Talk:Geometric algebra

Intro sentence is a barrage of jargon

I appreciate the effort that's gone into the article, and the desire for precision, but this intro paragraph is just not suitable for the first paragraph in a Wikipedia article as per MOS:INTRO. It's completely impenetrable to a non-expert.

In mathematics, the geometric algebra (GA) of a vector space with a quadratic form (usually the Euclidean metric or the Lorentz metric) is an algebra over a field, the Clifford algebra of a vector space with a quadratic form with its multiplication operation called the geometric product. The algebra elements are called multivectors, which contains both the scalars F and the vector space V.

I think this info could even make an appearance in paragraph 2, but paragraph 1 needs to be especially accessible to readers with no advanced mathematical training. It needs to be at a level more like the following, which could perhaps serve as a starting point for us to develop a better intro paragraph here:

In mathematics, a geometric algebra is a framework for describing properties of geometric objects in space, based on the central notions of the multivector and the geometric product. It provides an alternative to more widely-used approaches based on linear algebra or quaternions.

(Please excuse me if this isn't right; take it as a starting point for further refinement. I'm not an expert—only an amateur who has been learning about geometric algebra for the last couple of days.) --Doradus (talk) 13:43, 12 August 2021 (UTC)

Looking at the page's history, I notice that a year ago the intro was much more accessible. We seem to have injected additional jargon over the last few months.

The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which contains both the scalars F and the vector space V.

This is still a bit dense, but if the non-expert glosses over such jargon as "an algebra over a field", you immediately find out that GA is noted for is "geometric product" that operates on multivectors. This was a much better intro for the non-expert. --Doradus (talk) 13:53, 12 August 2021 (UTC)

Go ahead and move stuff around, just try not to lose any info in the process. You are right in principle, the "experts" do have a tendency to "jargonize" the articles. With some justification they will say that it is "math" and it is up to the casual user to learn it. The Clifford page (a GA is a CliffordA viewed more geometrically and less algebraically) is even worse.Selfstudier (talk) 14:13, 12 August 2021 (UTC)

The problem is not jargon, but vagueness and inaccuracy: there is no usable definition in the whole article. @Doradus: your suggestion is better than the current first paragraph, but not correct since an algebraic structure is not a framework. I would suggest the following first paragraph, but I am unable to certify its correctness. If it is correct, be free to use it in the article.

In mathematics, the geometric algebra (GA) of a quadratic space (a vector space with a non-degenerate quadratic form) is an associative algebra, more precisely a Clifford algebra, that is built from the quadratic space, and solves the universal problem of maps from vector spaces to Clifford algebras. Its operation is called the geometric product, and the GA contains the vector space and its exterior algebra as subspaces.
The geometric algebra allows carrying all operations of linear and multilinear algebra in a single algebraic structure (scalar product, exterior product, tensor product, determinant, tensor calculus, etc.).

D.Lazard (talk) 17:08, 12 August 2021 (UTC)

I've decided to be bold and craft an intro sentence much like the one from a year ago. I tried to retain all the info in the intro section, as per Selfstudier, but took a little of it out of that first sentence so it would be less daunting. I'm open to suggestions if anyone would like to make further changes. --Doradus (talk) 19:37, 14 August 2021 (UTC)

Reversion notation

Hi HongGong, in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1016190290&oldid=1016055619 (reverted) and then in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1030284754&oldid=1018780264 you changed the notation for reversion from dagger to tilde. I searched the talk page for an explanation, but could not find any. Your second edit comment merely says "The notation is correct." Notation is a convention. Since the dagger notation was defined in the article itself (just as the tilde notation now is), it was just as correct as the tilde notation (as long as it does not clash with other notation). I noticed that you've also changed a different article in a similar way (https://en.wikipedia.org/w/index.php?title=Rotor_(mathematics)&diff=1016188882&oldid=1001315838), with an edit comment of similar quality ("General cleanup"). Can you please explain what makes tilde better than dagger here? --RainerBlome (talk) 21:49, 3 December 2021 (UTC)

Meet

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)

Universal/general algebra

I undid a change by 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)

Unhelpful example?

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: $\mathbf {B} =\nabla \times \mathbf {A}$. 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)