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There seems to be a conflict of terminology with regards to the name biquaternion. Hamilton appears to have used the term to mean a quaternion with complex coefficients (i.e. C⊗H), while Clifford (in Preliminary Sketch of Biquaternions, 1873) uses the term to mean an algebra isomorphic to H⊕H, which follows the quaternions in the sequence of Clifford algebras:
The complexified quaternions are not isomorphic to Clifford's biquaternions. This page presently discusses Hamilton's notion, while the German version of the page discuss's Clifford's notion. Some mention should be made of the conflict. I'm not sure which term is more commonly used. -- Fropuff 21:43, 19 February 2006 (UTC)
Note that Hamilton used the term first (it appears in his 'Lectures on Quaternions', 1853, article 669, available at http://historical.library.cornell.edu/math/). Sangwine 21:50, 25 February 2007 (UTC)
Since the structure of Clifford biquaternions is demonstrably different than the classical twentieth century concept of biquaternions used to develop the relativity transformations, the works of the Clifford algebraists on their biquaternion need a separate space. Rgdboer 01:35, 23 February 2006 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:44, 10 November 2007 (UTC)
First congratulations on this fantastic new article. I hope to watch it expand with time. These would be good things to explain in this article.
Does a biquaterion have an inverse, all the time, some of the time or none of the time?
It has to at least some of the time I think because if the imaginary part is zero, it would have an inverse, but this is trivial.
Letting h equal the imaginary scalar of ordinary algebra, since i,j and k are already taken, it would seem to me that the simple biquaterion:
hi would be its own inverse.
When D ≠ 0, then q = u + vi + wj + xk has an inverse, otherwise it is singular. Rgdboer ( talk) 20:10, 26 November 2008 (UTC)
I found were Hamilton gives the general formula for the reciprocal of a bi-quaternion:
— Hobojaks ( talk) 20:36, 28 November 2008 (UTC)
If a biquaterion does not always have an inverse, why does making the imaginary scalar anti-commutative and anti-associative thus making an octonian change the system back into a division algebra?
Hobojaks ( talk) 18:19, 26 November 2008 (UTC)
Since biquaternions and 2 x 2 complex matrices are algebras with different bases, they are not exactly the same thing, though they are isomorphic algebras. In historical or educational contexts, reference to biquaternions is natural and valid. With the sophisticated modern view of spinor representation theory, contemporary authors need make no reference to biquaternions when dealing with 2 x 2 complex matrices. But this encyclopedia is for the general public and the policy WP:NPOV explicitly prohibits pejorative remarks such as:
that was contributed by a WP:User on January 14 . Therefore I am undoing the contribution. Rgdboer ( talk) 22:32, 24 January 2010 (UTC)
In reply to the comment;
The term "biquaternion" is archaic and no longer used much by mathematicians, because the algebra of biquaternions is isomorphic to the algebra of 2 by 2 complex matrices.
If you search the internet, then you will find this is not true. In particular, I reference you to this physics essay on biquaternions and the things they deduce in modern physics (pretty much all of it). http://arxiv.org/PS_cache/math-ph/pdf/0201/0201058v5.pdf
"To conclude this introduction, let us summarize our main point: If quaternions are used consistently in theoretical physics, we get a comprehensive and consistent description of the physical world, with relativistic and quantum effects easily taken into account. In other words, we claim that Hamilton’s conjecture, the very idea which motivated more then half of his professional life, i.e., the concept that somehow quaternions are a fundamental building block of the physical universe, appears to be essentially correct in the light of contemporary knowledge."
Secondly, I believe I now know why these equations are central to physics and physical reality (I discovered this just a few weeks ago). I have tried to explain this as simply as I am able (below) but I ask that those who better understand the mathematics of biquaternion wave functions check this to confirm it is true (I am a natural philosopher, I came to the solution not from mathematics but from the spherical standing wave structure of matter (WSM)).
