View Full Version : projective geometry in theoretical physics
<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\nI have read that Dirac used projective geometry in his derivations ...\nBut however in all physics books I have seen there are no use of\nprojective geometry .. Why ?\nIf in any case projective geometry is used in physics could anybody\nrecommend a book ?\nThank you in advance ...\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>I have read that Dirac used projective geometry in his derivations ...
But however in all physics books I have seen there are no use of
projective geometry .. Why ?
If in any case projective geometry is used in physics could anybody
recommend a book ?
Thank you in advance ...
per.vognsen@gmail.com
Dec10-04, 05:02 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>rst wrote:\n> I have read that Dirac used projective geometry in his derivations\n....\n> But however in all physics books I have seen there are no use of\n> projective geometry ..\n\nMost textbooks don\'t explicitly identify it as such. However, I\'m sure\nyou\'ve run into the idea that states are identified with rays (rather\nthan vectors) in a complex Hilbert space. That is, a pair of state\nvectors u and v in the complex Hilbert space are considered physically\nequivalent if and only if there exists a nonzero complex number lambda\nsuch that u = lambda v. But this is just another way of saying that the\ntrue state space is the projectivization of the Hilbert space,\nresulting in a complex projective space.\n\nWe can do calculations in this complex projective space by doing\ncalculations in the underlying Hilbert space as long as we always keep\nthe identification between complex-parallel vectors in mind. What\nphysicists usually do is work with unit vectors and then identify unit\nvectors that differ by a phase change, corresponding to a factor of a\nunit complex number (which can be represented as exp(it) for some real\nnumber t, this real number being interpreted as the phase difference).\nAnyway, I\'m sure you see how this restriction to unit vectors is\nnatural, given the usual probabilistic interpretation of the vectors as\nprobability amplitudes.\n\nI am not a physicist and so if any of this is a misrepresentation, I\'d\nlove to be corrected by someone more knowledgable.\n\nPer\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>rst wrote:
> I have read that Dirac used projective geometry in his derivations
....
> But however in all physics books I have seen there are no use of
> projective geometry ..
Most textbooks don't explicitly identify it as such. However, I'm sure
you've run into the idea that states are identified with rays (rather
than vectors) in a complex Hilbert space. That is, a pair of state
vectors u and v in the complex Hilbert space are considered physically
equivalent if and only if there exists a nonzero complex number \lambda
such that u = \lambda v. But this is just another way of saying that the
true state space is the projectivization of the Hilbert space,
resulting in a complex projective space.
We can do calculations in this complex projective space by doing
calculations in the underlying Hilbert space as long as we always keep
the identification between complex-parallel vectors in mind. What
physicists usually do is work with unit vectors and then identify unit
vectors that differ by a phase change, corresponding to a factor of a
unit complex number (which can be represented as \exp(it) for some real
number t, this real number being interpreted as the phase difference).
Anyway, I'm sure you see how this restriction to unit vectors is
natural, given the usual probabilistic interpretation of the vectors as
probability amplitudes.
I am not a physicist and so if any of this is a misrepresentation, I'd
love to be corrected by someone more knowledgable.
Per
<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\nper.vognsen@gmail.com wrote:\n> rst wrote:\n> > I have read that Dirac used projective geometry in his derivations\n> ...\n> > But however in all physics books I have seen there are no use of\n> > projective geometry ..\n>\n> Most textbooks don\'t explicitly identify it as such.\n\n\nCould you recommend a book about use projective geometry in physics ..?\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>per.vognsen@gmail.com wrote:
> rst wrote:
> > I have read that Dirac used projective geometry in his derivations
> ...
> > But however in all physics books I have seen there are no use of
> > projective geometry ..
>
> Most textbooks don't explicitly identify it as such.
Could you recommend a book about use projective geometry in physics ..?
Van www
Dec10-04, 10:17 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>\nper.vognsen@gmail.com wrote:\n> rst wrote:\n> > I have read that Dirac used projective geometry in his derivations\n> ...\n> > But however in all physics books I have seen there are no use of\n> > projective geometry ..\n>\n> Most textbooks don\'t explicitly identify it as such. However, I\'m sure\n> you\'ve run into the idea that states are identified with rays (rather\n> than vectors) in a complex Hilbert space. That is, a pair of state\n\n\nYes, I think that\'s about it. Wave functions have to be normalized\nto 1, I think.\nI would be interested if there is more to it too.\nVan\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>per.vognsen@gmail.com wrote:
> rst wrote:
> > I have read that Dirac used projective geometry in his derivations
> ...
