SU(2)

SU(2). In mathematics, specifically group theory, SU(2) is the Lie group of 2×2 unitary matrices with determinant 1. It is a fundamental object in theoretical physics and geometry, serving as the universal cover of the rotation group SO(3). Its Lie algebra, denoted su(2), is isomorphic to the cross product algebra of three-dimensional space.

Definition and basic properties

The group SU(2) consists of all complex matrices of the form that preserve the Hermitian inner product on ℂ². A standard parametrization uses the Pauli matrices, which form a basis for its Lie algebra. The topology of SU(2) is that of the 3-sphere, making it a simply connected compact space. Its center is the cyclic group of order two, and it is a double cover of the projective orthogonal group. The exponential map from su(2) onto SU(2) is surjective due to its compactness. This property is crucial for connecting infinitesimal generators to finite group elements in physical applications.

Representations

The representation theory of SU(2) is foundational in quantum mechanics. Its finite-dimensional, irreducible representations are labeled by a half-integer or integer spin quantum number. These representations are realized on spaces of homogeneous polynomials, as studied by Élie Cartan and Hermann Weyl. The fundamental representation acts on ℂ², with its generators given by the Pauli matrices. The Clebsch–Gordan coefficients describe the decomposition of tensor products of these representations, a process central to angular momentum coupling. The Peter–Weyl theorem guarantees the completeness of its matrix coefficients in *L*².

Relation to rotations and spin

There exists a homomorphism from SU(2) onto the rotation group SO(3), with kernel ±identity matrix. This 2:1 mapping underpins the connection between spinors and vectors in three-dimensional space. In quantum field theory, particles with half-integer spin, like the electron, are described by spinor fields transforming under SU(2). The quaternions, discovered by William Rowan Hamilton, provide an isomorphism between SU(2) and the unit quaternions. This relationship is exploited in computer graphics for 3D rotation interpolation and in the attitude control of spacecraft.

Applications in physics

SU(2) is the gauge group of the electroweak interaction in the Standard Model, where it is intertwined with the hypercharge group U(1). This Yang–Mills theory was formulated by Chen Ning Yang and Robert Mills. The Higgs mechanism breaks this symmetry, giving mass to the W and Z bosons. In quantum chromodynamics, although the gauge group is SU(3), isospin symmetry is an approximate SU(2) symmetry between the up quark and down quark. The group also appears in the description of magnetic resonance in nuclear magnetic resonance spectroscopy and MRI. Furthermore, the Bogoliubov transformation in superconductivity often utilizes SU(2) structures.

Generalizations and related groups

SU(2) is the simplest non-abelian compact Lie group and the first in the classical group families SU(*n*). Its complexification is the special linear group SL(2,C), which is the double cover of the restricted Lorentz group SO+(1,3). The quaternionic unitary group Sp(1) is isomorphic to SU(2). Higher-rank analogues include the symplectic group Sp(*n*), and its structure is generalized in the theory of root systems of type A₁. The quantum group SUq(2) is a key example in noncommutative geometry, studied by Vladimir Drinfeld and Michio Jimbo.

Category:Lie groups Category:Quantum mechanics Category:Theoretical physics