If you describe reality most simply (Occam’s razor) then you must describe reality in terms of only one substance, and since we all experience existing in one common space it is reasonable (consistent with science) to take this as our foundation (Hamilton thought similarly, why he devoted his life to developing maths for real objects in real 3D space).
Since the wave properties of light and matter are well known (the particle wave duality) and since we cannot add another separate substance, matter particles, to space, but we can have space vibrating (waves flowing through space), we can then use our biquaternion wave equations and see what we get based upon this most simple conception of physical reality. i.e.
“One thing, three dimensional space exists and has planar waves flowing through it in all directions.”
If we then apply biquaternion wave functions to this we find a most stunning thing. Space can actually vibrate in two completely different ways.
1. Space vibrates in all directions (background space, quantum field, vacuum fluctuations, Tao, Akasha Prana, ...)
2. Space vibrates radially around a central point, forming a scalar spherical standing wave. In this special case the biquaternions show how the vector / transverse wave components all cancel one another, leaving a scalar spherical standing wave. The wave center is what we see as matter 'particles'
The most profound thing is that the equations give us the Dirac equation – but now we can understand the cause of spin and antimatter. The biquaternion wave equations show us that there are four different phase arrangements for the transverse waves that cause them to cancel one another – and they create two pairs of scalar spherical standing waves that have opposite phase (matter and antimatter) and for each phase there are two phase arrangements that can construct it (the two spin states of the electron and positron). This is represented by the biquaternion multiplied by its complex conjugate (I assume this represents the waves flowing in opposite directions and thus opposite vectors) and the result is a scalar spherical standing wave (the vector / transverse wave components cancel).
And from this most simple science foundation for reality – just waves flowing through space in all directions - it seems (thanks to the work of many brilliant mathematicians) that you can then perfectly explain and unite quantum physics and Einstein’s relativity and exactly deduce all the central equations of modern physics. To confirm this you just need to search biquaternions and each subject area of physics and you will find the solutions, certainly Mendel Sach’s has done a lot in the area of general relativity.
What seems truly remarkable is that we have solved all the mathematics first, without ever understanding the amazingly simple physical reality behind it all that caused it. However, it should be acknowledged that Clifford was partly correct with his work "On the Space-Theory of Matter". He wrote;
"I hold: 1) That small portions of space are in fact analogous to little hills on a surface which is on the average flat, namely that the ordinary laws of geometry are not valid in them. 2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. 3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 4) That in this physical world nothing else takes place but this variation subject to the law of continuity." http://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter
Finally, with respect to the argument that quaternions are archaic because we now have more advanced maths that are more flexible and can work in infinite dimensions. What they forget is that reality determines the mathematics we must use to describe it (thus the current confusions of higher dimensions), and the reality we all experience is three dimensional space (where we can now deduce that the further 4th dimension of time is cause by the wave motion of this space). This is why biquaternions are so useful and important to mathematical physics (and to humanity), it seems their structure correctly represents the structure of physical reality. Geoffrey Haselhurst (14/05/2011) Haselhurst ( talk) 06:09, 14 May 2011 (UTC) —Preceding unsigned comment added by Haselhurst ( talk • contribs) 05:19, 14 May 2011 (UTC)
The tag asking for more inline citations has been removed. This is a mathematical article and assertions are verifiable. If a reader or editor requires further edification, please use this Talk space to make comments or raise questions. Rgdboer ( talk) 23:24, 20 January 2013 (UTC)
Without reading this entire article line by line, I was confused by the claim about the Linear representation that "each of these three arrays has a square equal to the negative of the identity matrix." Let's compute:
When I square the first array, I get h2 times the identity matrix:
When I first looked, I didn't see the Definition that h is "the square root of minus one".
By the way: Is "the square root of minus one" in this context the same as the imaginary unit? I assume so -- and if so, it might be nice to link to that Wikipedia on imaginary unit.
I"m making a change of this nature.
