The Forgotten Geometry: a New Path to Unification

I've been reading Peter Woit for a while, and while I can't say I understand anything about spinor geometry (which is what he's fascinated with - I first encountered the idea in Roger Penrose where I couldn't quite parse it either), I feel like spinors are

*kinda-sorta* linked to quaternions. Which Tom Bearden began raving about (with varing degrees of lucidity; usually less rather than more) from the mid-1970s. So that's somewhat interesting.

Because of Bearden, I've always been intrigued by quaternions, and particularly how much they are an example of "the unreasonable

*ineffectiveness* of mathematics in the physical sciences". At least as far as we know. Quaternions - in the mid-1800s, well before Einstein - gave us the idea of "a fourth dimension of space" which might or might not be the same as time. Quaternions gave us the "nabla/del" or div/grad/curl operator, which was hugely valuable in electromagnetic fields. But weirdly, quaternion del gives us - in one already-unified mathematical object - something that is the sum of delta, div, grad and curl.... and modern physics teaches us that we simply cannot add these quantities because they are of different types (specifically, because curl is a pseudovector while the others are vectors, but also vectors describing incompatible physical quantities). So that's super, super bizarre. Why does the earliest and simplest possible maths of 4D spacetime give us a "non-physical" object, but one that's

*so very close* to being useful, and with its

*components* - but not itself - (for example, 3D cross product, and 3D div/grad/curl) being extremely useful?

John Baez (not to be confused with his cousin Joan) will tell you that quaternions are physical, just that the physics they describe does not exist in our universe: they describe the "surface" of a four-dimensional sphere, or, the space of 4D stretches/rotations. The 4D component of the rotation in quaternions being almost but not quite exactly unlike the "4D rotation" in a Lorentz boost, which is hyperbolic rather than Euclidean geometry. All of this seems more than a little strange if we think that there ought to be some inherently "mathematically beautiful" element in physics. Quaternions are very elegant and beautiful... yet our universe very surprisingly "doesn't use them" - although it uses half of a quaternion, the complex number, all over the place. Is this true, or are we missing a very important clue in how our universe

*might* after all be using quaternions? And are spinors perhaps related to that clue?

I don't quite have either the maths or the physical intuition to fully expland and prove this argument, but I can't help but think that quaternions would look a lot more like Minkowski space if we thought of the 4D component as not measuring "time" but "1/time", which would be more like "frequency". A similar idea seems to appear in Wilbert Smith's weird book "The New Science", where he describes something called a "tempic field" which seems to have that same 1/t characteristic. He doesn't get as far, however, as describing how this might relate to quaternion maths.

Nate