Welcome and Thank You!
Hello and welcome to the third incarnation of the "hypercomplex" online discussion group!
My apologies for letting things slip over at Yahoo until it was almost too late. For all the thanks I owe to you here, I couldn't dare letting it fade away. Let me make a little "thank you!" list from the beginnings, on all things algebra. If I forget someone, please do call it out, this is important to me.
In 1993 I started to study physics in Heidelberg, and my "Algebra I" tutor was a lot of fun. Next to being a gifted billiard and pool player, he probed whether I would consider switching over to studying math. It scared me and I stayed with physics, however, I picked up a rather dry "Algebra II" book (Lambrecht) from which learned fascinating things, like the quaternions. I don't remember the name of my tutor, but thank you anyway!
Then in 2004 I was looking for an algebra with multiplicative modulus that is a 4th-degree form over the reals. I discovered the complex octonions in the published works by Kevin Carmody, who helped me in endless e-mails and online discussions, to learn not only the algebra concepts behind octonions, but also to question some of the assumptions behind it and to play with them. Thank you very much for all your help and patience!
Kevin also introduced me to the work of Charles Muses, a rather colorful 20th century person with interests in mathematics, religion, cybernetics, egyptology, and more. While none of his mathematical definitions and proofs seem to work out quite right (a situation he appears to have been fully OK with), it broadened my mind far enough to read more closely. He wrote somewhere, without proof, that the Dirac equation in physics can be written as a split-octonion product. It was enough for me to go ahead and provide that proof, which earned me my first formal algebra-related publication in 2006. A big thank you to him, and to those who found his ideas (and my follow-up) worth publishing.
Then John Shuster joined the hypercomplex group on Yahoo soon after me, and we've had a lot of fun discussions, chats, and work throughout the years, whether online, in e-mails, on the phone, and in person. We developed a good understanding of what Muses' "hypernumber" concept is (some approach to nondistributive algebras over a 2D real vector space, with types of exponentiation that give his "levels" a certain geometric structure). We got to publish two papers that follow-up on Muses work, and provide properly defined algebras which we believe each capture the spirit of what Muses originally intended. We also wrote a fun essay "1 + 1 = 2? A step in the wrong direction" for a 2012 FQXi essay contest. This was towards attempt to develop "lattice numbers", which have digits not just sequentially arranged as e.g. in binary representation of a number, but has digits all over some (crystalline-like) lattice points. We could show that you can make such a space Hausdorff, that you could define invertible nontrivial operations on certain lattices; but didn't succeed beyond the initial attempt to classify such a concept. That is not to say that they can't be made to work. To the contrary, I believe they can, but I also believe that the work that it takes to do so goes beyond my mathematical capabilities. John proposed a lot of constructs to me and to the online group over the years, out of which the "v" operation with "a v b := exp(log a log b)" keeps coming back. More recently, it allowed for some crazy ladder construction of field-like algebras. John, thank you for your friendship and patience, through fun times and also my capricious episodes. I couldn't have made any real work without you.
Also in the "hypercomplex" online group, I would like to thank Armahedi Mahzar, Marek Čtrnáct, and Rick Lockyer for inspiring discussion and sharing thoughts. Also, thanks for Mark Burgin for clarifying some of his work on "hypernumbers", which I reviewed in the group in detail.
Starting around 2008, I got in contact with various researchers in physics who use concepts from algebra one way or the other. I would like to thank Vladimir Dzhunushaliev, Merab Gogberashvili, Süleyman Demir, and Victor Mironov, for varying levels of discussion and collaboration, and exposure to their works.
Not sure exactly who I owe getting on the 2009 invitation distro for the "Mile High Conference on Nonassociative Mathematics" series in Denver, CO, but it was a big deal for me. After years working in isolation, exploring uncertain terrain far off the beaten path, I thank every single mathematician who took the time and prompted me towards areas I needed to study up on, learn better, no matter how basic and simple they were from the point of view of a trained mathematician. It may not seem like it, but I do listen closely and follow-up on feedback received. From little helping hints to long conversation, I would like to thank Jonathan Smith, Jonathan Hall, Petr Vojtechovsky, Shahn Majid, Jörg Feldfoss, Tevian Dray, Tony Sudbery, Geoffrey Dixon, Michael Duff, and ...
