Discussion:
Layman: Could it happen?
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R***@Lycos.com
2005-06-04 18:42:03 UTC
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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?
m***@yahoo.com
2005-06-14 07:49:00 UTC
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Post by R***@Lycos.com
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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