Re: How to Field Theorize AND 3-Dimensional Conformal Invariance [Archive]

tessel@um.bot

Apr7-04, 08:34 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>Attention please Shagird and Aaron!\n\nRegarding what I just said in a post submitted to the "How to Field\nTheorize from a Lagrangian" thread, -case in point-:\n\nOn Sun, 4 Apr 2004, Alexander Kubelsky asked:\n\n> The statement is that group of conformal trasformations in dimensions\n> higher than 2 have finite number of elements, and only in D=2 it have\n> infinite number of elements (analytic functions).\n> Where can I read about this in details ?\n\nA good and timely question!\n\nIn the "How to Field Theorize" thread, I claimed that too few physics\nstudents are yet exposed to the tremendously useful methods of Lie for\ncomputing symmetry groups of PDEs (and its application to finding\nparticular solutions of nonlinear systems of coupled PDEs in particular).\nIn the case at hand, Alexander is in effect asking for the "point symmetry\ngroup" of the Laplace equation in higher dimensions. This is given (with\nthe method for finding it via Lie\'s "determining equations") in several\nexcellent textbooks. I particularly recommend:\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\nI\'ve actually gone through the computation of the required group--- in\ngreat detail--- in a previous post to this group, but I\'m willing to do it\nagain. (Since realized later that I made a silly error in the very last\nstep!)\n\nBut first I challenge Aaron and Shagird to solve a much simpler problem:\nwhat is the symmetry group of the axisymmetric 3-dimensional Laplace\nequation? Namely, in cylindrical coordinates\n\nphi_(zz) + phi_(rr) + phi_r/r = 0\n\nwhere phi is a function of z,r only?\n\nI can also answer the question asked, of course! I\'ll do that later today\nor tomorrow if Aaron/Shagird don\'t beat me to the punch. The correct\n-answer- can be found in either of the two books above, but I can write\nout a full derivation. But first, interested readers should read "What is\na Vector Field?", archived in "Relativity on the World Wide Web", which\nyou can find at John Baez\'s website. This contains some neccessary\nbackground which I suspect not everyone here yet knows.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>Attention please Shagird and Aaron!

Regarding what I just said in a post submitted to the "How to Field
Theorize from a Lagrangian" thread, -case in point-:

On Sun, 4 Apr 2004, Alexander Kubelsky asked:

> The statement is that group of conformal trasformations in dimensions
> higher than 2 have finite number of elements, and only in D=2 it have
> infinite number of elements (analytic functions).
> Where can I read about this in details ?

A good and timely question!

In the "How to Field Theorize" thread, I claimed that too few physics
students are yet exposed to the tremendously useful methods of Lie for
computing symmetry groups of PDEs (and its application to finding
particular solutions of nonlinear systems of coupled PDEs in particular).
In the case at hand, Alexander is in effect asking for the "point symmetry
group" of the Laplace equation in higher dimensions. This is given (with
the method for finding it via Lie's "determining equations") in several
excellent textbooks. I particularly recommend:

author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}

author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}

I've actually gone through the computation of the required group--- in
great detail--- in a previous post to this group, but I'm willing to do it
again. (Since realized later that I made a silly error in the very last
step!)

But first I challenge Aaron and Shagird to solve a much simpler problem:
what is the symmetry group of the axisymmetric 3-dimensional Laplace
equation? Namely, in cylindrical coordinates

\phi_(zz) + \phi_(rr) + \phi_r/r =

where \phi is a function of z,r only?

I can also answer the question asked, of course! I'll do that later today
or tomorrow if Aaron/Shagird don't beat me to the punch. The correct
-answer- can be found in either of the two books above, but I can write
out a full derivation. But first, interested readers should read "What is
a Vector Field?", archived in "Relativity on the World Wide Web", which
you can find at John Baez's website. This contains some neccessary
background which I suspect not everyone here yet knows.

