The session featured, among other things, quaternions. Nevertheless, octonions also seem to be very interesting in the context of their non-associativity. The algebra of quaternions is non-commutative and associative, the algebra of octonions is non-commutative and non-associative.

However, let me give you a brief historical background.

Quaternions were introduced by Hamilton quite some time ago, in 1843. William Rowan Hamilton was very particular from early childhood. He learned a variety of unusual languages, including but not limited to Hebrew, and was interested in some aspects of theology and esotericism, as well as the natural sciences. He was perceived as eccentric and unusual. He was not understood by most people around him. Despite his broad interests, at one point quaternions became the main focus of Hamilton's work, his longtime obsession.

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space.

Complex numbers can be thought of as a generalization of the real numbers so that polynomial equations always have roots. We know that in the real domain, even an equation as simple as

*x*^{2} + a = 0 has no solution when

*a>0*.

This can be remedied by introducing imaginary numbers, which are the square roots of negative numbers. For this purpose, the imaginary unit

*i* must satisfy the condition

*i*^{2}=-1. The imaginary numbers can be added to the real numbers, resulting in complex numbers of the form

*a+bi*, where

*a,b* are real.

As is often the case with eccentrics, however, Hamilton was not satisfied with current methods of describing the world and wanted to rise above the physics of his day.

Hamilton sought a generalization of complex numbers to triples of numbers. He wanted such triples to be able to add and multiply by themselves. Multiplication was to be separable from addition, so that the rules of ordinary algebra could be applied. He also demanded that the absolute values of the numbers be multiplied when multiplying:

*|xy|=|x||y|*. In the case of the complex number

*z=a+bi* the absolute value equals

*|z|=(a*^{2}+b^{2})^{1/2}, in the case of triplets we would have under the root the sum of three squares. He was willing, however, to sacrifice the commutativity of the product, a step that was original and rather unpracticed before.

No wonder, then, that his attempts were for a long time met with a lack of understanding. When we do something for the first time and differently than others have done it before, we naturally meet with suspicion from the world.

For a long time, every morning when Hamilton came down to breakfast, his son would ask if he could already multiply triplets, to which the scientist would sadly reply that he couldn't and could only add them and subtract them.

It's not hard to imagine Hamilton's despair. He wanted so much to understand and eventually explain it all to the world, but he still faced numerous obstacles.

The solution, which was born in Hamilton's mind one October morning, consisted in a generalization that went one step further: instead of triples, consider quaternions of real numbers. Hamilton had just walked with his wife near Broome Bridge in Dublin, and to commemorate the moment, he engraved the laws of quaternion calculus on its stones. It takes as many as three extra dimensions:

*q = ae + bi + cj + dk*.

Here is an illustration commemorating this event:

View attachment 58160
But what would these quaternions be of use to us? Are they the bane of modern physics, or perhaps a hope for salvation? And the non-associative octonions?

I personally think of these strange creations mainly in the context of magnetic monopoles (see. e.g.

[quant-ph/9803002] On Quaternions and Monopoles). Modern physics says that the magnetic field is sourceless. This means that for a magnetic field to exist we need two poles. We know this well from school.

Although no experiment has demonstrated the existence of monopoles, a major theoretical premise points to it. Paul Dirac showed that the existence of even one magnetic monopole in the Universe explains the problem of the quantization of electric charge.

How then is it in fact? Is the magnetic field really sourceless? Or do magnetic monopoles represent one of the more significant steps towards Unified Field Theory (UFT)? UFT, in turn, leads even further and may one day unveil the mystery that in turn is my longtime obsession - time.

If you are interested in quaternions, I also recommend (The illustration I posted is from this book):

Quaternion Algebras | SpringerLink