The Pauli vector is defined by{{efn| The Pauli vector is a formal device. It may be thought of as an element of \ \mathcal M_2(\Complex) \otimes \R^3\ , where the
tensor product space is endowed with a mapping \ \cdot : \mathbb{R}^3 \times (\mathcal M_2(\Complex) \otimes \R^3) \to \mathcal M_2(\Complex)\ induced by the
dot product on \ \mathbb{R}^3 ~. }} \boldsymbol{\sigma} = \sigma_1 \boldsymbol{\hat{x}}_1 + \sigma_2 \boldsymbol{\hat{x}}_2 + \sigma_3 \boldsymbol{\hat{x}}_3, where \boldsymbol{\hat{x}}_1, \boldsymbol{\hat{x}}_2, and \boldsymbol{\hat{x}}_3 are an equivalent notation for the more familiar \boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}, and \boldsymbol{\hat{z}}. The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows: \begin{align} \boldsymbol{a} \cdot \boldsymbol{\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} More formally, this defines a map from \mathbb{R}^3 to the vector space of traceless Hermitian 2\times 2 matrices. This map encodes structures of \mathbb{R}^3 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 \ 2 \times 2\ Hermitian traceless matrix-valued dual vector, that is, an element of \ \mathrm{Mat}_{2\times 2}(\mathbb{C}) \otimes (\mathbb{R}^3)^*\ that maps \boldsymbol{a} \mapsto \boldsymbol{a} \cdot \boldsymbol{\sigma}
Completeness relation Each component of \boldsymbol{a} can be recovered from the matrix (see
completeness relation below) \frac{1}{2} \operatorname{tr} \Bigl[ \bigl(\ \boldsymbol{a} \cdot \boldsymbol{\sigma}\ \bigr)\ \boldsymbol{\sigma}\ \Bigr] = \boldsymbol{a} This constitutes an inverse to the map \boldsymbol{a} \mapsto \boldsymbol{a} \cdot \boldsymbol{\sigma}, making it manifest that the map is a bijection.
Determinant The norm is given by the determinant (up to a minus sign) \det\!\bigl(\ \vec{a} \cdot \vec{\sigma}\ \bigr)\ =\ -\vec{a} \cdot \vec{a}\ =\ -\left|\ \vec{a}\ \right|^2 ~. Then, considering the conjugation action of an \ \mathrm{SU}(2)\ matrix U on this space of matrices, : \ U * \vec a \cdot \vec \sigma\ :=\ U\ \vec a \cdot \vec \sigma\ U^{-1}\ , we find \ \det(U * \vec a \cdot \vec\sigma)\ =\ \det(\vec a \cdot \vec \sigma)\ , and that \ U * \vec a \cdot \vec \sigma\ is Hermitian and traceless. It then makes sense to define \ U * \vec a \cdot \vec\sigma\ =\ \vec a' \cdot \vec\sigma\ , where \ \vec a'\ has the same norm as \vec a, and therefore interpret U as a rotation of three-dimensional space. In fact, it turns out that the
special restriction on U implies that the rotation is orientation preserving. This allows the definition of a map \ R: \mathrm{SU}(2) \to \mathrm{SO}(3)\ given by : \ U * \vec a \cdot \vec \sigma\ =\ \vec a' \cdot \vec \sigma\ =:\ (R(U)\ \vec a) \cdot \vec \sigma\ , where \ R(U)\ \in\ \mathrm{SO}(3) ~. This map is the concrete realization of the double cover of \ \mathrm{SO}(3)\ by \ \mathrm{SU}(2)\ , and therefore shows that \ \mathrm{SU}(2)\ \cong\ \mathrm{Spin}(3) ~. The components of R(U) can be recovered using the tracing process above: : \ R(U)_{ij} = \frac{1}{2}\ \operatorname{tr}\!\left(\ \sigma_i U \sigma_j U^{-1}\ \right) ~.
