectors Template:Math and Template:Math (which commute with the Pauli matrices, i.e., Template:Math for each matrix Template:Math and vector component Template:Math (and likewise with Template:Math) yields

<math>~~ \begin{align}

 a_j b_k \sigma_j \sigma_k & = a_j b_k \left(i\varepsilon_{jk\ell}\,\sigma_\ell + \delta_{jk}I\right) \\
 a_j \sigma_j b_k \sigma_k & = i\varepsilon_{jk\ell}\,a_j b_k \sigma_\ell + a_j b_k \delta_{jk}I 

\end{align}.~</math>

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

If Template:Mvar is identified with the pseudoscalar Template:Math then the right hand side becomes <math> a \cdot b + a \wedge b </math>, which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as Template:Math, then Template:Math satisfies the commutation relation:<math display="block">\mathbf{J} \times \mathbf{J} = i\hbar \mathbf{J}</math>Or equivalently, the Pauli vector satisfies:<math display="block">\frac{\vec{\sigma}}{2} \times \frac{\vec{\sigma}}{2} = i\frac{\vec{\sigma}}{2}</math>

Some trace relations

The following traces can be derived using the commutation and anticommutation relations.

<math>\begin{align}

 \operatorname{tr}\left(\sigma_j                           \right) &= 0 \\
 \operatorname{tr}\left(\sigma_j \sigma_k                  \right) &= 2\delta_{jk} \\
 \operatorname{tr}\left(\sigma_j \sigma_k \sigma_\ell      \right) &= 2i\varepsilon_{jk\ell} \\
 \operatorname{tr}\left(\sigma_j \sigma_k \sigma_\ell \sigma_m \right) &= 2\left(\delta_{jk}\delta_{\ell m} - \delta_{j\ell}\delta_{km} + \delta_{jm}\delta_{k\ell}\right)

\end{align}</math>

If the matrix Template:Math is also considered, these relationships become

<math display=block>\begin{align}

                            \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} + 2 i \sum_{(\alpha \beta \gamma \mu)} \varepsilon_{0 \alpha \beta \gamma} \delta_{0 \mu}

\end{align}</math>

where Greek indices Template:Math and Template:Mvar assume values from Template:Math and the notation <math display="inline">\sum_{(\alpha \ldots)}</math> is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For

<math>\vec{a} = a\hat{n}, \quad |\hat{n}| = 1,</math>

one has, for even powers, Template:Math

<math>(\hat{n} \cdot \vec{\sigma})^{2p} = I ,</math>

which can be shown first for the Template:Math case using the anticommutation relations. For convenience, the case Template:Math is taken to be Template:Mvar by convention.

For odd powers, Template:Math

<math>\left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} \, .</math>

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

<math>\begin{align}

 e^{i a\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{align}</math>.

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

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

<math>e^{i a \sigma_1} = \begin{pmatrix} \cos a & i \sin a \\ i \sin a & \cos a \end{pmatrix}, \quad e^{i a \sigma_2} = \begin{pmatrix} \cos a & \sin a \\ - \sin a & \cos a \end{pmatrix}, \quad e^{i a \sigma_3} = \begin{pmatrix} e^{ia} & 0 \\ 0 & e^{-ia} \end{pmatrix}.</math>

Note that

<math>\det[i a(\hat{n} \cdot \vec{\sigma})] = a^2</math>,

while the determinant of the exponential itself is just Template:Math, which makes it the generic group element of SU(2).

