Non-physicist: Klein bottle [Archive]

David Kogan

May13-04, 06:46 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>I am not a physicist, so this question is purposefully vague: What\nsignificance, if any, does the Kline bottle have in string theory?\n\nThank you.\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 am not a physicist, so this question is purposefully vague: What
significance, if any, does the Kline bottle have in string theory?

Thank you.

Urs Schreiber

May13-04, 02:31 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>On Thu, 13 May 2004, David Kogan wrote:\n\n> I am not a physicist, so this question is purposefully vague: What\n> significance, if any, does the Kline bottle have in string theory?\n\nThe Klein bottle is one of the four Riemannian surfaces with\nEuler number 0. The other three are the torus, the cylinder\nand the Moebius strip, where the latter two have a boundary\nwhile the Klein bottle and the torus don\'t.\n\nYou may have heard how in particle physics physical processes\ncan be decomposed into sums of Feynman diagrams, which indicate\n"loops" that the particles trace out in spacetime.\n\nIt\'s similar in string theory, only that the diagrams here are\nsurfaces, due to the spatial extension of the string as compared\nto the 0-dimensional particle.\n\nFor perturbative calculations in string theory one hence has\nto sum over all processes where a string traces all possible\nRiemannian surface. At first order (at "one-loop") this are\nthe above surfaces of Euler number 0 for the open and\nthe closed string, respectively.\n\n(It is technically of importance that the torus and the\ncylinder are orientable, while the Klein bottle and\nthe Moebius strip are not, but you need not worry about\nthat at this point.)\n\nSo the short answer to your question is:\n\nThe significance of the Klein bottle in string theory is\nthat this is the surface traced out by a closed string\nwhich performs a loop in spacetime in such a way that when\ncoming back to its original position it has changed\nits orientation.\n\nWithout the change of orientation the same process instead\nyields the torus.\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 Thu, 13 May 2004, David Kogan wrote:

> I am not a physicist, so this question is purposefully vague: What
> significance, if any, does the Kline bottle have in string theory?

The Klein bottle is one of the four Riemannian surfaces with
Euler number . The other three are the torus, the cylinder
and the Moebius strip, where the latter two have a boundary
while the Klein bottle and the torus don't.

You may have heard how in particle physics physical processes
can be decomposed into sums of Feynman diagrams, which indicate
"loops" that the particles trace out in spacetime.

It's similar in string theory, only that the diagrams here are
surfaces, due to the spatial extension of the string as compared
to the 0-dimensional particle.

For perturbative calculations in string theory one hence has
to sum over all processes where a string traces all possible
Riemannian surface. At first order (at "one-loop") this are
the above surfaces of Euler number for the open and
the closed string, respectively.

(It is technically of importance that the torus and the
cylinder are orientable, while the Klein bottle and
the Moebius strip are not, but you need not worry about
that at this point.)

So the short answer to your question is:

The significance of the Klein bottle in string theory is
that this is the surface traced out by a closed string
which performs a loop in spacetime in such a way that when
coming back to its original position it has changed
its orientation.

Without the change of orientation the same process instead
yields the torus.

