Quaternions as a representation of four-vectors
in [Math]
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From: Peter Christensen on 2 Sep 2007 11:53 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
From: Peter Christensen on 2 Sep 2007 12:24 TR wrote: > > 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? Sorry that I write 3 different replies instead of just one. -I just came up with something a few times... I'm quite sure, that you will basically like my constraints anyway, as you do know a lot about SR and probably also QM: They are simply: E = gamma*m*c*2 p = gamma*m*v and then the broadenings from QM both in frequency and momentum: * deltaE = 1/deltaT * deltaPx = 1/deltaX (and so on for y and z) That's all, that I assume, so that should be fair enough... A thing that I have a problem with is however: 1) In a system moving with a constant velocity, then the momentum will always be in the same direction as the velocity. I agree with SR! 2) But what about GR, can we still assume that P = m*V (four-vectors) in the general case? Personally I don't like this approach, and I would really like to question it... PC
From: Tom Roberts on 2 Sep 2007 20:48 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
From: maxwell on 2 Sep 2007 21:01 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.
From: YBM on 2 Sep 2007 21:08
maxwell a �crit : > 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 ? |