I think he sets quite a high bar here. Like, just going through the basics of Noether’s Theorem is so far beyond what normally passes for popular science (which usually boils down to “the universe is really big you guys” and “quantum stuff is weird and nobody understands it”). Like, just using words like “conservation laws” and “Lagrangian” is risky already.
Personally, I would have liked it to dig deeper (as I already heard the basics of Noether’s theorem, but am not a physicist or have studied it in any great depth), but Quanta is not a scientific journal, it’s a pop-sci magazine. The article is a great intro.
I followed the link, that's what he means by 'cool stuff that's missing':
> In short:
> The key to Noether’s theorem is the requirement that we can freely reinterpret observables as symmetry generators, and vice versa — in a way that’s consistent with the action of symmetry generators on both observables and symmetry generators.
> In classical mechanics this is achieved by a hybrid structure: a Poisson algebra, whose elements are both observables and symmetry generators.
> In an algebraic approach to quantum theory, this requirement singles out complex quantum mechanics. i =√−1 turns observables into symmetry generators, and vice versa.
How would one explain this to the audience of Quanta Magazine?
First have a clear statement of the context. Second, define and explain all the obscure terminology and mathematical notation -- at least give good references.
Gee, in one discussion of Lagrangians, saw "configuration space" and "constraints". Okay, studied topological spaces, vector spaces, inner product spaces, measure spaces, Banach spaces, Hilbert spaces, probability spaces, but never saw a definition or explanation of a "configuration space".
"Constraints"? Kuhn-Tucker optimization theory has a lot on constraints, for one of the issues there was a question, and I solved and published it. Thought I had some background in "constraints", but the Lagrangian explanations didn't make clear what they meant by "constraints".
A "symmetry generator"? This is the first I ever heard of any such thing although ugrad math honors paper was on group representations.
Then there was "phase space": What does that have to do with "phase" in light waves and sound waves? Sounds like one word with two different meanings?
E.g., "Poisson algebra": Okay, there is the Poisson stochastic process, e.g., seems to get assumed in "half-life" calculations, and with more in
Erhan \c Cinlar,
{\it Introduction to Stochastic Processes,\/}
ISBN 0-13-498089-1,
Prentice-Hall,
Englewood Cliffs, NJ,
1975.\ \
and there is abstract algebra, e.g., groups, rings, fields, ... But a "Poisson algebra"?
Yeah I think that’s probably true and why I greatly admire efforts by people like Steven Strogatz and especially Sean Carroll who are leading the way from no-maths pop-sci to high school maths pop-sci where you know you don’t want to actually work with the maths but you can start to get an appreciation for the components of it and what the implications are.
Are you referring to his quote "If all of mathematics disappeared, physics would be set back by exactly one week." or his more general belief of if you can’t explain something in simple terms, you don’t understand it?
Maybe the issue is that some things are too complicated to be explained without maths and aren't accessible to lay people as they're so counter-intuitive.
Mostly the latter. Doing the maths doesn’t always mean understanding something. As a simple example it’s pretty easy to treat gravitation of a spherical object mathematically as a point source, but it was quite a conceptual leap, and it doesn’t mean you understand why you can treat it as a point source (and when you can’t).
I’m also reminded on the scene in “Severance” where the members of MDR are unable to explain their job to another department.
It's strange as I typically think the opposite - you don't really understand something until you can measure and quantify it. e.g. we don't have a handle on consciousness yet as we can't reliably measure it.
However, when it comes to a "deep" understanding, then maybe numbers aren't enough and we need to be able to intuit what's going on that creates those particular numbers.
