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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
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 ...
The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations.
For the leptons, the gauge group can be written SU(2) l × U(1) L × U(1) R. The two U(1) factors can be combined into U(1) Y × U(1) l, where l is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2) L × U(1) Y. A similar argument in the quark sector also gives the same result ...
Since SL(2, C) is simply connected, it is the universal covering group of the restricted Lorentz group SO + (1, 3). By restriction, there is a homomorphism SU(2) → SO(3). Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group ...
In this model, the Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the Standard Model subgroup below a very high energy scale called the grand unification scale.
Using the cross product as a Lie bracket, the algebra of 3-dimensional real vectors is a Lie algebra isomorphic to the Lie algebras of SU(2) and SO(3). The structure constants are f a b c = ϵ a b c {\displaystyle f^{abc}=\epsilon ^{abc}} , where ϵ a b c {\displaystyle \epsilon ^{abc}} is the antisymmetric Levi-Civita symbol .
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 ...