A Notion To Ponder

Why this module: The "structure group" $G$ of a bundle is a Lie group. For electromagnetism $G = U(1)$; for the weak and strong forces, $SU(2)$ and $SU(3)$. Connections take values in the Lie algebra — which is why the photon field $A_\mu$ is $\mathfrak{u}(1) \cong i\mathbb{R}$-valued (one real field), while gluons carry 8 components ($\dim \mathfrak{su}(3) = 8$).

Core idea

A Lie Group is a group that is also a smooth manifold. Its Lie Algebra is the tangent space at the identity, with a bracket recording the failure of commutativity to second order. The Exponential Map rebuilds group elements from algebra elements: $e^{i\theta}$ for $U(1)$ is the prototype.

Physics anchor: Every gauge transformation $\psi \to e^{i\theta(x)}\psi$ in Landau/QM is a point-dependent $U(1)$ element. Rotations vs. angular momentum operators = group vs. algebra.

Concepts to (re)work

• Lie Group — definition, matrix groups as the working examples
• Lie Algebra — tangent space at $e$, bracket, structure constants
• Exponential Map — matrix exponential, one-parameter subgroups
• U(1) — the star of this study; circle group, charge quantization
• SU(2) and SO(3) — the double cover, spin; the first nonabelian example
• Group Representation — needed for matter fields later (Module 10 - Representations and Matter Fields)
• Adjoint Representation — how $G$ acts on its own algebra; where curvature lives in nonabelian theories

Reading

• Sternberg - Lectures on Differential Geometry, Ch. V (Lie Groups)
• Woit - Quantum Theory, Groups and Representations — chapters on $U(1)$, $SU(2)$, Lie algebras (gentler than Sternberg; start here)

Self-check before moving on

Why is the Lie algebra of $U(1)$ one-dimensional? Why do representations of $U(1)$ carry an integer label while representations of $\mathbb{R}$ don't?