Vlasov Solver [work in progress]
-
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
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?
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
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?
Ars artis est celare artem.
-
- Posts: 191
- Joined: Thu Jun 05, 2008 3:57 am
- Location: Ithaca, NY
- Contact:
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.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?
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.
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
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.
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.
Ars artis est celare artem.
-
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
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?alexjrgreen wrote: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?
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?
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
"Subset" is the word.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?
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...
Ars artis est celare artem.
-
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
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.alexjrgreen wrote:"Subset" is the word.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?
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...
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
Doug goes here: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.
http://theworld.com/~sweetser/quaternio ... ation.html
which gives a flavour of what might be possible.
Ars artis est celare artem.
-
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
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.alexjrgreen wrote:Doug goes here: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.
http://theworld.com/~sweetser/quaternio ... ation.html
which gives a flavour of what might be possible.
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
An example of geometric algebra in Computational Chemistry
http://www.ti.inf.ethz.ch/ew/courses/GC ... review.pdf
http://www.ti.inf.ethz.ch/ew/courses/GC ... review.pdf
Ars artis est celare artem.
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
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
-
- Posts: 191
- Joined: Thu Jun 05, 2008 3:57 am
- Location: Ithaca, NY
- Contact:
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.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.
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.
-
- Posts: 815
- Joined: Thu Nov 13, 2008 4:03 pm
- Location: UK
Read Chapter 8 of Hestenes book...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.
Ars artis est celare artem.