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An extract on #suyolu

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives [ a , b ] + { a , b } = ( a b b a ) + ( a b + b a ) 2 i a b c c + 2 a b I = 2 a b {\displaystyle {\begin{aligned}\left[\sigma _{a},\sigma _{b}\right]+\{\sigma _{a},\sigma _{b}\}&=(\sigma _{a}\sigma _{b}-\sigma _{b}\sigma _{a})+(\sigma _{a}\sigma _{b}+\sigma _{b}\sigma _{a})\\2i\varepsilon _{abc}\,\sigma _{c}+2\delta _{ab}I&=2\sigma _{a}\sigma _{b}\end{aligned}}} so that, Contracting each side of the equation with components of two 3-vectors ap and bq (which commute with the Pauli matrices, i.e., apq = qap) for each matrix q and vector component ap (and likewise with bq), and relabeling indices a, b, c p, q, r, to prevent notational conflicts, yields a p b q p q = a p b q ( i p q r r + p q I ) a p p b q q = i p q r a p b q r + a p b q p q I . {\displaystyle {\begin{aligned}a_{p}b_{q}\sigma _{p}\sigma _{q}&=a_{p}b_{q}\left(i\varepsilon _{pqr}\,\sigma _{r}+\delta _{pq}I\right)\\a_{p}\sigma _{p}b_{q}\sigma _{q}&=i\varepsilon _{pqr}\,a_{p}b_{q}\sigma _{r}+a_{p}b_{q}\delta _{pq}I~.\end{aligned}}} Finally, translating the index notation for the dot product and cross product results in

In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, ij are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin . 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 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 S 2, they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space. For a spin particle, the spin operator is given by J=/2, 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 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 Gn is defined to consist of all n-fold tensor products of Pauli matrices.