Talk:Weyl–Brauer matrices

Where does the construction come from?

Something that would be useful would be some motivation as to how this construction comes about, ie why one might be led to it.

Also some signposting of what it is that leaves some latitude for other constructions, that lead to other types of spinors for the same Euclidean space, which have other properties and are not equivalent. Jheald (talk) 12:14, 28 April 2014 (UTC)

Dimensionality

I noticed Jheald changed the dimensionality for the statement of generalizing Pauli matrices: the "2" provided was the n=2k , k=1 and also n=2k+1, as the odd dimensionality constructions are always a trivial carryover of the lower even dimension, as described here. So, n=2 is actually less confusing to the reader than n=3. This is clearly detailed later in the article. I 'd rather that you change it back yourself, as I have no strong feelings on increasing confusion or decreasing it, since I cannot guess why a reader would opt for Euclidean, so, then, this stub, instead of Minkowski, for the main article of higher-dimensional gamma matrices. Cuzkatzimhut (talk) 13:55, 28 April 2014 (UTC)

The Pauli matrices are the matrices for describing 3D Euclidean space, even though they are 2x2 matrices.

Similarly, the "Construction" paragraph starts

"Suppose that V = Rⁿ is a Euclidean space of dimension n."

So it is the dimension of the Euclidean base space that n is being used to refer to (i.e. n=3 for the Pauli matrices), rather that the size (dimension) of the matrices.

Finally, note that Computer Scientists and Engineers tend to be more interested in Euclidean applications rather than Minkownskian applications (at least they are up until the point when they want to start embedding those Euclidean spaces in larger mixed-signature projective spaces) -- see our article Geometric algebra and its references and spin-out articles for material on this increasingly popular topic. Jheald (talk) 23:10, 28 April 2014 (UTC)

OK, If you feel so strongly about it, leave it like that, but I just know it will lead to confusion, if not grief-- I've seen this type of bad question asked again and again. My point was simple: the 2 Pauli matrices are just the gamma matrices for base space n=2 (and hence n=3, trivially, as per all spinor constructions the 3rd one is derived, as it is the product of two of them). 2 was never intended to represent the dimensionality of the spinor space the matrices act on, 2^⌊n/2⌋, of course. For n=2, the two coincide, but the student of Clifford algebras does not get duped. You are probably thinking of the Sylvester matrices of Generalizations of Pauli matrices, which may comfortably act on odd dimensions as well, but there are no Cliffords involved there. The dissonance is probably cultural. I will do my best to foil it, with the dimensionality statement I'll stick in. Cuzkatzimhut (talk) 00:14, 29 April 2014 (UTC)

In physics the Pauli matrices are identified with 3D - they can be identified with representations of vectors in the x, y, and z directions respectively, from which can be constructed matrix representations of the corresponding exterior algebra and the Clifford algebra. They were originally invented to represent spins aligned with each of these directions. Spinors are secondary quantities of questionable value, and "spinor space" even more so -- what's of primary interest is not the dimensionality of the spinor space, but the dimensionality of the underlying Euclidean base space, which is the real-world space that the objects are relevant to -- eg for constructing 2-sided representations of rotations, etc.

I suppose you could use the Pauli matrices to consider Clifford algebras over 2D base spaces, by forgetting about one of them. But overwhelmingly it is 3D space that they are associated with, and the representation of the algebra Cℓ_3,0 which is built over them, because that is the one that is so relevant to the everyday world. Jheald (talk) 01:23, 29 April 2014 (UTC)

Well, at least in the physics of the 20th century, the central object was the Dirac equation; and, in this construction, in this article, not in atomic physics, the reader is steered into understanding its generalized gamma matrices in arbitrary dimensions by first appreciating that 2, not 3, of the Pauli matrices provide the origin of the inductive construction of gammas in n=2, and thence getting them all. I cannot presume or could desire to focus readers and users elsewhere.

I fear the discussion has emptied itself of meaning and has devolved into subjectivity of focus. I'd let the user reveal their confusions first. Cuzkatzimhut (talk) 14:33, 29 April 2014 (UTC)

In this case, Cuzkatzimhut would be correct, and Jheald seems to be getting confused about the double covering of SO(3) by SU(2) which is not really relevant for the construction given here. At any rate, after skim-reading the article in it's current state, I did not spot any particular problems; it looks OK to me. Oh, well, the lead says "3d euclidean" and that is indeed confusing and poorly-stated, but maybe excusable (?) 67.198.37.17 (talk) 03:56, 9 May 2019 (UTC)

Oh, I see what the problem is. In several places, it says "just as in the three-dimensional case" which can be read as a gloss, if one already knows this material, but is hopelessly confusing if one is a newcomer. Which 3D case? n=3? Clearly not. So k=3? Clearly not that, either. So which 3D case is this talking about? I'm guessing that "just as in the three-dimensional case" means the k=1 case, but at this point, even if one already knows the material, confusion sets in. Edits are needed. I'm not going to be making them, but someone should fix this. 67.198.37.17 (talk) 04:12, 9 May 2019 (UTC)

Non-Euclidean construction?

This article presumes a Euclidean construction. Is there an analogous non-euclidean construction? Also, the original author of this article used the notation P and Q knowing full-well that these imply a symplectic structure (as he is the author of a canonical textbook on riemannian geometry) However, I'm too lazy to figure a more precise correspondence. What is it? I am left to guess ... the exterior algebra is a Hopf algebra. The Poisson algebra generalizes that. Am I supposed to be able to glue/solder the Weyl-Brauer matrices onto a Poisson manifold in some kind of canonical/tautological kind of way? Thus giving it some kind of metaplectic structure? Gamma matrices can be used as vielbeins on manifolds; is the goal of P,Q to not only vielbein them, but do so in a tautological/solder form kind of way? If I crack open my copy of Chari&Pressley Quantum groups, this doesn't just come flying out (the words "gamma", "clifford", "spin" do not appear in the index.) Not in Jurgen Fuchs affine lie algebras either. 67.198.37.16 (talk) 20:14, 17 November 2020 (UTC)