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• CommentRowNumber1.
• CommentAuthorIan_Durham
• CommentTimeMay 6th 2010
• (edited May 6th 2010)
It seems my interests are once again converging towards the nLab, at least in some small way, so I guess I'll venture forth from the "shadows" again.

Specifically, I have been reading some papers involving something that the authors are all calling "quantal" algebra (the algebra of "quantions"). Florin Moldoveanu gives a nice explanation of them in this article. Apparently it is some kind of *-algebra.

When I Google "quantal algebra," however, most of what comes up is directly related to its application to physics. So my question is, as mathematicians, are you aware of quantal algebra? If so, do mathematicians have a different name for it?
• CommentRowNumber2.
• CommentAuthorIan_Durham
• CommentTimeMay 7th 2010
*bump* (or am I being shunned? ;)
• CommentRowNumber3.
• CommentAuthorAndrew Stacey
• CommentTimeMay 7th 2010

It’s new to me.

• CommentRowNumber4.
• CommentAuthorIan_Durham
• CommentTimeMay 7th 2010
• (edited May 7th 2010)
To give it some context, it looks quite promising from the standpoint of unification theories in physics (this is a line of reasoning developed by Emile Grgin). I'd like to add some pages, but considering what happened recently I do not wish to cause further irritation. Thus I wanted to ask around and see if anyone else had knowledge of quantal algebra (the algebra of "quantions") and, if so, what they thought about adding some pages.
• CommentRowNumber5.
• CommentAuthorHarry Gindi
• CommentTimeMay 8th 2010
• (edited May 8th 2010)

Did you just find this paper browsing the arXiv? That's about as reliable as just searching google...

I'm not saying that this guy is necessarily a crank, but, ya know, you can never be too careful.

I'd also consider "quantal algebra" to be a naming conflict with "quantale", which is a certain type of algebraic structure (usually [always?] on a poset).

• CommentRowNumber6.
• CommentAuthorEric
• CommentTimeMay 8th 2010

I had a quick look, but, to be honest, the paper failed to keep my attention long enough to make any meaningful comments. I rarely read a paper from beginning to end even when it is extremely interesting because I end up chasing references halfway through. My spidey sensors were tingling though. The author claims an association to a university but uses a gmail address. This doesn’t mean I judged the paper because of this. I judged the paper based on a brief (but I think sufficient) gloss and THEN noticed this.

My initial thought was that it looked almost like a simple similarity transform of the usual spinor formulation of relativistic quantum mechanics. As you well know, there are few mysteries when it comes to (special) relativistic quantum mechanics and I didn’t see how this paper could get you any closer to general relativistic quantum mechanics.

Nevertheless, if you would like to write up some ideas, I suggest starting out at Ian Durham. Once you’ve got something that looks fairly polished you’re happy with, then start this conversation again. This is the procedure most of us follow when starting a new project or experimenting with new ideas, i.e. start on the personal webs and move stuff as it becomes appropriate. I’ve done this very thing on many occasions, start writing on my personal web and then ask for opinions before moving it over to the nLab. Sometimes I polish and move and THEN ask for opinion, but a good first step is to “polish” before moving to the nLab.

Just a thought…

• CommentRowNumber7.
• CommentAuthorIan_Durham
• CommentTimeMay 8th 2010
• (edited May 8th 2010)
Actually, the author of that particular paper is a friend of mine. He is partly associated with the University of Maryland in addition to his group in Bucharest (from which he is on leave). But he didn't come up with the idea. Emile Grgin came up with it quite awhile ago with Aage Peterson (one of Bohr's last students). Roman Jackiw, among others, has contributed to the field as well.

Given the response, though, I'm guessing this stuff isn't known in the mathematics community. It just seems a little odd that a mathematician hadn't found this type of algebraic structure before which is why I figured I'd ask around since I thought perhaps it had a different name in mathematical circles. The term 'quantal' is sometimes used as an adjective form of 'quantum.' I'm not sure why Grgin coined the term the way he did. They are, in some way, related to quaternions apparently.
• CommentRowNumber8.
• CommentAuthorEric
• CommentTimeMay 8th 2010

If two algebras have the same multiplication table, then I’m pretty sure they are the same algebra. That’s why I said it looks like it is related to the standard treatment by a similarity transformation. If you have an algebra with elements $e_i$ satisfying some multiplication table, then transform each element via

$f_i = S^{-1} e_i S$

the $f_i$ satisfy the same multiplication table and there is nothing new accomplished.

Although it is fun to consider “deformations” this way. Urs did this kind of fun stuff a lot many years ago. For example, we can consider

$f_i = exp(-\delta A) e_i exp(\delta A)$

and expand around small $\delta$.

Anyway, I still haven’t managed to get past the 5th page, but will give it a try time permitting.

• CommentRowNumber9.
• CommentAuthorIan_Durham
• CommentTimeMay 8th 2010
OK, so it seems the key point is on page 2 where he says that the "difference between a quantal algebra and a Lie-Jordan algebra is that a Lie-Jordan algebra has additional properties relating to its spectral properties." He cites this paper for evidence of that (haven't read that paper yet).

Comparing the multiplication tables, they don't appear to be the same, though they both appear to be obtainable via some transformation from the multiplication table (18). So I think the point is that they actually don't have the same multiplication table but are both sub-algebras of some more general algebra, assuming I'm reading it correctly.