Talk:Higher-dimensional gamma matrices

Naming

Since this article includes discussion of gamma matrices in 2 and 3 dimensions, would it not be better named "gamma matrices in dimensions other than 4" or similar? Stevvers (talk) 13:34, 15 April 2011 (UTC)

Separate article?

Well, these matrices are just a realization of gamma matrices. Why do they need a separate article? (Anonymous January 15th 2012)

Because the 4D case is crucial to QFT and deserves an article all to itself. The generic case is of more limited interest. 67.198.37.16 (talk) 21:41, 17 November 2020 (UTC)

Concerns

This article has no references, and the phrase "Higher-dimensional gamma matrices" on GBooks only returns Wikipedia scrapes and Cram101, which has been discussed at Wikipedia:Reliable_sources/Noticeboard/Archive_162#Cram101 and appears to also scrape Wikipedia. It seems likely that it is not notable. The introductory sentence "In mathematical physics, higher-dimensional gamma matrices are the matrices which satisfy the Clifford algebra $\{\Gamma _{a}~,~\Gamma _{b}\}=2\eta _{ab}I_{N}$" is meaningless. Deltahedron (talk) 20:20, 13 April 2014 (UTC)

Iti is not quite clear or meaningful which part of this is meaningless to you. On the face of it, it looks meaningful to anyone familiar with matrices, spinors, Clifford algebras and arbitrary dimensions. No reader unfamiliar with those could/should end up here, no? Cuzkatzimhut (talk) 18:49, 14 April 2014 (UTC)

A Clifford algebra is a type of set with a certain algebraic structure defined on it. No such structure was mentioned. It does not make sense to say that matrices "satisfy" an algebraic structure. The curly bracket notation is not defined, and could mean several different things. If it is intended to be a Poisson bracket, then there needs to be an explanation of how the Poisson algebra structure relates to the Clifford structure. If it is intended to be the anticommutator, then say so, and explain what relation it has to the Clifford algebra. All of this needs to be spelled out in an introductory paragraph. I think it is fair to use the term "meaningless" here. I am glad to see that some of these have now been fixed. Deltahedron (talk) 21:27, 14 April 2014 (UTC)

Thanks, I at least appreciate the cognitive dissonance involved: The link to Clifford algebra was off, now improved. In physics, Clifford algebra is nothing but the simple anticommutator relation, now suitably linked. This relation is obeyed, "satisfied" in standard physics usage, by the set of matrices discussed---I spatch-cocked something on representations there. Since "Gamma matrix" has an immediate, unambiguous, evocative meaning common to any first year physics graduate student, it never occurred to me there could be snags. I moved up the generalization bit of Gamma matrices to make it obvious that nobody should be expected to get anything out of this, unless they've already mastered the standard Gamma matrix culture. It is all about relativistically invariant fermion equations in arbitrary dimensions. Cuzkatzimhut (talk) 22:21, 14 April 2014 (UTC)

I have recast the introduction to make it clearer to the mathematician who is not a physicist what the context is and also to make the notation clearer. Deltahedron (talk) 06:47, 17 April 2014 (UTC)

Charge conjugation

The first paragraph is unclear: "Since the groups generated by Γ_a, −Γ_a^T, Γ_a^T are the same, it follows from Schur's lemma that there must exist a similarity transformation which connects them all. This transformation is generated by the charge conjugation matrix." There are four sequences of matrices referred to here, so presumably there are four groups being talked about. Is "same" supposed to mean "equal"? If so, use that term, otherwise the reader might think they were merely isomorphic. It's not clear that any form of Schur's lemma (disambiguation) applies here. What does it mean to say that there is a transformation that "connects" them? In what sense does one similarity transform connect four groups? Charge conjugation redirects to C-symmetry which does not help: no matrix is mentioned there. Finally, the text now proceeds on the basis that if groups are similar in this way then the similarity can be applied to the generators, and this simply isn't true. For example, the 2-dimensional orthogonal representation of the dihedral group of order 2n can be generated by a pair of matrices of order n, or one of order n and one of order 2. The groups are similar, indeed, identical, but the matrices are not, since similar matrices of finite order have the same order. Deltahedron (talk) 07:02, 17 April 2014 (UTC)

Apologies I lack the time right now to do it right. The Conjugation similarity permutes all Gammas to themselves, so "equal" groups. C-symmetry is the same symmetry group, although it only shows you how it acts on the spinor bases, and so the bispinor gamma matrices have to undergo this similarity transformation instead. Any quantum field theory text starts with these basic operations, as well as the references provided. Cuzkatzimhut (talk) 11:15, 17 April 2014 (UTC)