A more abstract version of formula Template:EquationNote for a general Template:Math matrix can be found in the article on matrix exponentials. A general version of Template:EquationNote for an analytic (at a and −a) function is provided by application of Sylvester's formula,<ref> Template:Cite book </ref>

<math>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}.</math>

The group composition law of Template:Math

A straightforward application of formula Template:EquationNote provides a parameterization of the composition law of the group Template:Math.Template:Efn One may directly solve for Template:Mvar in <math display=block>\begin{align}

 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{align}</math>

which specifies the generic group multiplication, where, manifestly, <math display=block>\cos c = \cos a \cos b - \hat{n} \cdot \hat{m} \sin a \sin b~,</math> the spherical law of cosines. Given Template:Mvar, then, <math display=block>\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).</math>

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to<ref>Template:Cite book In fact, however, the formula goes back to Olinde Rodrigues (1840), replete with half-angle: Template:Cite journal</ref>

<math display=block> 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). </math>

(Of course, when <math>\hat{n}</math> is parallel to <math>\hat{m}</math>, so is <math>\hat{k}</math>, and Template:Math.) Template:See also

Adjoint action

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle <math>a</math> along any axis <math>\hat n</math>: <math display=block>

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

</math>

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 <math display="inline">R_y\mathord\left(-\frac{\pi}{2}\right)\, \sigma_x\, R_y\mathord\left(\frac{\pi}{2}\right) = \hat{x} \cdot \left(\hat{y} \times \vec{\sigma}\right) = \sigma_z</math>.

Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index Template:Mvar in the superscript, and the matrix indices as subscripts, so that the element in row Template:Mvar and column Template:Mvar of the Template:Mvar-th Pauli matrix is Template:Math

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

<math>\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{k=1}^3 \sigma^k_{\alpha\beta}\,\sigma^k_{\gamma\delta} = 2\,\delta_{\alpha\delta} \,\delta_{\beta\gamma} - \delta_{\alpha\beta}\,\delta_{\gamma\delta}.</math>

As noted above, it is common to denote the 2 × 2 unit matrix by Template:Math so Template:Math The completeness relation can alternatively be expressed as <math display="block">\sum_{k=0}^3 \sigma^k_{\alpha\beta}\,\sigma^k_{\gamma\delta} = 2\,\delta_{\alpha\delta}\,\delta_{\beta\gamma} ~ .</math>

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 Template:Math as above, and then imposing the positive-semidefinite and trace Template:Math conditions.

For a pure state, in polar coordinates, <math display="block">\vec{a} = \begin{pmatrix}\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end{pmatrix},</math> the idempotent density matrix <math display="block">

 \tfrac{1}{2} \left(\mathbf{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}

</math>

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

Relation with the permutation operator

Let Template:Math be the transposition (also known as a permutation) between two spins Template:Math and Template:Math living in the tensor product space Template:Nowrap

<math>P_{jk} \left| \sigma_j \sigma_k \right\rangle = \left| \sigma_k \sigma_j \right\rangle .</math>

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

<math>P_{jk} = \frac{1}{2}\,\left(\vec{\sigma}_j \cdot \vec{\sigma}_k + 1\right) ~ .</math>

Its eigenvalues are thereforeTemplate:Efn 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 Template:Math matrices with unit determinant; its Lie algebra is the set of all Template:Math anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra <math>\mathfrak{su}_2</math> is the three-dimensional real algebra spanned by the set Template:Math. In compact notation,

<math> \mathfrak{su}(2) = \operatorname{span} \{\; i\,\sigma_1\, ,\; i\,\sigma_2\, , \; i\,\sigma_3 \;\}.</math>

As a result, each Template:Math 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 theory of SU(2)|representation of Template:Math]], as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is Template:Math so that

<math> \mathfrak{su}(2) = \operatorname{span} \left\{\frac{\,i\,\sigma_1\,}{2}, \frac{\,i\,\sigma_2\,}{2}, \frac{\,i\,\sigma_3\,}{2} \right\}.</math>

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

SO(3)

The Lie algebra <math> \mathfrak{su}(2)</math> is isomorphic to the Lie algebra <math> \mathfrak{so}(3)</math>, which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the Template:Math are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though <math> \mathfrak{su}(2)</math> and <math> \mathfrak{so}(3)</math> are isomorphic as Lie algebras, Template:Math and Template:Math are not isomorphic as Lie groups. Template:Math is actually a double cover of Template:Math, meaning that there is a two-to-one group homomorphism from Template:Math to Template:Math, see relationship between SO(3) and SU(2).

