Quaternions as a representation of four-vectors
in [Math]
|
From: varistor on 30 Aug 2007 15:54 On Aug 30, 9:49 pm, Peter Christensen <p...(a)peterchristensen.eu> wrote: > On 30 Aug., 21:28, varistor <w89q5o...(a)insurer.com> wrote: > > > > > Peter Christensen 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) > > > whay, are quaternions not ordinary complex numbers? > > > a complex number is a complex number, end of story > > > > 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)- Skjul tekst i anførselstegn - > > > - Vis tekst i anførselstegn - > > I'm afraid, that the whole story is a few kb longer: Read for example > this one:http://mathworld.wolfram.com/Quaternion.html > > They appear just to be interesting, because I would like to apply them > like this: > > 1: Time direction > j: X-direction > k: Y-direction > l: Z-direction > > This means like units in space-time. (Yet another 'silly' idea, but > why not try it?) becus is silly? > > Rgds, > PC
From: Peter Christensen on 31 Aug 2007 02:45 > 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
From: Peter Christensen on 31 Aug 2007 02:51 On 30 Aug., 21:54, varistor <w89q5o...(a)insurer.com> wrote: > On Aug 30, 9:49 pm, Peter Christensen <p...(a)peterchristensen.eu> > wrote: > > > > > > > On 30 Aug., 21:28, varistor <w89q5o...(a)insurer.com> wrote: > > > > Peter Christensen 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) > > > > whay, are quaternions not ordinary complex numbers? > > > > a complex number is a complex number, end of story > > > > > 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)- Skjul tekst i anførselstegn - > > > > - Vis tekst i anførselstegn - > > > I'm afraid, that the whole story is a few kb longer: Read for example > > this one:http://mathworld.wolfram.com/Quaternion.html > > > They appear just to be interesting, because I would like to apply them > > like this: > > > 1: Time direction > > j: X-direction > > k: Y-direction > > l: Z-direction > > > This means like units in space-time. (Yet another 'silly' idea, but > > why not try it?) > > becus is silly? No, because we scientists (If I may call myself that HeHe) always are trying to find something new and interesting. Even the ideas that might look 'silly' at a first glance should be tested... Why not? Rgds, PC
From: funk420 on 31 Aug 2007 03:27 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.
From: funk420 on 31 Aug 2007 03:44
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/ 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. Cheers - |