Cross-product The cross-product is given by the matrix commutator (up to a factor of \ 2\ i\ ) \left[\ \vec a \cdot \vec \sigma,\ \vec b \cdot \vec \sigma\ \right] = 2\ i\ \left( \vec a \times \vec b \right) \cdot \vec \sigma ~. In fact, the existence of a norm follows from the fact that \ \mathbb{R}^3\ 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 \ \vec a \cdot \vec \sigma\ are \ \pm |\vec{a}| ~. 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 \ (\vec a \cdot \vec \sigma)^2 - |\vec a|^2 = 0\ , since this can be factorised into \ (\vec a \cdot \vec \sigma - |\vec a|)(\vec a \cdot \vec \sigma + |\vec a|)= 0 ~. A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is
diagonalizable) means this implies \ \vec a \cdot \vec \sigma\ is diagonalizable with possible eigenvalues \ \pm |\vec a| ~. The tracelessness of \ \vec a \cdot \vec \sigma\ means it has exactly one of each eigenvalue. Its normalized eigenvectors are \psi_+ = \frac{1}{\sqrt{2|\vec{a}|(a_3+|\vec{a}|)}} \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} ~. These expressions become singular for \ a_3 \to -\left|\ \vec{a}\ \right| ~. They can be rescued by letting \vec{a} = \left|\ \vec{a}\ \right| \left( \epsilon,\ 0,\ -\left( 1 - \tfrac{\epsilon^2}{2} \right) \right)\ and taking the limit \ \epsilon \to 0\ , which yields the correct eigenvectors and of \ \sigma_z ~. Alternatively, one may use spherical coordinates \ \vec{a} = a\ \bigl(\ \sin \vartheta\ \cos \varphi,\ \sin \vartheta\ \sin \varphi,\ \cos\vartheta\ \bigr)\ to obtain the eigenvectors \ \psi_{+} = \left(\ \cos \tfrac{\vartheta}{2}, \; \sin \tfrac{\vartheta}{2}\ e^{+i\varphi}\ \right)\ and \ \psi_{-} = \left(\ -\sin \tfrac{\vartheta}{2}\ e^{-i\varphi}, \; \cos \tfrac{\vartheta}{2}\ \right) ~.
Pauli 4-vector The Pauli 4-vector, used in spinor theory, is written \ \sigma^\mu\ with components :\ \sigma^\mu = \bigl(\ I,\ \vec\sigma\ \bigr) ~. This defines a map from \ \mathbb{R}^{1,3}\ to the vector space of Hermitian matrices, :\ x_\mu \mapsto x_\mu\sigma^\mu\ , which also encodes the
Minkowski metric (with
mostly minus convention) in its determinant: :\ \det\bigl(\ x_\mu\sigma^\mu\ \bigr) = \eta(x,x) ~. This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector :\ \bar\sigma^\mu = \bigl(\ I, -\vec\sigma\ \bigr) ~. and allow raising and lowering using the Minkowski metric tensor. The relation can then be written \ x_\nu = \tfrac{1}{2} \operatorname{tr}\!\Bigl(\ \bar\sigma_\nu\bigl( x_\mu \sigma^\mu \bigr)\ \Bigr) ~. Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on \ \mathbb{R}^{1,3}\ ; in this case the matrix group is \ \mathrm{SL}( 2, \mathbb{C} )\ , and this shows \ \mathrm{SL}(2,\mathbb{C})\ \cong\ \mathrm{Spin}(1,3) ~. Similarly to above, this can be explicitly realized for \ S \in \mathrm{SL}(2,\mathbb{C})\ with components :\ \Lambda(S)^\mu{}_\nu = \tfrac{1}{2}\operatorname{tr}\!\left(\ \bar\sigma_\nu\ S\ \sigma^\mu\ S^{\dagger}\ \right) ~. In fact, the determinant property follows abstractly from trace properties of the \ \sigma^\mu ~. For \ 2\times 2\ matrices, the following identity holds: :\ \det(\ A + B\ )\ =\ \det(A)\ +\ \det(B)\ +\ \operatorname{tr}(A)\ \operatorname{tr}(B)\ -\ \operatorname{tr}(\ A\ B\ ) ~. That is, the 'cross-terms' can be written as traces. When \ A,B\ are chosen to be different \ \sigma^\mu\ , the cross-terms vanish. It then follows, now showing summation explicitly, \det\left(\sum_\mu x_\mu \sigma^\mu\right) = \sum_\mu \det\left(x_\mu\sigma^\mu\right). Since the matrices are \ 2 \times 2\ , this is equal to \ \sum_\mu x_\mu^2 \det(\sigma^\mu) = \eta(x,x) ~.