David Kogan

May15-04, 05:32 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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message\n> On Thu, 13 May 2004, David Kogan wrote:\n>\n> > I am not a physicist, so this question is purposefully vague: What\n> > significance, if any, does the Kline bottle have in string theory?\n>\n> The Klein bottle is one of the four Riemannian surfaces with\n> Euler number 0. The other three are the torus, the cylinder\n> and the Moebius strip, where the latter two have a boundary\n> while the Klein bottle and the torus don\'t.\n>\n> You may have heard how in particle physics physical processes\n> can be decomposed into sums of Feynman diagrams, which indicate\n> "loops" that the particles trace out in spacetime.\n>\n> It\'s similar in string theory, only that the diagrams here are\n> surfaces, due to the spatial extension of the string as compared\n> to the 0-dimensional particle.\n>\n> For perturbative calculations in string theory one hence has\n> to sum over all processes where a string traces all possible\n> Riemannian surface. At first order (at "one-loop") this are\n> the above surfaces of Euler number 0 for the open and\n> the closed string, respectively.\n>\n> (It is technically of importance that the torus and the\n> cylinder are orientable, while the Klein bottle and\n> the Moebius strip are not, but you need not worry about\n> that at this point.)\n>\n> So the short answer to your question is:\n>\n> The significance of the Klein bottle in string theory is\n> that this is the surface traced out by a closed string\n> which performs a loop in spacetime in such a way that when\n> coming back to its original position it has changed\n> its orientation.\n>\n> Without the change of orientation the same process instead\n> yields the torus.\n>\n\nCould the Klein bottle be the geometry of the Cosmos? The nonorientable\nsurface of the Klein bottle can be oriented by "defining" a string on the\nKlein bottle as a "cosmic event." Such as the cosmic microwave background\nradiation, the big bang, unification or a "now".\n\nYou said:\n> The significance of the Klein bottle in string theory is\n> that this is the surface traced out by a closed string\n> which performs a loop in spacetime in such a way that when\n> coming back to its original position it has changed\n> its orientation.\n\nThis would imply that the Cosmos is self-recursive and fractal, but the\n"change its orientation" that you refer to could also be related to a "now"\nseparating past and future.\n\nIn three-dimensions the surface of the Klein bottle passes through itself\ncreating a hole, but it does not in four-dimensions. There are three\ninvariant spatial dimensions, and a thermodynamic reversible temporal\ndimension, in (relativistic) space-time.\n\nThe hole in three dimensions could be a string that all other points/strings\nin space-time are using as orientation. The hole in three-dimensions could\nbe a "now" that keeps everything from happening at once. That the hole does\nnot exist in four-dimensions could by why time exhibits diferent behaviour\nthan the other three-dimensions.\n\nI realize all of this needs to be gotten mathematically, but the\n"extensible" surface of the Klein bottle seems to stand-out from other\nsurfaces. It has no volume, is nonorientable, lacks boundary and has no\n"inside" or "outside." It seems to have many of the same properties as a\npoint.\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>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message
> On Thu, 13 May 2004, David Kogan wrote:
>
> > I am not a physicist, so this question is purposefully vague: What
> > significance, if any, does the Kline bottle have in string theory?
>
> The Klein bottle is one of the four Riemannian surfaces with
> Euler number . The other three are the torus, the cylinder
> and the Moebius strip, where the latter two have a boundary
> while the Klein bottle and the torus don't.
>
> You may have heard how in particle physics physical processes
> can be decomposed into sums of Feynman diagrams, which indicate
> "loops" that the particles trace out in spacetime.
>
> It's similar in string theory, only that the diagrams here are
> surfaces, due to the spatial extension of the string as compared
> to the 0-dimensional particle.
>
> For perturbative calculations in string theory one hence has
> to sum over all processes where a string traces all possible
> Riemannian surface. At first order (at "one-loop") this are
> the above surfaces of Euler number for the open and
> the closed string, respectively.
>
> (It is technically of importance that the torus and the
> cylinder are orientable, while the Klein bottle and
> the Moebius strip are not, but you need not worry about
> that at this point.)
>
> So the short answer to your question is:
>
> The significance of the Klein bottle in string theory is
> that this is the surface traced out by a closed string
> which performs a loop in spacetime in such a way that when
> coming back to its original position it has changed
> its orientation.
>
> Without the change of orientation the same process instead
> yields the torus.
>

Could the Klein bottle be the geometry of the Cosmos? The nonorientable
surface of the Klein bottle can be oriented by "defining" a string on the
Klein bottle as a "cosmic event." Such as the cosmic microwave background
radiation, the big bang, unification or a "now".

You said:
> The significance of the Klein bottle in string theory is
> that this is the surface traced out by a closed string
> which performs a loop in spacetime in such a way that when
> coming back to its original position it has changed
> its orientation.

This would imply that the Cosmos is self-recursive and fractal, but the
"change its orientation" that you refer to could also be related to a "now"
separating past and future.

In three-dimensions the surface of the Klein bottle passes through itself
creating a hole, but it does not in four-dimensions. There are three
invariant spatial dimensions, and a thermodynamic reversible temporal
dimension, in (relativistic) space-time.

The hole in three dimensions could be a string that all other points/strings
in space-time are using as orientation. The hole in three-dimensions could
be a "now" that keeps everything from happening at once. That the hole does
not exist in four-dimensions could by why time exhibits diferent behaviour
than the other three-dimensions.

I realize all of this needs to be gotten mathematically, but the
"extensible" surface of the Klein bottle seems to stand-out from other
surfaces. It has no volume, is nonorientable, lacks boundary and has no
"inside" or "outside." It seems to have many of the same properties as a
point.