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It is not clear to me from this discussion: Are quarternions simply an alternative but equivalent formulation to vectors, at least as far as E&M is concerned, with different and beneficial numerical properties? Or does the quarternion formulation allow solutions that are contradicted by the vector formulation?

Art Carlson wrote:It is not clear to me from this discussion: Are quarternions simply an alternative but equivalent formulation to vectors, at least as far as E&M is concerned, with different and beneficial numerical properties? Or does the quarternion formulation allow solutions that are contradicted by the vector formulation?

Quaternions are not equivalent. Maxwell used them to develop his electromagnetic theory, but Oliver Heavyside found them too difficult to solve. Since the underlying complexity wasn't necessary for telegraph cable calculations, Heavyside (and Gibbs) developed a simpler version which we still use.

Art Carlson wrote:It is not clear to me from this discussion: Are quarternions simply an alternative but equivalent formulation to vectors, at least as far as E&M is concerned, with different and beneficial numerical properties? Or does the quarternion formulation allow solutions that are contradicted by the vector formulation?

From what I've been able to gather, the quaternion formulation is an alternative but equivalent formulation, which might have different and beneficial numerical properties.

A quick web search has turned up sites claiming that the quaternion formulation might lead to new science. As near as I can gather, this is because of a combination of all these assumptions/beliefs:

1. Maxwell initially published EM as 20 simultaneous quaternion equations
2. His editors, fearful that no one understood quaternions, pressured him to simplify, down to 4 vector equations. Some of this editing was done posthumously by Heaviside, the promoter of the "inferior" vector notation.
3. The suppressed equations must have predicted stuff the simplified version doesn't.
4. Maxwell didn't make any conceptual or mathematical errors in his papers.

Overall, that ranks quite high on the "crank" index, in my opinion.

Quaternions form a 4-dimensional normed division algebra over the real numbers. They and the complex numbers are the only two finite-dimensional division rings containing the real numbers as a proper subring (Frobenius theorem).

Some Physical Consequences of General Q-Covariance
D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser
Helvetica Physica Acta, Volume XXXV (1962) 328-329

Using quaternions, this paper arguably describes the "Higgs Mechanism" before Higgs and ElectroWeak Unification before Glashow, Salam and Weinberg.

alexjrgreen wrote:

Art Carlson wrote:It is not clear to me from this discussion: Are quarternions simply an alternative but equivalent formulation to vectors, at least as far as E&M is concerned, with different and beneficial numerical properties? Or does the quarternion formulation allow solutions that are contradicted by the vector formulation?

Quaternions are not equivalent. Maxwell used them to develop his electromagnetic theory, but Oliver Heavyside found them too difficult to solve. Since the underlying complexity wasn't necessary for telegraph cable calculations, Heavyside (and Gibbs) developed a simpler version which we still use.

So you're saying there are some phenomena allowed in the quaternion formulation which contradict the vector formulation. In other words, there is some experiment, where a particular result would falsify the vector formulation of Maxwell's equations, but be consistent with the quaternion formulation. What would that experiment be?

I guess I shouldn't be saying "formulation". We are talking about two different laws of nature. I suppose vector-E&M is a subset of quaternion-E&M, so that any solutioin of vector-E&M is also a solution of quaternion-E&M?

Art Carlson wrote:So you're saying there are some phenomena allowed in the quaternion formulation which contradict the vector formulation. In other words, there is some experiment, where a particular result would falsify the vector formulation of Maxwell's equations, but be consistent with the quaternion formulation. What would that experiment be?

I guess I shouldn't be saying "formulation". We are talking about two different laws of nature. I suppose vector-E&M is a subset of quaternion-E&M, so that any solutioin of vector-E&M is also a solution of quaternion-E&M?

"Subset" is the word.

Doug Sweetser has a quaternion treatment of EM here:
http://www.theworld.com/~sweetser/quate ... calem.html

and of the Schrödinger Equation here:
http://theworld.com/~sweetser/quaternio ... inger.html

Of course, the assumptions which Heavyside and Gibbs used to create the vector representation might be physically valid. Or not...

alexjrgreen wrote:

Art Carlson wrote:So you're saying there are some phenomena allowed in the quaternion formulation which contradict the vector formulation. In other words, there is some experiment, where a particular result would falsify the vector formulation of Maxwell's equations, but be consistent with the quaternion formulation. What would that experiment be?

