SU(2), SO(3), and U(1): Key Symmetry Groups in Physics

Table of Contents

1. Introduction
2. Overview of SU(2), SO(3), and U(1)
3. U(1): The Simplest Lie Group
4. U(1) in Physics: Electromagnetism and Phase Invariance
5. SO(3): The Rotation Group in Three Dimensions
6. Properties and Structure of SO(3)
7. Representations of SO(3) and Angular Momentum
8. SU(2): The Spin Group and Its Significance
9. SU(2) vs SO(3): Double Cover and Topology
10. SU(2) Representations and Spin
11. Pauli Matrices and SU(2) Generators
12. Embedding SU(2), SO(3), and U(1) in Field Theory
13. Gauge Symmetries and the Standard Model
14. Group Manifolds and Global Properties
15. Conclusion

1. Introduction

SU(2), SO(3), and U(1) are three fundamental Lie groups that appear repeatedly across physics. They describe symmetries in quantum mechanics, gauge theories, and classical systems. Understanding their structure and differences is critical to theoretical physics.

2. Overview of SU(2), SO(3), and U(1)

• U(1): Group of complex numbers with unit magnitude
• SO(3): Group of all rotations in 3D space
• SU(2): Group of 2×2 unitary matrices with determinant 1

All are compact, continuous Lie groups, but their topological and algebraic properties differ significantly.

3. U(1): The Simplest Lie Group

U(1) is defined as:

\[
U(1) = \{ e^{i\theta} \mid \theta \in [0, 2\pi) \}
\]

• Abelian: group elements commute
• Topologically a circle \( S^1 \)
• Lie algebra: \( \mathfrak{u}(1) \cong \mathbb{R} \)

4. U(1) in Physics: Electromagnetism and Phase Invariance

• Governs phase symmetry in quantum mechanics
• Global U(1): conservation of electric charge (via Noether’s theorem)
• Local U(1): leads to electromagnetic gauge theory
• Appears in the Standard Model as part of \( SU(3) \times SU(2) \times U(1) \)

5. SO(3): The Rotation Group in Three Dimensions

SO(3) consists of real orthogonal \( 3 \times 3 \) matrices with determinant +1:

\[
SO(3) = \{ R \in \mathbb{R}^{3 \times 3} \mid R^T R = I, \ \det R = 1 \}
\]

• Non-Abelian
• Describes rigid body rotations in classical mechanics

6. Properties and Structure of SO(3)

• 3 parameters (e.g., Euler angles)
• Not simply connected: \( \pi_1(SO(3)) = \mathbb{Z}_2 \)
• Every rotation corresponds to an axis and angle

7. Representations of SO(3) and Angular Momentum

Quantum angular momentum operators form an \( \mathfrak{so}(3) \) algebra:

\[
[J_i, J_j] = i\hbar \epsilon_{ijk} J_k
\]

• Representations labeled by integer \( l \): \( l = 0, 1, 2, \dots \)

8. SU(2): The Spin Group and Its Significance

SU(2) consists of \( 2 \times 2 \) unitary matrices with determinant 1:

\[
SU(2) = \left\{
U = \begin{bmatrix} a & b \ -b^* & a^* \end{bmatrix} \mid |a|^2 + |b|^2 = 1
\right\}
\]

• Non-Abelian
• Topology: 3-sphere \( S^3 \)
• Fundamental in spin and weak interactions

9. SU(2) vs SO(3): Double Cover and Topology

• SU(2) double covers SO(3): \( SU(2)/\mathbb{Z}_2 \cong SO(3) \)
• SU(2) is simply connected: \( \pi_1(SU(2)) = 0 \)
• SO(3) is not simply connected: \( \pi_1(SO(3)) = \mathbb{Z}_2 \)

Implication: spin-½ particles (e.g., electrons) described by SU(2), not SO(3)

10. SU(2) Representations and Spin

• Irreducible representations labeled by half-integers: \( j = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots \)
• Spin-½: two-dimensional representation
• Spin-1: three-dimensional (vector) representation, matches SO(3)

11. Pauli Matrices and SU(2) Generators

Generators of SU(2) are:

\[
T_i = \frac{1}{2} \sigma_i, \quad \text{where } \sigma_i \text{ are the Pauli matrices}
\]

\[
\sigma_1 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}, \quad
\sigma_2 = \begin{bmatrix} 0 & -i \ i & 0 \end{bmatrix}, \quad
\sigma_3 = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}
\]

12. Embedding SU(2), SO(3), and U(1) in Field Theory

• SU(2): weak interaction gauge symmetry
• U(1): electromagnetic gauge symmetry
• SO(3): spatial symmetry of angular momentum

Combined in electroweak theory via:
\[
SU(2)L \times U(1)_Y \to U(1){\text{EM}}
\]

13. Gauge Symmetries and the Standard Model

• SU(2) gauge bosons: \( W^+, W^-, W^0 \)
• U(1) boson: \( B \)
• Mixing gives photon and \( Z^0 \) boson

Electroweak unification is rooted in the algebra of SU(2) and U(1).

14. Group Manifolds and Global Properties

• U(1): circle \( S^1 \)
• SU(2): 3-sphere \( S^3 \)
• SO(3): 3D projective space \( \mathbb{RP}^3 \)

These properties affect quantization and topological configurations.

15. Conclusion

SU(2), SO(3), and U(1) are foundational symmetry groups in physics. U(1) governs phase invariance and electromagnetism. SO(3) encodes rotational symmetry, while SU(2) is central to spin, angular momentum, and the weak interaction.

Understanding their algebraic and topological properties is essential for any physicist delving into quantum mechanics, particle theory, or gauge symmetries.