Quaternions as a representation of four-vectors
in [Math]
|
From: Peter Christensen on 3 Sep 2007 14:34 On 31 Aug., 09:44, funk420 <funk...(a)yahoo.com> wrote: > On Aug 30, 3:30 pm, Peter Christensen <p...(a)peterchristensen.eu> > wrote: > > > > > On 30 Aug., 20:46,funk420<funk...(a)yahoo.com> wrote: > > > > On Aug 30, 2:15 pm, Peter Christensen <p...(a)peterchristensen.eu> > > > wrote: > > > > > More about these 4-dimensionsl numbers here:http://en.wikipedia.org/wiki/Quaternion > > > > (Please notice, that I prefer to use j, k and l, because I would like > > > > to keep the symbol 'i' free, so that I can still use ordinary complex > > > > numbers without confusing people) > > > > > As I can understand from wikipedia, the idea of using there 'hyper- > > > > complex' number simply as a replacement for the four-vectors in > > > > physics (I'm thinking about special relativity) is new. (Sorry if I'm > > > > wrong.) > > > > > I just realised, that this complex structure can give some really nice > > > > results, when the usual four-vectors are replaced with these (very > > > > nice) structures . > > > > > To try to be abstract from the physics, and focus on the math, here is > > > > a brief summary: > > > > > In physics these two types of socalled four-vectors are very often > > > > used: > > > > > Position four-vector: R = (ct,x,y,z) where c is the speed of light, t > > > > is coordinate time and (x,y,z) is a spatial position. IMHO, things > > > > works much better, if we instead use c*t+j*x+k*y+l*z, where j, k and l > > > > are the the quaternion parameters as defined above in the reference. > > > > > Another very important four-vector is the socalled momentum four- > > > > vector P = (E/c,p_x,p_y,p_z). Where E is the energy, c is the speed of > > > > light constant and the p's are the momentum in the different > > > > directions of space. Again a formulation with quaternions is much more > > > > elegant: P = E/c + j*p_x+k*p_y+l*p_z. > > > > > Before the use of quaternions, we had to use vectors and multiply > > > > these vectors with matrices when going from one physical system to > > > > another. With the quaternions things are just so much easier, as I > > > > will just show in these examples: > > > > > Position four-vector in one spatial direction: > > > > > R = ct + j*x > > > > > When transformed with the Lorentz-transform: > > > > > R' = ct'+j*x' = 1/(1-(v/c)^2) (ct+j*x) > > > > > or the other way around: > > > > > R = ct+j*x = (1-(v/c)^2) (ct'+j*x') > > > > > -Just so much easier than usual with vectors and matrix. (v is the > > > > velocity of our particle I just have to add...) > > > > > The same with the momentum four-vector: > > > > > P' = E'/c + j*p' = 1/(1-(v/c)^2) * (E/c + j*p) > > > > > Or the other way around: > > > > > P = E/c + j*p = (1-(v/c)^2) * (E'/c + j*p') > > > > > So these hyper-complex numbers do definately have application in > > > > physics. > > > > > An interesting area for research, IMHO. -It's usefull for much more > > > > than just 3D rotations in computer-graphics. I think, that these > > > > numbers are very relevant for both the work with the Poincaré group > > > > and quantum Mechanics in general. Simply math when it's most > > > > interesting.. :-) > > > > > Rgds, > > > > Peter Christensen > > > > > (Copenhagen, Denmark) > > > > You might also be interested in the other hypercomplex 4D algebra, > > > which is commutative. > > > There is an interesting book about its potential use in relativity > > > theory: > > > > Davenport(1), C. M., A Commutative Hypercomplex Calculus with > > > Applications to Special Relativity (Privately published, Knoxville, > > > Tennessee, 1991) > > > > You might also be interested in some information here: > > > >http://home.usit.net/~cmdaven/hyprcplx.htm > > > > The relativity page looks to be erased but you can find it here: > > > >http://web.archive.org/web/20061010203358/http://home.usit.net/~cmdav.... > > > > Any comments appreciated!- Skjul tekst i anførselstegn - > > > > - Vis tekst i anførselstegn - > > > Hi, > > > You really hit the subject that I was interested in. Even though I > > still haven't got the book you were talking about (of course not), and > > I still haven't read the links, then I would like to say "thanks, very > > interesting"... > > > The case is, that I've recently got really interested in these hyper- > > complex numbers, after I read something about them, and today I > > basically haven't been doing anything else than sitting and 'testing > > them out' with some various calculations. Very interesting. I will > > have some comments later. > > > So far, I use a combination of complex numbers and the hyper-complex > > numbers like this (notice, I use i as the usual complex unit and j, k > > and l for the hyper-complex numbers) > > > "1" coordinate time > > "j" x-position > > "k" y-position > > "l" z-position > > > "i" energy > > "i*j" momentum in the x-direction > > "i*k" momentum in the y-direction > > "i*l" momentum in the z-direction > > > But ok, so what? - The point