Thanks, DavidMCEddy ( talk) 09:10, 14 December 2015 (UTC)
The following was removed as unreferenced:
CA texts don't highlight biquaternions so why should this article direct to CA ? Compared to composition algebras, CA,s have weak foundation. — Rgdboer ( talk) 02:43, 1 February 2017 (UTC)
The issue of coinage is pertinent to this article since proponents of Geometric Algebra assert use of namespace bivector for a 2-vector in multilinear algebra, diminishing the claim to the namespace by those attentive to literature in mathematics using the term who are now compelled to revert to bivector (complex) to express the vector part of a biquaternion. You are encouraged to remove the script notations of your team, as the tenure at Bivector is but one piece of kompromat that can be displayed. Any editor, understanding what is at stake, can do it for you. — Rgdboer ( talk) 21:47, 30 August 2017 (UTC)
I have removed the above reference again as I was unable to find it or where it was published, so am not persuaded it is a reliable source. If it’s online then there should be a link. If it’s in a journal there should be a DOI. Then anyone can check it’s reliability or otherwise. The Youtube videos are certainly not a reliable source, and links to Youtube do not belong in articles.-- JohnBlackburne words deeds 07:58, 18 August 2017 (UTC)
Dear JohnBlackburne,
I have checked the regulations on videos, and there is no restriction against chalkboard presentations from youtube. Could you please demonstrate otherwise?
Also, as I have already mentioned above, the material in these videos has been published in PRD, a well known peer reviewed journal in particle physics. The content is the same in the video as what has already been published, but the videos provide a visual way to present the content.
Throughout our discussion, you have offered many reasons for why you feel this link should be removed. This includes stating that the links are not references (they are), stating that they are not good links (they are), or stating that youtube is not acceptable due to advertising, etc (however I have not been able to find anything banning youtube in the regulations). This makes it difficult to understand your motivation.
Theor-phys ( talk) 12:49, 18 August 2017 (UTC)
Video based on a PhD thesis?
That's just pushing it too far. I'll remove it. YohanN7 ( talk) 10:08, 28 August 2017 (UTC)
I raised the question (about possible self-promotion) at Wikipedia talk:WikiProject Mathematics#Talk:Biquaternion. YohanN7 ( talk) 10:46, 29 August 2017 (UTC)
I think I found a solution everybody can live with. YohanN7 ( talk) 13:07, 1 September 2017 (UTC)
At the end of the section /info/en/?search=Biquaternion#Associated_terminology, there is a mention of a "complex light cone" method that has superseded the biquaternions. Can someone explain what this is? Cheers. — Preceding unsigned comment added by 82.13.14.165 ( talk) 05:42, 27 May 2019 (UTC)
The notation of and to denote the 2 different conjugations is bad; they can't even be distinguished if the font is small. As a replacement, I would suggest these ones:
,
,
also let .
These ones will match operations on biquanternions represented as commutators of gamma matrices (shown to the very right above):
,
,
;
where and . 150.135.165.6 ( talk) 06:45, 15 April 2023 (UTC)
I was looking into the reference article on the table addition made on February 20 added by an IP (not a WP user). The link https://www.naturalspublishing.com/files/published/e71f3zs34zg62q.pdf has this table, but it simply defers to a reference to another article [12] S. J. Sangwine, T. A. Ell, N. Le Bihan, Adv. Appl. Clifford Algebras 21, 607-636 (2011).
I couldn't find that specific published article online, but I did find a well referenced article by the same authors from the prior year 2010 arXiv:1001.0240v1 [math.RA] 1 Jan 2010 Fundamental representations and algebraic properties of biquaternions. The table in that is much different (no way to get from that to the table in this WP Biquaternion), but looks right. Other references I find to biquaternion tables seem closer to this and I am concerned the one on the WP page might be wrong (or unsubstantiated) without further published source detail.
