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In mathematical physics and mathematics, the Pauli matrices are a set of three complex number, complex matrix (mathematics), matrices which are Hermitian matrix, Hermitian and Unitary matrix, unitary. Usually indicated by the Greek (alphabet), Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. $\begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end$ 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 (physics), 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 matrix, Hermitian, and together with the identity matrix (sometimes considered as the zeroth Pauli matrix ), the Pauli matrices form a Basis (linear algebra), basis for the real vector space of Hermitian matrices. This means that any Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex number, complex -dimensional Hilbert space. In the context of Pauli's work, represents the observable corresponding to spin along the th coordinate axis in three-dimensional Euclidean space $\mathbb^3.$ The Pauli matrices (after multiplication by to make them skew-Hermitian, anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices form a basis for the real Lie algebra $\mathfrak\left(2\right)$, which Exponential map (Lie theory), exponentiates to the special unitary group SU(2)#n = 2, SU(2). The Algebra over a field, algebra generated by the three matrices is isomorphic to the Clifford algebra of , and the (unital associative) algebra generated by is effectively identical (isomorphic) to that of quaternions ($\mathbb$).

Algebraic properties

All three of the Pauli matrices can be compacted into a single expression: :$\sigma_j = \begin \delta_ & \delta_ - i\,\delta_\\ \delta_ + i\,\delta_ & -\delta_ \end$ where the solution to is the "imaginary unit", and is the Kronecker delta, which equals +1 if and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of , in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations. The matrices are Involutory matrix, ''involutory'': :$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\,\sigma_1 \sigma_2 \sigma_3 = \begin 1 & 0 \\ 0 & 1 \end = I$ where is the identity matrix. The determinants and trace of a matrix, traces of the Pauli matrices are: :$\begin \det \sigma_j &~=\, -1\,, \\ \operatorname \sigma_j &~=~~~\; 0 ~. \end$ From which, we can deduce that each matrix has eigenvalues +1 and −1. With the inclusion of the identity matrix, (sometimes denoted ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt operator, Hilbert–Schmidt) of the Hilbert space of real number, real Hermitian matrices, $\mathcal_2\left(\mathbb\right)$, and the Hilbert space of all complex number, complex matrices, $\mathcal_\left(\mathbb\right)$.

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, and . The corresponding Normalisable wavefunction, normalized eigenvectors are: :$\begin \psi_ &= \frac\sqrt \begin 1 \\ 1 \end \; , & \psi_ &= \frac\sqrt \begin 1 \\ -1 \end \; , \\ \psi_ &= \frac\sqrt \begin 1 \\ i \end \; , & \psi_ &= \frac\sqrt \begin 1 \\ -i \end \; , \\ \psi_ &= \begin 1 \\ 0 \end \; , & \psi_ &= \begin 0 \\ 1 \end ~. \end$

Pauli vector

The Pauli vector is defined by :$\vec = \sigma_1 \hat_1 + \sigma_2 \hat_2 + \sigma_3 \hat_3 ~,$ where $\; \hat_1, \hat_2, \, \text \, \hat_3 \;$ are an equivalent notation for the more familiar $\; \hat, \hat, \, \text \, \hat \, ; \;$ the subscripted notation $\, \hat_1, \hat_2, \hat_3 \,$ is more compact than the old $\, \hat, \hat, \hat \,$ form. The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows, :$\begin \vec \cdot \vec &= \left\left(a_k \hat_k\right\right) \cdot \left\left(\sigma_\ell \hat_\ell \right\right) = a_k \sigma_\ell \hat_k \cdot \hat_\ell \\ \\ &= a_k \sigma_\ell \delta_ = a_k \sigma_k \\ \\ &= ~ a_1 \; \begin 0 & 1 \\ 1 & 0 \end ~ + ~ a_2 \;i\begin 0 & -1 \\ 1 &\;\; 0 \end ~ + ~ a_3 \; \begin 1 & 0 \\ 0 & -1 \end ~ = ~ \begin a_3 & a_1 - i a_2 \\ a_1 + i a_2 & -a_3 \end \end$ using Einstein notation, Einstein's summation convention. Further, :$\det \bigl\left( \vec \cdot \vec \bigr\right) = -\vec \cdot \vec = -\left, \vec\^2,$ its eigenvalues being $\pm , \vec,$, and moreover (see #completeness_anchor, § completeness relation, below) :$\frac \operatorname \Bigl\left( \bigl\left( \vec \cdot \vec \bigr\right) \vec \Bigr\right) = \vec ~.$ Its normalized eigenvectors are :$\psi_+ = \frac\begin a_3 + , \vec, \\ a_1 + ia_2 \end; \qquad \psi_- = \frac\begin ia_2 - a_1 \\ a_3 + , \vec, \end ~ .$

