8.4 Representations of SU(2) and SO(3)

SU(2) and SO(3) are key compact Lie groups in physics and math. They describe rotations and angular momentum, with SU(2) being the double cover of SO(3). Their representations are crucial for understanding particle spin and 3D rotations.

These groups showcase important properties of compact Lie groups, like finite-dimensional irreducible representations. The relationship between SU(2) and SO(3) highlights the role of topology in group theory and has led to powerful mathematical tools used across various fields.

Classifying SU(2) and SO(3) Representations

SU(2) and SO(3) Group Definitions

• SU(2) is the group of 2x2 unitary matrices with determinant 1 (Pauli matrices)
• SO(3) is the group of 3x3 orthogonal matrices with determinant 1, representing rotations in 3D space (rotation matrices)
• SU(2) and SO(3) are examples of non-abelian compact Lie groups

Irreducible Representations of SU(2)

• Labeled by half-integers j = 0, 1/2, 1, 3/2, ..., and have dimension 2j+1
• Representation space is spanned by basis vectors |j,m⟩, where m = -j, -j+1, ..., j-1, j
• Generators of SU(2) in the j-representation are given by the angular momentum operators Jx, Jy, and Jz
• Angular momentum operators satisfy the commutation relations [Jx, Jy] = iJz, [Jy, Jz] = iJx, and [Jz, Jx] = iJy

Irreducible Representations of SO(3)

• Labeled by integers l = 0, 1, 2, ..., and have dimension 2l+1
• Representation space is spanned by spherical harmonics Ylm(θ,φ), where m = -l, -l+1, ..., l-1, l
• Generators of SO(3) in the l-representation are given by the angular momentum operators Lx, Ly, and Lz
• Angular momentum operators satisfy the same commutation relations as the SU(2) generators

SU(2) vs SO(3) Representations

Relationship between SU(2) and SO(3)

• SU(2) is the double cover of SO(3), meaning there is a two-to-one homomorphism from SU(2) to SO(3)
• Homomorphism is surjective but not injective, as there are two elements of SU(2) mapped to each element of SO(3)
• For each integer l-representation of SO(3), there is a corresponding j = l/2 representation of SU(2)
• For each half-integer j-representation of SU(2), there is no corresponding representation of SO(3)

Similarities in Representation Theory

• Generators of both SU(2) and SO(3) satisfy the same commutation relations
• Eigenvalues of the generators differ by a factor of 2 between SU(2) and SO(3) representations
• Clebsch-Gordan coefficients describe the coupling of two SU(2) representations
• Wigner 3j-symbols describe the coupling of three SO(3) representations and are related to the Clebsch-Gordan coefficients

Applications of SU(2) and SO(3) Representation Theory

Quantum Mechanics and Particle Physics

• SU(2) representation theory describes angular momentum and spin of particles
• Pauli matrices are the generators of SU(2) in the j = 1/2 representation (spin-1/2 particles)
• SU(2) is used to classify particles according to their spin and study gauge theories with SU(2) symmetry (weak interaction in the Standard Model)

Classical Mechanics and Rotational Symmetry

• SO(3) representation theory is crucial for studying rotational symmetry
• Spherical harmonics form a basis for the l-representations of SO(3) and are eigenfunctions of the angular momentum operators
• Spherical harmonics are used to describe angular dependence in various physical problems (electrostatics, quantum mechanics)

Computer Graphics and Robotics

• SU(2) and SO(3) representation theory is applied to describe rotations and orientations in 3D space
• Quaternions, which are related to SU(2), are used to represent rotations in computer graphics and robotics
• Rotation matrices, which are elements of SO(3), are used to describe the orientation of objects in 3D space

Importance of SU(2) and SO(3) as Compact Lie Groups

Simplicity and Foundational Role

• SU(2) and SO(3) are among the simplest non-abelian compact Lie groups
• Their representation theories serve as a foundation for understanding the representation theory of more complex Lie groups (SU(3), SO(n) for n > 3)
• Well-understood representation theories provide a framework for studying other compact Lie groups

Properties of Compact Lie Group Representations

• Compactness of SU(2) and SO(3) ensures that their irreducible representations are finite-dimensional and unitary
• Finite-dimensionality and unitarity make these representations particularly suitable for applications in physics and mathematics
• Compactness also leads to the complete reducibility of representations, simplifying their analysis

Topological Considerations and Covering Groups

• Relationship between SU(2) and SO(3) exemplifies the concept of covering groups
• Topology plays a crucial role in the representation theory of Lie groups
• Understanding the relationship between SU(2) and SO(3) provides insight into the connection between group structure and topology

Development of Mathematical Tools

• Representation theory of SU(2) and SO(3) has led to the development of powerful mathematical tools
• Clebsch-Gordan coefficients, Wigner 3j-symbols, and the Wigner-Eckart theorem have found applications in various branches of physics and mathematics
• These tools have been extensively studied and have contributed to the understanding of symmetries in physical systems