:: Jgmoxness ( talk) 20:05, 8 May 2024 (UTC)
A Biquaternion multiplication table Naturalspublishing is shown below:
Opinions requested. — Rgdboer ( talk) 01:57, 9 May 2024 (UTC)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||
|
There seems to be a conflict of terminology with regards to the name biquaternion. Hamilton appears to have used the term to mean a quaternion with complex coefficients (i.e. C⊗H), while Clifford (in Preliminary Sketch of Biquaternions, 1873) uses the term to mean an algebra isomorphic to H⊕H, which follows the quaternions in the sequence of Clifford algebras:
The complexified quaternions are not isomorphic to Clifford's biquaternions. This page presently discusses Hamilton's notion, while the German version of the page discuss's Clifford's notion. Some mention should be made of the conflict. I'm not sure which term is more commonly used. -- Fropuff 21:43, 19 February 2006 (UTC)
Note that Hamilton used the term first (it appears in his 'Lectures on Quaternions', 1853, article 669, available at http://historical.library.cornell.edu/math/). Sangwine 21:50, 25 February 2007 (UTC)
Since the structure of Clifford biquaternions is demonstrably different than the classical twentieth century concept of biquaternions used to develop the relativity transformations, the works of the Clifford algebraists on their biquaternion need a separate space. Rgdboer 01:35, 23 February 2006 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:44, 10 November 2007 (UTC)
First congratulations on this fantastic new article. I hope to watch it expand with time. These would be good things to explain in this article.
Does a biquaterion have an inverse, all the time, some of the time or none of the time?
It has to at least some of the time I think because if the imaginary part is zero, it would have an inverse, but this is trivial.
Letting h equal the imaginary scalar of ordinary algebra, since i,j and k are already taken, it would seem to me that the simple biquaterion:
hi would be its own inverse.
When D ≠ 0, then q = u + vi + wj + xk has an inverse, otherwise it is singular. Rgdboer ( talk) 20:10, 26 November 2008 (UTC)
I found were Hamilton gives the general formula for the reciprocal of a bi-quaternion:
— Hobojaks ( talk) 20:36, 28 November 2008 (UTC)
If a biquaterion does not always have an inverse, why does making the imaginary scalar anti-commutative and anti-associative thus making an octonian change the system back into a division algebra?
Hobojaks ( talk) 18:19, 26 November 2008 (UTC)
Since biquaternions and 2 x 2 complex matrices are algebras with different bases, they are not exactly the same thing, though they are isomorphic algebras. In historical or educational contexts, reference to biquaternions is natural and valid. With the sophisticated modern view of spinor representation theory, contemporary authors need make no reference to biquaternions when dealing with 2 x 2 complex matrices. But this encyclopedia is for the general public and the policy WP:NPOV explicitly prohibits pejorative remarks such as:
that was contributed by a WP:User on January 14 . Therefore I am undoing the contribution. Rgdboer ( talk) 22:32, 24 January 2010 (UTC)
In reply to the comment;
The term "biquaternion" is archaic and no longer used much by mathematicians, because the algebra of biquaternions is isomorphic to the algebra of 2 by 2 complex matrices.
If you search the internet, then you will find this is not true. In particular, I reference you to this physics essay on biquaternions and the things they deduce in modern physics (pretty much all of it). http://arxiv.org/PS_cache/math-ph/pdf/0201/0201058v5.pdf
"To conclude this introduction, let us summarize our main point: If quaternions are used consistently in theoretical physics, we get a comprehensive and consistent description of the physical world, with relativistic and quantum effects easily taken into account. In other words, we claim that Hamilton’s conjecture, the very idea which motivated more then half of his professional life, i.e., the concept that somehow quaternions are a fundamental building block of the physical universe, appears to be essentially correct in the light of contemporary knowledge."
Secondly, I believe I now know why these equations are central to physics and physical reality (I discovered this just a few weeks ago). I have tried to explain this as simply as I am able (below) but I ask that those who better understand the mathematics of biquaternion wave functions check this to confirm it is true (I am a natural philosopher, I came to the solution not from mathematics but from the spherical standing wave structure of matter (WSM)).