... John Huerta for your huge help and patience, for helping me discover that what I write in an e-mail usually arrives unintelligible at the other end, for prompting questions and pointing out areas of math I need to study up on, and in general to help me realize the need to spend some time preparing a particular question before asking. I'm still guilty of writing bad e-mails; but when forced to first prepare my thoughts in some self-contained presentation- or paper-style write-up, I'm now able to communicate with you and others.
Alright - a lot of thanks! And now there's more work to do. I've backed up all the old messages ( https://bitbucket.org/jenskoeplinger/yahoo-hypercomplex-2019-backup/ ) and like to thank Arma and John S again for spending a lot of work compiling message extracts. I know it's quirky to extract information from the Yahoo backup format, but at least it won't go away. Happy browsing :)
Then looking forward, I plan on loosely moderating this continued "hypercomplex" group here, in the old spirit of "feel free to post anything, be respectful, if you have a question or something doesn't quite work then we'll try and work towards answering or fixing it". Please don't expect me (or anyone else) to answer to every of your post; but if the work is important to you, then please post it and try to bring it into as-compact-as-possible form and leave it at some place that stays. Group posts here stay, and are online-searchable, just like the only Yahoo groups. No wonder, check out who the founding father of groups.io is :)
From my end, I had started to develop a concept of quantum mechanics as the "square root of Bayesian inference", using multivalued logarithm in the complexes, quaternions, and aiming at octonions. I've written about this online last year. Two things have happened that are currently holding it up:
For one, a friend of mine brought up a line of reasoning in the physics literature that may be suitable towards bringing a solid foundation to the "complexified probability amplitudes" concept (e.g. from https://arxiv.org/abs/0907.0909). As intriguing as this and the authors' follow-on work is, we believe there are issues in these works that we would want to understand and disentangle first. This is taking most of my time the last 6 months. Whatever outcome has to be solid academic work, because we're commenting on existing published work by established academics. We like their work, no doubt, and believe it may have consequence - in some shape of fashion. But we cannot agree with that paper in its published form (after all, Phys. Rev. A in 2010). You bet that's laborious ... not only do we need full clarity on our view on that paper, but also on all follon-on work - at least by these authors - that cite that paper, each time repeating the original claim, without further comment or notice. It'll be good at the end, we believe.
And the other thing that's holding "sqrt(Bayes)" up: The accidental discovery of "quatquats" has me stumped, some four-dimensional quaternion-like algebra over the reals, with quaternary product that has composition and quaternary-group property. In the sense of https://en.wikipedia.org/wiki/N-ary_group , there's a specific type of quatquat that lacks a multiplicative identity, and the algebra cannot be reduced to conventional quaternion algebra with binary products anymore. I wrote about this here online, too, but over past year more properties have surfaced. This might actually be something for the group here, since it's incomplete, not really well understood yet, but imaginative. There is so much that just doesn't work anymore when you cannot have a binary product anymore: Groups, bilinearity, ... everything is defined using binary operations. It feels a little bit like mathematical freefall, having put it all into question. The carrot-on-a-stick, however, is that general forms of exponentiation in one variable x, given constants a, b, c, d, could be written as
f(x | a, b, c, d ) = (ax)^b c^(dx)
g(x | a, b, c, d ) = (ax)^bx c^(dx)
That's a little bit like multiplication, where get a handle on expressions (ax)(bx) through rules like commutativity associativity, alternativity, and flexibility. In exponentiation, this gets a bit more complicated since it does matter a lot whether the x is in the base or the exponent. In a truly quaternary product, where all four factors are equal in respect to one another, it seems that beauty can be restored for exponentiation as well ... but what does "commutativity" or "associativity" mean in a quaternary product? ... Well, I told you it's more a dream that fact at this point :)