> > But however in all physics books I have seen there are no use of
> > projective geometry ..
>
> Most textbooks don't explicitly identify it as such. However, I'm sure
> you've run into the idea that states are identified with rays (rather
> than vectors) in a complex Hilbert space. That is, a pair of state
Yes, I think that's about it. Wave functions have to be normalized
to 1, I think.
I would be interested if there is more to it too.
Van
antimatter33@yahoo.com
Dec10-04, 10:18 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>\nWell, there is direct projective geometry, in two ways (one and two\nhalves);\n\n1) Quantum mechanics is projective - we don\'t worry about the phase of\nthe wave function, so the components of the state vector have a\nhomogeneous aspect\n\n2a) Kaluza-Klein theory is - accidentally - based on something called\nby its creators the "projective geometry of paths". See the literature\nfrom the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.\n\n2b) From the perspective of projective geometry, the difference between\nMinkowski (affine) geometry and Euclidean (affine) geometry is that the\nformer is characterized by a real number (c=1) and the latter by an\nimaginary one (c=i). So relativity is a nearly perfect realization of\nthe idea of "projective metric" in the sense of Klein\'s program. (Not\nthe same Klein as 2a).\n\nHowever, what never appear in physics (as far as I can tell) are\nprojective invariants, that is, 4-point linear fractional invariants\n(cross ratios). I\'ve always thought that this was odd. If one could\nmake a thoroughgoing projective physics, the problem with infinities of\nall kinds would go completely away.\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>Well, there is direct projective geometry, in two ways (one and two
halves);
1) Quantum mechanics is projective - we don't worry about the phase of
the wave function, so the components of the state vector have a
homogeneous aspect
2a) Kaluza-Klein theory is - accidentally - based on something called
by its creators the "projective geometry of paths". See the literature
from the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.
2b) From the perspective of projective geometry, the difference between
Minkowski (affine) geometry and Euclidean (affine) geometry is that the
former is characterized by a real number (c=1) and the latter by an
imaginary one (c=i). So relativity is a nearly perfect realization of
the idea of "projective metric" in the sense of Klein's program. (Not
the same Klein as 2a).
However, what never appear in physics (as far as I can tell) are
projective invariants, that is, 4-point linear fractional invariants
(cross ratios). I've always thought that this was odd. If one could
make a thoroughgoing projective physics, the problem with infinities of
all kinds would go completely away.
-drl
<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>I remember now Dirac used projective geometry in STO ...\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>I remember now Dirac used projective geometry in STO ...
Arnold Neumaier
Dec11-04, 03:04 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>antimatter33@yahoo.com wrote:\n\n> However, what never appear in physics (as far as I can tell) are\n> projective invariants, that is, 4-point linear fractional invariants\n> (cross ratios).\n\nThis is because these are invariants of the projective line (1D),\nor rather its automorphism group PSL(1), and QM on the projective line\nis trivial - a single qbit, Fullfledged QM needs an infinite-dimensional\nprojective space, ans since there is a distinguisehd metric, the invariance\ngroup is the group of unitary transformations, not an infinite-dimensional PSL.\n\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>antimatter33@yahoo.com wrote:
> However, what never appear in physics (as far as I can tell) are
> projective invariants, that is, 4-point linear fractional invariants
> (cross ratios).
This is because these are invariants of the projective line (1D),
or rather its automorphism group PSL(1), and QM on the projective line
is trivial - a single qbit, Fullfledged QM needs an infinite-dimensional
projective space, ans since there is a distinguisehd metric, the invariance
group is the group of unitary transformations, not an infinite-dimensional PSL.
Arnold Neumaier
Strong_Field
Dec11-04, 03:08 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><antimatter33@yahoo.com> wrote in message\nnews:1102641916.890653.237360@c13g2000cwb .googlegroups.com...\n>\n>\n> If one could\n> make a thoroughgoing projective physics, the problem with infinities of\n> all kinds would go completely away.\n\nWhat are these infinities? How many are there? Why do they exist?\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><antimatter33@yahoo.com> wrote in message
news:1102641916.890653.237360@c13g2000cwb.googlegr oups.com...
>
>
> If one could
> make a thoroughgoing projective physics, the problem with infinities of
> all kinds would go completely away.
