Sometimes on this blog I’ll write on more technical subjects and assume a bit more familiarity with mathematics. I’ll try to label posts accordingly if that’s not your thing. Let me know what you think. These might be for me as much as for anyone else!
Today I have a question that might seem really esoteric and out of the blue, probably because it is. But the thought process below has really clarified physics in general for me over the past few weeks, and it has surprisingly broad connections that I hope to discuss more in the future. Here we go:
Isn't a field simply an object in a space with one higher dimension?
What? Let me explain. I've been thinking about how the atmospheric science community tends to use "object" conceptual frameworks some of the time and "field" conceptual frameworks some of the time. For example, we tend to think of clouds as objects with well-defined boundaries, moving, evolving, and interacting with their environment. We also have field representations, especially in weather simulations, such as the wind field: every point in space has a wind vector. Are these concepts compatible? If they aren’t, should we prefer one to the other? The above question is my starting point into this space.
First, we need a working definition for an object: An object is a set of points. There are no additional properties of an object beyond what points are included and where they are. Any object may be specified by delineating the space it occupies. Scientific questions about cloud objects, for example, are answered by describing the properties of the set of cloudy points: the area covered by cloud points, the altitude of the cloud points, etc.
This might be somewhat counterintuitive, because objects have other properties, like color, right? I don't think so; fields have properties. We often carve up fields into discrete objects, and so properties like "color" are conflated with the object itself. But color is a field property (the electromagnetic field, specifically).
This is the typical definition of a field: it is a space full of points, and every point has a value. For example, a one dimensional field would be visualized using a plot like this:
Ahh, but now, I can see an object: a spiky line. But it's living in a two-dimensional space! That is what I meant at the start. A field, which is a region full of points, each with a value, can be represented as an object living a space with one higher dimension. The extra dimension is needed to plot the field values. In the plot above, the additional dimension is the vertical.
So, you might say that fields and objects are the same thing, and they simply live in different dimensions. Well, I have convinced myself now that it's not so simple.
The reason is that the units are different.
What does that mean, specifically? Think of the set of possible coordinate transformations. This is the range of ways in which you could look at the field without changing the field itself, or technically, the set of transformations which simply change your coordinate vectors. In particular, it's helpful to think about rotations. I can rotate the field in the spatial directions, and this would be equivalent to moving your head around and looking at the same thing from a different side. But if we try our object-in-higher-dimension viewpoint, there is one thing that can't be done: to rotate the object in the direction of the field coordinate. For example, think of a temperature field. You can exchange the x-axis for the y-axis by changing your view. You can't change the T-axis for the x-axis because doing so would amount to more than a simple coordinate vector rotation. The x → y transformation changes your view of the physical thing, while the x → T conversion would change the physical thing itself.
So if we attempt to view a 3D temperature field as a sort of an object in a higher-dimensional space, then the set of possible coordinate transformations is restricted relative to the set of possible coordinate transformations some ordinary 4D object would enjoy.
The set of possible coordinate transformations is called a symmetry group. A symmetry exists when you can change one thing without affecting other things. And here, a coordinate transformation is something you can change without affecting the nature of the physical object itself.
Since our "4D temperature object" (3D temperature field) has a restricted set of possible coordinate transformations, it therefore lives in a different type of space than an ordinary 4D object—a space defined by the restricted symmetry group.
Mathematically, the symmetry group places restrictions on the Jacobian matrix which is used to transform vectors under coordinate system changes. Specifically, in the temperature field example (x,y,z,T), if we disallow rotations into the temperature direction, the Jacobian must have zeros in the entries above and to the left of T:
According to Claude, this idea is well-trodden in topology. A space such as this is called a “fiber bundle”, a name I quite like because it paints a picture that emphasizes the one "special" direction (T) which can't be "rotated out of". I imagine something like this:
Also Muse’s The 2nd Law album cover (except if the fibers were straight):
So, we have the answer to the question we started with: Isn't a field simply an object in a space with one higher dimension?
No. Or, at least, not a "normal" space in one higher direction: it is a fibered space, a higher-dimensional space with an unusual symmetry group.
Note that so far we have only discussed scalar fields. You could also wonder about vector fields. I'm already less tempted to picture a vector field as a higher dimensional object, because you now have to double the number of dimensions rather than simply add one. But maybe you're one of those crazy people that can visualize very high-dimensional spaces.
Anyway, a vector field has an additional constraint. To illustrate, consider a rotation. In order to leave the physical system unchanged, the rotation has to occur in the exact same way along the spatial dimensions as it does along the field value dimensions. You must preserve both the field magnitudes and the field vector directions. This would place additional constraints on the Jacobian.
So that's coordinate transformations, which don't actually change anything about the stuff actually there in the real world. What about physical transformations? How would the system evolve in time, according to the laws of physics?
Transformations in time may be represented as a matrix which increments the system forward in time. This matrix does not have the same constraints as the Jacobian above, because physical laws will allow the physical thing itself to change. For example, consider a nonuniform temperature field. Heat will diffuse in order to reduce temperature gradients. In a sense, this might look a bit like a rotation between the temperature direction and one of the spatial axes: a "pile" of temperature is getting spread out spatially.
If we were previously talking about the properties of the space and its symmetries, now that we are discussing the physical laws we should discuss their symmetries. For example, one such symmetry is that the same physical laws apply at every time. In the above framing, this means that the evolution matrix is invariant.
According to a remarkable insight due to Emmy Noether, the fact that the physical laws have symmetry is equivalent to the fact that conservation laws exist. Conservation laws, such as the conservation of energy, are probably the most useful fact of the universe that humans have ever discovered. They underlie nearly all of physics and therefore all technology that has ever been invented.
To be honest, I was hoping that the framework I presented above would lead to some improved intuition about why Noether’s theorem is true. Unfortunately I do not yet see it, and I probably need to work through the math much more than I have so far. I do have a strong intuition that this would be a fruitful research direction for the atmospheric sciences, because there is a key symmetry of the atmosphere which has not yet been fully taken advantage of: scale invariance.
It’s always rewarding and a bit exciting when one research idea (here, objects vs. fields) ends up turning into another research idea that was originally separate (scale invariance as symmetry). More to be learned!
So, for you: What errors did I make here?