What is the difference between a section of a complex line bundle and a complex-valued function?

Question

In quantum mechanics the wave function is complex-valued function. However in some approach it is seen as section of a complex line bundle. What is the difference between a section of a complex line bundle and a complex-valued function?

Answer 1

A complex-valued function is a single function defined globally over all of spacetime (or whatever physical system you're considering). A section of a complex line bundle locally looks like an ordinary function $M \to \mathbb{C}$, but there is not necessarily a single coordinate chart that covers all of the spacetime. Instead, you need to specify the wavefunction via different functions in different regions of spacetime, along with a set of rules for how to translate between the different functions in the regions where their domains overlap. It's extremely similar to the way that not all manifolds can be covered by a single coordinate chart, but you can always make a global coordinate system by piecing together local functions defined on multiple patches.

The situation is rather subtle, because with fiber bundles there are several distinct notions of "topological non-triviality:"

1. The base manifold itself (i.e. spacetime) can have nontrivial topology (e.g. a torus rather than a disk).
2. There is also a more severe type of topological non-triviality where, very roughly speaking, there exists a collection of maps, one from each patch to the fiber, that are individually valid local sections, but that can't "fit together" into a smooth global section.
3. There is an even more severe notion, where the fiber bundle is so "twisted" topologically that there are no global sections, even ones defined separately over multiple patches. For example, the hairy ball theorem says that no circle bundle with fiber $S^1$ over the sphere $S^2$ admits a global section.

These three notions are not independent: any fiber bundle whose base manifold is contractible is trivial according to sense #2. Also, sense #3 is strictly stronger than sense #2 for general fiber bundles, but for principal bundles they are equivalent.

In physics, nontrivial sections of a complex line bundle usually come up in the context of E&M in the presence of magnetic monopoles (which usually come about because some nonabelian gauge symmetry is spontaneously broken). Pointlike magnetic monopoles are like topological point defects in the domain of definition of the gauge potential $A_\mu(x)$, so they render the domain of $A_\mu$ topologically nontrivial. So as long as you're working in the usual situation where there are no magnetic monopoles, you can get away with just using a single wavefunction $M^n \to \mathbb{C}$ to describe $n$ electrically charged particles.