APS Talk: Revisiting Lorentz Transformations Using Quaternions
Channel: Doug Sweetser
Date: 2019-10-15

This was the last talk in the 2019 Fall New England American Physical Society meeting in Warwick, RI (33 people in the audience).

Near the start, I pointed out I have a new approach to gravity which was not the focus of the presentation, but that I should mention. To get others to actively think about the details of the proposal, I handed out a homework contest with cash prizes which you can view as a pdf, http://dougsweetser.github.io/Q/Stuff/pdfs/PGT_homework.pdf or visit a web page: http://dougsweetser.github.io/Q/Gravity/PGT/personal_gravity_theory/.

Relativity - both special and general - uses three interrelated ideas: transformation laws, invariance principles, and symmetries. When there are no boosts, ruler relativity is why two observers agree upon a distance: x² + y² = x'² + y'² = R². The transformation law was known to Rodrigues in the 1840s: R'=uRu*. Special relativity is a tiny upgrade to ruler relativity, taking care of observers in constant motion. The Lorentz invariant interval is t² - R² = t'² - R'². 

The first effort to develop a Lorentz transformation using real-valued quaternions was done in 1910 and 1911. These efforts failed, only working if one used complex-valued quaternions. A hundred years later, Dr. Kharinov and myself discovered a way to do Lorentz boosts by adding on a pair of terms where the triple product put the u's on one side:

R' = u R u* + 1/2( (u u R)* - (u* u* R)* )

There are three physics Ph.D.s who say this in fact does do a Lorentz boost, but do not see how a rotation is possible. A fourth Ph.D. thought a quaternion representation of the Lorentz group was reasonable since one can construct the Lorentz group using Dirac matrices which are only a factor of i different from quaternions. 

If one picks out a very particular member of the unit quaternions (a double cover of all rotations), one where the scalar is zero u = (0, a, b, c) & |u|=1, then the second and third terms cancel leaving the 3D spatial rotation equation known to Rodrigues. One member of SU(2) is all that is needed to do all rotations.

There are 6 degrees of freedom in the Lie algebra so(3, 1) used to generate the Lie group SO(3, 1) for the Lorentz group. Doesn't that appear to be in conflict with a one parameter quaternion expression? To do physics, one needs both space-time - where-when things are - and a tangent space, energy-momentum being an example. Angles appear as ratios in space-time, while boosts can live independently in energy-momentum.

Think of how one would represent the groups SU(2) and SO(3, 1) as an animation in space-time. An animation has a beginning, a middle, and an end. The beginning for SU(2) is (-1, 0, 0, 0) as a point. The point expands to a sphere that reaches its maximum at time t=0. It then shrinks to a point at (1, 0, 0, 0) after which there is nothing.

The Lorentz group is a mirror of the above animation. It starts out as far back in the past as possible and is huge. As the past approaches now, the sphere shrinks, reaching its minimum value at t=0. If the interval is scaled to unity, then the Lorentz group and SU(2) share points for this one moment. The Lorentz group then marches out to infinity getting ever larger.

iPhython notebooks concerning this controversy are available on github: https://github.com/dougsweetser/Quaternion_Lorentz_Transformations