[SOLVED] Re: "The other quaternions": how useful are 2x2 real matrices to [Archive]

Arnold Neumaier

Apr14-04, 08:36 AM

<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>>The essential difference between the two is that, for the quaternions,\n>>the squares of all three of the non-real unit vectors i,j,k, are -1.\n>\n> If Hamilton would have chosen a righthanded system, this would be +1 I\n> 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">  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

Doug Sweetser

Apr15-04, 04:03 AM

<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\n\nHello:\n\nCould someone remind me of how the difference in choice of handedness\naffects the algebra of quaternions? I cannot find the reference I had\non the issue. Hamilton\'s way of writing them is only one\nrepresentation.\n\nBy the way, Gauss was the first to discover quaternions. It was in one\nof his notebooks.\n\n\ndoug\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>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

Danny Ross Lunsford

Apr15-04, 11:24 AM

<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>Vaughan Pratt wrote:\n\n> While I\'m familiar with some (but perhaps not all) of the applications\n> of the geometry of the quaternions to physics, I haven\'t encountered\n> any for the corresponding geometry of the 2x2 matrices. I\'d therefore\n> greatly appreciate hearing about their applications to physics as a\n> noncommutative geometry. Commonplace physical phenomena especially\n> welcome.\n\nA similar issue arises with the Dirac algebra on spacetime - that is\nCL(3,1) has a real representation in terms of 4x4 real matrices\n(Majorana representation) (likewise CL(2,2)), while CL(1,3) is 2x2\nquaternions. In the physical world, the Majorana representation and its\nspinors don\'t seem to correspond to real particles. So, while there may\nbe some special configuration of, say, photons with a special\npolarization that have CL(1,1) as a basis, one shouldn\'t necessarily\nexpect a general correspondence with physical systems.\n\nIn the Dirac schema the Majorana particle is a neutral Fermion (massive\nor not) that is its own antiparticle, and this latter is what makes it\nunphysical (that is, it\'s not really a lepton). So the real physics (and\nmystery) is living in the idea of the unit pseudoscalar on whatever\nClifford algebra comes under consideration. If Gauss were alive he\'d say\n"The true metaphysics of gamma_5 is hard" :)\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>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

Danny Ross Lunsford

Apr15-04, 11:39 AM

<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> Could someone remind me of how the difference in choice of handedness\n> affects the algebra of quaternions? I cannot find the reference I had\n> on the issue. Hamilton\'s way of writing them is only one\n> 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> By the way, Gauss was the first to discover quaternions. It was in one\n> 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">  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

Doug Sweetser

Apr16-04, 02:28 AM

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

Email me at CS not Boole

Apr16-04, 05:51 AM

<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\nIn article <c5kvnh\\$vc1\\$1@pcls4.std.com>,\nDoug Sweetser <sweetser@alum.mit.edu> wrote:\n>Could someone remind me of how the difference in choice of handedness\n>affects the algebra of quaternions? I cannot find the reference I had\n>on the issue. Hamilton\'s way of writing them is only one\n>representation.\n\nA vector is a quaternion with a zero real part, and a unit vector x is\none satisfying x^2 = -1, i.e. quaternion multiplication supplies a\nnorm.\n\nPick any two unit vectors x,y such that 1,x,y,xy are linearly\nindependent. (So the multiplication supplies not just a norm but an\ninner product.) The set of coordinate systems of the form x,y,xy\nobtainable in this way is closed under rigid rotations. The set of\nthose of the form x,y,-xy (or y,x,xy, or x,y,yx) is also so closed, and\nconsists of the reflections of the other set about an odd number of\naxes. Call these two sets respectively the right- and left-handed\nsystems.\n\nThe beauty of this definition is that it depends only on the operations\nof the algebra of quaternions, and is independent of any notational\nconventions Hamilton or anyone else might use.\n\nIs handedness defined and/or useful for the enumeration i,1,j? (I\'m\nasking, I don\'t know.) If either, then I believe there can be only two\ninternally consistent nontrivial conventions, describable as follows.\nTake the 6 oriented necklaces containing beads 1,i,j,k (one of each)\nand take the handedness of the necklace obtained by deleting the bead\nopposite 1 to be always equal to that of the necklace obtained by\ndeleting 1. The other convention is to make them always not equal.\nReasoning: the position of 1 in (e.g.) i,1,j should not affect\nhandedness (1 is not physical and should commute with everything), only\nthe order of i and j. (Then there\'s the two trivial conventions: make\nall triples containing 1 right-handed, or its opposite convention, but\nI would become suddenly and violently ill if there were any\nmathematical reason for having to choose between those two\nconventions.)\n\nWhat I don\'t see is why the order of i and j should affect anything.\nIs there in fact a reason to define handedness of i,1,j? For example\nis there a natural homology of the quaternions that entails a\nhandedness?\n\n>By the way, Gauss was the first to discover quaternions. It was in one\n>of his notebooks.\n\nAnother area where I feel rather in the dark. I\'d be interested in\nknowing a little more detail about exactly what was in the notebook.\nHe wrote in an 1846 letter to Schumacher about Lobachevsky\'s geometry\nthat he had "had the same conviction for 54 years," i.e. since the age\nof 15, but it is one thing to have a conviction and quite another to\nhave the courage of that conviction to come out with it. Did he keep\nquiet about his ideas on nonstandard geometries because he thought they\nwouldn\'t be accepted, or because of a concern the authorities might\ntreat him like Galileo for such spatial heresies, or because he didn\'t\nthink the world was ready for it yet (he tended to act that way\nconcerning his use of complex numbers in number theory, which he was in\nthe habit of translating away), or because he was having difficulty\ngetting the details to work out right? Hamilton famously had trouble\nfiguring out the quaternions, until it hit him in a flash that the real\naxis should be a *separate* dimension from the three physical\ndimensions, commuting with everything while the physical dimensions had\nto anticommute. (Having that sudden insight on such a fundamental\nquestion as the algebra of 3-space must have been fun!)\n\nVaughan Pratt\n--\nDon\'t contact me at pratt@boole.stanford.edu, substitute cs for boole instead.\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <c5kvnh$vc1$1@pcls4.std.com>,
Doug Sweetser <sweetser@alum.mit.edu> 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.

Danny Ross Lunsford

Apr19-04, 01:52 PM

<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[....]\n\n"It must be true, I found it on the internet!"\n\nIf Gauss had acutally invented non-commutative algebra, he would get\ncredit for it, insofar as it behind most all of particle physics and a\nlot of relativity.\n\nThe true story of quaternions is well-known. The key idea only occured\nto Hamilton after 10 years of long hard thinking.\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>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