Is a complex scalar field on a curved manifold $M$ just a complex-valued function on $M$?

Question

I used to think of real and complex scalar fields as just functions, both on a flat and curved manifolds. Let's take a complex scalar field $\Phi$. In physics, usually we think of it as a map $\Phi: M \to \mathbb{C}$. However, from Lecture $26$ of Prof. Frederic Schuller's lecture series on `Geometrical anatomy of theoretical physics', I learnt that a quantum wave function is not actually a complex-valued function on the base manifold. Instead, the vague idea is that, it is a section of a $\mathbb C$-vector bundle over $M$. The wave function $\Psi \in \Gamma(E) $, where $E$ is an associated bundle to the frame bundle (a principal bundle). Sections of associated bundles can be represented by $\mathbb C$ valued functions on the total space of the principal bundle. What all this leads to is that on a curved manifold $M$ or in non-Cartesian coordinates in a flat manifold, the derivative operator on the wave function is changed to some kind of covariant derivative using connections on the frame bundle.

I understand neither associated bundles nor principal bundles. Above, I tried to write what is mentioned in the lecture.

My question is, what really are complex scalar fields on a curved manifold? Are they just complex valued functions on the manifold or are they something else? The related question is, what is the right derivative operator for such fields? Do we need some covariant derivative defined with some connection or is it just the partial derivative? I am talking about quantum as well as classical fields.

I have read the answer to the question What is the difference between a section of a complex line bundle and a complex-valued function? but it doesn't clear my confusion fully.

Answer 1

My question is, what really are complex scalar fields on a curved manifold? Are they just complex valued functions on the manifold or are they something else?

The question really poses a false dichotomy between "functions" and "sections of bundles": A complex-valued function $f : M\to\mathbb{C}$ equivalently is a section of the trivial $\mathbb{C}$-fibered bundle $M\times \mathbb{C}$ just by $\bar{f} : M\to M\times \mathbb{C}, x\mapsto (x,f(x))$. Hence the notion of "section of a bundle" includes all regular "functions on a manifold" and is a true generalization of it. Saying that fields are sections of bundles does not destroy your earlier belief that fields are functions on a manifold - every function on a manifold defines a section of a trivial bundle.

Such generalizations become necessary in physics at various points where it turns out the "fields" physics wants to deal with cannot be phrased as one single function on the entire manifold without contradictions. Many examples involve gauge fields (see e.g. this answer of mine explaining the bundle viewpoint there) in some form or another, and plenty of useful physics can be done without needing this additional abstraction.

Really, different physical situations and questions call for different formalizations. Sometimes it's fine to just treat all fields as globally defined functions, sometimes it's not and you need to switch to a framework where they are sections of bundles - just like much of classical mechanics works without ever thinking about manifolds and just assuming we're on $\mathbb{R}^n$ but at some point you might find you have to consider the notion of space(time) as a manifold.

Likewise, the "right" derivative operator is the one that you can gainfully use to model the physics of the system under consideration correctly! There is no inherently different notion of differentation on sections of bundles as opposed to complex-valued functions - but the bundle viewpoint is indeed necessary to properly formalize what is happening mathematically when we replace the ordinary derivative $\partial_\mu$ by a covariant derivative $\partial_\mu + A_\mu$ for a gauge field $A_\mu$. Whether you need that formalization or you can just take $\partial_\mu + A_\mu$ and apply it to "fields" without caring about whether or not those fields "are" functions or bundle sections or whatever depends on what you're trying to do!

It's not that "really" a field is a section of a bundle and not a function or that space(time) "really" is a manifold and not just $\mathbb{R}^n$ - those are just different theoretical frameworks operating at different levels of abstraction and with different scopes of what kind of questions they answer. Don't insist on trying to fit all possible mathematical formulations of physical theories into one theory where they're supposed to be valid all together all the time. That's not how it works.

Answer 2

I am not really sure of what I am saying, but I think this is a correct waving of hands.

A complex number can be represented by $re^{i\theta}$, or equivalently a $2$D arrow. A complex valued function can be represented by a field of arrows, sort of like an electric field can, or fur on a cat.

To be continuous, the fur must be combed so the magnitude and direction change smoothly. This is not a problem in $\mathbb{R}^n$. But on a spherical cat, there is always a point where the fur points toward or away. Most people have such a point on their head. So no single continuous complex function can cover a sphere.

But you can choose two different functions that cover different parts of the sphere and join smoothly.

This works nicely with gauge theories where you can represent a field with different functions. You can find one gauge that is continuous in one region of space, and another that works where the first would be discontinuous.