## [SOLVED] Re: &quot;The other quaternions&quot;: how useful are 2x2 real matrices to

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nDoug Sweetser wrote:\n\n&gt;&gt;The essential difference between the two is that, for the quaternions,\n&gt;&gt;the squares of all three of the non-real unit vectors i,j,k, are -1.\n&gt;\n&gt; If Hamilton would have chosen a righthanded system, this would be +1 I\n&gt; believe.\n\nThis is like saying that if Gauss had chosen a different convention\nfor the complex numbers then i^2=1.\n\nYou\'d be _much_ more careful about what you post.\nIn a couple of years you\'ll probably be ashamed\nof what you posted in the last few months.\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Doug Sweetser wrote:

>>The essential difference between the two is that, for the quaternions,
>>the squares of all three of the non-real unit vectors i,j,k, are -1.

>
> If Hamilton would have chosen a righthanded system, this would be $+1 I$
> believe.

This is like saying that if Gauss had chosen a different convention
for the complex numbers then $i^2=1$.

You'd be _much_ more careful about what you post.
In a couple of years you'll probably be ashamed
of what you posted in the last few months.

Arnold Neumaier

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Hello: Could someone remind me of how the difference in choice of handedness affects the algebra of quaternions? I cannot find the reference I had on the issue. Hamilton's way of writing them is only one representation. By the way, Gauss was the first to discover quaternions. It was in one of his notebooks. doug



Vaughan Pratt wrote: > While I'm familiar with some (but perhaps not all) of the applications > of the geometry of the quaternions to physics, I haven't encountered > any for the corresponding geometry of the 2x2 matrices. I'd therefore > greatly appreciate hearing about their applications to physics as a > noncommutative geometry. Commonplace physical phenomena especially > welcome. A similar issue arises with the Dirac algebra on spacetime - that is CL(3,1) has a real representation in terms of 4x4 real matrices (Majorana representation) (likewise CL(2,2)), while CL(1,3) is 2x2 quaternions. In the physical world, the Majorana representation and its spinors don't seem to correspond to real particles. So, while there may be some special configuration of, say, photons with a special polarization that have CL(1,1) as a basis, one shouldn't necessarily expect a general correspondence with physical systems. In the Dirac schema the Majorana particle is a neutral Fermion (massive or not) that is its own antiparticle, and this latter is what makes it unphysical (that is, it's not really a lepton). So the real physics (and mystery) is living in the idea of the unit pseudoscalar on whatever Clifford algebra comes under consideration. If Gauss were alive he'd say "The true metaphysics of $\gamma_5$ is hard" :) $$-drl$$

## [SOLVED] Re: &quot;The other quaternions&quot;: how useful are 2x2 real matrices to

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Doug Sweetser wrote:\n\n&gt; Could someone remind me of how the difference in choice of handedness\n&gt; affects the algebra of quaternions? I cannot find the reference I had\n&gt; on the issue. Hamilton\'s way of writing them is only one\n&gt; representation.\n\nIt has nothing at all to do with it - quaternions as such are not tied\nto space geometry. But, if you make j correspond to the z-axis and k to\nthe y-axis, then you\'ve adopted a left-handed convention.\n\n&gt; By the way, Gauss was the first to discover quaternions. It was in one\n&gt; of his notebooks.\n\nI don\'t know where you "discovered" this fact but I\'d bet my last penny\non it being false. I suppose we must now add Hamilton to the growing\nlist of post-modernist bashees.\n\n-drl\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Doug Sweetser wrote:

> Could someone remind me of how the difference in choice of handedness
> affects the algebra of quaternions? I cannot find the reference I had
> on the issue. Hamilton's way of writing them is only one
> representation.

It has nothing at all to do with $it -$ quaternions as such are not tied
to space geometry. But, if you make j correspond to the z-axis and k to
the y-axis, then you've adopted a left-handed convention.

> By the way, Gauss was the first to discover quaternions. It was in one
> of his notebooks.