Quaternions

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The real linear span of Template:Math is isomorphic to the real algebra of quaternions, <math>\mathbb{H}</math>, represented by the span of the basis vectors <math> \left\{\; \mathbf{1}, \, \mathbf{i}, \, \mathbf{j}, \, \mathbf{k} \;\right\} .</math> The isomorphism from <math>\mathbb{H}</math> to this set is given by the following map (notice the reversed signs for the Pauli matrices): <math display=block>

 \mathbf{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.

</math>

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,<ref> Template:Cite book </ref>

<math>

 \mathbf{1} \mapsto I, \quad
 \mathbf{i} \mapsto i\,\sigma_3 \, , \quad
 \mathbf{j} \mapsto i\,\sigma_2 \, , \quad
 \mathbf{k} \mapsto i\,\sigma_1 ~ .

</math>

As the set of versors Template:Math forms a group isomorphic to Template:Math, Template:Mvar gives yet another way of describing Template:Math. The two-to-one homomorphism from Template:Math to Template:Math may be given in terms of the Pauli matrices in this formulation.

Physics

Classical mechanics

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters.<ref name=Goldstein-1959> Template:Cite book </ref> The matrix Template:Mvar corresponding to the position <math>\vec{x}</math> of a point in space is defined in terms of the above Pauli vector matrix,

<math>P = \vec{x} \cdot \vec{\sigma} = x\,\sigma_x + y\,\sigma_y + z\,\sigma_z .</math>

Consequently, the transformation matrix Template:Math for rotations about the Template:Mvar-axis through an angle Template:Mvar may be written in terms of Pauli matrices and the unit matrix as<ref name=Goldstein-1959/>

<math>Q_\theta = \boldsymbol{1}\,\cos\frac{\theta}{2} + i\,\sigma_x \sin\frac{\theta}{2} .</math>

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-1/2|spin Template:1/2]] particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, Template:Math are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin Template: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 Template:1/2 particles is that they must be rotated by an angle of 4Template:Mvar 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 Template:Math they are actually represented by orthogonal vectors in the two-dimensional complex Hilbert space.

For a spin Template:1/2 particle, the spin operator is given by Template:Math, 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 Template:Section link. 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.<ref>Template:Cite journal</ref>

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Template:Math is defined to consist of all Template:Mvar-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

<math>\mathsf{\Sigma}_k = \begin{pmatrix} \mathsf{\sigma}_k & 0 \\ 0 & \mathsf{\sigma}_k \end{pmatrix} .</math>

It follows from this definition that the <math>\ \mathsf{ \Sigma }_k \ </math> matrices have the same algebraic properties as the Template:Mvar matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence <math>\ \mathsf{\Sigma}_k\ </math> needs to be replaced by Template:Mvar, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Template:Math are also antisymmetric. Hence there are only six independent matrices.

The first three are the <math>\ \Sigma_{k\ell} \equiv \epsilon_{jk\ell}\mathsf{\Sigma}_j .</math> The remaining three, <math>\ -i\ \Sigma_{0k} \equiv \mathsf{\alpha}_k\ ,</math> where the [[Dirac equation|Dirac Template:Math matrices]] are defined as

<math>

\mathsf{\alpha}_k = \begin{pmatrix}

0   &   \mathsf{\sigma}_k \\
\mathsf{\sigma}_k   &   0

\end{pmatrix} . </math>

The relativistic spin matrices Template:Math are written in compact form in terms of commutator of gamma matrices as

<math>\Sigma_{\mu\nu} = \frac{i}{2} \bigl[ \gamma_\mu, \gamma_\nu \bigr] .</math>

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 ".

See also

• Algebra of physical space
• Spinors in three dimensions
• Gamma matrices
  • Template:Section link
• Angular momentum
• Gell-Mann matrices
• Poincaré group
• Generalizations of Pauli matrices
• Bloch sphere
• Euler's four-square identity
• For higher spin generalizations of the Pauli matrices, see Template:Section link
• Exchange matrix (the first Pauli matrix is an exchange matrix of order two)
• Split-quaternion

References

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book