"T. Essel" (hiding somewhere in cyberspace)

tessel@um.bot

Apr12-04, 10:05 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>Last week I foolishly promised to post here the full derivation of the Lie\nalgebra of the conformal group on R^3, as the Lie algebra of the "point\nsymmetries" of the three-dimensional Laplace equation. Alas, I don\'t have\ntime to do that after all, but the diligent reader can study the textbook\nI mentioned and work the exercise on the Laplace equation.\n\nLet me just say what the answer is:\n\nThe full Lie algebra of the point symmetries of the three-dimensional\nLaplace equation is "trivially" infinite dimensional. The generators can\nbe given as follows:\n\n(a) three translations:\n\nX1 = @/@x\n\nX2 = @/@y\n\nX3 = @/@z\n\n(b) three rotations:\n\nX4 = y @/@x - x @/@y\n\nX5 = y @/@z - z @/@y\n\nX6 = z @/@x - x @/@z\n\n(c) 1 dilation:\n\nX7 = x @/@x + y @/@y + z @/@z\n\n(d) 3 "Moebius rotations":\n\nX8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u\n\nX9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u\n\nX10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u\n\n(e) scalar multiple of u:\n\nX11 = u @/@u\n\n(f) addition of another harmonic function to u:\n\nXg = g @/@u\n\nwhere g is any harmonic function, i.e. g_(xx) + g_(yy) + g_(zz) = 0.\n\nHere, the family of generators (f) accounts for the fact that the algebra\nis trivially infinite dimensional. The symmetry algebra of a -linear- PDE\n(with dependent variable u) always contains (e)-(f) because linearity\nmeans that you can always combine linearly two solutions to get a third,\nindependently of doing anything with the independent variables. So the\nremaining generators, if any, are usually the interesting/informative\nones.\n\nIn this case, (a)-(d) generate a ten dimensional subalgebra which acts on\nthe space of independent variables, namely R^3. This is the Lie algebra\nof the conformal group of R^3, aka the Lie algebra of the\nthree-dimensional Moebius group acting on R^3 (treated as the\nstereographic image of S^3 minus a pole). You can delete the @/@u terms\nfrom (d) if you like, since we are only interested in the action of the\nMoebius group on R^3. Then our ten dimensional algebra obviously defines\na ten dimensional Lie group acting on R^3, as promised.\n\nThe generators (a)-(c) are presumably familiar. The last three appear in\nthe textbook exercise I cited -without- the initial ones. Of course,\nsubtracting the appropriate translation from the generators yields the\ngenerators given in the textbook exercise. Since those generators are\nvery slightly simpler than mine, why did I bother to include the initial\nones? Because my generators are precise analogues of the "Moebius\nrotation" of S^2 (stereographically projected to R^2) which has flowlines\ngiven by two families of circles nested about +/- i (in complex notation),\ndegenerating to the real axis.\n\nFor example,\n\nY8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z\n\n-fixes pointwise- the circle x^2 + y^2 = 1 (the stereographic image in R^3\nof a great circle on S^3) and leaves invariant (but doesn\'t fix pointwise)\nthe axis x = y = 0 (the stereographic image of the "opposite" great circle\non S^3). Indeed, Y8 leaves invariant the stereographic images of -the\nfull family of Hopf tori- on S^3 which are nested around these two great\ncircles, but "rolls" each one along its "longitudes".\n\nTo see the analogy, take a section by any plane through the axis x = y =\n0. The Hopf tori now appear as nested circles, and the flow "rotates"\nthese in the manner familiar from books like\n\nauthor = {Konrad Knopp},\ntitle = {Elements of the theory of functions},\npublisher = {Dover},\nyear = 1952}\n\nauthor = {Tristham Needham},\ntitle = {Visual Complex Analysis},\npublisher = {Oxford University Press},\nyear = 1997}\n\nboth of which contain excellent discussions of the two dimesional Moebius\ngroup acting on R^2 (treated as the stereographic image of S^2 minus a\npole).\n\nNote that (b)(d) together come from the six rotations in E^4 which fix\npointwise one of the six coordinate two-planes. In the same way, the two\ndimensional Moebius group acting on R^2 has one generator which is an\nordinary E^2 rotation in one coordinate two-plane, namely z = 0, plus two\nmore coming from the other two coordinate two-planes.\n\n(BTW, the textbook exercise might also have something like u @/@u tacked\nonto some of the other generators; I forget. If so, this can be removed\nusing the generator X11 = u @/@u, so the generators I gave really do\ndefine the same algebra.)\n\n> But first I challenge Aaron and Shagird to solve a much simpler problem:\n> what is the symmetry group of the axisymmetric 3-dimensional Laplace\n> equation?\n\nApparently they didn\'t take me up on this--- or else they didn\'t see the\nchallenge. (My point was that computing point symmetries would seem to be\nan essential skill for field theorizing from a Lagrangian--- specifically\nfor verifying the existence of wave solutions for a free field, which is\npresumably desirable for "updating" the field as the source is\nrearranged--- so I/m sure I don\'t know how physicists tie their field\ntheoretical shoelaces in the morning!)\n\n> Namely, in cylindrical coordinates\n>\n> phi_(zz) + phi_(rr) + phi_r/r = 0\n>\n> where phi is a function of z,r only?\n\nNote that X3, X4, X7, X8 above preserve the axis x = y = 0. Rewrite them\nin the cylindrical chart:\n\nX3 = @/@z\n\nX4 = @/@u\n\nX7 = @/@z + @/@r\n\nX8 = (1 + z^2 - r^2) @/@z + r @/@r + zu @/@u\n\nThese generate the four dimensional subalgebra of the full Lie algebra\nwhich preserves the "axisymmetrical" condition. One can verify directly\nfrom the determining equations that this is indeed the Lie algebra of\npoint symmetries of the axisymmetric Laplace equation.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>Last week I foolishly promised to post here the full derivation of the Lie
algebra of the conformal group on R^3, as the Lie algebra of the "point
symmetries" of the three-dimensional Laplace equation. Alas, I don't have
time to do that after all, but the diligent reader can study the textbook
I mentioned and work the exercise on the Laplace equation.