What are these infinities? How many are there? Why do they exist?
me@privacy.net
Dec13-04, 11:25 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>proportional to the amount of pressure or discomfort to which the\nsociety subjects people. That is certainly not the case. There is good\nreason to believe that many primitive societies subjected people to\nless pressure than the European society did, but European society\nproved far more efficient than any primitive society and always won\nout in conflicts with such societies because of the advantages\nconferred by technology.\n\n26. (Paragraph 147) If you think that more effective law enforcement\nis unequivocally good because it suppresses crime, then remember that\ncrime as defined by the system is not necessarily what YOU would call\ncrime. Today, smoking marijuana is a "crime," and, in some places in\nthe U.S.., so is possession of ANY firearm, registered or not, may be\nmade a crime, and the same thing may happen with disapproved methods\nof child-rearing, such as spanking. In some countries, expression of\ndissident political opinions is a crime, and there is no certainty\nthat this will never happen in the U.S., since no constitution or\npol\n\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>proportional to the amount of pressure or discomfort to which the
society subjects people. That is certainly not the case. There is good
reason to believe that many primitive societies subjected people to
less pressure than the European society did, but European society
proved far more efficient than any primitive society and always won
out in conflicts with such societies because of the advantages
conferred by technology.
26. (Paragraph 147) If you think that more effective law enforcement
is unequivocally good because it suppresses crime, then remember that
crime as defined by the system is not necessarily what YOU would call
crime. Today, smoking marijuana is a "crime," and, in some places in
the U.S.., so is possession of ANY firearm, registered or not, may be
made a crime, and the same thing may happen with disapproved methods
of child-rearing, such as spanking. In some countries, expression of
dissident political opinions is a crime, and there is no certainty
that this will never happen in the U.S., since no constitution or
pol
<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>social evolution that are not under rational human control.\n\n107. The fifth principle is a consequence of the other four.\n\n108. To illustrate: By the first principle, generally speaking an\nattempt at social reform either acts in the direction in which the\nsociety is developing anyway (so that it merely accelerates a change\nthat would have occurred in any case) or else it only has a transitory\neffect, so that the society soon slips back into its old groove. To\nmake a lasting change in the direction of development of any important\naspect of a society, reform is insufficient and revolution is\nrequired. (A revolution does not necessarily involve an armed uprising\nor the overthrow of a government.) By the second principle, a\nrevolution never changes only one aspect of a society; and by the\nthird principle changes occur that were never expected or desired by\nthe revolutionaries. By the fourth principle, when revolutionaries or\nutopians set up a new kind of society, it never works o\n\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>social evolution that are not under rational human control.
107. The fifth principle is a consequence of the other four.
108. To illustrate: By the first principle, generally speaking an
attempt at social reform either acts in the direction in which the
society is developing anyway (so that it merely accelerates a change
that would have occurred in any case) or else it only has a transitory
effect, so that the society soon slips back into its old groove. To
make a lasting change in the direction of development of any important
aspect of a society, reform is insufficient and revolution is
required. (A revolution does not necessarily involve an armed uprising
or the overthrow of a government.) By the second principle, a
revolution never changes only one aspect of a society; and by the
third principle changes occur that were never expected or desired by
the revolutionaries. By the fourth principle, when revolutionaries or
utopians set up a new kind of society, it never works o
antimatter33@yahoo.com
Dec14-04, 05:35 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>Strong_Field wrote:\n\n> > If one could\n> > make a thoroughgoing projective physics, the problem with\ninfinities of\n> > all kinds would go completely away.\n>\n> What are these infinities? How many are there? Why do they exist?\n\nBecause like thermodynamics, which is inherently free of infinities,\nwhen one deals with homogeneous functions of degree 1, there is an\nimplied independence from scales. I\'ve never seen projective\nthermodynamics but something like it must exist. Assuming something\nlike that, the formula\n\nS = k log W\n\nwhich looks exactly like a Klein-Laguerre projective measure, would\ncause one to think a lot about the scaling behavior of the number of\nstates, that is, the density of states.\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>Strong_Field wrote:
> > If one could
> > make a thoroughgoing projective physics, the problem with
infinities of
> > all kinds would go completely away.
>
> What are these infinities? How many are there? Why do they exist?
Because like thermodynamics, which is inherently free of infinities,
when one deals with homogeneous functions of degree 1, there is an
implied independence from scales. I've never seen projective
thermodynamics but something like it must exist. Assuming something
like that, the formula
S = k[/itex] log W
which looks exactly like a Klein-Laguerre projective measure, would
cause one to think a lot about the scaling behavior of the number of
states, that is, the density of states.