Commutation relations

The Pauli matrices obey the following commutator, commutation relations: :$\left[\sigma_j, \sigma_k\right] = 2 i \varepsilon_\,\sigma_\ell ~ ,$ and anticommutator, anticommutation relations: :$\ = 2 \delta_\,I ~ .$ where the structure constant is the Levi-Civita symbol, Einstein summation notation is used, is the Kronecker delta, and is the identity matrix. For example,

Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives :$\begin \left\left[\sigma_j, \sigma_k\right\right] + \ &= \left(\sigma_j \sigma_k - \sigma_k \sigma_j \right) + \left(\sigma_j \sigma_k + \sigma_k \sigma_j\right) \\ 2i\varepsilon_\,\sigma_\ell + 2 \delta_I &= 2\sigma_j \sigma_k \end$ so that, tensor contraction, 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 :$~~ \begin a_j b_k \sigma_\ell \sigma_k & = a_j b_k \left\left(i\varepsilon_\,\sigma_\ell + \delta_I\right\right) \\ a_j \sigma_j b_k \sigma_k & = i\varepsilon_\,a_j b_k \sigma_\ell + a_j b_k \delta_I \end ~.~$ Finally, translating the index notation for the dot product and cross product#Index notation for tensors, cross product results in 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.

Some trace relations

The following traces can be derived using the commutation and anticommutation relations. :$\begin \operatorname\left\left(\sigma_j \right\right) &= 0 \\ \operatorname\left\left(\sigma_j \sigma_k \right\right) &= 2\delta_ \\ \operatorname\left\left(\sigma_j \sigma_k \sigma_\ell \right\right) &= 2i\varepsilon_ \\ \operatorname\left\left(\sigma_j \sigma_k \sigma_\ell \sigma_m \right\right) &= 2\left\left(\delta_\delta_ - \delta_\delta_ + \delta_\delta_\right\right) \end$ If the matrix is also considered, these relationships become :$\begin \operatorname\left\left(\sigma_\alpha \right\right) &= 2\delta_ \\ \operatorname\left\left(\sigma_\alpha \sigma_\beta \right\right) &= 2\delta_ \\ \operatorname\left\left(\sigma_\alpha \sigma_\beta \sigma_\gamma \right\right) &= 2 \sum_ \delta_ \delta_ - 4 \delta_ \delta_ \delta_ + 2i\varepsilon_ \\ \operatorname\left\left(\sigma_\alpha \sigma_\beta \sigma_\gamma \sigma_\mu \right\right) &= 2\left\left(\delta_\delta_ - \delta_\delta_ + \delta_\delta_\right\right) + 4\left\left(\delta_ \delta_ \delta_ + \delta_ \delta_ \delta_\right\right) - 8 \delta_ \delta_ \delta_ \delta_ + 2 i \sum_ \varepsilon_ \delta_ \end$ where Greek indices and assume values from and the notation $\sum_$ is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For :$\vec = a\hat, \quad , \hat, = 1,$ one has, for even powers, :$\left(\hat \cdot \vec\right)^ = I$ 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, :$\left\left(\hat \cdot \vec\right\right)^ = \hat \cdot \vec \, .$ Matrix exponential, Matrix exponentiating, and using the Taylor series#List of Maclaurin series of some common functions, Taylor series for sine and cosine, :$\begin e^ &= \sum_^\infty \\ &= \sum_^\infty + i\sum_^\infty \\ &= I\sum_^\infty + i \left(\hat\cdot \vec\right) \sum_^\infty\\ \end$. In the last line, the first sum is the cosine, while the second sum is the sine; so, finally, which is quaternions and spatial rotation#Using quaternion as rotations, analogous to Euler's formula, extended to quaternions. Note that :$\det\left[i a\left(\hat \cdot \vec\right)\right] = a^2$, 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 exponential#via Laurent series, matrix exponentials. A general version of for an analytic (at ''a'' and −''a'') function is provided by application of Sylvester's formula, :$f\left(a\left(\hat \cdot \vec\right)\right) = I\frac + \hat \cdot \vec \frac ~.$

