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Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how com

Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how come nobody has proof of the Riemann hypothesis, arguably the hardest of the Math problems, and according to him this is because there is no appropriate tools available at the moment to proof it. I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis. Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number. For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2]. [1] Maxwell’s eight equations as one quaternion equation: https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwell-s-eight-equations-as-one-quaternion https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwe... [2] A derivation of the quaternion Maxwell’s equations using geometric algebra: https://peeterjoot.com/2018/03/05/a-derivation-of-the-quaternion-maxwells-equations-using-geometric-algebra/ https://peeterjoot.com/2018/03/05/a-derivation-of-the-quater...

nathan_compton 2y ago
Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible. As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.
- teleforce 2y ago
  Please read my comments properly, I'm saying that currently engineers are not familiar with quaternion because they have been exposed to vector calculus in their formal education not physicists. About thirty years ago in my EM class we have had a combined class of engineers and physicists, for some unknown reasons the lecturer was a microwave engineer. Thanks to physicists like David Hestenes that are more physicists now who are familiar with quaternion but for engineers it's still very much a minority thanks to Oliver Heaviside who really hated quaternions and popularized the inferior vector notations rather than superior quaternion versor notations. As I've also mentioned in my comments the quaternion is necessary in order to fully describe polarization in EM, and there other comments in this post that upholds Heaviside vector can provide the exact representation of EM that is not correct. Heaviside vector representation is the simplication of the more comprehensive quaternion representation but do not mislead to say otherwise i.e the same thing. For Riemann hypothesis, I just providing my intuition that whoever want to proof it need to have quaternion in their toolbox while Terence commented that whoever want to proof it needs a proper set of tools but he did not mention the exact tool just merely saying that current tools are inadequate. For me whatever the set of tools that will be used to proof Riemann hypothesis, one of them will be most probably quaternion.

nathan_compton 2y ago

Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible. As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.

teleforce 2y ago
Please read my comments properly, I'm saying that currently engineers are not familiar with quaternion because they have been exposed to vector calculus in their formal education not physicists. About thirty years ago in my EM class we have had a combined class of engineers and physicists, for some unknown reasons the lecturer was a microwave engineer. Thanks to physicists like David Hestenes that are more physicists now who are familiar with quaternion but for engineers it's still very much a minority thanks to Oliver Heaviside who really hated quaternions and popularized the inferior vector notations rather than superior quaternion versor notations. As I've also mentioned in my comments the quaternion is necessary in order to fully describe polarization in EM, and there other comments in this post that upholds Heaviside vector can provide the exact representation of EM that is not correct. Heaviside vector representation is the simplication of the more comprehensive quaternion representation but do not mislead to say otherwise i.e the same thing. For Riemann hypothesis, I just providing my intuition that whoever want to proof it need to have quaternion in their toolbox while Terence commented that whoever want to proof it needs a proper set of tools but he did not mention the exact tool just merely saying that current tools are inadequate. For me whatever the set of tools that will be used to proof Riemann hypothesis, one of them will be most probably quaternion.