Timeline for Representing the Dirac equation in spacetime algebra without leftover indices
Current License: CC BY-SA 4.0
10 events
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| Jun 2, 2023 at 21:31 | comment | added | MadMax | A side note: Lorentz transformations act on the left-side of the algebraic spinor, while the left-ideal definition depends on the idempotent acting on the right-side of the algebraic spinor. There is no "Lorentz-violating axis" whatsoever, whether it's "seemingly" or not. | |
| Jun 2, 2023 at 21:24 | comment | added | MadMax | Well, there are lot of questions that Clifford algebra $Cl(1,3)$ alone can not answer. As you rightfully pointed out: "There's no explanation of why the left-chiral symmetry is dynamically broken but the right-chiral symmetry is explicitly broken". The last time I checked, the standard model can not answer this question either. And for that matter, there are a lot other questions the Clifford algebra can not answer: for instance, out of so many able and talented American politicians why the 2024 presidential election is a duel between two 80+ old dudes ;) | |
| Jun 2, 2023 at 21:06 | comment | added | benrg | Electroweak unification certainly doesn't pop out. There's nothing to connect this to the electroweak theory beyond the existence of an SU(2) or U(2) group action on the spinors. There's no explanation of why the left-chiral symmetry is dynamically broken but the right-chiral symmetry is explicitly broken. You still have to pick a seemingly Lorentz-violating axis for the latter, and the former suggests that the Higgs field breaks Lorentz symmetry also (in an undetectable way). Besides all that, the SM gauge group isn't SU(2) or U(2), but S(U(2)×U(3)). | |
| Jun 2, 2023 at 20:29 | comment | added | MadMax | Actually, in the updated version, you can get rid of the "index" all together (without the right-sided idempotent projection as in eq. 23). Then the left-chiral part of the index-free spinor $\Psi$ describes the electron-neutrino weak $SU(2)$ doublet. In other words, the electroweak unification pops out of the Clifford algebra $Cl(1,3)$ automatically. Is that neat! | |
| Jun 2, 2023 at 20:20 | comment | added | MadMax | The updated version is a major improvement, since the "index" in the ideal definition is a bonus and has the concrete meaning of differentiating a electron from a neutrino via right-sided idempotent projections (see eq. 23 in referenced link), while in the original version the "index" does not have this physical meaning. | |
| Jun 2, 2023 at 19:50 | comment | added | benrg | But this version of the Dirac equation still references specifically enumerated indices in the definition of the ideal. You can again argue that it's Lorentz invariant because the choice of indices doesn't matter, but it seems to be no improvement on the other form. | |
| Jun 2, 2023 at 19:19 | history | edited | MadMax | CC BY-SA 4.0 |
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| Jun 2, 2023 at 19:14 | history | edited | MadMax | CC BY-SA 4.0 |
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| Jun 2, 2023 at 17:47 | history | edited | MadMax | CC BY-SA 4.0 |
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| Jun 2, 2023 at 14:20 | history | answered | MadMax | CC BY-SA 4.0 |