If you describe reality most simply (Occam’s razor) then you must describe reality in terms of only one substance, and since we all experience existing in one common space it is reasonable (consistent with science) to take this as our foundation (Hamilton thought similarly, why he devoted his life to developing maths for real objects in real 3D space).
Since the wave properties of light and matter are well known (the particle wave duality) and since we cannot add another separate substance, matter particles, to space, but we can have space vibrating (waves flowing through space), we can then use our biquaternion wave equations and see what we get based upon this most simple conception of physical reality. i.e.
“One thing, three dimensional space exists and has planar waves flowing through it in all directions.”
If we then apply biquaternion wave functions to this we find a most stunning thing. Space can actually vibrate in two completely different ways.
1. Space vibrates in all directions (background space, quantum field, vacuum fluctuations, Tao, Akasha Prana, ...)
2. Space vibrates radially around a central point, forming a scalar spherical standing wave. In this special case the biquaternions show how the vector / transverse wave components all cancel one another, leaving a scalar spherical standing wave. The wave center is what we see as matter 'particles'
The most profound thing is that the equations give us the Dirac equation – but now we can understand the cause of spin and antimatter. The biquaternion wave equations show us that there are four different phase arrangements for the transverse waves that cause them to cancel one another – and they create two pairs of scalar spherical standing waves that have opposite phase (matter and antimatter) and for each phase there are two phase arrangements that can construct it (the two spin states of the electron and positron). This is represented by the biquaternion multiplied by its complex conjugate (I assume this represents the waves flowing in opposite directions and thus opposite vectors) and the result is a scalar spherical standing wave (the vector / transverse wave components cancel).
And from this most simple science foundation for reality – just waves flowing through space in all directions - it seems (thanks to the work of many brilliant mathematicians) that you can then perfectly explain and unite quantum physics and Einstein’s relativity and exactly deduce all the central equations of modern physics. To confirm this you just need to search biquaternions and each subject area of physics and you will find the solutions, certainly Mendel Sach’s has done a lot in the area of general relativity.
What seems truly remarkable is that we have solved all the mathematics first, without ever understanding the amazingly simple physical reality behind it all that caused it. However, it should be acknowledged that Clifford was partly correct with his work "On the Space-Theory of Matter". He wrote;
"I hold: 1) That small portions of space are in fact analogous to little hills on a surface which is on the average flat, namely that the ordinary laws of geometry are not valid in them. 2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. 3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 4) That in this physical world nothing else takes place but this variation subject to the law of continuity." http://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter
Finally, with respect to the argument that quaternions are archaic because we now have more advanced maths that are more flexible and can work in infinite dimensions. What they forget is that reality determines the mathematics we must use to describe it (thus the current confusions of higher dimensions), and the reality we all experience is three dimensional space (where we can now deduce that the further 4th dimension of time is cause by the wave motion of this space). This is why biquaternions are so useful and important to mathematical physics (and to humanity), it seems their structure correctly represents the structure of physical reality. Geoffrey Haselhurst (14/05/2011) Haselhurst ( talk) 06:09, 14 May 2011 (UTC) —Preceding unsigned comment added by Haselhurst ( talk • contribs) 05:19, 14 May 2011 (UTC)
The tag asking for more inline citations has been removed. This is a mathematical article and assertions are verifiable. If a reader or editor requires further edification, please use this Talk space to make comments or raise questions. Rgdboer ( talk) 23:24, 20 January 2013 (UTC)
Without reading this entire article line by line, I was confused by the claim about the Linear representation that "each of these three arrays has a square equal to the negative of the identity matrix." Let's compute:
When I square the first array, I get h2 times the identity matrix:
When I first looked, I didn't see the Definition that h is "the square root of minus one".
By the way: Is "the square root of minus one" in this context the same as the imaginary unit? I assume so -- and if so, it might be nice to link to that Wikipedia on imaginary unit.
I"m making a change of this nature.
