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Ben Sprott (May 25 2024 at 13:39):Let me apologize in advance for coming back here if you have had to deal with me in the past. I am an armchair category theorist who tends to misuse advanced topics without much knowledge. I also tend to go on fishing expeditions just to see if those advanced topics seem useful to me.
I tried posting on math overflow but it was closed. I hesitate to do this, but I am going to include a "preprint". I am doing this because it gives some background to my question. It has several problems including an incorrect assumption about something called cyclic lists. They are comonads, not monads.
Here is what I have been working on:
I have been playing around with monads and comonads for a while. My background is in quantum theory, but I have since moved on into using monads for data science (think list and multiset). I have been going over my physics applications again and thinking about the holographic principle.
I find that the holographic principle is a bit hard to pin down formally at the kind abstraction you expect for a Category Theory treatment of things. I was given some links in a Math overflow post and was struck by a few things.
It looks like mathematical physicists tend to handle this principle on a “scale” that makes it rather opaque. They phrase it in terms of quantum field theories. Those are wildly complicated theories and a simpler picture would be nice. For instance, could one have a basic demonstration of the principle for just one qubit? The idea here is that you have a “theory” on the inside which is standard finite dimensional Hilbert spaces in 3D space, but then on the surface you have another theory that handles the data. There should be a map between them and the obvious choice for that map to translate between them is a functor.
The data-theory you have on the surface would expose just the readouts of, say, each of two arms of a Stern Gerlach and the readouts are either 1 or 0. It could be just one readout that says Left or Right. This is a bit like the drawings you see in the standard literature on the holographic principle where the surface of a sphere is covered with many “bit readouts”. You will want some kind of spherical symmetry on the surface data-theory and there might be a way to do this. I have been working on a data model of quantum theory and you can see a paper about it here. Essentially, you are looking for data symmetries that give rise to (co)monads. My choice for this structure requires monad composition, which is notoriously difficult. In my paper, I propose the composition of the pointed cyclic list comonad with the multiset monad. I actually didn't know much about cyclic lists when I wrote it, but had some help recently (here, in fact at Zulip).
What I am proposing is the following. For the surface data-theory, you take something like the Kleisli category of the monad defined by the composition of cyclic list and multiset monad. Then on the inside you have the full spatially grounded (read 3D) theory of a Stern Gerlach. Then there is a functor between them and this is an example of the holographic principle for a simple quantum system.
Does anyone see how this is completely wrong? Is there a way to do this, to demonstrate the holographic principle with the simplest quantum system?
Jean-Baptiste Vienney (May 25 2024 at 14:34):Here are three comments on your "paper":
On the math side: The cool thing in your document is a cyclic list monad. If you could define it in the usual way we write mathematics and prove that it is a monad, in the usual way we're writing proofs, it would be nice. If there is distributive law between this cyclic list monad and the multiset monad, it would also be nice if you could explain it in the standard way mathematics is written by mathematicians. If there is another commutative law between this monad and a distribution monad, please explain it in this same rigorous language.
On the modelization side: You're combining an angle, multiple experiments and a probabilistic behaviour so I agree, the idea of modelling the output data of the experiment as a multiset of "probablistic cyclic lists" doesn't look absurd. Now you would have to use the usual equations used to model the Stern-Gerlach experiment and show that you can indeed describe the output data as a multiset of "probabilistic cyclic lists" if you want to convince anyone. Once more, if you want to write the equations for some physics experiment in another way, you must show that the way you write them agrees with the usual equations physicists use.
On the physics side: If you succeed to present the math and the modelization correctly, if you claim that you are not just working on the way of presenting the data but get some additional understanding of the physics from this, you must explain it in a clear way. I can't really comment on this because I'm not a physicist.
Jean-Baptiste Vienney (May 25 2024 at 14:38):As to the problem of knowing if there is a cyclic list monad or comonad, you could very well have the two. There is a list monad and a list comonad if you work in the category of sets and relations and not only sets and functions. The same could be verified in terms of cyclic lists.
Jean-Baptiste Vienney (May 25 2024 at 14:45):I don't think your "paper" is completely absurd but the way you express yourself, with vague reasonings is no longer very much in fashion today. It looks a bit like the reasonings of the first philosophers, such as Thales who said that magnets have a soul because they make other objects move and that only things with a soul can make stuff moving. Thales was a great guy because reasoning on nature was an interesting idea at the time. It was a new thing and even done incorrectly it was an improvement in philosophy and science. But if you tried to publish Thales thoughts in a scientific journal today, it would probably be considered as bullshit by the referees.
Ben Sprott (May 25 2024 at 15:27):Thank you for your critique! It is very good for me to get this kind of input. I will read it tonight. I see you work with Richard Blute. I was a student of Panangaden. I showed something like this to Richard a few years ago. He is a great guy!