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 : \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} so that, {{Equation box 1 |indent =: |equation = ~~ \sigma_j \sigma_k = \delta_{j k}I + i\varepsilon_{j k \ell}\,\sigma_\ell ~ .~ |cellpadding= 6 |border |border colour = #0073CF |bgcolor=#F9FFF7 }}
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 ) yields :~~ \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} ~. Finally, translating the index notation for the
dot product and
cross product results in {{NumBlk||{{Equation box 1 |indent =: |equation = ~~\Bigl(\vec{a} \cdot \vec{\sigma}\Bigr)\Bigl(\vec{b} \cdot \vec{\sigma}\Bigr) = \Bigl(\vec{a} \cdot \vec{b}\Bigr) \, I + i \Bigl(\vec{a} \times \vec{b}\Bigr) \cdot \vec{\sigma}~~ |cellpadding= 6 |border |border colour = #0073CF |bgcolor=#F9FFF7 }} | }} If is identified with the pseudoscalar then the right hand side becomes \ a \cdot b + a \wedge b\ , which is also the definition for the product of two vectors in geometric algebra. If we define the spin operator as then satisfies the commutation relation:\ \mathbf{J} \times \mathbf{J} = i\ \hbar \mathbf{J}\ Or equivalently, the Pauli vector satisfies:\ \frac{\vec{\sigma}}{2} \times \frac{\vec{\sigma}}{2} = i\ \frac{\vec{\sigma}}{2} ~.
Some trace relations The following traces can be derived using the commutation and anticommutation relations. :\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} ~. If the matrix \sigma_0 = \mathbb{I} is also considered, these relationships become \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} ~. where Greek indices \alpha, \beta, \gamma and \mu assume values from \{0, x, y, z\} and the notation \sum_{(\alpha \ldots)} is used to denote the sum over the
cyclic permutation of the included indices.
Exponential of a Pauli vector For :\vec{a} = a\ \hat{n}, \quad \left|\ \hat{n}\ \right| = 1\ , one has, for even powers, :\ (\hat{n} \cdot \vec{\sigma})^{2p} = I\ , 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, :\ \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, :\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} ~. In the last line, the first sum is the cosine, while the second sum is the sine; so, finally, {{NumBlk||{{Equation box 1 |indent =: |equation = ~~e^{i\ a\left(\hat{n} \cdot \vec{\sigma}\right)} = I\ \cos{a} + i\ (\hat{n} \cdot \vec{\sigma}) \sin{a} ~~ |cellpadding= 6 |border |border colour = #0073CF |bgcolor=#F9FFF7 }} | }} which is
analogous to
Euler's formula, extended to
quaternions. In particular, 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^{i\ a} & 0 \\ 0 & e^{-i\ a} \end{pmatrix} ~. Note that :\det\!\left[\ i\ a\ \left(\hat{n} \cdot \vec{\sigma} \right)\ \right] = a^2\ , 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 and ) function is provided by application of
Sylvester's formula, :\ 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 A straightforward application of formula provides a parameterization of the composition law of the group .{{efn| The relation among derived here in the representation holds for
all representations of , being a
group identity. Note that, by virtue of the standard normalization of that group's generators as
half the Pauli matrices, the parameters correspond to
half the rotation angles of the rotation group. That is, the Gibbs formula linked amounts to \ \hat k \tan \tfrac{c}{2} = (\hat n\ \tan \tfrac{a}{2} + \hat m\ \tan \tfrac{b}{2} - \hat m\ \times \hat n\ \tan \tfrac{a}{2} ~ \tan \tfrac{b}{2} )/(1 - \hat m\cdot \hat n\ \tan \tfrac{a}{2} ~ \tan \tfrac{b}{2} ) ~. }} One may directly solve for in \begin{align} e^{i\ a\left(\hat{n} \cdot \vec{\sigma}\right)}\ e^{i\ b\ \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^{i\ c\ \left(\hat{k} \cdot \vec{\sigma} \right) }\ , \end{align} which specifies the generic group multiplication, where, manifestly, \ \cos c = \cos a\ \cos b\ -\ \hat{n} \cdot \hat{m}\ \sin a\ \sin b\ , the
spherical law of cosines. Given , then, \ \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 \ 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 \ \hat{n}\ is parallel to \ \hat{m}\ , so are \ \hat{k}\ and
Adjoint action It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle a along any axis \hat n: 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)) ~ . 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 \ 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 ~.