Thanks, DavidMCEddy ( talk) 09:10, 14 December 2015 (UTC)
The following was removed as unreferenced:
CA texts don't highlight biquaternions so why should this article direct to CA ? Compared to composition algebras, CA,s have weak foundation. — Rgdboer ( talk) 02:43, 1 February 2017 (UTC)
The issue of coinage is pertinent to this article since proponents of Geometric Algebra assert use of namespace bivector for a 2-vector in multilinear algebra, diminishing the claim to the namespace by those attentive to literature in mathematics using the term who are now compelled to revert to bivector (complex) to express the vector part of a biquaternion. You are encouraged to remove the script notations of your team, as the tenure at Bivector is but one piece of kompromat that can be displayed. Any editor, understanding what is at stake, can do it for you. — Rgdboer ( talk) 21:47, 30 August 2017 (UTC)
I have removed the above reference again as I was unable to find it or where it was published, so am not persuaded it is a reliable source. If it’s online then there should be a link. If it’s in a journal there should be a DOI. Then anyone can check it’s reliability or otherwise. The Youtube videos are certainly not a reliable source, and links to Youtube do not belong in articles.-- JohnBlackburne words deeds 07:58, 18 August 2017 (UTC)
Dear JohnBlackburne,
I have checked the regulations on videos, and there is no restriction against chalkboard presentations from youtube. Could you please demonstrate otherwise?
Also, as I have already mentioned above, the material in these videos has been published in PRD, a well known peer reviewed journal in particle physics. The content is the same in the video as what has already been published, but the videos provide a visual way to present the content.
Throughout our discussion, you have offered many reasons for why you feel this link should be removed. This includes stating that the links are not references (they are), stating that they are not good links (they are), or stating that youtube is not acceptable due to advertising, etc (however I have not been able to find anything banning youtube in the regulations). This makes it difficult to understand your motivation.
Theor-phys ( talk) 12:49, 18 August 2017 (UTC)
Video based on a PhD thesis?
That's just pushing it too far. I'll remove it. YohanN7 ( talk) 10:08, 28 August 2017 (UTC)
I raised the question (about possible self-promotion) at Wikipedia talk:WikiProject Mathematics#Talk:Biquaternion. YohanN7 ( talk) 10:46, 29 August 2017 (UTC)
I think I found a solution everybody can live with. YohanN7 ( talk) 13:07, 1 September 2017 (UTC)
At the end of the section /info/en/?search=Biquaternion#Associated_terminology, there is a mention of a "complex light cone" method that has superseded the biquaternions. Can someone explain what this is? Cheers. — Preceding unsigned comment added by 82.13.14.165 ( talk) 05:42, 27 May 2019 (UTC)
The notation of and to denote the 2 different conjugations is bad; they can't even be distinguished if the font is small. As a replacement, I would suggest these ones:
,
,
also let .
These ones will match operations on biquanternions represented as commutators of gamma matrices (shown to the very right above):
,
,
;
where and . 150.135.165.6 ( talk) 06:45, 15 April 2023 (UTC)
I was looking into the reference article on the table addition made on February 20 added by an IP (not a WP user). The link https://www.naturalspublishing.com/files/published/e71f3zs34zg62q.pdf has this table, but it simply defers to a reference to another article [12] S. J. Sangwine, T. A. Ell, N. Le Bihan, Adv. Appl. Clifford Algebras 21, 607-636 (2011).
I couldn't find that specific published article online, but I did find a well referenced article by the same authors from the prior year 2010 arXiv:1001.0240v1 [math.RA] 1 Jan 2010 Fundamental representations and algebraic properties of biquaternions. The table in that is much different (no way to get from that to the table in this WP Biquaternion), but looks right. Other references I find to biquaternion tables seem closer to this and I am concerned the one on the WP page might be wrong (or unsubstantiated) without further published source detail.
:: Jgmoxness ( talk) 20:05, 8 May 2024 (UTC)
A Biquaternion multiplication table Naturalspublishing is shown below:
Opinions requested. — Rgdboer ( talk) 01:57, 9 May 2024 (UTC)