The group composition law of

A straightforward application of formula provides a parameterization of the composition law of the group . One may directly solve for in :$\begin e^ e^ &= I\left\left(\cos a \cos b - \hat \cdot \hat \sin a \sin b\right\right) + i\left\left(\hat \sin a \cos b + \hat \sin b \cos a - \hat \times \hat ~ \sin a \sin b \right\right) \cdot \vec \\ &= I\cos + i \left\left(\hat \cdot \vec\right\right) \sin c \\ &= e^, \end$ which specifies the generic group multiplication, where, manifestly, :$\cos c = \cos a \cos b - \hat \cdot \hat \sin a \sin b~,$ the spherical law of cosines. Given , then, :$\hat = \frac\left\left(\hat \sin a \cos b + \hat \sin b \cos a - \hat\times\hat \sin a \sin b\right\right) ~.$ Consequently, the composite rotation parameters in this group element (a closed form of the respective Baker–Campbell–Hausdorff formula, BCH expansion in this case) simply amount to :$e^ = \exp \left\left( i\frac \left\left(\hat \sin a \cos b + \hat \sin b \cos a - \hat\times\hat ~ \sin a \sin b\right\right) \cdot \vec\right\right) ~.$ (Of course, when $\hat$ is parallel to $\hat$, so is $\hat$, and .)

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle , :$e^ ~ \vec~ e^ = \vec \cos \left(2a\right) + \hat \times \vec ~\sin \left(2a\right)+ \hat ~ \hat \cdot \vec ~ \left(1 - \cos \left(2a\right)\right)~ .$

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 :$\vec_\cdot\vec_\equiv \sum_^3 \sigma^k_\,\sigma^k_ = 2\,\delta_ \,\delta_ - \delta_\,\delta_~.$ : Proof: The fact that the Pauli matrices, along with the identity matrix , form an orthogonal basis for the Hilbert space of all 2 × 2 complex number, complex matrices means that we can express any 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 :: $\operatorname\left\left( \sigma^j\,\sigma^k \right\right) = 2\,\delta_$ : where "" denotes the trace (linear algebra), trace, and hence that :: $\begin c &= \tfrac\,\operatorname \, M\,,\quad \ a_k = \tfrac\,\operatorname\,\sigma^k\,M ~. \\\left[3pt\right] \therefore ~~ 2\,M &= I\,\operatorname\, M + \sum_k \sigma^k\,\operatorname\, \sigma^k M ~, \end$ : which can be rewritten in terms of matrix indices as :: $2\, M_ = \delta_\,M_ + \sum_k \sigma^k_\,\sigma^k_\,M_~,$ : where Einstein notation, 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. As noted above, it is common to denote the 2 × 2 unit matrix by so The completeness relation can alternatively be expressed as :$\sum_^3 \sigma^k_\,\sigma^k_ = 2\,\delta_\,\delta_ ~ .$ The fact that any Hermitian complex number, 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 state (physics), mixed states’ density matrix, (Positive semidefinite 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 as above, and then imposing the positive-semidefinite and Trace (linear algebra), trace conditions. For a pure state, in polar coordinates, $\vec = \begin\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end$, the idempotent density matrix :$\tfrac\,\left\left(\,\mathbf + \vec \cdot \vec\,\right\right) = \begin \cos^2\left\left(\frac\right\right) & e^\sin\left\left(\frac\right\right)\cos\left\left(\frac\right\right) \\ e^\sin\left\left(\frac\right\right)\cos\left\left(\frac\right\right) & \sin^2\left\left(\frac\right\right) \end$ acts on the state eigenvector $\,\begin\cos\left\left(\frac\right\right) & e^\,\sin\left\left(\frac\right\right) \end \,$ with eigenvalue +1, hence it acts like a projection (linear algebra), projection operator.

Relation with the permutation operator

Let be the transposition (mathematics), transposition (also known as a permutation) between two spins and living in the tensor product space :$P_ \left, \sigma_j \sigma_k \right\rangle = \left, \sigma_k \sigma_j \right\rangle ~.$ This operator can also be written more explicitly as Exchange interaction#Inclusion of spin, Dirac's spin exchange operator, :$P_ = \frac\,\left\left(\vec_j \cdot \vec_k + 1\right\right) ~ .$ Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

SU(2)

The group SU(2) is the Lie group of unitary matrix, unitary matrices with unit determinant; its Lie algebra is the set of all anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra $\mathfrak_2$ is the 3-dimensional real algebra linear span, spanned by the set . In compact notation, :$\mathfrak\left(2\right) = \operatorname \~.$ As a result, each can be seen as an Lie group#The Lie algebra associated with a Lie group, 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 , as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is so that :$\mathfrak\left(2\right) = \operatorname \left\~.$ As SU(2) is a compact group, its Cartan decomposition is trivial.

