The goal of physics

post by Jim Pivarski (jim-pivarski) · 2023-09-02T23:08:02.125Z · LW · GW · 4 comments

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In grad school, I was a teaching assistant for a course called, Why the Sky is Blue. It was a qualitative introduction to physics for non-majors, covering a lot of the same topics as Physics I, such as forces, conservation of energy and momentum, electric charges and magnetic fields, in less detail, with not much math. The actual question about why the sky is blue was saved for the end. As the course dragged on and the students (who expected no math, rather than not much math) started to complain, "Are we ever going to find out why the sky is blue?" I watched the schedule slip and wondered the same thing.

We skipped some sections and managed to wedge it into the last lecture: finally, we were talking about why the sky is blue! "The sky is blue because of Rayleigh scattering." Okay, that's not an answer we hadn't defined Rayleigh scattering, there wasn't time for it, so we said that air molecules absorb and re-radiate—effectively changing the direction of—blue light more than red light. Red light goes straight through the atmosphere, and blue light bounces around, making the whole sky glow blue. Conversely, sunrises and sunsets are red because you're looking at the light that has gone straight through a larger wedge of atmosphere. It lost most of its blue on the way to your eye.

Pretty good explanation, for not being able to say

$$I=I_0\frac{1+{\cos }^2\theta }{2R^2}{\left(\frac{2\pi }{\lambda }\right)}^4{\left(\frac{n^2-1}{n^2+2}\right)}^2{\left(\frac{d}{2}\right)}^6$$

(the $1/{\lambda }^4$ part affects small-$\lambda$ blue light more than large-$\lambda$ red light). We also showed pictures like this sunset:

to demonstrate the effect of straight-through red light and bouncing-around blue light.

So in the end, "Why is the sky blue?"

Answer: "Because sunsets are red!"

"And why are sunsets red...?"

It was understandably unsatisfying. One thing was only explained in terms of another thing. But even if we had the time to get into detail about Rayleigh scattering, they could reasonably ask, "Why does light scatter according to that formula?" We could go deeper and explain Lord Rayleigh's proof in terms of Maxwell's equations. And whyfore Maxwell's equations? Well, quantum electrodynamics, which is a quantum field theory with a local $U\left(1\right)$ gauge symmetry, which is to say that every point in space has an extra degree of freedom, similar to a fourth spatial dimension except that this dimension can't be rotated with normal space like the other three, this dimension is connected to itself as a circle instead of being infinite (that's what the $U\left(1\right)$ means), and neighboring points in 3D space try to minimize differences in this extra parameter, which leads to waves. The explanatory power is breathtaking: you can actually derive that photons must exist, if you assume that there's this $U\left(1\right)$ symmetry laying around. But why is there a $U\left(1\right)$ symmetry?

Modern physics seems to be obsessed with symmetries. Even this $U\left(1\right)$ symmetry is explained in terms of a more fundamental $SU\left(2\right)\times U\left(1\right)$ (different $U\left(1\right)$) and the Higgs mechanism. Physicists seem to be holding their tongues, avoiding saying, "This is the most basic thing," by saying, "This one thing is actually a manifestation that other thing." Answering the question, "Why do photons exist?" with "Because space has an internal $U\left(1\right)$ symmetry" is a bit like saying, "The sky is blue because sunsets are red."

Symmetry explanations collapse our description of the world onto a smaller description. They say that one thing is mathematically derivable from the other, maybe in both directions, but they don't say why either is there at all. Perhaps that's an unanswerable question, and the symmetry language is a way of acknowledging the limitation.

To show what I mean, consider a universe that consists of nothing but a point in the exact center of a perfect circle. (I've been feeling free to say, "Consider a universe..." ever since a lecture in which the professor wrote down "$\psi \left(\text{universe}\right)$" and said, "The goal of physics is to find the wavefunction for the universe," to a round of nervous laughter and shuffling of feet.)

This is a two-dimensional world, so we can locate points in this space using Cartesian coordinates $x$ and $y$.

However, because of the rotational symmetry, $x$ and $y$ say too much. Any point $(x,y)$ is utterly indistinguishable from another point $(x^{\prime },y^{\prime })$ in which

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

for any $\varphi$. Let's rephrase this in terms of polar coordinates $r=\sqrt{x^2+y^2}$ and $\varphi ={\tan }^{-1}\left(y/x\right)$:

Now we can see that $\varphi$ is a superfluous coordinate. Remember that the circle is perfectly smooth (not made of granular atoms) and it's perfectly centered on the dot. If motion along the direction of $\varphi$ can't be observed in any way, then it's not a physical thing; there are no experiments that could reveal that we are moving in $\varphi$.

This universe really just consists of two dots in a one-dimensional space that is bounded on one end. Or further, that nothing exists in this universe except a distance, a single number that quantifies the distance between those two dots (with nothing to compare that distance to).

Unconvinced? Suppose we went in the other direction—instead of removing superfluous coordinates, let's add superfluous coordinates. Given the two-dimensional circle, suppose I say that there's another coordinate, $z$, and we're actually looking at an infinite cylinder.

I could keep adding superfluous coordinates, but there are no constraints on what I can make up. The only description that is unique is the one that shrink-wraps around the matter to be described. (Or, if not unique, one of a small set of equivalent representations.) The uniqueness (or near-uniqueness) makes it interesting. Or, rather, the lack of freedom makes it interesting: mathematics is more interesting than random symbols on a page because it has constraints.

That's why physics is on a unification kick: we make the most progress when we describe one thing in terms of another, so that they don't seem to be two distinct things anymore. If you think of "Why is the sky blue?" and "Why are sunsets red?" as two, unrelated questions, the world seems more complex and arbitrary than if these are known to be two aspects of the same question.

This comes back to a previous post, in which I argued that causes (and reasons-why in general) are a different thing from physical reality [LW · GW]. In physical reality, there's a deep connection between the blue sky and red sunsets, but neither causes the other, nor are they caused by a third thing. Even though it's conventional to say things like, "The blue sky and red sunset are both caused by Rayleigh scattering," this sentence doesn't make sense if we take it literally.

Literally, that would mean, "Without Rayleigh scattering, there would be no blue sky or red sunset" (for an exclusive cause; modify as necessary for probabilistic and multiple causes). We would imagine a world with Rayleigh scattering and a world without Rayleigh scattering, and only the world with Rayleigh scattering has a blue sky and red sunset. That would be fine if we didn't think any further, about how "no Rayleigh scattering" would imply that Maxwell's equations are wrong (assuming the mathematical derivation has no errors), and a world with something different from Maxwell's equations would be a very different world.

These things might have been imaginable when physics was in its infancy, before so many connections between things were found that modifying one thing implies changes in many other things, except at the frontiers, like extremely high energies or strong gravitational fields. But now that we're at this point, in which everything seems to be derivable from the Standard Model and General Relativity, there isn't much wiggle room, except at the frontiers where these two theories are expected to not apply.

Nevertheless, it is meaningful to talk about causes, particularly in realms that are less well-understood, such as human behaviors. This is why I separate talk about causes (and other reasons-why) from what really happens, and put them in different reality-boxes [LW · GW]. We even use different verb tenses to describe these things: subjunctive for causes—what would be, if the cause happens—and simple present for physical reality.

Causes and reasons-why are a layer on top of physical reality, things we say about physical reality based on our incomplete understanding and how we imagine it could be different—in principle, if not in fact. It's constrained, too: we can't say just anything is a cause. Even more loosely defined reasons-why, such as narratives and mythology, have ways in which they can be wrong. And as long as it's possible to be wrong, it can be interesting!

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