Complex Conformal Group

Pauca Sed Matura

<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>Hi all,\n\nFirst of all, some apologies.\n1) Sorry if this question falls below the academic level of this\nnewsgroup!\n2) Sorry for the length!\n\nI\'m having some difficulty reconciling an explicit isomorphism between\n\n1) the group of proper (orientation-preserving) conformal\ntransformations C(1,3) of compactified complex Minkowski space (CM#)\n2) the complex projective linear group PGL(4,C)\n\ngiven in some lecture notes I have been given.\n\n--------------------------------------------------------------------------------\nFirst, notation and preliminaries:\n\nLet z_1, z_2, w_1, w_2 be complex double null co-ordinates such that\nthe metric on CM is\n\nds^2 = 2(dz_1 dz_2 - dw_1 dw_2)\n\nand the orientation/volume 4-form is\nn = dw_1 ^ dw_2 ^ dz_1 ^ dz_2\n\n(We can then recover real Minkowski space, real Euclidean space, and\nother interesting spaces by imposing conditions relating these\ncoordinates to each other.)\n\nA (proper) conformal transformation is an orientation preserving map\ndenoted P from CM# -> CM# such that ds_new^2 = omega^2 ds_old^2, and\nn_new = omega^4 n_old.\n\nInfinitesmal conformal transformations are given by conformal Killing\nvectors, where the Lie derivative must be a multiple of the metric. The\nspace of such conformal Killing vectors is 15-dimensional. It can be\ndecomposed into 4 translation parameters, 6 rotation/Lorentz\nparameters, 1 dilation parameter, and 4 \'special\' conformal\ntransformation parameters.\n\n--------------------------------------------------------------------------------\nOK, here\'s the setup (phew!):\n\nWe can write a point of CM uniquely as the skew-symmetric degenerate\nmatrix:\n\n0 z_1 z_2 - w_1 w_2 -w_1 z_2\nz_1 z_2 - w_1 w_2 0 -z_1 w_2\nw_1 z_1 0 1\n-z_2 -w_2 -1 0\n\nwhich we label x^{a b}.\n\nFurther, unless x^{2 3} = 0, every skew-symmetric degenerate 4*4 matrix\nmust be of this form up to a scalar multiple.\n\nAlso, epsilon_abcd dx^{a b} dx^{c d} = lambda (dz_1 dz_2 - dw_1 dw_2)\nwhere epsilon_abcd is the 4-dimensional alternating symbol and lambda\nis some scalar.\n\n--------------------------------------------------------------------------------\nNow here are the claims I don\'t really understand.\n\nCLAIM 1: \'Hence\' if b denotes a general member of GL(4,C) then the\ntransformation x -> b x b^t induces a conformal transformation of CM#.\nFurthermore, every proper conformal transformation arises in this way.\n\nCLAIM 2: We can regard the six elements of x^{a b} as homogenous\ncoordinates on CP5. When we do so, we can identify CM# = {x^[a b x^c d]\n= 0} in CP5, the Klein quadric. Here the [] brackets denote\nantisymmetrization in the usual way.\n\nCLAIM 3: We can identify null geodesics in CM# with those projective\nlines in CP5 which lie in CM#.\n\nCan anyone help me shed light?\n\nThanks!\n\nPauca Sed Matura\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>Hi all,

First of all, some apologies.
1) Sorry if this question falls below the academic level of this
newsgroup!
2) Sorry for the length!

I'm having some difficulty reconciling an explicit isomorphism between

1) the group of proper (orientation-preserving) conformal
transformations C(1,3) of compactified complex Minkowski space (CM#)
2) the complex projective linear group PGL(4,C)

given in some lecture notes I have been given.

--------------------------------------------------------------------------------
First, notation and preliminaries:

Let [itex]z_1, z_2, w_1, w_2[/itex] be complex double null co-ordinates such that
the metric on CM is

[tex]ds^2 = 2(dz_1 dz_2 - dw_1 dw_2)[/tex]

and the orientation/volume 4-form is
[itex]n = dw_1 ^ dw_2 ^ dz_1 ^ dz_2[/itex]

(We can then recover real Minkowski space, real Euclidean space, and
other interesting spaces by imposing conditions relating these
coordinates to each other.)

A (proper) conformal transformation is an orientation preserving map
denoted P from CM# -> CM# such that [itex]ds_new^2 = \omega^2 ds_old^2,[/itex] and
[itex]n_{new} = \omega^4 n_{old}[/itex].

Infinitesmal conformal transformations are given by conformal Killing
vectors, where the Lie derivative must be a multiple of the metric. The
space of such conformal Killing vectors is 15-dimensional. It can be
decomposed into 4 translation parameters, 6 rotation/Lorentz
parameters, 1 dilation parameter, and 4 'special' conformal
transformation parameters.

--------------------------------------------------------------------------------
OK, here's the setup (phew!):

We can write a point of CM uniquely as the skew-symmetric degenerate
matrix:

[tex]z_1 z_2 - w_1 w_2 -w_1 z_2z_1 z_2 - w_1 w_2-z_1 w_2w_1 z_1[/itex] 1
[itex]-z_2 -w_2 -1[/tex]

which we label [itex]x^{a b}[/itex].

Further, unless [itex]x^{2 3} = 0,[/itex] every skew-symmetric degenerate 4*4 matrix
must be of this form up to a scalar multiple.

Also, [itex]\epsilon_abcd dx^{a b} dx^{c d} = \lambda (dz_1 dz_2 - dw_1 dw_2)[/itex]
where [itex]\epsilon_abcd[/itex] is the 4-dimensional alternating symbol and [itex]\lambda[/itex]
is some scalar.

--------------------------------------------------------------------------------
Now here are the claims I don't really understand.

CLAIM 1: 'Hence' if b denotes a general member of GL(4,C) then the
transformation [itex]x -> b x b^t[/itex] induces a conformal transformation of CM#.
Furthermore, every proper conformal transformation arises in this way.

CLAIM 2: We can regard the six elements of [itex]x^{a b}[/itex] as homogenous
coordinates on CP5. When we do so, we can identify CM# [itex]= {x^[a b x^c d]= 0}[/itex] in CP5, the Klein quadric. Here the [] brackets denote
antisymmetrization in the usual way.

CLAIM 3: We can identify null geodesics in CM# with those projective
lines in CP5 which lie in CM#.

Can anyone help me shed light?

Thanks!

Pauca Sed Matura

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