A Notion To Ponder

Definition. A Fiber Bundle $P \xrightarrow{\pi} M$ with a smooth, free right action of a Lie Group $G$ on $P$ whose orbits are exactly the fibers: $\pi^{-1}(x) \cong G$ as a $G$-space. Local trivializations are required to be $G$-equivariant.

The crucial subtlety. Each fiber is a copy of $G$ with no distinguished identity element — a $G$-torsor. You can multiply a fiber point by a group element (move along the fiber), but you cannot ask "which point is $e$?" without choosing a Local Trivialization.

Intuition. The bundle of "conventions." Fiber over $x$ = all possible choices of internal reference frame at $x$. For frame bundles: all bases of $T_xM$. For EM: all phase conventions at $x$.

Key theorem. $P$ is trivial $\iff$ $P$ admits a global section. (A section picks an identity in every fiber, i.e., a global gauge.) Contrast vector bundles, which always have the zero section.

Physics anchor. Gauge freedom in Landau §18 = the torsor structure: nature specifies no preferred phase convention, so only convention-independent statements are physical. The gauge field is not a field on spacetime at all — it's a connection on $P$.

Related: Frame Bundle · Associated Bundle · Gauge Transformation · Hopf Fibration · Module 07 - Principal Bundles

Sources: Sternberg Ch. VII.