Thursday, November 29, 2018

"Fantastic Quaternions": how a mathematical algebra generates the particles and forces of physics, and even the (libertarian) moral idea of "personal responsibility"



The YouTube site NumberPhile (in the UK) offers a video explaining the mathematical basis of elementary particles in physics. It’s “Fantastic Quaternions” with Dr. James Grime.


The quaternion is a number system with four dimensions that extends what we normally call complex numbers (based on two). It is possible to define consistent operations and form an algebra.  The rules, as with complex, correspond to the way trigonometric functions work when moving objects in a lattice. In fact, Grime says that a “complex number” really should have been called a “compound number”.'

It isn’t possible to define an algebra with just three dimensions.  On the other hand, there  is also an octonion, which generalizes somewhat for eight dimensions but some properties (like commutative law) get lost. The octonion corresponds to the perfect “crystal” in space-time, and can be projected onto a quaternion without loss of information, leading to a “quasi-crystal”.  But projection onto three spatial dimensions requires a fourth dimension that requires honoring a causality concept – time.  That is said to require sentient consciousness, that can make choices and be held morally responsible for choices that change the information content in the lattice – because the causality aspect of time is irreversible.

OK, this doesn’t all quite explain the personality of Donald Trump, or of me, for that matter.  But it’s a good start.

The operations of quaternions correspond to the behaviors of various sub-atomic particles. They also help explain the forces in nature (weak and strong forces especially).  Yet in the past there have been proposals for “weakless” universes, which could be more stable.

The octonion (8 dimensions) raises a question: does any lattice generating an "algebra" need to be based on a number of dimensions that is an integer power of 2?  Probably, but I won’t look for a proof right now.
  
When I was a graduate student in mathematics in the 1960s at KU, real analysis and complex analysis were separate year-long courses.  The Liouville Theorem got asked about in my Master’s orals in Jan. 1968.  I don’t recall quaternions being mentioned then.  The algebra portion of the degree included groups, rings, and fields.  The idea of a one-dimension space(reals) has political significance, because it can be well-ordered (like people being “right-sized”, which the Chinese want to do with their social credit score – something we’ll come back to later).

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