I guess I shouldn't be saying "formulation". We are talking about two different laws of nature. I suppose vector-E&M is a subset of quaternion-E&M, so that any solutioin of vector-E&M is also a solution of quaternion-E&M?

"Subset" is the word.

Doug Sweetser has a quaternion treatment of EM here:
http://www.theworld.com/~sweetser/quate ... calem.html

and of the Schrödinger Equation here:
http://theworld.com/~sweetser/quaternio ... inger.html

Of course, the assumptions which Heavyside and Gibbs used to create the vector representation might be physically valid. Or not...

Where does Sweetser say anything other than that he used quaternions to generate an equivalent formulation of E&M and Schödinger's equation? Maybe quaternions can inspire somebody to make a brilliant extension some day, and maybe the numerical properties are advantageous, but I don't see any new physics yet.

Art Carlson wrote:Where does Sweetser say anything other than that he used quaternions to generate an equivalent formulation of E&M and Schödinger's equation? Maybe quaternions can inspire somebody to make a brilliant extension some day, and maybe the numerical properties are advantageous, but I don't see any new physics yet.

Doug goes here:
http://theworld.com/~sweetser/quaternio ... ation.html

which gives a flavour of what might be possible.

alexjrgreen wrote:

Art Carlson wrote:Where does Sweetser say anything other than that he used quaternions to generate an equivalent formulation of E&M and Schödinger's equation? Maybe quaternions can inspire somebody to make a brilliant extension some day, and maybe the numerical properties are advantageous, but I don't see any new physics yet.

Doug goes here:
http://theworld.com/~sweetser/quaternio ... ation.html

which gives a flavour of what might be possible.

Thanks. I would have to do some real work on this before I could ask any more intelligent questions. I'll just say my crank-meter has been pretty quiet. It looks like I could follow his derivation if I worked at it, and he doesn't seem to get carried away with wild claims. The MIT email address helps, too. A few references would have made it better.

An example of geometric algebra in Computational Chemistry
http://www.ti.inf.ethz.ch/ew/courses/GC ... review.pdf

Art references:

David Hestenes (1966). Space-Time Algebra, Gordon & Breach.

This was the excellent little book that I found most useful. It has derivations for Maxwell, Schrodinger, Dirac and GR equations all using geometric algebra machinery.

Quaternions are a sub-algebra of the space-time algebra. S-T algebra is more powerful but has necessarily more overhead than quaternions, but less than tensor calculus and more intuitive, I found.

Here's an updated version on-line (pdf):
http://modelingnts.la.asu.edu/pdf/SpaceTimeCalc.pdf

icarus wrote:This was the excellent little book that I found most useful. It has derivations for Maxwell, Schrodinger, Dirac and GR equations all using geometric algebra machinery.

Quaternions are a sub-algebra of the space-time algebra. S-T algebra is more powerful but has necessarily more overhead than quaternions, but less than tensor calculus and more intuitive, I found.

For all I've read of GA from Hestenes and others, I haven't heard it's proponents pushing the idea that using GA instead of vectors/tensors/matrices, etc yields new physics, just a more intuitive, more geometric interpretation/formulation of existing physics.

I have seen such claims from quaternion-devotees.

The main thing that strikes me as favorable about GA initially is that it doesn't confuse polar and axial vectors. The GA analog of the vector cross product doesn't yield vectors, but rather a "bivector", which form their own subspace of in GA. Whereas the cross product is only defined in 3 dimensions, bivectors are defined for all dimensions, and have C(n,2) basis bivectors in an n-dimensional GA.

Quaternions are a subalgebra of 3-D GA, but not (3,1)-D GA which I assume you'd use for a Minkowski-signature GA.

blaisepascal wrote:For all I've read of GA from Hestenes and others, I haven't heard it's proponents pushing the idea that using GA instead of vectors/tensors/matrices, etc yields new physics, just a more intuitive, more geometric interpretation/formulation of existing physics.

Read Chapter 8 of Hestenes book...

alexjrgreen: "Read Ch 8 of Hestnes book"

Which Hestnes book is that?