Let me just say what the answer is:

The full Lie algebra of the point symmetries of the three-dimensional
Laplace equation is "trivially" infinite dimensional. The generators can
be given as follows:

(a) three translations:

X1 = @/@x

X2 = @/@y

X3 = @/@z

(b) three rotations:

X4 = y @/@x - x @/@y

X5 = y @/@z - z @/@y

X6 = z @/@x - x @/@z

(c) 1 dilation:

X7 = x @/@x + y @/@y + z @/@z

(d) 3 "Moebius rotations":

X8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u

X9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u

X10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u

(e) scalar multiple of u:

X11 = u @/@u

(f) addition of another harmonic function to u:

Xg = g @/@u

where g is any harmonic function, i.e. g_(xx) + g_(yy) + g_(zz) = .

Here, the family of generators (f) accounts for the fact that the algebra
is trivially infinite dimensional. The symmetry algebra of a -linear- PDE
(with dependent variable u) always contains (e)-(f) because linearity
means that you can always combine linearly two solutions to get a third,
independently of doing anything with the independent variables. So the
remaining generators, if any, are usually the interesting/informative
ones.

In this case, (a)-(d) generate a ten dimensional subalgebra which acts on
the space of independent variables, namely R^3. This is the Lie algebra
of the conformal group of R^3, aka the Lie algebra of the
three-dimensional Moebius group acting on R^3 (treated as the
stereographic image of S^3 minus a pole). You can delete the @/@u terms
from (d) if you like, since we are only interested in the action of the
Moebius group on R^3. Then our ten dimensional algebra obviously defines
a ten dimensional Lie group acting on R^3, as promised.

The generators (a)-(c) are presumably familiar. The last three appear in
the textbook exercise I cited -without- the initial ones. Of course,
subtracting the appropriate translation from the generators yields the
generators given in the textbook exercise. Since those generators are
very slightly simpler than mine, why did I bother to include the initial
ones? Because my generators are precise analogues of the "Moebius
rotation" of S^2 (stereographically projected to R^2) which has flowlines
given by two families of circles nested about +/- i (in complex notation),
degenerating to the real axis.

For example,

Y8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z

-fixes pointwise- the circle x^2 + y^2 = 1 (the stereographic image in R^3
of a great circle on S^3) and leaves invariant (but doesn't fix pointwise)
the axis x = y = (the stereographic image of the "opposite" great circle
on S^3). Indeed, Y8 leaves invariant the stereographic images of -the
full family of Hopf tori- on S^3 which are nested around these two great
circles, but "rolls" each one along its "longitudes".

To see the analogy, take a section by any plane through the axis x = y =
. The Hopf tori now appear as nested circles, and the flow "rotates"
these in the manner familiar from books like

author = {Konrad Knopp},
title = {Elements of the theory of functions},
publisher = {Dover},
year = 1952}

author = {Tristham Needham},
title = {Visual Complex Analysis},
publisher = {Oxford University Press},
year = 1997}

both of which contain excellent discussions of the two dimesional Moebius
group acting on R^2 (treated as the stereographic image of S^2 minus a
pole).

Note that (b)(d) together come from the six rotations in E^4 which fix
pointwise one of the six coordinate two-planes. In the same way, the two
dimensional Moebius group acting on R^2 has one generator which is an
ordinary E^2 rotation in one coordinate two-plane, namely z = 0, plus two
more coming from the other two coordinate two-planes.