[itex]-drl
Van www
Dec14-04, 10:18 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\nantimatter33@yahoo.com wrote:\n> Well, there is direct projective geometry, in two ways (one and two\n> halves);\n>\n> 1) Quantum mechanics is projective - we don\'t worry about the phase\nof\n> the wave function, so the components of the state vector have a\n> homogeneous aspect\n>\n> 2a) Kaluza-Klein theory is - accidentally - based on something called\n> by its creators the "projective geometry of paths". See the\nliterature\n> from the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.\n>\n> 2b) From the perspective of projective geometry, the difference\nbetween\n> Minkowski (affine) geometry and Euclidean (affine) geometry is that\nthe\n> former is characterized by a real number (c=1) and the latter by an\n> imaginary one (c=i). So relativity is a nearly perfect realization of\n> the idea of "projective metric" in the sense of Klein\'s program. (Not\n> the same Klein as 2a).\n>\n> However, what never appear in physics (as far as I can tell) are\n> projective invariants, that is, 4-point linear fractional invariants\n> (cross ratios). I\'ve always thought that this was odd. If one could\n> make a thoroughgoing projective physics, the problem with infinities\nof\n> all kinds would go completely away.\n>\n> -drl\n\nI would say that Euclidean geometry is elliptic in that it\nhas a metric g = diag(1,1,1) in 3D,\nwhile Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),\nand integrates time into things.\n\nVan\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>antimatter33@yahoo.com wrote:
> Well, there is direct projective geometry, in two ways (one and two
> halves);
>
> 1) Quantum mechanics is projective - we don't worry about the phase
of
> the wave function, so the components of the state vector have a
> homogeneous aspect
>
> 2a) Kaluza-Klein theory is - accidentally - based on something called
> by its creators the "projective geometry of paths". See the
literature
> from the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.
>
> 2b) From the perspective of projective geometry, the difference
between
> Minkowski (affine) geometry and Euclidean (affine) geometry is that
the
> former is characterized by a real number (c=1) and the latter by an
> imaginary one (c=i). So relativity is a nearly perfect realization of
> the idea of "projective metric" in the sense of Klein's program. (Not
> the same Klein as 2a).
>
> However, what never appear in physics (as far as I can tell) are
> projective invariants, that is, 4-point linear fractional invariants
> (cross ratios). I've always thought that this was odd. If one could
> make a thoroughgoing projective physics, the problem with infinities
of
> all kinds would go completely away.
>
> -drl
I would say that Euclidean geometry is elliptic in that it
has a metric g = diag(1,1,1) in 3D,
while Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),
and integrates time into things.
Van
antimatter33@yahoo.com
Dec14-04, 10:18 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\nArnold Neumaier wrote:\n\n> > However, what never appear in physics (as far as I can tell) are\n> > projective invariants, that is, 4-point linear fractional\ninvariants\n> > (cross ratios).\n>\n> This is because these are invariants of the projective line (1D),\n> or rather its automorphism group PSL(1), and QM on the projective\nline\n> is trivial - a single qbit, Fullfledged QM needs an\ninfinite-dimensional\n> projective space, ans since there is a distinguisehd metric, the\ninvariance\n> group is the group of unitary transformations, not an\ninfinite-dimensional PSL.\n\nI don\'t really understand this point. First, I\'m referring also the the\nreal geometry of the world, not just the ray-nature of QM. Second, in\nthe real geometry of the world, we have at least two direct examples of\nprojective ideas (Euclidean and Minkowski space and their ideal\ndomains). Third, we have the very interesting fact that we don\'t\nexperience dilations, although the spaces of our "experience" are -\napparently - Euclidean and Minkowskian *affine* geometry, and allow\ndilations. I strongly suspect the missing projective invariants are\nsomehow associated with these missing dilations.\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>Arnold Neumaier wrote:
> > However, what never appear in physics (as far as I can tell) are
> > projective invariants, that is, 4-point linear fractional
invariants
> > (cross ratios).
>
> This is because these are invariants of the projective line (1D),
> or rather its automorphism group PSL(1), and QM on the projective
line
> is trivial - a single qbit, Fullfledged QM needs an
infinite-dimensional
> projective space, ans since there is a distinguisehd metric, the
invariance
> group is the group of unitary transformations, not an
infinite-dimensional PSL.
I don't really understand this point. First, I'm referring also the the
real geometry of the world, not just the ray-nature of QM. Second, in
the real geometry of the world, we have at least two direct examples of
projective ideas (Euclidean and Minkowski space and their ideal
domains). Third, we have the very interesting fact that we don't
experience dilations, although the spaces of our "experience" are -
apparently - Euclidean and Minkowskian *affine* geometry, and allow
dilations. I strongly suspect the missing projective invariants are
somehow associated with these missing dilations.