A possible source would be Frè, Pietro Giuseppe Frè (2012). Gravity, a Geometrical Course: Volume 1: Development of the Theory and Basic Physical Applications. Springer-Verlag. ISBN 9400753608. I note this section follows page 315 quite closely. However that source makes the same mistake: or perhaps it would be fairer to say, that the explanation is not a proof although it resembles one. The group G generated by the gamma matrices is finite, and the gamma matrices constitute a representation of G. Schur's lemma applies if the representation is irreducible, that is, if there is no non-trivial subspace invariant under the whole group. This may well be true but has not been stated, let alone proved. However, let's assume it for the moment. Frè goes on to say "two irreducible representations of the same group with the same dimension and defined over the same vector space must be equivalent". This is not correct. It is perfectly possible to have inequivalent representations of a group of the same dimension: for example, all the irreducible representations of an abelian group are of dimension 1 on C¹ but they are far from being equivalent. So again, let's assume that the representations are equivalent. It is still not true that the generators must be similar term by term as I explained above for the dihedral group.

It very well might be true that there is a single matrix that conjugates each Γ to the corresponding −Γ, and one that conjugates each Γ to the corresponding −Γ^⊤, but the discussion in the article, and in the book I cited, does not prove it, and it would be better for now to leave it in the article as a bare assertion. Finally I note that, although not well explained in the article, the charge conjugation matrices C_± in fact do not always exist (which already should have suggested a problem with the "proof"). Both exist in even dimension, and only one in odd dimension. Deltahedron (talk) 12:03, 17 April 2014 (UTC)

Symmetry properties

This section suddenly refers to $\Gamma _{a_{1}\dots a_{n}}$ without explanation: we have only seen a single subscript up to now. I assume that

$\Gamma _{a_{1}\dots a_{n}}={\frac {1}{n!}}\sum _{\pi \in S_{n}}\epsilon (\pi )\Gamma _{a_{\pi }(1)}\cdots \Gamma _{a_{\pi }(n)}$

as one would expect from the Clifford algebra? (The corresponding formula in Fre appears to be garbled.)

Yes, you need 2^d distinct matrices (de Wit & Smith, Appendix E). Cuzkatzimhut (talk) 13:53, 17 April 2014 (UTC)

I wrote 2^d products, which is correct as it is the number of unordered n-tuples of d things summed over all n from 0 to d, and what is written above, and what is written inbn the linked notes. Why replace that with N², which is not correct? $N=2^{[d/2]}$ so N² is equal to 2^d only when d is even. Deltahedron (talk) 19:13, 17 April 2014 (UTC)

OK, I'll leave it alone... The deWit et al appendix above explains it... I just put down the number of independent matrices, as exemplified in the table that follows, so, then, 4 for d=3, 16 for d=4 and 5, etc... you might want the reader going on to the table to be able to check the numbers... You might try your tack---too many cooks spoil the broth. This is a graceless article to refurbish.... Cuzkatzimhut (talk) 19:24, 17 April 2014 (UTC)

If 2^d is the number of distinct terms and N² is the number of independent terms then by all means say so. Deltahedron (talk) 19:36, 17 April 2014 (UTC)

OK, corrected "independent" above to "distinct". The point is that, for even d, the one that matters, the two terms amount to the same. The γ_chir of an even d becomes the γ_d+1 for d+1, whereas its γ_chir, the product of all d+1 γ_as, is proportional to the identity, and thus omitted: So, for d=4, the γ_chir is the same as the γ₅ of d=5, and there is no in depended chir for d=5, as the identity is already there. So odd dimensions are parasitic cousins of the even ones below them. I assume all this preamble is a clarification of the table that flows, and it should not generate more questions than it answers. While I would not have organized this jubilantly misbegotten article this way, I can still get the useful parts out of it, and would rather not dwell on small fussy points... But, then again, I would not be in the focus group of its readers. The de Wit et al appendix should be more than adequate to address questions. Cuzkatzimhut (talk) 20:13, 17 April 2014 (UTC)

Chiral gamma

The term Γ_chir appears in the table under Symmetry properties without previous definition. Deltahedron (talk) 19:34, 17 April 2014 (UTC)

As per above, it is the product of all d Γ_as with a suitable normalization... hopefully correct, in the ref the tables came from... de Wit et al call it ${\tilde {\gamma }}$. In even dimension, it anticommutes with all Γ_as ; while in odd it commutes with them so it is (taken to be) the identity, and so not independent.Cuzkatzimhut (talk) 10:37, 18 April 2014 (UTC)