(BTW, the textbook exercise might also have something like u @/@u tacked
onto some of the other generators; I forget. If so, this can be removed
using the generator X11 = u @/@u, so the generators I gave really do
define the same algebra.)

> But first I challenge Aaron and Shagird to solve a much simpler problem:
> what is the symmetry group of the axisymmetric 3-dimensional Laplace
> equation?

Apparently they didn't take me up on this--- or else they didn't see the
challenge. (My point was that computing point symmetries would seem to be
an essential skill for field theorizing from a Lagrangian--- specifically
for verifying the existence of wave solutions for a free field, which is
presumably desirable for "updating" the field as the source is
rearranged--- so I/m sure I don't know how physicists tie their field
theoretical shoelaces in the morning!)

> Namely, in cylindrical coordinates
>
> \phi_(zz) + \phi_(rr) + \phi_r/r =
>
> where \phi is a function of z,r only?

Note that X3, X4, X7, X8 above preserve the axis x = y = . Rewrite them
in the cylindrical chart:

X3 = @/@z

X4 = @/@u

X7 = @/@z + @/@r

X8 = (1 + z^2 - r^2) @/@z + r @/@r + zu @/@u

These generate the four dimensional subalgebra of the full Lie algebra
which preserves the "axisymmetrical" condition. One can verify directly
from the determining equations that this is indeed the Lie algebra of
point symmetries of the axisymmetric Laplace equation.

"T. Essel" (hiding somewhere in cyberspace)

Danny Ross Lunsford

Apr14-04, 03:17 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>tessel@um.bot wrote:\n\n> Last week I foolishly promised to post here the full derivation of the Lie\n> algebra of the conformal group on R^3, as the Lie algebra of the "point\n> symmetries" of the three-dimensional Laplace equation. Alas, I don\'t have\n> time to do that after all, but the diligent reader can study the textbook\n> I mentioned and work the exercise on the Laplace equation.\n\n> (d) 3 "Moebius rotations":\n>\n> X8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u\n>\n> X9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u\n>\n> X10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u\n\nThese are the 3-d analogues of the "special conformal transformations"\nof spacetime\n\nX\'m = (Xm - X^2 Am) / (1 - 2AmXm + X^2 A^2)\n\nIt amounts to the sequence\n\n1) Inversion X -> X / X^2\n\n2) Translation X -> X - A\n\n3) Re-inversion X -> X / X^2\n\nThe fascinating thing is that finite translations are embedded in this.\nThere is something very deep hiding here. Can we replace the quadratic\nform implicit in 1 and 3 (XmXm = 1) by any quadratic form?\n\nThe above interpretation should generalize to any dimension, so one\nshould be able to define conformality in Dim=N by orthogonality in\nDim=N+2. In general\n\nN(N-1)/2 + (2N + 1) = (N^2 + 3N + 2)/2 = (N+2)(N+1)/2\n\nThis conjecture contains the Lorentz group as the "orthogonal cover" of\nthe conformality of the complex plane :) Since we know this is\nassociated with 2-spinors, we wonder if 2N-spinors have a similar\ngeometric origin.\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>tessel@um.bot wrote:

> Last week I foolishly promised to post here the full derivation of the Lie
> algebra of the conformal group on R^3, as the Lie algebra of the "point
> symmetries" of the three-dimensional Laplace equation. Alas, I don't have
> time to do that after all, but the diligent reader can study the textbook
> I mentioned and work the exercise on the Laplace equation.

> (d) 3 "Moebius rotations":
>
> X8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u
>
> X9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u
>
> X10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u

These are the 3-d analogues of the "special conformal transformations"
of spacetime

X'm = (Xm - X^2 Am) / (1 - 2AmXm + X^2 A^2)

It amounts to the sequence

1) Inversion X -> X / X^2

2) Translation X -> X - A

3) Re-inversion X -> X / X^2

The fascinating thing is that finite translations are embedded in this.
There is something very deep hiding here. Can we replace the quadratic
form implicit in 1 and 3 (XmXm = 1) by any quadratic form?

The above interpretation should generalize to any dimension, so one
should be able to define conformality in Dim=N by orthogonality in
Dim=N+2. In general

N(N-1)/2 + (2N + 1) = (N^2 + 3N + 2)/2 = (N+2)(N+1)/2

This conjecture contains the Lorentz group as the "orthogonal cover" of
the conformality of the complex plane :) Since we know this is
associated with 2-spinors, we wonder if 2N-spinors have a similar
geometric origin.