I don't know where you "discovered" this fact but I'd bet my last penny
on it being false. I suppose we must now add Hamilton to the growing
list of post-modernist bashees.

$$-drl$$



Danny Ross Lunsford wrote: > It has nothing at all to do with $it -$ quaternions as such are not tied > to space geometry. But, if you make j correspond to the z-axis and k > to the y-axis, then you've adopted a left-handed convention. That is the issue I was referring too. >> By the way, Gauss was the first to discover quaternions. It was in >> one of his notebooks. > > I don't know where you "discovered" this fact but I'd bet my last > penny on it being false. I was surprised to read it too. Here is what some googling revealed: from http://www.infoplease.com/ce6/people/A0820346.html: Gauss was extremely careful and rigorous in all his work, insisting on a complete proof of any result before he would publish it. As a consequence, he made many discoveries that were not credited to him and had to be remade by others later; for example, he anticipated Bolyai and Lobachevsky in non-Euclidean geometry, Jacobi in the double periodicity of elliptic functions, Cauchy in the theory of functions of a complex variable, and Hamilton in quaternions. However, his published works were enough to establish his reputation as one of the greatest mathematicians of all time. It was only noticed many years after the fact. Hamilton should never be mentioned solo anyway, since Rodriguez discovered them independently, and with a purpose in mind, namely rotations. These are just historical tales. Hamilton gets so much more space because he wrote about the process of this particular discovery, which is not common. Few facts are even known about Rodriguez. He may have been a banker of Spanish decent, but my memory is vague here too. doug



In article , Doug Sweetser wrote: >Could someone remind me of how the difference in choice of handedness >affects the algebra of quaternions? I cannot find the reference I had >on the issue. Hamilton's way of writing them is only one >representation. A vector is a quaternion with a zero real part, and a unit vector x is one satisfying $x^2 = -1, i$.e. quaternion multiplication supplies a norm. Pick any two unit vectors x,y such that 1,x,y,xy are linearly independent. (So the multiplication supplies not just a norm but an inner product.) The set of coordinate systems of the form x,y,xy obtainable in this way is closed under rigid rotations. The set of those of the form $x,y,-xy (or y,x,xy, or x,y,yx)$ is also so closed, and consists of the reflections of the other set about an odd number of axes. Call these two sets respectively the right- and left-handed systems. The beauty of this definition is that it depends only on the operations of the algebra of quaternions, and is independent of any notational conventions Hamilton or anyone else might use. Is handedness defined $and/or$ useful for the enumeration i,1,j? (I'm asking, I don't know.) If either, then I believe there can be only two internally consistent nontrivial conventions, describable as follows. Take the 6 oriented necklaces containing beads 1,i,j,k (one of each) and take the handedness of the necklace obtained by deleting the bead opposite 1 to be always equal to that of the necklace obtained by deleting 1. The other convention is to make them always not equal. Reasoning: the position of 1 in (e.g.) i,1,j should not affect handedness (1 is not physical and should commute with everything), only the order of i and j. (Then there's the two trivial conventions: make all triples containing 1 right-handed, or its opposite convention, but I would become suddenly and violently ill if there were any mathematical reason for having to choose between those two conventions.) What I don't see is why the order of i and j should affect anything. Is there in fact a reason to define handedness of i,1,j? For example is there a natural homology of the quaternions that entails a handedness? >By the way, Gauss was the first to discover quaternions. It was in one >of his notebooks. Another area where I feel rather in the dark. I'd be interested in knowing a little more detail about exactly what was in the notebook. He wrote in an 1846 letter to Schumacher about Lobachevsky's geometry that he had "had the same conviction for 54 years," i.e. since the age of 15, but it is one thing to have a conviction and quite another to have the courage of that conviction to come out with it. Did he keep quiet about his ideas on nonstandard geometries because he thought they wouldn't be accepted, or because of a concern the authorities might treat him like Galileo for such spatial heresies, or because he didn't think the world was ready for it yet (he tended to act that way concerning his use of complex numbers in number theory, which he was in the habit of translating away), or because he was having difficulty getting the details to work out right? Hamilton famously had trouble figuring out the quaternions, until it hit him in a flash that the real axis should be a *separate* dimension from the three physical dimensions, commuting with everything while the physical dimensions had to anticommute. (Having that sudden insight on such a fundamental question as the algebra of 3-space must have been fun!) Vaughan Pratt -- Don't contact me at pratt@boole.stanford.edu, substitute cs for boole instead.



Doug Sweetser wrote: [....] "It must be true, I found it on the internet!" If Gauss had acutally invented non-commutative algebra, he would get credit for it, insofar as it behind most all of particle physics and a lot of relativity. The true story of quaternions is well-known. The key idea only occured to Hamilton after 10 years of long hard thinking. $$-drl$$