-drl
Eugene Shubert
Dec14-04, 10:18 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"rst" <rustamkm@yahoo.com> wrote in message\nnews:1102592282.014070.134990@f14g2000cwb .googlegroups.com...\n> If in any case projective geometry is used in physics could anybody\n> recommend a book ?\n> Thank you in advance ...\n\nYes.\n\nSee http://homepage.ntlworld.com/stebla/Whitehead.html\nand http://homepage.ntlworld.com/stebla/papers/Euclid.pdf\n\nEugene Shubert\nhttp://www.everythingimportant.org\n\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>"rst" <rustamkm@yahoo.com> wrote in message
news:1102592282.014070.134990@f14g2000cwb.googlegr oups.com...
> If in any case projective geometry is used in physics could anybody
> recommend a book ?
> Thank you in advance ...
Yes.
See http://homepage.ntlworld.com/stebla/Whitehead.html
and http://homepage.ntlworld.com/stebla/papers/Euclid.pdf
Eugene Shubert
http://www.everythingimportant.org
Eugene Stefanovich
Dec14-04, 10:18 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\nrst wrote:\n> per.vognsen@gmail.com wrote:\n>\n>>rst wrote:\n>>\n>>>I have read that Dirac used projective geometry in his derivations\n>>\n>>...\n>>\n>>>But however in all physics books I have seen there are no use of\n>>>projective geometry ..\n>>\n>>Most textbooks don\'t explicitly identify it as such.\n>\n>\n>\n> Could you recommend a book about use projective geometry in physics ..?\n\nI would recommend\n\nC.Piron, "Foundations of Quantum Physics" (W. A. Benjamin, Reading, 1976)\n\nwhere quantum mechanics is formulated in the language of\n(quantum) logic and projective geometry.\n\nEugene Stefanovich.\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>rst wrote:
> per.vognsen@gmail.com wrote:
>
>>rst wrote:
>>
>>>I have read that Dirac used projective geometry in his derivations
>>
>>...
>>
>>>But however in all physics books I have seen there are no use of
>>>projective geometry ..
>>
>>Most textbooks don't explicitly identify it as such.
>
>
>
> Could you recommend a book about use projective geometry in physics ..?
I would recommend
C.Piron, "Foundations of Quantum Physics" (W. A. Benjamin, Reading, 1976)
where quantum mechanics is formulated in the language of
(quantum) logic and projective geometry.
Eugene Stefanovich.
Cl.Massé
Dec14-04, 10:18 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"rst" <rustamkm@yahoo.com> a écrit dans le message de\nnews:1102592282.014070.134990@f14g2000cwb.goog legroups.com...\n\n> I have read that Dirac used projective geometry in his derivations ...\n> But however in all physics books I have seen there are no use of\n> projective geometry .. Why ?\n\nIt is used mainly in the theory of the magnetic monopole, not a so\nstandard topic.\n\n--\n~~~~ clmasse on free dot F-country\nLiberty, Equality, Profitability.\n\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>"rst" <rustamkm@yahoo.com> a écrit dans le message de
news:1102592282.014070.134990@f14g2000cwb.googlegr oups.com...
> I have read that Dirac used projective geometry in his derivations ...
> But however in all physics books I have seen there are no use of
> projective geometry .. Why ?
It is used mainly in the theory of the magnetic monopole, not a so
standard topic.
--
~~~~ clmasse on free dot F-country
Liberty, Equality, Profitability.
Strong_Field
Dec16-04, 08:05 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><antimatter33@yahoo.com> wrote in message\nnews:1102992489.565019.175450@z14g2000cwz .googlegroups.com...\n> Strong_Field wrote:\n>\n> > > If one could\n> > > make a thoroughgoing projective physics, the problem with\n> infinities of\n> > > all kinds would go completely away.\n> >\n> > What are these infinities? How many are there? Why do they exist?\n>\n> Because like thermodynamics, which is inherently free of infinities,\n> when one deals with homogeneous functions of degree 1, there is an\n> implied independence from scales. I\'ve never seen projective\n> thermodynamics but something like it must exist. Assuming something\n> like that, the formula\n>\n> S = k log W\n>\n> which looks exactly like a Klein-Laguerre projective measure, would\n> cause one to think a lot about the scaling behavior of the number of\n> states, that is, the density of states.\n>\n> -drl\n\nMaybe I should have asked a different question. Are infinities in physics\ncaused by a particular choice of geometry? Would they disappear if different\ngeometry is used?\n\n>\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><antimatter33@yahoo.com> wrote in message
news:1102992489.565019.175450@z14g2000cwz.googlegr oups.com...
