Module 10 — Representations and Matter Fields
Why this module: Bundles + connections give the force fields. Representation theory (Woit's territory) supplies the matter fields. This module closes the triangle between your three sources.
Core idea
Given a Principal Bundle $P \to M$ with group $G$ and a Group Representation $\rho: G \to GL(V)$, the Associated Bundle $E = P \times_\rho V$ is a Vector Bundle whose sections are "$V$-valued fields transforming in the representation $\rho$." A connection on $P$ induces a Covariant Derivative on every associated bundle at once — one gauge field couples to all matter uniformly.
Physics anchor (Woit): For $G = U(1)$, irreducible representations are labeled by $n \in \mathbb{Z}$: $\rho_n(e^{i\theta}) = e^{in\theta}$. A charge-$n$ field is a section of $P \times_{\rho_n} \mathbb{C}$. Electric charge is a representation label. Minimal coupling $\partial_\mu \to \partial_\mu - i n e A_\mu$ (Landau §17) is nothing but the induced covariant derivative in the rep $\rho_n$.
Concepts
- Group Representation — revisit with bundle eyes
- Associated Bundle — the central construction
- Adjoint Representation — curvature of a nonabelian connection is an ad-valued 2-form; for $U(1)$, ad is trivial, so $F$ is an honest 2-form. This is why EM is simpler than Yang–Mills.
- U(1) and SU(2) and SO(3) — the working examples
- Yang-Mills Theory — matter in nonabelian representations (isospin doublets, color triplets)
Reading
- Woit - Quantum Theory, Groups and Representations — $U(1)$ reps, $SU(2)$ reps, and the gauge-theory chapters
- Your own paper referencing Woit — reread it after this module and note what reads differently
Concepts in this module
Study tasks
- Read the concept notes
- Exercise: build $P \times_\rho V$ explicitly as $(P \times V)/G$ and check its transition functions are $\rho(t_{\alpha\beta})$
- Exercise: verify a section of $P \times_\rho V$ = an equivariant function $\phi: P \to V$ with $\phi(p \cdot g) = \rho(g^{-1})\phi(p)$
- Exercise: derive $\nabla_\mu = \partial_\mu - ineA_\mu$ for the charge-$n$ rep from the general covariant derivative formula
- Reread your paper; add a "My Notes" entry on how its use of Woit connects to associated bundles