## Particle - String Duality (was: Layman: Could it happen?)

<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>Rozmonth@Lycos.com wrote:\n\n&gt; Could a theory like Loop Quantum Gravity (Except more simplified, so\n&gt; that it\'s all geometry, no property on the nodes and such.) give rise\n&gt; to string theory if we assumed that energy was a geometric property of\n&gt; space?\n\nLet me rephrase your question a little differently: could there be a\nnatural, but not yet fully seen or exploited, correspondence between\nstring theory and other more traditional foundations (a\' la\nSchroedinger-Heisenberg)?\n\nConsider the relativistic particle, and assume it has positive spin.\nAt the semi-classical level its motion may be described by the\nDirac-Kemmer Hamiltonian\nH = alpha.(pc) + beta (mc^2)\nwhere alpha = (alpha^1, alpha^2, alpha^3) and beta belong to the\nalgebra generated by the Kemmer matrices.\n\n(For spin 1/2, these reduce to the familiar expressions involving the\nDirac matrices; spin 1 Kemmer matrices are 10x10; things get more\ncomplex for spin 3/2 and above).\n\nIn the Heisenberg picture, the equations of motion describe a\n(quantized) lightlike helical worldline.\n\nA central mystery, if you\'re only looking from within the particle\nframe of mind, is where this bizarre behavior comes from. There\'s\nnothing within the classical theory (the geodesic law) which mandates\nanything like this behavior for particle-like singularities.\n\nBut when you move up 1 dimension, things change. Though the\n1-singularities in classical theory describe nice straight worldlines,\nthe singularities of dimension 2 or above follow the generalization of\nthe geodesic law, which identifies the corresponding n-brane as a\nharmonic map.\n\nFor the 2-brane, in the form of an open string, Thierre was the first\nto published a closed solution. This solution is consists of a 2-sheet\nwhich is generated as the locus of the midpoints (in a suitable\ncoordinate representation) taken from -- a helical worldline.\n\nThis suggests the a more general duality between particles of positive\nspin and strings; the particles\' worldline arising as a suitable\naveraging of the strings\' coordinates and, a\' la Thierre, the string\nbeing generated by a natural geometric construction from the particle\nworldline.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Rozmonth@Lycos.com wrote:

> Could a theory like Loop Quantum Gravity (Except more simplified, so
> that it's all geometry, no property on the nodes and such.) give rise
> to string theory if we assumed that energy was a geometric property of
> space?

Let me rephrase your question a little differently: could there be a
natural, but not yet fully seen or exploited, correspondence between
string theory and other more traditional foundations (a' la
Schroedinger-Heisenberg)?

Consider the relativistic particle, and assume it has positive spin.
At the semi-classical level its motion may be described by the
Dirac-Kemmer Hamiltonian
$H = \alpha$.(pc) $+ \beta (mc^2)$
where $\alpha = (\alpha^1, \alpha^2, \alpha^3)$ and $\beta$ belong to the
algebra generated by the Kemmer matrices.

(For spin $1/2,$ these reduce to the familiar expressions involving the
Dirac matrices; spin 1 Kemmer matrices are 10x10; things get more
complex for spin 3/2 and above).

In the Heisenberg picture, the equations of motion describe a
(quantized) lightlike helical worldline.

A central mystery, if you're only looking from within the particle
frame of mind, is where this bizarre behavior comes from. There's
nothing within the classical theory (the geodesic law) which mandates
anything like this behavior for particle-like singularities.

But when you move up 1 dimension, things change. Though the
1-singularities in classical theory describe nice straight worldlines,
the singularities of dimension 2 or above follow the generalization of
the geodesic law, which identifies the corresponding n-brane as a
harmonic map.

For the 2-brane, in the form of an open string, Thierre was the first
to published a closed solution. This solution is consists of a 2-sheet
which is generated as the locus of the midpoints (in a suitable
coordinate representation) taken from -- a helical worldline.

This suggests the a more general duality between particles of positive
spin and strings; the particles' worldline arising as a suitable
averaging of the strings' coordinates and, a' la Thierre, the string
being generated by a natural geometric construction from the particle
worldline.

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