Talk:Geometric algebra

In the article the Grassmann exterior product is - be it between quotes - referred to as the "meet". Allthough both concepts use the same symbol, in my opinion there is no relation between the two. Madyno (talk) 16:46, 2 December 2022 (UTC)

:It says that the meet is the dual of the Grassmann exterior product. -Bryan Rutherford (talk) 17:44, 2 December 2022 (UTC)

I undid a change by {{u|Jasper Deng}}, relating to defining the algebra as what Vaz&daRocha and Lounesto appear to term "universal". I'm not objecting to the idea that the properties as stated do not necessarily define a universal Clifford algebra in this sense (and we do want the universal case), but we should provide a clearer definition, rather than some wording that someone may only understand if they are familiar with universal properties. Lounesto does this by requiring that Cl(V, g) is not generated by any proper subspace of V, while it is by the whole of V (Lounesto 2001 Clifford Algebras and Spinors p. 190). This seems to be more understandable to a reader. One possible shortfall in this is that he does not seem to cover the case of a degenerate quadratic form in this claim (though I suspect that it still applies). Vaz&al prove that requiring that dim(Cl(V, g)) = 2^{dim V} is sufficient for it to be universal (Vaz & da Rocha 2016 p. 58). Maybe we can use these ideas? —Quondum 14:46, 2 March 2024 (UTC)

:My point with those edits is that just giving those properties is not a definition in itself. I'm perfectly fine if there's a better yet still elementary way to word it because most casual readers won't understand abstract algebra here. I suggest adding the "not generated by a proper subspace but by the whole space condition" and the sentence "It can be shown that these conditions uniquely characterize the geometric product".--Jasper Deng (talk) 19:38, 2 March 2024 (UTC)

:: I'll spend a bit of time on it. I'll model it on your suggestion initially, which I like. I might include footnotes that may need trimming/rework. —Quondum 21:33, 2 March 2024 (UTC)

::: I've tried to do this. It turned out that the condition that I gave above was sufficient only in the case that the quadratic form is nondegenerate, which is rather unsatisfying: some useful GAs have degenerate quadratic forms. To keep the less abstract part uncomplicated, I've chosen to not call it "axioms", only "properties", and hence not needing them to definitively define the geometric product. The result is a bit clumsy; feel free to refine. —Quondum 01:05, 3 March 2024 (UTC)

The lead currently contains a statement "... pseudovector quantities of vector calculus normally defined using a cross product, such as ... the magnetic field". I am not familiar with the magnetic field "normally" being defined in this way, although I'm aware if the Biot–Savart law, which is not only a cross product. The point is that few readers will find this helpful; examples such as this are meant to be familiar. Maybe this particular example (magnetic field) might be better removed? —Quondum 02:04, 13 March 2024 (UTC)

:The magnetic field B is the curl of the magnetic potential A, which is the cross product with the del operator: $\backslash mathbf\; B\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\; A$. BrtSaw (talk) 04:53, 31 August 2024 (UTC)

:: The cross product is not defined on operators in general, and the "del operator" is ill-defined in general. The 'del-cross' notation for the curl operator is merely a handy mnemonic. (The nabla/del symbol is used with a well-defined meaning in geometric calculus, but its similar combinations with binary operators are again simply mnemonic notations.) —Quondum 14:45, 31 August 2024 (UTC)

My understanding is that there are two different communities who study the same object but with different notation. Is this sufficient for the same object to have two different pages?

I'm willing to believe that there is some principled reason for this, but I would like to see it spelled out explicitly, even if just in the talk page. Cooljeff3000 (talk) 16:34, 5 January 2025 (UTC)

:I suppose because it is and isn't {{tq|the same subject}} at the same time. Yes GAs are specific examples of CAs but the focus is on low dimensional applications over the reals, how much of what you see in this article is in the CA page? So no fork, right? Selfstudier (talk) 17:20, 5 January 2025 (UTC)

The Pfaffian expansion looks interesting, but as far as I can tell, it's based on two papers (by a fellow named Wilmot) that received almost no citations in the 37 years since publication and that essentially appear to have been a thesis of some sort. Of course, this doesn't mean it's wrong --- and it's precisely the sort of expansion I had been looking for, which makes me grateful for Wilmot's effort. However, it probably shouldn't be presented as if it is some standard textbook result. Also, the notation is homegrown. In the form it is written in this wikipedia article, the formula makes no sense. A Pfaffian takes a skew-symmetric matrix as an argument (it's just sqrt(det(M) when the matrix is real), but the argument in the formula is not a matrix. It's a list of scalars. There's some sort of missing step or notation from the papers that didn't make its way into the wikipedia article. Also, the formula and surrounding sentences seem to be lifted directly from a recent paper "Construction of exceptional Lie algebra G2 and non-associative algebras using Clifford algebra" by that same Wilmot. At the very least, the meaning of the expression should be clarified --- in particular, what matrix the Pfaffian has as an argument. It also probably would be a good idea to caveat this in some way since it seems to be one person's project and, as far as I can tell, nobody else is using it. Again, this doesn't mean it's wrong --- it could be a brilliant result that unjustifiably has languished in obscurity --- but it does raise concerns about whether it has been thoroughly vetted. Unfortunately, I lack access to the original papers and cannot work through the details myself --- and the later paper simply refers to those other papers as canon rather than explaining the notation in a self-contained fashion. 73.186.196.76 (talk) 19:29, 9 June 2025 (UTC)