I am looking to the "A Class of Models with the Potential to Represent Fundamental Physics" document and I find it lacking of mentions to electromagnetism.
Some sentences I found: "In traditional physics, local gauge invariance already occurs in classical theories (such as electromagnetism), and it is notable that for us it appears to arise from considering multi‐way systems." and "electric/gauge charges: counts of local hyperedge configurations". Later on there is some discussion on the size of the electron.
As seen that the model can explain mass, momentum, and gravity in a relatively nice way, I wonder if electromagnetism would also have a similar explanation. More specifically, could Lorentz force and Maxwell's equations emerge somehow? Or are they expected to be tied to specific rules.
Keep in mind, electromagnetism is a special case of a gauge theory. We are not currently trying to find electromagnetism itself, which will likely be rule-specific (although we will search for it in the future). We are now just trying to understand how gauge theories would work in our models in general.
Electromagnetism has been unified with gravity by Kaluza Theory. It's just Einstein's Equation in 5D without an Energy-Momentum-Tensor: Rab=0. You get that way Maxwell too.
That Kaluza-Klein theory feels very appealing. I am not an expert and I am not quite grasping many aspects of it, but if just by adding a fifth dimension the electromagnetism follows then it is a argument for electromagnetism not being rule specific.
There seem to be several possible compactifications on Kaluza-Klein. And since we are discussing a discrete model we should have more candidates. If we have an structure like the Cartesian product of three large cycles (the three spatial dimensions) and some graph G. For what kind of graphs G we would get the electromagnetism from Kaluza-Klein? I guess that G should have low diameter, so that that 'dimension' is not observable as a spacelike dimension.
Adding a fifth dimension and setting 5D-Ricci-Tensor Rab=0, the electromagnetic differential geometry operators div and rot are included and electromagnetism follows.
I am working with a version, where we have 3 spatial dimensions and 2 two of other qualities: time and mass/charge (spacelike). Additional to the ten gravity potentials there are the four potentials Ai of the electromagnetic 4-Vector and the gravity-'constant' as a potential. A small diameter is not neccessary for the 'classical' Kaluza theory.
Here is a classic paper from 1994 that might be of interest with regard to Kaluza-Klein theory. https://arxiv.org/abs/hep-th/9410046
Thanks. I have yet much to read and process. Currently, I am reading Jonathan's paper on relativity and I got a related question to this matter. When computing dimensionality, it is used that if the size of a ball of radius r grows as N=a*r^n then the dimension is n. And also that r^(n+2) terms are associated to curvature. Could it be that any non-zero r^(n+1) term would cause a Kaluza-Klein-like effect? It would not be a fifth dimension, but it seems related. And we can extend to question to fractional terms such as r^(n+0.3).
In the Curvature section of the technical introduction https://www.wolframphysics.org/technical-introduction/limiting-behavior-and-emergent-geometry/curvature/ there is a little discussion on how there is not any apriori restriction on which kind of function is N(r). Yet it only addresses dimension and curvature, without hinting possible meanings of other terms.
"it is used that if the size of a ball of radius r grows as N=a*r^n then the dimension is n."
That is in general only an indication (only true in the absence of cutvature).
It differs a little bit when there is curvature an r is not small compared to a geodesic curve.
"And also that r^(n+2) terms are associated to curvature."
Where did you read that? But nevertheless, only "associated", only a hint where it goes to.
Curvature is a very complicated thing. Usually you need a metric in a space.
For example take a x,y,z space which is curved an embedded in a 4D space: x²+y²+z²+u²=R².
You can get an impression of the curvature by comparing PI r² with r²=x²+y² with the size
of the area included in the circle and you have to do it in the xy, yz and zx plane.
For 5D you get even more coordinate planes ...
The total number of algebraically independent components of the curvature tensor
is in N dimensions 1/12 n²(n²-1) n=3--> 6, n=4-->20, n=5-->50
(bottom of https://www.mathpages.com/rr/appendix/appendix.htm)
I am aware that curvature is a complicated thing and I am simplifying a lot. Also, we need to be very careful about what are we calling dimension. As we are considering a (hyper)graph model instead of a vector space we do not have a natural concept of dimension. But if this model is to represent the real world, then the actual dimensions we see everyday have to emerge from the model somehow. Studying the function of the growing of a ball is a cheap option for addressing it. Other option would be to study how many pairwise orthogonal geodesics can be incident on a single point, but that has the caveat of possible 'short' dimensions.
In those approaches above, and possibly any other, the metric used is fundamental. This is also hard is (hyper)graphs. The option of just counting edges in a path makes difficult to give place to the Euclidean-like metric of our everyday. Jonathan Gorard uses something with parallel transport that I do not quite understand yet (something about random walks pondered by a more basic distance I think).
Then, as you pointed to the Kaluza-Klein theory, I got thinking. We are supposed to add a fifth dimension, but dimensions are sorta meaningless in this context. We can force it by explicitly considering our graph to be some Cartesian product of other graphs. But the dimension also gives form to the growing function of the balls, so perhaps instead of trying to add an extra dimension it is better if we just try to relate its implications on the ball growing function with its implications on electromagnetism. And as Wolfram and Jonathan point, the r^n component is the 'principal' component of its size on a 'standard' n-dimensional space, and the r^(n+2) component is 'hinted' by the curvature. Of course that knowing this scalar component cannot give the full curvature tensor, and it is hard anyway to define such a tensor for a graph. But in a graph the balls do not need to grow as anr^n+a_(n+2)r^(n+2)+a(n+4)*r^(n+4)+O(r^(n+6)); we can also have a r^(n+1) term. As I have not read comment from Wolfram or Jonathan about this term I see possible for it to have some relevance.
When we have a time plus 3D space and add a short dimension for Kaluza's sake we get that balls grow as 4-dimensional at first, until the short dimension is completely 'absorbed' (not sure what word should I use for this); followed by the normal 3-dimensional growth. If we instead have a growing as r^4+r^5, it would be actually he other way around, since for large r, r^5 grows faster. But you also said that a small diameter is not necessary for Kaluza-Klein to apply, so I think this merits attention.
The question was "Where would magnetism fit in this model?".
Kaluza theory says: "It will fit!"
So an advice could be: first get Minkowski-Space and Einstein-Equations done,
before including Electromagnetism.
I think, that a light cone should be only one causal knot (!)
(for a photon time stands still and everything is beyond event horizon,
but it transports an interaction between generation point and absorption point).