> Strong_Field wrote:
>
> > > If one could
> > > make a thoroughgoing projective physics, the problem with
> infinities of
> > > all kinds would go completely away.
> >
> > What are these infinities? How many are there? Why do they exist?
>
> Because like thermodynamics, which is inherently free of infinities,
> when one deals with homogeneous functions of degree 1, there is an
> implied independence from scales. I've never seen projective
> thermodynamics but something like it must exist. Assuming something
> like that, the formula
>
> S = k log W
>
> which looks exactly like a Klein-Laguerre projective measure, would
> cause one to think a lot about the scaling behavior of the number of
> states, that is, the density of states.
>
> -drl
Maybe I should have asked a different question. Are infinities in physics
caused by a particular choice of geometry? Would they disappear if different
geometry is used?
>
antimatter33@yahoo.com
Dec17-04, 07:48 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>Strong_Field wrote:\n\n> Maybe I should have asked a different question. Are infinities in\nphysics\n> caused by a particular choice of geometry? Would they disappear if\ndifferent\n> geometry is used?\n\nNot as such, but PG allows one to consistently deal with infinity in a\ncontinuum.\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>Strong_Field wrote:
> Maybe I should have asked a different question. Are infinities in
physics
> caused by a particular choice of geometry? Would they disappear if
different
> geometry is used?
Not as such, but PG allows one to consistently deal with infinity in a
continuum.
-drl
Arnold Neumaier
Dec17-04, 07:51 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>antimatter33@yahoo.com wrote:\n> Arnold Neumaier wrote:\n>\n>\n>>>However, what never appear in physics (as far as I can tell) are\n>>>projective invariants, that is, 4-point linear fractional\n>\n> invariants\n>\n>>>(cross ratios).\n>>\n>>This is because these are invariants of the projective line (1D),\n>>or rather its automorphism group PSL(1), and QM on the projective\n>\n> line\n>\n>>is trivial - a single qbit, Fullfledged QM needs an\n>\n> infinite-dimensional\n>\n>>projective space, ans since there is a distinguisehd metric, the\n>\n> invariance\n>\n>>group is the group of unitary transformations, not an\n>\n> infinite-dimensional PSL.\n>\n> I don\'t really understand this point. First, I\'m referring also the the\n> real geometry of the world, not just the ray-nature of QM.\n\nreal geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.\nThey have their own symmetry groups - I don\'t see how the group PSL(1) and\ncross ratios could be relevant for real 3D or 4D geometry.\nProjective geomentry only enters when viewing 3D objects from 2D cameras,\nand this is well understood.\n\n> Second, in\n> the real geometry of the world, we have at least two direct examples of\n> projective ideas (Euclidean and Minkowski space and their ideal\n> domains).\n\nThere is nothing projective in Euclidean and Minkowski space.\nProjective transformations change the metric.\n\n> Third, we have the very interesting fact that we don\'t\n> experience dilations, although the spaces of our "experience" are -\n> apparently - Euclidean and Minkowskian *affine* geometry, and allow\n> dilations. I strongly suspect the missing projective invariants are\n> somehow associated with these missing dilations.\n\nEven augmenting the group of Euclidean motions by dilations leaves you\nfar away from the projective group!\n\nSpeculation in physics should be constrained by the requirements of\nreality and the demands of compatible mathematics.\n\n\nArnold Neumaier\n\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>antimatter33@yahoo.com wrote:
> Arnold Neumaier wrote:
>
>
>>>However, what never appear in physics (as far as I can tell) are
>>>projective invariants, that is, 4-point linear fractional
>
> invariants
>
>>>(cross ratios).
>>
>>This is because these are invariants of the projective line (1D),
>>or rather its automorphism group PSL(1), and QM on the projective
>
> line
>
>>is trivial - a single qbit, Fullfledged QM needs an
>
> infinite-dimensional
>
>>projective space, ans since there is a distinguisehd metric, the
>
> invariance
>
>>group is the group of unitary transformations, not an
>
> infinite-dimensional PSL.
>
> I don't really understand this point. First, I'm referring also the the
> real geometry of the world, not just the ray-nature of QM.
real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
They have their own symmetry groups - I don't see how the group PSL(1) and
cross ratios could be relevant for real 3D or 4D geometry.
Projective geomentry only enters when viewing 3D objects from 2D cameras,
and this is well understood.
> Second, in
> the real geometry of the world, we have at least two direct examples of
> projective ideas (Euclidean and Minkowski space and their ideal
> domains).