-drl

tessel@tum.bot

Apr19-04, 03:19 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>On Wed, 14 Apr 2004, Danny Ross Lunsford wrote:\n\n> > (d) 3 "Moebius rotations":\n> >\n> > X8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u\n> >\n> > X9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u\n> >\n> > X10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u\n>\n> These are the 3-d analogues of the "special conformal transformations"\n> of spacetime\n>\n> X\'m = (Xm - X^2 Am) / (1 - 2AmXm + X^2 A^2)\n>\n> It amounts to the sequence\n>\n> 1) Inversion X -> X / X^2\n>\n> 2) Translation X -> X - A\n>\n> 3) Re-inversion X -> X / X^2\n\nYes indeed. In fact, IIRC this observation is part of the exercise I\n"cited", in the textbook by Bluman and Kumei. But my characterization is\nsimpler, much more direct, and it focuses attention on the Moebius action.\nBut I agree that your decomposition is also valuable, pretty much for the\nreason you state.\n\n> The fascinating thing is that finite translations are embedded in this.\n> There is something very deep hiding here.\n\nDeep but well known :-/\n\nIn the past I\'ve often discussed how decompositons into\n\northoreflection, translation, orthoreflection\n\narise naturally in transformation group geometry--- so do other\ndecompositions.\n\nSee also Coxeter, Geometry and the books of Yaglom = Iaglom, plus my past\nposts on half-angles in spinorial double covers, including the\nthree-dimensonal Lie group E(2), for example, a semidirect product which I\ncontrasted with SO(3).\n\nPoint being that inversions are very closely analogous to orthoreflections\n(e.g. both are involutions), so clearly your decomposition fits nicely\ninto a general pattern.\n\n> Can we replace the quadratic form implicit in 1 and 3 (XmXm = 1) by any\n> quadratic form?\n\nSorry, I don\'t understand the question. But it sounds like you are asking\nessentially the natural followup I had in mind: how can we understand why\nE^n conformal groups arise from the Moebius actions by generalized Lorentz\ngroups, in the manner I stated? The key word here is "Moebius"! (This is\nsurely standard material, but I happen not to have any citation--- maybe\nsomeone else does?)\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Wed, 14 Apr 2004, Danny Ross Lunsford wrote:

> > (d) 3 "Moebius rotations":
> >
> > X8 = (1 + x^2 - y^2 - z^2) @/@x + 2 xy @/@y + 2 xz @/@z - xu @/@u
> >
> > X9 = (1 + y^2 - x^2 - z^2) @/@y + 2 xy @/@x + 2 yz @/@z - yu @/@u
> >
> > X10 = (1 + z^2 - x^2 - y^2) @/@z + 2 xz @/@x + 2 yz @/@y - zu @/@u
>
> These are the 3-d analogues of the "special conformal transformations"
> of spacetime
>
> X'm = (Xm - X^2 Am) / (1 - 2AmXm + X^2 A^2)
>
> It amounts to the sequence
>
> 1) Inversion X -> X / X^2
>
> 2) Translation X -> X - A
>
> 3) Re-inversion X -> X / X^2

Yes indeed. In fact, IIRC this observation is part of the exercise I
"cited", in the textbook by Bluman and Kumei. But my characterization is
simpler, much more direct, and it focuses attention on the Moebius action.
But I agree that your decomposition is also valuable, pretty much for the
reason you state.

> The fascinating thing is that finite translations are embedded in this.
> There is something very deep hiding here.

Deep but well known :-/

In the past I've often discussed how decompositons into

orthoreflection, translation, orthoreflection

arise naturally in transformation group geometry--- so do other
decompositions.

See also Coxeter, Geometry and the books of Yaglom = Iaglom, plus my past
posts on half-angles in spinorial double covers, including the
three-dimensonal Lie group E(2), for example, a semidirect product which I
contrasted with SO(3).

Point being that inversions are very closely analogous to orthoreflections
(e.g. both are involutions), so clearly your decomposition fits nicely
into a general pattern.

> Can we replace the quadratic form implicit in 1 and 3 (XmXm = 1) by any
> quadratic form?

Sorry, I don't understand the question. But it sounds like you are asking
essentially the natural followup I had in mind: how can we understand why
E^n conformal groups arise from the Moebius actions by generalized Lorentz
groups, in the manner I stated? The key word here is "Moebius"! (This is
surely standard material, but I happen not to have any citation--- maybe
someone else does?)

"T. Essel" (hiding somewhere in cyberspace)