Local Trivialization
Definition. For a Fiber Bundle $\pi: E \to M$, a diffeomorphism $\Phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times F$ with $\mathrm{pr}_1 \circ \Phi_\alpha = \pi$ — a local identification of the bundle with a product.
Intuition. The bundle-theoretic analogue of a coordinate chart. A chart flattens a piece of a manifold onto $\mathbb{R}^n$; a trivialization "untwists" a piece of a bundle into a product. Neither is canonical; the physics/geometry must not depend on the choice.
The dictionary entry that matters most. On a Principal Bundle, choosing a local trivialization = choosing a gauge. All of Landau's formulas involving $A_i$ are written in a trivialization; that's what "in a particular gauge" means geometrically. A Gauge Transformation is precisely a change of local trivialization, and gauge-dependence = trivialization-dependence = "not a real geometric object."
Test question. Why can't you generally choose one trivialization covering all of $M$? (Answer: that would make the bundle trivial — Mobius Band as a Bundle, Hopf Fibration, Dirac Monopole are counterexamples.)
Related: Transition Functions · Gauge Transformation · Section of a Bundle · Module 05 - Fiber Bundles - First Definitions
Sources: Sternberg Ch. VII.
Linked from
- Fiber Bundle
- Gauge Transformation
- Landau & Lifshitz Vol. 2 — The Classical Theory of Fields
- Möbius Band as a Bundle
- Module 01 — Smooth Manifolds Refresher
- Module 04 — Landau Reread: Electromagnetism as Geometry
- Module 05 — Fiber Bundles: First Definitions
- Module 07 — Principal Bundles
- Module 09 — Electromagnetism as a U(1) Gauge Theory
- Principal Bundle
- Section of a Bundle
- Transition Functions