Timeline for What is the relationship between the Dirac algebra and the Clifford algebra?
Current License: CC BY-SA 4.0
18 events
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| Mar 17, 2021 at 16:40 | vote | accept | JustWannaKnow | ||
| Mar 14, 2021 at 7:37 | comment | added | user347489 | That sounds correct to me @IamWill . You should check R. Shaw's book I mentioned in an edit. As far as I recall that was a really nice book. | |
| Mar 14, 2021 at 1:07 | comment | added | JustWannaKnow | Does it sound okay? | |
| Mar 14, 2021 at 1:07 | comment | added | JustWannaKnow | user347489 in summary, I can use the definition on my post (with arbitrary $V$ and using the universal property) and prove its existence by using the quotient of the tensor algebra with the ideal generatedby $u\otimes u - B(u,v)$ as Greub does. This proves existence. If $V$ is finite-dimensional and $\mathbb{K}$ has characteristic different from 2, taking a symmetric bilinear form with signature $(p,q)$ leads to $\mathcal{Cl}(V,\Phi)$, which is essentially what Greub constructed but with some additional hypothesis. Finally, take $B$ to be the Minkowski inner product to get $\mathcal{Cl}_{1,3}$ | |
| Mar 13, 2021 at 20:57 | history | edited | user347489 | CC BY-SA 4.0 |
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| Mar 13, 2021 at 20:20 | comment | added | user347489 | BTW, I edited the definition of the Clifford algebra to contain a factor of two. This will help with the translation between blinear forms and quadratic forms. | |
| Mar 13, 2021 at 20:20 | history | edited | user347489 | CC BY-SA 4.0 |
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| Mar 13, 2021 at 20:18 | comment | added | user347489 | @IamWill It's a nondegenerate, symmetric bilinear form. This guarantees you can diagonalize the matrix to one having $p$ ones and $q$ minus ones along the diagonal. | |
| Mar 13, 2021 at 20:11 | comment | added | user347489 | @IgorKhavkine Thanks for that short clarification! This post originally was meant to be just a comment, since the OP contains so many questions. I left that comment in there because I don't know the representation theoretic part of this story well enough to explain it here. Those notes by Trautman's seem to contain the whole story, though. | |
| Mar 13, 2021 at 15:55 | comment | added | JustWannaKnow | user347489, what is your definition of an "orthogonal bilinear form"? Does it mean that $V$ has a basis $\{v_{\alpha}\}_{\alpha \in I}$ where $B(v_{\alpha},v_{\beta}) = 0$ whenever $\alpha \neq \beta$? (I guess you are not thinking of finite-dimensional vector spaces necessarily, right?) | |
| Mar 13, 2021 at 15:53 | comment | added | JustWannaKnow | @IgorKhavkine I think I'm getting the idea. I'm going to work out the details! | |
| Mar 13, 2021 at 15:18 | comment | added | Andreas Blass | @IamWill The two ideals agree as long as the characteristic of the underlying field is not $2$. In characteristic $2$, though, Greub's ideal that you quoted is larger and, as far as I know, is the preferred version. | |
| Mar 13, 2021 at 15:07 | comment | added | Igor Khavkine | Why is this a partial answer? It seems to cover everything. Ah, the connection to Dirac matrices. The $\gamma$-matrix identities are precisely those needed by the universal property, so they define a representation of $Cl_{1,3}$ on $\mathbb{C}^4$. If the representation is faithful, then the $\gamma$-matrices generate an isomorphic copy of $Cl_{1,3}$. That is the case (see Trautman's note), so you could have taken it as the definition, which is what had happened historically. | |
| Mar 13, 2021 at 14:09 | comment | added | JustWannaKnow | An additional comment: I just checked the construction on Greub's book and he uses the ideal $u\otimes u -B(u,u)$ apparently. Is this equivalent to your approach? | |
| Mar 13, 2021 at 12:36 | comment | added | JustWannaKnow | Also, If you have other references to suggest, where this construction using quotient space is done, I'd appreciate. | |
| Mar 13, 2021 at 12:36 | comment | added | JustWannaKnow | Your partial answer is already very useful! So, in short: I can keep my definition using quadratic forms as it is and show that the construction via quotient spaces provide a particular example where the definition holds (i.e. it proves the existence). I guess this is what Greub does. After, I have to choose $B$,$V$ and so on to construct the Dirac algebra. Is that correct? I still need to understand the details about the second part, on how to construct the Dirac algebra from the general quotient construction tho. | |
| Mar 13, 2021 at 11:23 | history | edited | user347489 | CC BY-SA 4.0 |
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| Mar 13, 2021 at 10:55 | history | answered | user347489 | CC BY-SA 4.0 |