SO(3)

The Lie algebra is isomorphism, isomorphic to the Lie algebra , which corresponds to the Lie group Rotation group SO(3), SO(3), the group (mathematics), group of rotations in three-dimensional space. In other words, one can say that the are a realization (and, in fact, the lowest-dimensional realization) of ''infinitesimal'' rotations in three-dimensional space. However, even though and are isomorphic as Lie algebras, and are not isomorphic as Lie groups. is actually a Double covering group, double cover of , meaning that there is a two-to-one group homomorphism from to , see Rotation group SO(3)#Connection between SO(3) and SU(2), relationship between SO(3) and SU(2).

Quaternions

The real linear span of is isomorphic to the real algebra of quaternions $\mathbb$, represented by the span of the basis vectors $~ \left\ ~.$ The isomorphism from $\mathbb$ to this set is given by the following map (notice the reversed signs for the Pauli matrices): :$\mathbf \mapsto I, \quad \mathbf \mapsto - \sigma_2\sigma_3 = - i\,\sigma_1, \quad \mathbf \mapsto - \sigma_3\sigma_1 = - i\,\sigma_2, \quad \mathbf \mapsto - \sigma_1\sigma_2 = - i\,\sigma_3.$ Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order, :$\mathbf \mapsto I, \quad \mathbf \mapsto i\,\sigma_3 \, , \quad \mathbf \mapsto i\,\sigma_2 \, , \quad \mathbf \mapsto i\,\sigma_1 ~ .$ As the set of versors forms a group isomorphic to , gives yet another way of describing . The two-to-one homomorphism from to may be given in terms of the Pauli matrices in this formulation.

Physics

Classical mechanics

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix corresponding to the position $\vec$ of a point in space is defined in terms of the above Pauli vector matrix, :$P = \vec \cdot \vec = x\,\sigma_x + y\,\sigma_y + z\,\sigma_z ~.$ Consequently, the transformation matrix for rotations about the -axis through an angle may be written in terms of Pauli matrices and the unit matrix as :$Q_\theta = \boldsymbol\,\cos\frac + i\,\sigma_x \sin\frac ~.$ 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 (physics), spin of a spin-1/2, spin particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on theory of relativity, non-relativistic particles with spin . The mathematical formulation of quantum mechanics, states of the particles are represented as two-component Spinors in three dimensions, spinors. In the same way, the Pauli matrices are related to the Isospin, isospin operator. An interesting property of spin particles is that they must be rotated by an angle of 4 in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space. For a spin particle, the spin operator is given by , the fundamental representation of representation theory of SU(2), 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 operator#Angular momentum, ladder operators. They can be found in Rotation group SO(3)#A note on Lie algebra. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. Also useful in the quantum mechanics of multiparticle systems, the general Pauli group is defined to consist of all -fold tensor products of Pauli matrices.

Relativistic quantum mechanics

In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as :$\mathsf_k = \begin \mathsf_k & 0 \\ 0 & \mathsf_k \end.$ It follows from this definition that the $\; \mathsf_k \;$ matrices have the same algebraic properties as the matrices. However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence $\mathsf_k$ needs to be replaced by the generator of representation theory of the Lorentz group#The (1/2, 0) ⊕ (0, 1/2) spin representation, Lorentz transformations on spinors. By the antisymmetry of angular momentum, the are also antisymmetric. Hence there are only six independent matrices. The first three are the $\; \Sigma_\equiv \epsilon_\mathsf_j ~.$ The remaining three, $\;-i\,\Sigma_ \equiv \mathsf_k\;,$ where the Dirac equation, Dirac matrices are defined as :$\mathsf_k = \begin 0 & \mathsf_k\\ \mathsf_k & 0\end~.$ The relativistic spin matrices are written in compact form in terms of commutator of gamma matrices as :$\Sigma_ = \frac\left\left[\gamma_\mu, \gamma_\nu\right\right]~.$

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".