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Why should fields be thought of as sections on a bundle? In particular, what is the problem with thinking of them as functions (with additional conditions on target domain and smoothness) on spacetime?

I could come up with one example where there is a clear difference - sections on the Möbius strip with base space the unit circle and fibres the interval $[-1,1]$. Here a smooth section is forced to take the value $0$ on at least one point of the circle, while general smooth functions on the unit circle do not have to satisfy this.

This example seemed to me however not very physically motivated - are there physical examples where this difference between sections and smooth functions on the base space becomes more pronounced? Or is there some other motivation for defining fields as sections over bundles? For what it is worth, I have encountered this concept only during my studies of general relativity, maybe this distinction only becomes apparent in QFT?

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    $\begingroup$ I think that bundles are more appropriate when it comes to gauge fields and covariant derivatives. $\endgroup$
    – md2perpe
    Aug 12, 2021 at 21:38
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    $\begingroup$ For example the Aharonov-Bohm effect made it clear that a topologically nontrivial structure needed. $\endgroup$
    – NDewolf
    Aug 12, 2021 at 22:49
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    $\begingroup$ The conceptual/mathematical issue is that on a manifold you don't have just one tangent space, you have one at each point, so a vector field can't be a regular function - it doesn't have a single tangent space as codomain. $\endgroup$
    – Javier
    Aug 13, 2021 at 0:49
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    $\begingroup$ See e.g. physics.stackexchange.com/a/167160/50583, physics.stackexchange.com/a/317273/50583 - the questions themselves are not duplicates of this one, but the answers are pretty much what I would write as an answer here. $\endgroup$
    – ACuriousMind
    Aug 13, 2021 at 8:07

2 Answers 2

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As far as experiments have been able to tell, spacetime is topologically trivial. Fiber bundles over topologically trivial spacetime are necessarily trivial, so sections are just functions. But if we had some reason to consider spacetimes with nontrivial topology, then nontrivial fiber bundles (where we need sections instead of just functions) would become relevant. I'll mention two good reasons for considering spacetimes of nontrivial topology: one relatively straightforward, and one speculative. Both are active research topics today.

A straightforward reason

QFT is hard. Perturbative methods (small-coupling expansions, represented by Feynman diagrams) can only get us so far in our understanding of QFT. Different-looking QFTs can turn out to be physically equivalent, and a theory's predictions at low energy can look very different than the structure that was used to define it at high energy. The classic example is quantum chromodynamics (QCD), which is defined in terms of quark and gluon fields, but at lower energies, QCD correctly predicts that we should only see mesons and baryons. Understanding 't Hooft anomalies can help us diagnose which different-looking QFTs are equivalent to each other, and which QFTs can arise as low-energy effective models of other QFTs.

An 't Hooft anomaly is an obstruction to promoting a theory's rigid symmetry to a gauge symmetry. (For a taste of what the study of 't Hooft anomalies involves, see Witten (2015), Anomalies Revisited, https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2015/6-2.00-2.30-Edward-Witten.pdf). Understanding a theory's 't Hooft anomalies is more informative than merely understanding its symmetries. This extra information often reaches farther than perturbation theory can see, and it becomes even more enlightening when we consider QFTs formulated on a whole family of spacetime topologies instead of on just one. That's a reason for considering topologically nontrivial spacetimes, which in turn requires considering nontrivial fiber bundles, where most sections are not mere functions.

A speculative reason

Spacetime is almost certainly an emergent phenomenon. Even though it's a fundamental ingredient in the two established pillars of modern physics, quantum field theory and general relativity, string theory (the most promising path toward reconciling those two pillars with each other) suggests that spacetime as we know it is only an approximation. Any QFT that arises as a low-energy approximation to string theory might need to be definable in spacetimes of nontrivial topology, even if such topologies aren't directly relevant at lower energies where quantum gravity effects are negligible. That means they can't have any 't Hooft anomalies in the symmetries that need to be gauged, in spacetimes of any topology — at least any topology that is consistent with any other important restrictions, like admitting a spin structure.

The Standard Model of Particle Physics satisfies that condition: its gauged symmetry group doesn't have any (known) 't Hooft anomalies on the relevant family of spacetimes. This is stated below equation 3.5 on page 24 in arXiv:1808.00009, which says that the standard model defines a consistent quantum theory in any background, of any topology. That's probably not just a random coincidence.

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The distinction between functions and sections is only contentful when dealing with a non trivial bundle. One might then wonder when non trivial bundles actually arise in physics. The most famous example is that of a magnetic monopole. However, how did that non trivial bundle "get there"? Well, the answer is, with our current paradigms, spontaneous symmetry breaking. You could have some larger non abelian symmetry group which is broken to U(1). Given some fairly general conditions on the gauge group, you could possibly have monopole configurations. These are, in a sense, not "true" non trivial bundles, but low energy "effective" non trivial bundles, where the ground state has come to rest in some topologically protected non trivial configuration.

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  • $\begingroup$ Do you mean spontaneous symmetry breaking to U (1) like for example by Higgs potential? $\endgroup$
    – JanG
    Dec 22, 2023 at 16:23

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