My time to apologize for not appreciating your bewilderment: The chiral matrix further down, for the d+2 case, was simply wrong, now corrected---my eye was gliding over it, assuming it was the customary right expression. It was missing the final 2 Γs... I now defined it for d, before it is tabulated, as per your question, and repeated it correctly in the recursive construction further down. (I hope my conventions for it are consistent.) Cuzkatzimhut (talk) 23:12, 21 April 2014 (UTC)

Example of an explicit construction in chiral base

Chiral base is not defined in the article, and indeed not mentioned elsewhere. Its only other appearances in Wikipedia appear to be in organic chemistry. In the first formula, σ matrices make an undefined first appearance. Presumably these are the Pauli matrices? In the second formula, a quantity s appears but is not explained. Deltahedron (talk) 08:24, 18 April 2014 (UTC)

It is the Weyl basis in the Gamma matrix article, the introduction to this one, statutorily, where its better name, this, is given: well-worth wikilinking here. Yes, the σs are the Pauli matrices, as per the gamma matrix article, again. I assume s is the sign factor made explicit below. Cuzkatzimhut (talk) 10:37, 18 April 2014 (UTC)

Period eight

Just a passing thought. Is the periodicity in the construction in any way related to Bott periodicity or the Radon–Hurwitz numbers? Deltahedron (talk) 08:28, 18 April 2014 (UTC)

Have been uninterested in this level of mathematical sophistication to confirm or deny, but I have seen rants on the Bott periodicity theorem in this connection before... The answer should be out there and be known to mathematical stringers. Cuzkatzimhut (talk) 10:37, 18 April 2014 (UTC)

See Classification of Clifford algebras, in particular the section on Bott periodicity. The algebra of Gamma matrices is a representation of a corresponding Clifford algebra, so it's reasonable to expect the Bott periodicity of the Clifford algebras to manifest itself. That's not precisely an answer to your question, because your question was specifically about the construction; so a more explicit answer could probably be constructed. But yes, it would seem likely that there is a close connection. Jheald (talk) 21:10, 22 April 2014 (UTC)

Signature of the spaces?

I see that the article limits itself to Clifford algebras of signature (1, n-1). But Clifford algebras of different signatures are also of interest -- for example with all real dimensions, Cl_(n,0); or Clifford algebras with more than one 'countersphere' dimension, Cl_(n,m). How much of the article carries over to such cases? How similar a construction is there in such cases for matrices to represent the underlying set of 'vector-like' basis elements? Does the article need to be specialised to spaces with one and only one time-like dimension? Jheald (talk) 21:10, 22 April 2014 (UTC)

Of course, the article can be extended, mutatis mutandis, to all these general cases, with great care... The prefactors and phases and all will have to change to keep suit. If you were up to it, it might be helpful to add a new section, but I suspect I'm done tweaking this for now, myself. In practical terms, I think it should be up to the user to do it themselves, once the basic pattern is illustrated by example here. After all, this appears like a quick reference manual for conventional extended space-time model-building, and the majority of mainstream applications to higher-dimensional supergravities, for instance, are covered here. Anyone into recondite speculative model-building should be able to adapt all this on their own. But if there were a sufficiently compact and pithy formulation to save readers time, why.... Cuzkatzimhut (talk) 22:35, 22 April 2014 (UTC)

Gamma groups

Many properties of higher-dimensional gamma matrices, including their role in chiral symmetry, can be retrieved when the gamma elements are presented as members of an abstract group, neither with referring to matrices nor to Clifford algebras. The anticommutation rules are expressed only with the group operation xref>Petitjean, Michel (2020). "Chirality of Dirac spinors revisited". Symmetry. 12 (4): 616. doi:10.3390/sym12040616. xx/ref>. However I do not know in which paragraph that could be relevant in this article, if it is: a new one? Just a reference to add? 2A01:CB04:A01:3F00:F8BF:62E3:B385:825F (talk) 11:12, 8 October 2020 (UTC)

Wow! Reading now. I'll try to add it by starting a new section; please come back and correct/expand. This is nice, because effing chirality is an effing pain in the buttinski for everything. 67.198.37.16 (talk) 22:32, 17 November 2020 (UTC)

I just added this and featured it prominently. Minus signs are the bane of practical calculations, and physicists are infuriating when it comes to that, throwing in random factors of i and random hermitian conjugations wherever they feel like, making it impossible to figure out what forms are split-real, what's complex, what's compact, what's non-compact without making 23 and one-half sign errors along the way. The treatment of CPT symmetries is particularly appalling. So the kind of approach taken in the above reference is appreciated. 67.198.37.16 (talk) 05:13, 18 November 2020 (UTC)