My letters to Warren Siegel [Physics] about the 26 simple groups and the possibility that they represent symmetries of the 26 curled up dimensions. [This exchange was in 2016]

Dear Professor Siegel,

I am just a beginner but still I just wanted to ask a question.

 I am wondering if we start with Emmy Noether's theorem and put groups of fractional symmetry in the Lagrangian of QFT (Quantum Field Theory). I mean to say I have been fascinated by the idea of fractional derivatives and higher order symmetries for  awhile. So we have from Noether that for every symmetry you can put into the Lagrangian a conservation law why not just postulate symmetries and thus higher conservation laws up to any order?  What I am thinking of is not the same as translational symmetry of fractional charges like quarks. 

This might sound like a ridiculous suggestion but sometimes this kind of idea gives results. Originally it was Leibniz himself who thought of fractional derivatives but he did not think the results would interesting so he did not pursue the idea. It turned out there are some interesting results. 

I would have liked to have thought of some examples.

I am really sorry if this sounds stupid. I really like learning about QFT but I admit I am just struggling at this point.

Sincerely,

Avraham Rosenblum 

The answer of Dr Siegel quoted the above letter and added :I'm not sure what symmetries you're thinking of, but in general if you impose too much symmetry you find that only a free theory can satisfy it.

Then at the end he added: If by fractional derivative you mean some arbitrary noninteger power of the differential operator, the result is nonlocal (does not depend on just infinitesimally nearby points).  Locality is a basic physical property that field theory requires.  It follows from special relativity & causality.

Then I wrote another letter: Dear Professor Siegel,

What I am thinking is that according to the number of dimensions there are, we would have the same number of conservation laws. So for our little world we have conservation of energy and mass, electric charge, etc. In string theory we get some crumbled up dimensions for the normal 26. So what I would like to find are groups to put into the Lagrangian that will correspond to each conservation law for a different quantity. I still need to think about what kinds of groups I am looking for. But the most obvious would be those 26 simple groups I was reading about when I was studying group theory. 

Does any of this make any sense?

Sincerely, Avraham Rosenblum

Answer of Dr Siegel: I'm not sure what you're saying.

If you compactify some dimensions into a symmetric space, you'll get the symmetry of that space.

E.g., if you compactify some extra N dimensions into a submicroscopic sphere, you'll get the rotational group for those N dimensions, i.e., the orthogonal group O(N+1).

It will appear as an "internal" symmetry with respect to the uncompactified dimensions (i.e., not affecting them directly).

[So I am not thinking of the regular symmetries but some new kinds,[i.e., new kinds of conservation laws].  I do not know how they would be made manifest in our 4 dimensional world.

After all that discussion I thought of another point. That is the Feynman integral what really matters is not all the trajectories but rather just the ones that have no derivative. So even if you have lots of complicated homotopies as you go to higher dimension, still not all the closed circles matter. That is because the smooth ones that have a derivative cancel out in the final result. So what matter is the lines that are continuous but are shaped in the way that every point makes a sharp corner with the next point. Also the last dimension matters because with no other dimension to go into it would lack a derivative.