Editing Pauli matrices (section)

== Physics ==

=== Classical mechanics ===
{{Main|Quaternions and spatial rotation}}

In [[classical mechanics]], Pauli matrices are useful in the context of the Cayley-Klein parameters.<ref name=Goldstein-1959>
{{cite book
 |last=Goldstein |first=Herbert
 |year=1959
 |title=Classical Mechanics
 |pages=109–118
 |publisher=Addison-Wesley
}}
</ref> The matrix {{mvar|P}} 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 {{math|''Q{{sub|θ}}''}} for rotations about the {{mvar|x}}-axis through an angle {{mvar|θ}} 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 (physics)|spin]] of a [[spin-1/2|spin {{1/2}}]] particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, {{math|''iσ{{sub|j}}''}} 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 {{1/2}}. 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 {{1/2}} particles is that they must be rotated by an angle of 4{{mvar|π}} 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]] {{math|''S''{{sup|2}},}} they are actually represented by [[orthogonal]] vectors in the two-dimensional complex [[Hilbert space]].

For a spin {{1/2}} particle, the spin operator is given by {{math|1='''''J''''' = {{sfrac|''ħ''|2}}'''''σ'''''}}, the [[fundamental representation]] of [[representation theory of SU(2)|SU(2)]]. By taking [[Kronecker product]]s of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting [[spin operator]]s 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 {{section link|Rotation group SO(3)#A note on Lie algebras}}. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.<ref>{{Cite journal | doi=10.3842/SIGMA.2014.084 |last1=Curtright|last2=Fairlie|last3=Zachos |first1=T L |first2=D B |first3=C K|author-link=Thomas Curtright|author-link2=David Fairlie|author-link3=Cosmas Zachos|year=2014|title=A compact formula for rotations as spin matrix polynomials| journal =SIGMA| volume=10| page=084|arxiv=1402.3541 |bibcode=2014SIGMA..10..084C |s2cid=18776942}}</ref>

Also useful in the [[quantum mechanics]] of multiparticle systems, the general [[Pauli group]] {{math|''G{{sub|n}}''}} is defined to consist of all {{mvar|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
:<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 {{mvar| σ{{sub|k}} }} 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 {{mvar|Σ{{sub|μν}} }}, 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 {{math|Σ''{{sub|μν}}''}} 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 {{math|''α{{sub|k}}''}} 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 {{math|Σ''{{sub|μν}}''}} 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 gate]]s 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'' ".