Post Awp8INaUdmRlov4sFM by johncarlosbaez@mathstodon.xyz
 (DIR) More posts by johncarlosbaez@mathstodon.xyz
 (DIR) Post #Awp8IFkTmlgJVZgjhY by johncarlosbaez@mathstodon.xyz
       2025-08-04T10:28:21Z
       
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       In 1843 Hamilton invented the quaternions: a 4-dimensional number system ℍ where you can divide.  Quaternions have a 3d 'vector' part and a 1d 'scalar' part, but they were invented before vectors.   So, throughout the 1800s, scientists used them a lot.   Maxwell wrote a version of his equations using quaternions!   And in the USA, for a while quaternions were the only graduate-level mathematics course.They're less popular now, but still fun.  Recently I noticed that the hydrogen atom has a neat quaternionic description.In complex analysis we single out certain specially nice functions from the complex numbers to the complex numbers called 'analytic' functions.   There are whole courses on these.   But you can also define analytic functions from ℍ to ℍ.  Not many people think about these - but some good mathematical physicists have, like Frenkel and Sudbery.Here's what I noticed:If we take into account the fact that electrons have spin ½, bound states of the hydrogen atom - states where the electron doesn't fly off - correspond to analytic functions from ℍ - {0} to ℍ.    We're combining quantum mechanics and quaternions in a fun way here.   Quanternion mechanics!   🤪 And since the quaternions are 4-dimensional, this formulation meshes very nicely with the mysterious 4d rotational symmetry of the hydrogen atom. For more, go here:https://johncarlosbaez.wordpress.com/2025/08/04/the-kepler-problem-part-9/(1/2)
       
 (DIR) Post #Awp8IHBSRwr7xXJnOK by bartholin@fops.cloud
       2025-08-04T10:47:48.487712Z
       
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       @johncarlosbaez what about octonions
       
 (DIR) Post #Awp8INaUdmRlov4sFM by johncarlosbaez@mathstodon.xyz
       2025-08-04T10:44:14Z
       
       0 likes, 0 repeats
       
       You'll notice I said bound states of the hydrogen atom correspond to analytic functions from ℍ - {0} to ℍ.    That's because these functions may 'blow up' - approach infinity - as we approach the quaternion 0.These functions may also approach infinity for very large quaternions.   In fact there's a nice symmetry here.  To make this evident, we can take the quaternions and add on an extra point called ∞.   This gives a space called the 'quaternionic projective line' or ℍP¹ for short.  It's a 4-sphere, with 0 as the south pole and ∞ as the north pole.   The quaternions q with |q| = 1 form the equator of this 4-sphere.  This equator is a 3-sphere.   All this is just like what people often do with the complex numbers.  They take the complex plane and add on an extra point called ∞.  This gives a space called the 'complex projective line' or ℂP¹.  It's a 2-sphere, an ordinary sphere, with 0 as the south pole and ∞ as the north pole.   It's also called the 'Riemann sphere'.  See the picture!Anyway, 'm saying that bound states of the hydrogen atom are analytic functions from ℍP¹ - {0,∞} to ℍ.   You can take any one of these states and write it as the sum of two: one that blows up only at 0, and one that only blows up at ∞.   This has an interesting physical meaning, which I'll talk about on my blog.  (I haven't yet, because I just noticed this.)(2/2)
       
 (DIR) Post #AwqwYLSvUIq2jqIxAO by weekend_editor@mathstodon.xyz
       2025-08-04T13:23:59Z
       
       1 likes, 0 repeats
       
       @bartholin @johncarlosbaez For that, you want Cohl Furey and her octonionic versions of QM and the standard model:https://en.wikipedia.org/wiki/Cohl_Furey
       
 (DIR) Post #AwqwYzbUFtSBDycM40 by johncarlosbaez@mathstodon.xyz
       2025-08-04T14:17:49Z
       
       1 likes, 0 repeats
       
       @weekend_editor @bartholin - or read my massive website on octonions:https://math.ucr.edu/home/baez/octonions/and my blog posts and talks on the octonions and the Standard Model:https://math.ucr.edu/home/baez/standard/