It is probably just a ridiculous suggestion but here goes anyway. 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 or 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. The same goes for higher orders of acceleration. The third order came up in [I forget where maybe Lorenz Abraham's theorem. I can't recall off hand.]
Further I would like to suggest the order of symmetry will have some proportionality constant with the number of dimensions.
I really would not say anything but it is that sometimes some idea pops into my head that even to me sounds silly and then after a few years I find out that it really was a good idea.
Dr. Warren Siegel answered this:
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. The same goes for higher orders of acceleration. The third order came up in [I forget where maybe Lorenz Abraham's theorem. I can't recall off hand.]
Further I would like to suggest the order of symmetry will have some proportionality constant with the number of dimensions.
I really would not say anything but it is that sometimes some idea pops into my head that even to me sounds silly and then after a few years I find out that it really was a good idea.
Dr. Warren Siegel answered this:
"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."
And as for the fractional derivative he answered this:
"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."
What I am thinking is that according to the number of dimensions you have got, you 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, [i.e., sporadic groups].
Dr Siegel answered to me:
What I am thinking is that according to the number of dimensions you have got, you 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, [i.e., sporadic groups].
Dr Siegel answered to me:
"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 clearly I need to do some more learning and thinking. What Dr Siegel was saying I think was that all I had gotten to was the regular Orthogonal groups.
[What I am trying to do here is to put any (or all) of the sporadic groups into the Lagrangian. That is all. Nothing more. But by doing so I am hoping to get a new conservation law for each group. Then I am hoping that each law will show up in one of the crunched up dimensions of String Theory.]
Dr Siegel is in at SUNY (State University of NY) at Stony Brook, and at the time I wrote this I was trying to learn his book Fields.
Dr Siegel is in at SUNY (State University of NY) at Stony Brook, and at the time I wrote this I was trying to learn his book Fields.
