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

Email me at CS not Boole

Apr13-04, 05:51 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>Taking "dimension" to mean that of the underlying vector space, there\nare only two four-dimensional real Clifford algebras, namely the\nquaternions and the 2x2 real matrices. The only other name I know for\nthe quaternions is the Clifford algebra CL(0,2). However the 2x2 real\nmatrices have a serious split personality and many aliases: CL(2,0),\nthe split Clifford algebra CL(1,1), the split quaternions, the unipodal\nnumbers, the complex hyperbolic numbers, the hyperbolic complex\nnumbers, etc.\n\nThe essential difference between the two is that, for the quaternions,\nthe squares of all three of the non-real unit vectors i,j,k, are -1.\nThe 2x2 real matrices on the other hand have only one such, namely i.\n\nWhile I\'m familiar with some (but perhaps not all) of the applications\nof the geometry of the quaternions to physics, I haven\'t encountered\nany for the corresponding geometry of the 2x2 matrices. I\'d therefore\ngreatly appreciate hearing about their applications to physics as a\nnoncommutative geometry. Commonplace physical phenomena especially\nwelcome.\n\nVaughan Pratt\n--\nDon\'t contact me at pratt@boole.stanford.edu, substitute cs for boole instead.\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>Taking "dimension" to mean that of the underlying vector space, there
are only two four-dimensional real Clifford algebras, namely the
quaternions and the 2x2 real matrices. The only other name I know for
the quaternions is the Clifford algebra CL(0,2). However the 2x2 real
matrices have a serious split personality and many aliases: CL(2,0),
the split Clifford algebra CL(1,1), the split quaternions, the unipodal
numbers, the complex hyperbolic numbers, the hyperbolic complex
numbers, etc.

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.
The 2x2 real matrices on the other hand have only one such, namely i.

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.

Vaughan Pratt
--
Don't contact me at pratt@boole.stanford.edu, substitute cs for boole instead.

Doug Sweetser

Apr14-04, 03:16 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>Hello Vaughan:\n\nSince 1997, it has been a hobby of mine to find uses in physics for\nquaternions. You can check out my collection at the URL below.\n\nOne thing I know from experience is that it is tough to communicate any\nfindings because it is not in the lingo of physicists. Let me give one\nexample that I like, Newton\'s 2nd law in polar coordinates for a\ncentral force in a plane. If you get out a freshman physics book, they\nwill explain this one in two or three pages. Here is the couple of\nline explanation using quaternions for the acceleration:\n\nA = (Cos(B), 0, 0, -Sin(B)) (d/dt,0,0,0)^2 (t, r Cos(B), r Sin(B), 0)\n\n= (0, L^2/m^2 r^3 + d^2 r/dt^2, 2 L (dr/dt) /m r^2, 0)\n\nwhere dB/dt = L/m r^2\nd^2 B/dt^2 = 0\n\nAs if this were Hebrew, start on the right with polar coordinates in\nthe x-y plane. Look at the Newtonian acceleration by hitting it with\ntwo time derivatives. The chain rule makes a bunch of terms with sines\nand cosines. Fortunately the second time derivative of the angle drop.\nNow rotate with the left most term to wipe the blackboard of trig\nfunctions. For me, it does make a difference that a few lines of\nmultiplication can take the place of an essay.\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\nIf Hamilton would have chosen a righthanded system, this would be +1 I\nbelieve.\n\n> The 2x2 real matrices on the other hand have only one such, namely i.\n\nThe deep difference is that quaternions are a division algebra, and the\n2x2 real matrices are not.\n\n> I\'d therefore greatly appreciate hearing about their applications to\n> physics as a noncommutative geometry. Commonplace physical phenomena\n> especially welcome.\n\nGood luck in your search,\ndoug\nquaternions.com\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>Hello Vaughan:

Since 1997, it has been a hobby of mine to find uses in physics for
quaternions. You can check out my collection at the URL below.

One thing I know from experience is that it is tough to communicate any
findings because it is not in the lingo of physicists. Let me give one
example that I like, Newton's 2nd law in polar coordinates for a
central force in a plane. If you get out a freshman physics book, they
will explain this one in two or three pages. Here is the couple of
line explanation using quaternions for the acceleration:

A = (Cos(B), 0, 0, -Sin(B)) (d/dt,0,0,0)^2 (t, r Cos(B), r Sin(B), 0)

= (0, L^2/m^2 r^3 + d^2 r/dt^2, 2 L (dr/dt) /m r^2, 0)

where dB/dt = L/m r^2
d^2 B/dt^2 =

As if this were Hebrew, start on the right with polar coordinates in
the x-y plane. Look at the Newtonian acceleration by hitting it with
two time derivatives. The chain rule makes a bunch of terms with sines
and cosines. Fortunately the second time derivative of the angle drop.
Now rotate with the left most term to wipe the blackboard of trig
functions. For me, it does make a difference that a few lines of
multiplication can take the place of an essay.

> 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.

> The 2x2 real matrices on the other hand have only one such, namely i.

The deep difference is that quaternions are a division algebra, and the
2x2 real matrices are not.

> I'd therefore greatly appreciate hearing about their applications to
> physics as a noncommutative geometry. Commonplace physical phenomena
> especially welcome.

Good luck in your search,
doug
quaternions.com