Yeah, it's a reference to how "category theory" generalizes "group theory". So if transformations in a group are called "symmetries" then you might call the transformations in a category "generalized symmetries" or "non-invertible symmetries" as in this article.

Erm the whole idea of a symmetry is that it is a group invariant. If they're using the term in some more general way, it would help if they said what the new way was. The article is otherwise almost completely uninformative. What on earth is a "generalized symmetry", especially one that still has something like a conservation law? Does it have applications in math as well as physics? E.g. topologists are always looking for new invariants.

I can understand that popularizations have to gloss over the math, but they usually at least identify the important points even if they don't get into the weeds of explaining the details. This seems like more of a sleight of hand.

Read the original paper: https://arxiv.org/abs/1412.5148

Apparently the QM and condensed matter people use the term "defect" for a (possibly non-invertible) transformation (semigroup or similar), and a "symmetry" is an "invertible" "defect".

https://physics.stackexchange.com/questions/561118/why-shoul...

Think of a symmetry as "something that remains the same" when you do something to a system.

Imagine a pair of denim pants, one could say it has the property of being made of denim, and when you bend the pants or move them around the room, they keep being made of denim. Anything you do to the pants that you can undo, like moving them around or bending them, those are invertible symmetries. Now imagine I rip the pants in half: they're still made of denim, but this action is not so easy to invert. So there are actions I can perform on these pants that are hard or impossible to undo, but still maintain the property of them being made of denim, these are non-invertible symmetries.

The example given in the article is a system being split into a superposition in a way that's hard or impossible to undo, sort of like the pants being torn in half, but just because the system is now in a superposition doesn't mean there weren't properties preserved along the way.

Great explanation, thanks.

That just looks like a conserved property. I know those are linked to symmetries, but they're not identical.

Is this something like the physical operation to invert it doesn't exist, even though it's logically possible? The "pants" example, and maybe the superposition example, makes it sound like they're uninvertible for entropy reasons, which seems interesting if it's not an illusion.

Right, they're not identical, but they're generally paired up so neatly via Noether's theorem that they're treated as the same thing.

In seems like physicists are finding more general conserved quantities associated with more general spaces of physical operations such that not every action is reversible, but there are still conserved quantities associated to the underlying systems, AKA symmetries.

I don't have any solid physical intuition as to what makes an operation irreversible in this context, though, whether it's related to entropy or not.

Interesting. Thanks!