Quaternions as a representation of four-vectors
in [Math]
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From: Peter Christensen on 30 Aug 2007 14:15 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)
From: funk420 on 30 Aug 2007 14:46 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/~cmdaven/special.htm Any comments appreciated!
From: varistor on 30 Aug 2007 15:28 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)
From: Peter Christensen on 30 Aug 2007 15:30 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
From: Peter Christensen on 30 Aug 2007 15:49
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?) Rgds, PC |