The Useful Part of Physics

[December 3, 2022]

Miscellaneous thoughts:

A physics undergraduate degree isn’t very useful in real life except for, like, pretending to know stuff on a blog. But it does give you a sort of mathematical intuition that’s pretty rare, which probably everyone ought to acquire from their education, and that almost everyone isn’t getting. That core skill ought to not require studying Lagrangians or spinors or whatever to learn; we should really figure out how to directly teach it to everyone so that it might become universal in public discourse.

The essential skill is basically “estimation”, except if you call it that and try to put it into schools, you end up with stupid word problems written by people who think “estimation” means “rounding”. The better name is something like “numeracy”, but for functions instead of numbers: generally “getting” how functions work and relate to each other. A core skill of this is estimation, specifically “estimation with Taylor Series” aka “first-order approximation”.

An example first-order approximation is approximating \(\frac{1}{1-x} \approx 1 + x\) for tiny \(x \ll 1\), based on its Taylor series. But maybe you don’t need to know about Taylor series at all to learn it? I could imagine introducing the small-\(x\) approximations of each standard function in a pre-calculus class and having people memorize and apply them to problems without ever knowing the word “derivative”.

\[\begin{aligned} \frac{1}{1-x} &\approx 1 + x \\ \sqrt{1 + x} &\approx 1 + \frac{1}{2} x \\ e^{x} &\approx 1 + x \\ \ln(1 + x) &\approx x \\ \sin(x) &\approx x \\ \cos(x) &\approx 1 \text{ (or } 1-\frac{x^2}{2} \text{) } \end{aligned}\]

They would certainly be more useful than trig identities are.

Today it’s pointless to put a non-trivial equation into a book or news article because, well, people’s eyes glaze over, but also because most people have no skill at deriving intuition from equations. Why would they? But the sort of facility you get from this kind of approximation teaches you tosay: “what does this equation do for tiny \(x\)? what about for giant \(x\)? Huh, okay.” which is what physics involves every day.

Even if you never apply approximation to an equation after you finish school, it still confers a sort of intuition for “how functional relationships work” that is useful all the time. There’s no reason that skill needs to be trained only in a couple academic fields. I sure wish some non-mathematical professionals, such as… doctors?… had it.

… of course real mathematical intuition is something that needs to be all over the curriculum in general, ideally replacing most of the memorized algorithms that we teach today. Also we should focus on mental math, especially in young students, instead of paper algorithms, and stop using graphing calculators entirely.

“You’ll always have a computer to do this, so why do it by hand”, they said. Is that so? I bet most people wouldn’t know where to go to graph a function as an adult to save their lives, unless they still have their TI-83+ laying around. (Easiest way is Google, by the way.)

Dunno know where people got the idea that the point of education is to learn facts. The point should be to make you smart. Then you can know all the facts you want. The school of facts just makes you “dumb, but full of facts”. Harrumph.

I guess it’s on me to demonstrate that first-order approximation is actually useful, and I don’t have a great example in mind. Sorry. But once you get in the habit of thinking in order-approximations you kinda keep doing that the rest of your life, and it… feels… good? You say things like “to first order, stopping to fix my phone should have no effect on how long this road trip takes” (“… unless there’s an unknown ‘interaction term’, like maybe once my phone is working it’s more fun to get distracted singing along to music so on average I drive slower…”, and other examples of totally unnecessary but actually pretty valid physics jargon.)

Incidentally, this half-baked thought came from the following rabbit hole that led to a pleasing example of such an approximation:

  1. This blog post about heavy sterile right-handed neutrinos as a dark matter candidate in a new paper, which linked to
  2. The Wikipedia article on The See-saw Mechanism, which is the mathematical phenomenon which might explain why there would be a very-heavy neutrino for each of the known very-light ones, and which has a wonderful name.

The mechanism is that a matrix of the form \(\begin{pmatrix} 0 & M \\ M & B \end{pmatrix}\) has eigenvalues

\[\lambda_{\pm} = \frac{B \pm \sqrt{B^2 + 4 M^2}}{2}\]

Which have \(\lambda_+ \times \lambda_- = -M^2\), aka, if one goes up, the other goes down. Hence the “see-saw”.

In particular, since \(\sqrt{B^2 + 4 M^2} = B \sqrt{1 + \frac{4M^2}{B^2}}\), if \(B \gg M\) then

\[\frac{4 M^2}{B^2} \ll 1\]

and so, unpacking the square root to its first-order term:

\[\begin{aligned} \frac{B \pm B \sqrt{1 + \frac{4 M^2}{B^2}}}{2} &\approx \frac{B \pm B(1 + \frac{1}{2} \frac{4 M^2}{B^2})}{2}\\ &= \frac{B \pm (B + \frac{2M^2}{B})}{2} \end{aligned}\]

Which gives

\[\begin{aligned}\lambda_+ &\approx B + \frac{M^2}{B} \approx B \\ \lambda_- &\approx - \frac{M^2}{B}\end{aligned}\]

And (read the article or whatever) these \(\lambda\) end up related to the masses of the neutrinos.

This particular calculation is a bit more complex than anything you’d probably ask a high-schooler to do. But there’s really nothing especially difficult about it. It’s just not something we show anybody else how to do. What’s up with that?