There is nothing projective in Euclidean and Minkowski space.
Projective transformations change the metric.
> Third, we have the very interesting fact that we don't
> experience dilations, although the spaces of our "experience" are -
> apparently - Euclidean and Minkowskian *affine* geometry, and allow
> dilations. I strongly suspect the missing projective invariants are
> somehow associated with these missing dilations.
Even augmenting the group of Euclidean motions by dilations leaves you
far away from the projective group!
Speculation in physics should be constrained by the requirements of
reality and the demands of compatible mathematics.
Arnold Neumaier
Arnold Neumaier
antimatter33@yahoo.com
Dec19-04, 07:19 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>Arnold Neumaier wrote:\n\n> real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.\n> They have their own symmetry groups - I don\'t see how the group PSL(1) and\n> cross ratios could be relevant for real 3D or 4D geometry.\n\nOk, give two events, they determine a line, which intersects the light\ncone twice, thus 4 collinear points, so there is the cross-ratio\n\n> Projective geomentry only enters when viewing 3D objects from 2D cameras,\n> and this is well understood.\n\nSee above.\n\n> There is nothing projective in Euclidean and Minkowski space.\n> Projective transformations change the metric.\n\nEuclid: (x - iy)(x + iy) = 0\nMinkowski: (x - ct)(x + ct) = 0\n\n> > Third, we have the very interesting fact that we don\'t\n> > experience dilations, although the spaces of our "experience" are -\n> > apparently - Euclidean and Minkowskian *affine* geometry, and allow\n> > dilations. I strongly suspect the missing projective invariants are\n> > somehow associated with these missing dilations.\n>\n> Even augmenting the group of Euclidean motions by dilations leaves you\n> far away from the projective group!\n\nThis misses the point entirely. E and M are *affine*, not projective,\ngeometries. We don\'t experience (at least naively) one characteristic\nfeature of affine space, the dilations. So the space of experience is\n*not* affine.\n\n> Speculation in physics should be constrained by the requirements of\n> reality and the demands of compatible mathematics.\nRelativity = quadratic forms, any way you cut it.\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>Arnold Neumaier wrote:
> real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
> They have their own symmetry groups - I don't see how the group PSL(1) and
> cross ratios could be relevant for real 3D or 4D geometry.
Ok, give two events, they determine a line, which intersects the light
cone twice, thus 4 collinear points, so there is the cross-ratio
> Projective geomentry only enters when viewing 3D objects from 2D cameras,
> and this is well understood.
See above.
> There is nothing projective in Euclidean and Minkowski space.
> Projective transformations change the metric.
Euclid: (x - iy)(x + iy) =
Minkowski: (x - ct)(x + ct) =
> > Third, we have the very interesting fact that we don't
> > experience dilations, although the spaces of our "experience" are -
> > apparently - Euclidean and Minkowskian *affine* geometry, and allow
> > dilations. I strongly suspect the missing projective invariants are
> > somehow associated with these missing dilations.
>
> Even augmenting the group of Euclidean motions by dilations leaves you
> far away from the projective group!
This misses the point entirely. E and M are *affine*, not projective,
geometries. We don't experience (at least naively) one characteristic
feature of affine space, the dilations. So the space of experience is
*not* affine.
> Speculation in physics should be constrained by the requirements of
> reality and the demands of compatible mathematics.
Relativity = quadratic forms, any way you cut it.
-drl
Arnold Neumaier
Dec20-04, 09:59 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>antimatter33@yahoo.com wrote:\n> Arnold Neumaier wrote:\n>\n>>real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.\n>>They have their own symmetry groups - I don\'t see how the group PSL(1) and\n>>cross ratios could be relevant for real 3D or 4D geometry.\n>\n> Ok, give two events, they determine a line, which intersects the light\n> cone twice, thus 4 collinear points, so there is the cross-ratio\n\nWhat should be its physical interpretation???\n\n\n>>>Third, we have the very interesting fact that we don\'t\n>>>experience dilations, although the spaces of our "experience" are -\n>>>apparently - Euclidean and Minkowskian *affine* geometry, and allow\n>>>dilations. I strongly suspect the missing projective invariants are\n>>>somehow associated with these missing dilations.\n>>\n>>Even augmenting the group of Euclidean motions by dilations leaves you\n>>far away from the projective group!\n>\n> This misses the point entirely. E and M are *affine*, not projective,\n> geometries. We don\'t experience (at least naively) one characteristic\n> feature of affine space, the dilations. So the space of experience is\n> *not* affine.\n\nDilation invariance is broken by the existence of objects, which provide\na mass and length scale. Thus one would not expect to experience it.\nAnyway, this has nothing to do with projective geometry.\n\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>antimatter33@yahoo.com wrote:
> Arnold Neumaier wrote:
>
>>real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
>>They have their own symmetry groups - I don't see how the group PSL(1) and
>>cross ratios could be relevant for real 3D or 4D geometry.
