Quaternions as a representation of four-vectors
in [Math]
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From: maxwell on 2 Sep 2007 21:26 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.
From: Peter Christensen on 3 Sep 2007 04:49 On 3 Sep., 02:48, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > maxwell wrote: > > The [Lorentz] transforms only form a group in 1D space > > This is simply not true. The Lorentz group is a group of transforms on > (3+1)-dimension spacetime. Yep, 3 in space and one in time, therefore the name spacetime. :-) (one would not be enough, I mean) > > The > > LT's also get messy when the two inertial reference frames do not > > initially align exactly at common origins of space & time. > > Not really. Yes, ELEMENTARY books only present the homogeneous Lorentz > transform in 1+1 dimension, but the formulas for the complete Poincare' > group in 3+1 dimensions are well known, and not very complicated. Yes again. But one thing is going from the 1+1 presentation to a 3+1 representation. Have a look here: http://en.wikipedia.org/wiki/Lorentz_transform#Matrix_form Another thing is that we need two things more: * Translations * Rotations First, when our physical laws apply to a point in spacetime, then of course they should ALSO apply to other points in spacetime. (Otherwise the laws would be worthless) -I mean Bill and Bob should get the same results even though they live at different places and different times. That's the spacetime translations. -There are 4 totally: 3 spatial and one in time Second there are the rotations, actually 3 of them in a 3D space. It's really not so strange, or abstract, as it sounds. Just take an example as an aeroplane in the air. It can turn in three different ways in it's 3D space: First it can turn from left to right. Then it can select to go up in hight or the opposite. At last it can roll around it's main axis. -It doesn't really take a professor in group theory to see this. Third, we have 3 socalled boosts. That's really the homogenous Lorents Transform that was the real subject of the posting. It causes 3 socalled boosts, and it doesn't really matter if you use Newtonian mechanics or 'Einsteinian mechanics'. But there is a difference in the transformation itself: Today we ALWAYS use the Lorentz Transformation if v is comparable with c. The Poincaré group have these 10 dimensions. And it's well understood, TR is completely right. -And recently Perelman (http://en.wikipedia.org/wiki/Grigori_Perelman) improved the understanding even further, by the way. -But if you try to do a whole Poincaré transformation, then I'm not sure if I would say "Not very complicated"... :-) 'Pet C'
From: Peter Christensen on 3 Sep 2007 05:12 On 3 Sep., 03:08, YBM <ybm...(a)nooos.fr> wrote: > maxwell a écrit : > >From France I C... > > > On Sep 2, 5:48 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > >> maxwell wrote: > >>> The [Lorentz] transforms only form a group in 1D space > >> This is simply not true. The Lorentz group is a group of transforms on > >> (3+1)-dimension spacetime. > > >>> The > >>> LT's also get messy when the two inertial reference frames do not > >>> initially align exactly at common origins of space & time. > >> Not really. Yes, ELEMENTARY books only present the homogeneous Lorentz > >> transform in 1+1 dimension, but the formulas for the complete Poincare' > >> group in 3+1 dimensions are well known, and not very complicated. > > >> Tom Roberts > > > Exactly my point, Tom. One can always apply group theory to any > > vector in 3D but that is just being pedantic. If the so-called > > ELEMENTARY textbooks were to show the full Poincare transforms, people > > would realize that the elegant simplicity of SR is not so 'elegant' > > after all. > > So what ? Elegance is not on formulas but on structures. > > Did you ever write down the complete 3+1 galilean transformations ?- Skjul tekst i anførselstegn - Scientists should probably avoid words like 'elegant', or even worse 'beautiful' in general, IMHO. Usefull or not usefull would prob be better. -But who knows what is..? I must just say, that I do not feel the slightest bit better than others. I also just work on the socalled 'beautifull things' while having this strange intuition that they MUST be usefull somewhere. Is it silly to think like that, or is it just a 'good' intuition? PC (Denmark)
From: Owen on 3 Sep 2007 06:28 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) We can also develop hypercomplex numbers as complex-complex numbers. C1 a+bi, where a,b are real and i^2=-1. C2 (a+bi)+(c+di)j, where a,b,c,d are real and: i^2=j^2=-1, ij=ji=k, ik=ki=-j, jk=kj=-i, k^2=+1, ~(i=j), ~(i=k), ~(j=k). The usual functions that apply to complex numbers also apply to complex-complex numbers with the addition that the reciprocal function is denied for zero and zero divisors. (a+bi)+(c+di)j=(a+bi+cj+dk). C3 ((a+bi)+(c+di)j)+((e+fi)+(g+hi)j)l ...=(a+bi+cj+dk)+(e+fi+gj+hk)l ...=(a+bi+cj+dk+el+fm+gn+ho). Where a,b,c,d,e,f,g,h are real and: i^2=j^2=l^2=-1, ij=ji=k, il=li=m, jl=lj=n, kl=lk=o, and so on for any complex-complex number of dimention 2^n where n is a natural number.
From: Peter Christensen on 3 Sep 2007 10:35
On 3 Sep., 12:28, Owen <owenhol...(a)rogers.com> wrote: > On Aug 30, 2:15 pm, Peter Christensen <p...(a)peterchristensen.eu> > wrote: > .... > We can also develop hypercomplex numbers as complex-complex numbers. > > C1 a+bi, where a,b are real and i^2=-1. > > C2 (a+bi)+(c+di)j, where a,b,c,d are real and: i^2=j^2=-1, ij=ji=k, > ik=ki=-j, jk=kj=-i, k^2=+1, ~(i=j), ~(i=k), ~(j=k). > > The usual functions that apply to complex numbers also apply to > complex-complex numbers with the addition that the reciprocal function > is denied for zero and zero divisors. > (a+bi)+(c+di)j=(a+bi+cj+dk). > > C3 ((a+bi)+(c+di)j)+((e+fi)+(g+hi)j)l > ..=(a+bi+cj+dk)+(e+fi+gj+hk)l > ..=(a+bi+cj+dk+el+fm+gn+ho). > > Where a,b,c,d,e,f,g,h are real and: i^2=j^2=l^2=-1, ij=ji=k, il=li=m, > jl=lj=n, kl=lk=o, and so on for any complex-complex number of > dimention 2^n where n is a natural number. Thanks, I'm interested B Rgds, PC |