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These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere S 3, and SU(2) is the universal cover of ...
Representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space. Relativistic spin is described by the representation theory of SL 2 (C), a supergroup of SU(2), which in a similar way covers SO + (1;3), the relativistic version of the rotation
The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons. Gauge theories are also important in explaining gravitation in the theory of general relativity.
Thus, for example, the adjoint representation of su(2) is the defining representation of so(3). Examples. If G is abelian of dimension n, ...
Of course, this is not the case in general – the standard model includes the non-Abelian SU(2) and SU(3) groups (such groups lead to what is called a Yang–Mills gauge theory). We need to introduce three gauge fields corresponding to each of the subgroups SU(3) × SU(2) × U(1) .
Let Γ be a finite subgroup of SO(3), the three-dimensional rotation group.There is a natural homomorphism f of SU(2) onto SO(3) which has kernel {±I}. [4] This double cover can be realised using the adjoint action of SU(2) on the Lie algebra of traceless 2-by-2 skew-adjoint matrices or using the action by conjugation of unit quaternions.
The traditional Pauli matrices are the matrix representation of the () Lie algebra generators , , and in the 2-dimensional irreducible representation of SU(2), corresponding to a spin-1/2 particle. These generate the Lie group SU(2) .