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 : \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 proof | proof = The fact that the Pauli matrices, along with the identity matrix , form an orthogonal basis for the Hilbert space of all
complex matrices \ \mathcal{M}_{2,2}(\mathbb{C})\ over \ \mathbb{C}\ , means that we can express any complex matrix as M = c\ I + \sum_k a_k\ \sigma^k where is a complex number, and is a 3-component, complex vector. It is straightforward to show, using the properties listed above, that \operatorname{tr}\left( \sigma^j\,\sigma^k \right) = 2\ \delta_{jk} where "" denotes the
trace, and hence that \begin{align} c &={} \tfrac{1}{2}\ \operatorname{tr}\, M\ ,\begin{align}&& a_k &= \tfrac{1}{2}\ \operatorname{tr}\ \sigma^k\ M \end{align} ~.\\[3pt] \therefore ~~ 2\,M &= I\,\operatorname{tr}\, M + \sum_k \sigma^k\,\operatorname{tr}\, \sigma^k M\ , \end{align} which can be rewritten in terms of matrix indices as 2\ M_{\alpha\beta} = \delta_{\alpha\beta}\ M_{\gamma\gamma} + \sum_k \sigma^k_{\alpha\beta}\ \sigma^k_{\gamma\delta}\ M_{\delta\gamma}\ , where
summation over the repeated indices is implied and . Since this is true for any choice of the matrix , the completeness relation follows as stated above.
Q.E.D. }} As noted above, it is common to denote the 2 × 2 unit matrix by so The completeness relation can alternatively be expressed as \ \sum_{k=0}^3 \sigma^k_{\alpha\beta}\ \sigma^k_{\gamma\delta} = 2\ \delta_{\alpha\delta}\ \delta_{\beta\gamma} ~. 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 {{math|{
σ,
σ,
σ,
σ}}} as above, and then imposing the positive-semidefinite and
trace conditions. For a pure state, in polar coordinates, \vec{a} = \begin{pmatrix}\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end{pmatrix}, the
idempotent density matrix \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} acts on the state eigenvector \ \begin{pmatrix}\cos\left(\frac{\ \theta\ }{2}\right) & e^{+i\phi}\ \sin\left(\frac{\ \theta\ }{2}\right) \end{pmatrix}\ with eigenvalue +1, hence it acts like a
projection operator.
Relation with the permutation operator Let be the
transposition (also known as a permutation) between two spins and living in the
tensor product space :P_{jk} \left| \sigma_j \sigma_k \right\rangle = \left| \sigma_k \sigma_j \right\rangle . This operator can also be written more explicitly as
Dirac's spin exchange operator, :\ P_{jk} = \frac{1}{2}\ \left(\vec{\sigma}_j \cdot \vec{\sigma}_k + 1 \right) ~. Its eigenvalues are therefore{{efn| Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is \left(\begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right) ~ . }} 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) ==