Editing Pauli matrices (section)

=== 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.