Quaternions as a representation of four-vectors
in [Math]
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From: Peter Christensen on 31 Aug 2007 12:23 On 31 Aug., 09:27, funk420 <funk...(a)yahoo.com> wrote: > On Aug 31, 2:45 am, Peter Christensen <p...(a)peterchristensen.eu> > wrote: > > > > 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) > > > Just forgot to ask: If you have the book, could you give me the ISBN > > number too (Then it's much easier to find). Thanks... > > > Rgds, > > PC > > I found it with bookfinder.com > ISBN 0962383708 > > but looks expensive from those shops.. > > I found it in a physics library. Good luck. Hi, I just say Thanks... PC
From: Tom Roberts on 1 Sep 2007 22:04 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
From: Peter Christensen on 2 Sep 2007 08:04 On 2 Sep., 04:04, 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. OK, I understand. > > 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). Ok, you are right, I should really say dR or deltaR. You can get the impression, that I mix up the position 4-vector with the displacement 4-vector. I really know this problem, but I just forget... > 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.... Ok, they are probably quite complicated to use. Maybe a whole new 'quaternion-geometry' is needed. (I do NOT claim to have such, I must say). But I know that many famous mathematicians and physicists have worked on the quaternions... > > 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. I must say yes, different is probably the right word > 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? Can't say, I just got interested in those hyper-complex numbers. > 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. OK > 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.... Yep, I'm reading about them at the moment. As I understand it there are many apps for quaterions (groups), but there are quite little interest for octonions, as I've understood so far. I just liked their 8D and symmetries because I would like to make a phase-space distribution in 4 spatial and 4 'momentum-energy' dimensions. -Just an idea, so far nothing more... http://en.wikipedia.org/wiki/Octonions I also have some 'unusual ideas' (to try to say it in a humble way) conserning the following things: * The collapse of the wavefunction (spontaneous in spacetime?) * Normalisation of wavefunctions 'in time' (just trying with some gaussian distributions...) * A sort of phase-space distribution (something like Wignerfunctions, ALSO for (time,energy). That's probably the reason why I suddenly like these old and rejected structures once again. -While most physicists doesn't... (But these things doesn't really belong here is SM/SPR, I know) Ps. I do understand, that you are an expert in SR (one of only a few in the group), so thanks for the reply... By the way, I do NOT have any disagreements with SR, so I belong to the (few % in SPR is my impression ?) on usenet who are supporting it.. :-) B Rgds, PC
From: Peter Christensen on 2 Sep 2007 11:36 > 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.... As I've understood progress has been made with 'quaternion geometry'. Again, as I've understood, this book should be good (I can't tell for sure yet, because I've just ordered it today) http://www.amazon.com/Quaternions-Rotation-Sequences-Applications-Aerospace/dp/0691102988/ref=pd_ecc_rvi_cart_2/102-9078927-2975308 -At least 5 stars on average from 26 users looks promising... But I'm trying to import something from a different area of phys. Right on that one... Rgds, PC
From: maxwell on 2 Sep 2007 11:39
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. |