The 200 missing quaternion equations from Maxwell's Treatis on Electromagnetism, Edition 1

06 Nov 2009
Posted by doug

Hello:

A significant driver of traffic to my web site quaternions.com comes from a persistent story about the 200 quaternion equations that appeared in Maxwell's "Treatise on Electromagnetism", the first edition. By the third edition, there were no quaternion equations. In this post, I will report what I found.

After enough emails about the missing 200 equations, I thought I would look on the web for them (this was the late 1990's). Someone had scanned and put them up on the web. As I remembered it, the quaternion expressions where all "pure quaternions", or what I would call the 3-vector part of a quaternion. At the time, I thought that was an empty way to use quaternions. The main point of special relativity is that what is a pure quaternion in one inertial reference frame is not a pure quaternion in a different inertial reference frame. EM is inherently a relativistic theory, so one had best not start with non-relativistic terms. We cannot fault Maxwell on this point since relativity was build by Einstein's analysis of EM 40 years later.

Here it is a decade later. I get similar questions. And I was able to find the original source again. I can see that pure quaternions are used exclusively. What this indicates to me is that it is impossible to learn anything different by writing things in pure quaternions from the far more common vector notation. A fellow named Col. Tom Bearden claims that the deletion of this section - all of 3 pages - is one grand scientific conspiracy. It looks like Bearden thinks 20 equations were "lost", while the number I here cited more is 200. No matter, any information written in a pure quaternion form can 100% faithfully be represented using vectors.

Clicking around, I was able to confirm that Maxwell did keep his scalars separate from his 3-vectors. If you are into this old stuff - and I am not - read André Waser's paper, On the Notation of Maxwell’s Field Equations.

People in the late 1800s figured out how to write the Maxwell equations using complex-valued quaternions, also known as biquaternions. Complex-valued quaternions are not a division algebra. To me, that is an enormous loss. The most important numbers in all of physics are division algebras: the real and complex numbers. It seams obvious to me to use a 4D division algebra for 4D problems in spacetime. I looked into a good number of papers where someone claimed to have written the Maxwell equations using quaternions, but that was someone else's summary. The author invariably needed the fourth i (quaternions already have i, j, and k). The magic fourth i commutes with all, unlike the other 3. Biquaternions to this day strike me as bogus.

In the late 1990s, I felt compelled to represent the Maxwell equations using only real valued quaternions. I have an inch and a half stack of papers of failed efforts: there was always a sign wrong somewhere. I eventually found the combination of even ((a+b)/2) and odd ((a-b)/2) operators to do the trick:



The homogeneous equations are a combination of even and odd operators, while the source sequations need to double up on the odds and double up on the evens. Nice.

Maxwell had nothing like this in his amazing Treatise (and it is amazing, since Jackson's modern work looks like a variation on Maxwell). For a few years, I thought I was the first person to have done this, yet a fellow amateur named Peter Jack was the first, beating me by 1 year. I was preparing to go to the second and last meeting on quaternions in physics in Rome in 1999. The slide with this equation looked too complicated. I was wondering why I had to throw so many things away. What was the stuff I was throwing away? I guessed it might be gravity! That was an inspired leap, caused by the complexity of the expression for the Maxwell equations shown above.

Here is a summary of the next decade of work. Professional physicists are trained on tensors, not quaternions. I observed that writing equations using quaternions meant professionals didn't listen. Therefore I learned how to write everything I had done with quaternions using tensors. The last few years I was struggling with the current coupling equation. That drove me back to quaternions and on to hypercomplex numbers.

One important realization in my training is that one needs to work with an action, all the ways energy can be exchanged within a unit of volume. From the action, one can derive the field equations using the Euler-Lagrange equations. One can also calculate the stress-energy tensor, and head out into quantum field theory. All amateurs I have looked at work with field equations. That is not good enough, one needs to work with actions and derive the field equations from the action.

A message from the standard model of physics is that EM is not enough: one needs both the weak and the strong force. Folks who worry about the great scientific conspiracy ignore the standard model, staying focused on EM. That is understandable: the standard model is not easy to understand or do calculations with. The group sitting in the middle of the standard model, SU(2), is known as the unit quaternions! That name alone suggests writing equations with quaternions will lead to a way to represent the unit quaternion symmetry of the standard model. My latest work does exactly that. Maxwell naturally did not deal with these issues that physicists developed almost a century later.

Maxwell was a smarter cat than I will ever be. I live in a later time, with a vast accumulation of new knowledge. On a technical level, I don't think anything was lost by writing pure quaternions as 3-vectors in the transition from the first edition of Maxwell's Treatise to the third. My research suggests adapting quaternions to an action with all the symmetries of the standard model may be productive.



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