Talk:Pin group

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There is more to it. From Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155v4. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. {{cite journal}}: Invalid |ref=harv (help)

...so just as Pin(n) is a double cover of O(n), Spin(n) is a double cover of SO(n). Together with a French dirty joke which we shall not explain, this analogy is the origin of the terms ‘Pin’ and ‘pinor’.

We shall need to find out what that dirty French joke is. YohanN7 (talk) 19:04, 16 June 2015 (UTC)

Recent edits based on Roelfs and De Keninck

@Tbuli: I have concerns about the recently added content. It is not explained what is an "oriented reflection" is. The term is not standard. It is also confusing; in ordinary geometric language, all reflections are orientation-reversing and all rotations are orientation-preserving. To say "oriented reflection" is likely to confuse rather than clarify. I don't think the difference between $x$ and $-x$ in a Clifford algebra should be conflated with the concept of orientation.

The content added is based on https://arxiv.org/abs/2107.03771. Wikipedia normally considers preprints to not be reliable sources. Even if/when it is peer-reviewed, Wikipedia cautions against basing large passages on primary research papers (WP:RSPRIMARY), and prefers secondary sources such as review articles or textbooks, especially for a well-established concept like the pin group. I do not see that the wider mathematical literature has accepted the views expressed in the preprint.

For these reasons, I respectfully disagree about the content. You have not done anything dishonorable or unreasonable; thank you for your efforts in contributing to this article, an article which certainly is in need of improvement. Adumbrativus (talk) 10:15, 20 November 2021 (UTC)

@Adumbrativus: Thank you for expressing your concerns about the content I added, that can only help it to improve further. This is my first contribution so thanks for the pointers. My goal with these edits was to provide some geometric interpretation to the Pin groups, which is currently rather formal and I think this is a shame, Pin groups lent themselves to a geometric narrative just as easily as the orthogonal groups.

I see your point about how the term orientation is perhaps not commonly used in this way, and indeed often in the way you just did. The standard way you are revering to, says something about the orientation of the object being reflected/rotated, whereas the way I've used it here says something about the orientation of the mirror itself by distinguishing the front and back sides of the mirrors being reflected in. This use of the term also has a long history in established text books, it can e.g. be found in texts like "Geometric Algebra for Computer Scientists" by L. Dorst et al., "Clifford Algebra to Geometric Calculus" by David Hestenes, or "Geometric Algebra for Physicists" by Doran et al. These texts all use the sign of the normal vector to the hyperplane of the reflection to make the distinction between its front and back. The identification between Pin group elements and combinations of reflections also isn't new, this can be found in e.g. "An introduction to Clifford algebras and spinors" by Rocha Jr., Roldão da Vaz Jr., Jayme. But all these texts still use vectors to represent the normal vectors to the hyperplanes being reflected in, while the cited preprint is to the best of my knowledge the first one that just identifies vectors with hyperplanes directly. This makes the geometric intuition I wanted to add to this article even easier to grasp IMHO, while at the same time it is not such a big change that I thought it would be a problem to base it on a preprint. Afterall, nothing has changed mathematically, it is just a geometry centred narrative, which I think is worth presenting in this article.

I would also like to essentially redo some of the images of that preprint such that they can be added to this article, which should help clarify this use of the word orientation as simply marking the back and front of mirrors, would this be a sufficient explanation in your view?

Thanks again for entering this discussion, I hope we can improve the article somewhat by the end of it. — Preceding unsigned comment added by Tbuli (talk • contribs) 13:57, 22 November 2021 (UTC)

@Tbuli: Certainly a hyperplane has two sides and is orientable. That's not the same as a reflection being oriented. And I guess what I don't see is how it explains the structure of the pin group. A reflection is a linear transformation of Rⁿ. For O(n), we can picture Rⁿ, and reflections acting on it, and compositions of those reflections. That's all standard. If we then start talking about reflections plus one more bit of information ("oriented reflections"), that leaves a lot unexplained. How does the extra bit of information "compose" when multiplying? How does it still form a group? Is it unique? No? How does it end up being not unique, with two different Pin_±(n) groups for a single O(n)? I don't see how these are being explained geometrically. A geometric interpretation has to be more than just using geometric-sounding words while basic non-trivial features still ultimately come from algebra.

On the other hand, Spin(n) generally does not have an n-dimensional faithful representation; the smallest faithful representation has dimension growing exponentially as a function of n, if I recall correctly. The same would be true for Pin_±(n) too. That seems like a good reason why (abstract) spaces of spinors/pinors end up being needed, and why picturing things in Rⁿ is missing something essential.

In general I share your appreciation for geometric / visual-minded intuition. For example, I admire Tristan Needham's books, or some of what Penrose and Rindler were doing in Spinors and Space-time with Spin(1, 3) and SL(2, C) and light cones and the Riemann sphere. I just don't think the attempt is successful in this instance. Adumbrativus (talk) 11:04, 26 November 2021 (UTC)

@Adumbrativus: Association of the sign with the double cover of the orthogonal group is also well understood. For example, let me share this quotation from John C. Baez (https://math.ucr.edu/home//baez/octonions/octonions.pdf) which is right above the quotation shared above by YohanN7:

...To see how this works, first let Pin(n) be the group sitting inside Cliff(n) that consists of all products of unit vectors in Rn. This group is a double cover of the orthogonal group O(n), where given any unit vector v ∈ Rn, we map both ±v ∈ Pin(n) to the element of O(n) that reflects across the hyperplane perpendicular to v. Since every element of O(n) is a product of reflections, this homomorphism is indeed onto. Next, let Spin(n) ⊂ Pin(n) be the subgroup consisting of all elements that are a product of an even number of unit vectors in Rn. An element of O(n) has determinant 1 iff it is the product of an even number of reflections, so just as Pin(n) is a double cover of O(n), Spin(n) is a double cover of SO(n). Together with a French dirty joke which we shall not explain, this analogy is the origin of the terms ‘Pin’ and ‘pinor’.

Apart from the fact that we still need to find out what this dirty French joke is, this is exactly what I have also added. And similar statements can be found e.g. in the reference to Vaz & Da Rocha. So no, the difference between Pin and the orthogonal group can really be visualised within Rn. The essential bit that orthogonal groups are missing which you referred to is precisely this: by ignoring the sign we can not distinguish the two directions of rotation, which Spin groups can. In order to cover all rotations in both directions, the group elements should have 720 degrees of freedom. This is perfectly geometrical, there is nothing algebraic about that. To illustrate this, I have now also included the image from the preprint in the article. (It was released under a CC licence so that shouldn't be a problem. If it is, I'll make a new one expressing the same idea.) Tbuli (talk) 16:04, 12 December 2021 (UTC)

I have edited the article to help clarify the content I agree with, in the continued hope of finding common ground. In particular, I certainly agree it that it's worth mentioning that $-vwv^{-1}$ is a reflection, and worth mentioning Cartan–Dieudonné, and I agree with everything in the passage you quoted from John Baez. I continue to object to the other additions for reasons stated above, and WP:RS and WP:UNDUE. I left out the mention of only k ≤ n reflections being needed (the number is not that important of a detail for the purposes here), as well as the mention of Conf(p,q), but I have no strong opinion on these and don't object to them being re-added. Adumbrativus (talk) 12:31, 14 December 2021 (UTC)