is just that I'm VERY interested in these > > hyper-complex numbers. > > > Best regards, > > Peter Christensen > > Glad you are interested! I also was studying these for a time. > > I came across one other person working with this 4D complex algebra > (w/ commutative multiplication, not a group because there is more than > one point with no inverse), that is Dominic Rochon at the University > of Quebec, who calls them "bicomplex". > > see e.g.http://www.3dfractals.com/manuscripts_bicomplex_dynamics.php > > For other pretty pictures you can check out > > http://www.javaspider.com/jfract/ Thanks... > The Davenport treatment is much more complete but he comes up with > some unusual physics predictions such as a modified Lorentz > transformation that includes contraction also in the directions > perpendicular to the relative motion.. which I recall was > incompatible with some experimental result but I don't remember > which. Looks interesting... > Cheers - You too... PC
From: Peter Christensen on 3 Sep 2007 14:38
On 3 Sep., 03:26, maxwell <s...(a)shaw.ca> wrote: > On Sep 2, 8:53 am, Peter Christensen <p...(a)peterchristensen.eu> wrote: > > > On 2 Sep., 17:39, maxwell <s...(a)shaw.ca> wrote: > > > > On Sep 1, 7:04 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > > > > > Peter Christensen wrote: > > > > > As I can understand from wikipedia, the idea of using there 'hyper- > > > > > complex' number simply as a replacement for the four-vectors in > > > > > physics (I'm thinking about special relativity) is new. (Sorry if I'm > > > > > wrong.) > > > > > This was already old and essentially rejected when I was in school >30 > > > > years ago. Tensor approaches have proved to be MUCH more effective, and > > > > of MUCH wider applicability. > > > > > > Position four-vector: R = (ct,x,y,z) [...] > > > > > One problem with this approach is that this is not really a 4-vector > > > > (though it sometimes masquerades as one in elementary books). > > > > > The basic attractiveness of quaternions is that their norm is naturally > > > > the same as the norm of a 4-vector in the Minkowski coordinates of SR. > > > > But AFAIK they are completely unable to sensibly and simply handle other > > > > coordinates -- we physicists often use spherical and cylindrical > > > > coordinates, and quaternions no longer have the same norm naturally.. And > > > > then there are the curved manifolds of GR.... > > > > > > Position four-vector in one spatial direction: > > > > > R = ct + j*x > > > > > When transformed with the Lorentz-transform: > > > > > R' = ct'+j*x' = 1/(1-(v/c)^2) (ct+j*x) > > > > > -Just so much easier than usual with vectors and matrix. > > > > > Hmmm. Count degrees of freedom in the transform and you'll find they are > > > > the same as the matrix method (of course!). Not really "simpler", merely > > > > different. > > > > > Please remember the world has 3+1 dimensions, not 1+1. So write the > > > > Lorentz transform in an arbitrary direction: you'll find it gets > > > > complicated and quite messy (with a hyper-complex constraint equation to > > > > ensure it is a valid Lorentz transform) -- Not really "easier" is it? > > > > > BTW theoretical physicists almost never perform a Lorentz transform -- > > > > they work with invariants most of the time so there is no need. > > > > Experimenters, of course, often use them because the theory is usually > > > > expressed in the center-of-mass frame, but the detectors are in the > > > > laboratory frame, and those are often not at all the same. > > > > > Note, however, that quaternions and especially octonions have other > > > > applications in physics. Not as "4-vectors" but because of their group > > > > properties. I am not an expert in this.... > > > > > Tom Roberts > > > > Sorry, Tom, you just cut your own throat. It is exactly the fact that > > > the world is 3D in space that destroys the simplicity of the Lorentz > > > transformation. The transforms only form a group in 1D space while > > > electrons always move in 3D due to the so-called Lorentz force. The > > > LT's also get messy when the two inertial reference frames do not > > > initially align exactly at common origins of space & time. All the > > > applications of quaternions in physics since 1900 have used bi- > > > quaternions.- Skjul tekst i anførselstegn - > > > > - Vis tekst i anførselstegn - > > > May I just about about these bi-quaterions. -What's that, do you have > > some link? (Thanks in advance) > > > Rgds, > > PC > > Hi Peter, two great sites for info on quaternions are: > http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html > & arXiv:math-ph/0201058 > The authors of this last tribute to W. R. Hamilton have also created a > massive bibliography that is available at: > arXiv:math-ph/0510059 > Good luck, you will find you will be very pleased with your efforts > with quats. I also think so. I ordered 3 books about them over Internet yesterday, so I hope I will 'like them'. Interesting that already Hamilton was working with the quats. Rgds, PC |