>
> Ok, give two events, they determine a line, which intersects the light
> cone twice, thus 4 collinear points, so there is the cross-ratio
What should be its physical interpretation???
>>>Third, we have the very interesting fact that we don't
>>>experience dilations, although the spaces of our "experience" are -
>>>apparently - Euclidean and Minkowskian *affine* geometry, and allow
>>>dilations. I strongly suspect the missing projective invariants are
>>>somehow associated with these missing dilations.
>>
>>Even augmenting the group of Euclidean motions by dilations leaves you
>>far away from the projective group!
>
> This misses the point entirely. E and M are *affine*, not projective,
> geometries. We don't experience (at least naively) one characteristic
> feature of affine space, the dilations. So the space of experience is
> *not* affine.
Dilation invariance is broken by the existence of objects, which provide
a mass and length scale. Thus one would not expect to experience it.
Anyway, this has nothing to do with projective geometry.
Arnold Neumaier
antimatter33@yahoo.com
Dec22-04, 05:55 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>Arnold Neumaier wrote:\n\n> > Ok, give two events, they determine a line, which intersects the light\n> > cone twice, thus 4 collinear points, so there is the cross-ratio\n>\n> What should be its physical interpretation???\n\nThis one is simple - log XR ~ proper time between the events. This is\nthe standard embedding of a (pseudo) Pythagorean metric inside\nprojective geometry, using a given quadratic form (in this case, a\ndegenerate one = affine space).\n\nLikewise - in Euclidean geometry the "light cone" is better known as\nthe "circular points at infinity" with equation\n\nx^2 + y^2 + z^2 + ... = 0\n\nGive two points, their connecting line intersects this "ideal domain"\nin two points, form XR, take log, multiply by i, result is the\nPythagorean distance between the points.\n\nSomehow the observer posits the ideal domain - in one case, the light\ncone, in the other (with c->inf), "spatial infinity", that is, where\nparallel rails meet! The "thoroughgoing projective physics" would be\ninvariant under changing this domain.\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>Arnold Neumaier wrote:
> > Ok, give two events, they determine a line, which intersects the light
> > cone twice, thus 4 collinear points, so there is the cross-ratio
>
> What should be its physical interpretation???
This one is simple - log XR ~ proper time between the events. This is
the standard embedding of a (pseudo) Pythagorean metric inside
projective geometry, using a given quadratic form (in this case, a
degenerate one = affine space).
Likewise - in Euclidean geometry the "light cone" is better known as
the "circular points at infinity" with equation
x^2 + y^2 + z^2 + .[/itex].. =
Give two points, their connecting line intersects this "ideal domain"
in two points, form XR, take log, multiply by i, result is the
Pythagorean distance between the points.
Somehow the observer posits the ideal domain - in one case, the light
cone, in the other (with c->inf), "spatial infinity", that is, where
parallel rails meet! The "thoroughgoing projective physics" would be
invariant under changing this domain.
[itex]-drl
antimatter33@yahoo.com
Jan9-05, 07:47 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>\nVan www wrote:\n\n> I would say that Euclidean geometry is elliptic in that it\n> has a metric g = diag(1,1,1) in 3D,\n> while Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),\n> and integrates time into things.\n\nActually both are affine, because the fundamental quadric is\ndegenerate, e.g. in the plane\n\nMinkowski (x-ct)(x+ct) = 0 (light cone)\nEuclid (x-iy)(x+iy) = 0 (circular points at infinity)\n\nBTW for the person asking about books, the classic is\n"Nicht-Euklidische Geometrie" by Felix Klein, which however is in\nGerman (until I get around to translating it :)\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>Van www wrote:
> I would say that Euclidean geometry is elliptic in that it
> has a metric g = diag(1,1,1) in 3D,
> while Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),
> and integrates time into things.
Actually both are affine, because the fundamental quadric is
degenerate, e.g. in the plane
Minkowski (x-ct)(x+ct) = (light cone)
Euclid (x-iy)(x+iy) = (circular points at infinity)
BTW for the person asking about books, the classic is
"Nicht-Euklidische Geometrie" by Felix Klein, which however is in
German (until I get around to translating it :)
-drl
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