Everyone's introduction to quaternions always has the same math one-liner:



This blog will dive into an issue that has bothered me for years, namely that the line assume the only coordinate system that can be used for quaternions are both normal and orthogonal.  It is the later property that stumped me, and I made a small step forward I will report on.

[Clarification: I tweaked the title because the focus of this blog was intended to be quite small.  While scalars and vectors are used in a dizzying array of areas in many dimensions, quaternions are constrained to work in one plus three dimensions.  One could say one plus three equals four, which is true, but I say one plus three because with quaternions, the one has different math properties from the three.  What happened in the comments was the first two folks admitted they were looking for a blog with a different subject, one that might have me linking quaternions to objects that transform like 4-vectors.  That really is beyond the scope of this blog which doesn't confront how anything transforms.

I spent the day with the family at the zoo, idly thinking about physics, but not enough to blog about it yet.  Since it is Memorial Day, a day when one should think about those who gave their lives for our liberty, I considered a war story from my family.

In this blog I will derive the Einstein field equations starting from the Hilbert action.  Since there are only two terms in the Hilbert action, one of which is left alone, there is not that much to do.  Well, there is always way more to do - how well is this step really understood?  Where does that factor come from?  What kinds of variations could one do?  The core of this blog is an extensive translation of the wikipedia page on the Einstein-Hilbert action written in my own personal style, doing minor variations so the steps made more sense to me.  Go there if my style confuses you for a step or two.

This blog required programming.  The basic idea is to have four degrees of freedom for the four parameters living inside a four dimensional quaternion.  Four for loops did the trick.

Since 1930, efforts have been made to get our best theory for gravity - general relativity (GR) - to work with our best theory for atoms - quantum field theory (QFT) in the forms quantum electrodynamics (QED) and quantum chromodynamics (QCD).  In this comparatively short blog, I will frame the struggle and let people provide their own speculations in the comments (mine will reside there too, so it will be easy to skip if you would like :-)


Consider three physicists: Feynman, Weinberg, and Hawking.