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Stream: community: general

Topic: condensed matter physics

fosco (Jul 23 2025 at 18:22):

It was a pleasant surprise to learn it actually gets used within my field!

Hah! I can finally ask... what is condensed matter physics and how is category theory relevant to it?

fosco (Jul 23 2025 at 18:24):

What choices of $f,g$ in calculating the integral $\int g(x)e^{f(x)} dx$ define condensed matter physics?

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John Baez (Jul 24 2025 at 07:54):

I can't resist answering: condensed matter physics is the study of matter that's 'condensed', most notably solids and liquids. An old physics definition of 'solid' says that solids are crystals. This would exclude glasses and 'amorphous' solids like charcoal, which are often made of small crystals randomly piled next to each other. Crystals are mathematically much easier to understand than these more random solids, and liquids. Condensed matter physics tries to understand all these things.

John Baez (Jul 24 2025 at 07:57):

A fundamental tool of condensed matter physics is statistical mechanics. Classical statistical mechanics was snidely summarized by that passage you quoted (with the wrong sign in the exponential, and some missing factors, so that mathematicians who take this literally will seem like fools to actual physicists.) Quantum statistical mechanics is more important, but it uses similar expressions, where the integral is replaced by a trace.

Who wrote that passage? It reminds me of V. I. Arnold's explanation of mathematics as the branch of physics where experiments are cheap.

John Baez (Jul 24 2025 at 07:58):

@Ruby Khondaker (she/her) will surely be able to fill in a lot more details.

Ruby Khondaker (she/her) (Jul 24 2025 at 08:10):

Yes I think solids and liquids is accurate - the way I often introduce condensed matter to newcomers is saying that it’s like condensed milk :P

Indeed, path integral methods are very prevalent in condensed matter via statistical mechanics and statistical field theory - Feynman diagrams get used all the time! There’s a cool story about the similarities between high-energy physics and condensed matter physics.

In terms of the relevance of category theory - there are two main ways I’ve seen so far (though I’m relatively new to the field so I’m sure there are more). The first is topology - condensed matter is a setting where topological effects produce interesting nontrivial physics. Phenomena like the fractional quantum Hall effect, or anyons. It turns out that topological field theories are good tools to describe these, which can be thought of categorically as functors whose domain is an appropriate cobordism category. The braiding present within the category itself describes some of the most interesting parts of the physics!

The second way I’m familiar with is through symmetry. Typically one imagines that group theory is the natural setting to discuss this, but somehow the materials nature produces end up being stranger. It’s helpful to consider so-called “higher form symmetries” which may or may not be invertible; these are usually implemented by topological operators, which depend on the homeomorphism type of some submanifold. Category theory provides a useful tool to organise these symmetries and study how they compose in a setting where invertibility isn’t guaranteed!

John Baez (Jul 24 2025 at 08:24):

Nice. It's probably good to warn fosco (or anyone like fosco) that the fractional Hall effect only shows up in some carefully designed systems: thin films of materials at extremely low temperatures. The "extremely low temperature" part lets us use beautiful math: at higher temperatures things get more random and complicated. The "thin film" part lets use math like anyons and 3d TQFT that are relevant only when space is 2-dimensional.

John Baez (Jul 24 2025 at 08:27):

More generally, condensed matter physicists who like some particular beautiful branch of math (like 2-group symmetries, etc. etc.) typically have to work very hard to find real-world systems described by that math. Most condensed matter physicists work in the reverse way, where they take real-world systems that are easier to find (like condensed milk, or quartz crystals, or silicon, or graphite, etc.) and figure out the math needed to describe those.

Ruby Khondaker (she/her) (Jul 24 2025 at 08:29):

Yeah it’s very important to clarify that the physics takes precedent here, not the math; category theory is important in condensed matter insofar as it is useful, irrespective of how pretty people find it.

John Baez (Jul 24 2025 at 08:31):

But of course there are lots of people like me who claim they are interested in condensed matter physics, who are mainly interested in beautiful mathematical structures and hoping they can be found in the real world.

Ruby Khondaker (she/her) (Jul 24 2025 at 08:31):

In my case I think my interest is more genuine with condensed matter itself, and it’s a nice bonus that I get to use cool math from time to time :)

fosco (Jul 24 2025 at 08:35):

so, you're essentially telling me that condensed milk is mathematically described by a functor out of a cobordism category?

fosco (Jul 24 2025 at 08:35):

If yes, I can finally say I understand what the hell is "milk"

fosco (Jul 24 2025 at 08:35):

it's just an object of the category of...

Ruby Khondaker (she/her) (Jul 24 2025 at 08:36):

As far as I’m aware, no, haha

fosco said:

so, you're essentially telling me that condensed milk is mathematically described by a functor out of a cobordism category?

John Baez (Jul 24 2025 at 09:11):

TQFTs and cobordism categories are only useful for describing very specially prepared systems like thin films of graphene at temperatures close to absolute zero. More ordinary systems, like condensed milk at room temperature, are described by mathematics that's much more complicated. (Higher) category theory is relevant here too, but it will take a few decades to clarify this. One step in this direction is the work on factorization algebras.

Ruby Khondaker (she/her) (Jul 24 2025 at 09:12):

Wait ordinary systems need harder mathematics?

fosco (Jul 24 2025 at 09:14):

Ruby Khondaker (she/her) said:

Wait ordinary systems need harder mathematics?

"Aristotle said a bunch of stuff that was wrong. Galileo and Newton fixed things up. Then Einstein broke everything again. Now, we’ve basically got it all worked out, except for small stuff, big stuff, hot stuff, cold stuff, fast stuff, heavy stuff, dark stuff, turbulence, and the concept of time."

John Baez (Jul 24 2025 at 09:21):

I find such jokes a bit tiresome since they neglect what we do understand, which is a hell of a lot.

John Baez (Jul 24 2025 at 09:24):

For example we understand a lot about liquids (like milk); I was just saying we have not yet brought higher category theory to bear on them.

Ruby Khondaker (she/her) (Jul 24 2025 at 09:24):

I am enjoying the repeated references to condensed milk :P

Morgan Rogers (he/him) (Jul 24 2025 at 09:59):

It's a pretty sweet example

fosco (Jul 24 2025 at 10:40):

image.png

So, a prefactorization algebra is a copresheaf on a topological space $M$, which has some... monoidal property when regarded as a functor $(Op(M),\sqcup,\varnothing)\to (Vect,\otimes,k)$? Not exactly, but the idea seems to be that

Notification Bot (Jul 24 2025 at 10:45):

23 messages were moved here from #community: general > Introduce yourself! by Morgan Rogers (he/him).

Amar Hadzihasanovic (Jul 24 2025 at 10:48):

Clark Barwick gave his CT keynote on a new abstract approach to factorization algebras, he has a note about it on his website

Amar Hadzihasanovic (Jul 24 2025 at 10:50):

He explained his vision of the subject as being, possibly, a kind of "formal theory of dissimilarity" in the same vein as homotopy theory is a "formal theory of similarity"

fosco (Jul 24 2025 at 10:54):

Cool!

Federica Pasqualone (Jul 24 2025 at 14:35):

You can also read about it from my thesis @fosco , and all the other talks I gave. The material is accessible here: www.federicapasqualone.com