alistair
Nov2-04, 11:27 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>What difference would it make to string theory if the strings had a\nsmall width?\nAm I right in thinking that they are treated mathematically as being\ninfinitely thin?\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>What difference would it make to string theory if the strings had a
small width?
Am I right in thinking that they are treated mathematically as being
infinitely thin?
Lubos Motl
Nov2-04, 03:30 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 Tue, 2 Nov 2004, alistair wrote:\n\n> What difference would it make to string theory if the strings had a\n> small width? Am I right in thinking that they are treated\n> mathematically as being infinitely thin?\n\nActually it seems as a good question, at least to me. Yes, the strings in\nperturbative string theory are treated as infinitely thin objects, and the\nconfiguration is literally described by an embedding of this thin string\ninto spacetime - i.e. by functions X^i(sigma) that only depend on one\ncoordinate. The image of this map in spacetime is an infinitesimally thin\nstring.\n\nWhen you turn on the interactions, the strings start to split and join,\nand in some sense it must mean that the conformal field theory based on\nX^i(sigma) is just an approximation, something that becomes invalid if the\nstring coupling is nonzero.\n\nIf you imagine that the strings are literally 3-dimensional tubes, for\nexample, you will not be able to derive a meaningful theory - Heisenberg\nand Dirac tried this, and it did not work. The only loophole of their\nno-go theorems seem to be the relativistic string, which is 1-dimensional.\n\nNevertheless, the strings become "different" if the coupling is finite.\n\nIn some cases, we know what the strings become. In "type IIA" or "E8 x E8\nheterotic" string theory, the strings become two-dimensional membranes in\nM-theory, wrapped on a circle (or stretched between the two domain walls -\nthe boundaries of spacetime). The membranes are still infinitely thin, in\na sense, but they are two-dimensional - they are extended in 2 dimensions\ninstead of one, unlike the strings.\n\nWell, the description purely in terms of infinitely thin membranes does\nnot really work too well, and one must discretize it to get Matrix theory,\nand so forth. This makes the local geometry "fuzzy", in a sense.\n\nIt must be morally true that the concept of the infinitesimal string is\njust an approximation valid in perturbative string theory where "g" is\ninfinitesimally small: not only spacetime geometry, but also the existence\nof a string is a derived approximate notion. Well, we know that the\nprevious sentence is definitely true - the whole perturbative string\ntheory is just an approximation. But by saying the sentence in this way, I\nalso implicitly propose that we should try to look for the framework that\n"generalizes" the description of string theory at a finite value of the\ncoupling, in which the strings become "M", something more general than a\nstring, a fattened fuzzy modified generalized string whose dimension can\noscillate, but whose theory is algebraically analogous to the\ntwo-dimensional CFTs that describe the infinitesimally thin strings.\n\nLook at the formula for the S-matrix amplitudes calculated from a CFT, and\nimagine that you replace the detailed components by something else. The\npath integral will not be over 2-dimensional CFTs, but some general fields\n- imagine the configuration space of large matrices. There will be a new\naction, and there will be vertex operators that won\'t be 2-dimensional\nintegrals anymore, but rather traces or something more general, and\nint d^2\\sigma exp(i.k.X(z)) can become \\Tr exp(i.k.X) where X is a\nmatrix.\n\nDo you know how to generalize the argument that could show that the\nS-matrix calculated along these lines will be unitary? The conformal\nsymmetry may be generalized to something else that can remove the\nunphysical polarizations - i.e. the N-scaling symmetry. Do we know how to\nconstruct the analogue of the stress energy tensor for the symmetry under\nscaling of N? Is there some replacement for the central charge, and some\ncounting of the critical dimension?\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\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>On Tue, 2 Nov 2004, alistair wrote:
> What difference would it make to string theory if the strings had a
> small width? Am I right in thinking that they are treated
> mathematically as being infinitely thin?
Actually it seems as a good question, at least to me. Yes, the strings in
perturbative string theory are treated as infinitely thin objects, and the
configuration is literally described by an embedding of this thin string
into spacetime - i.e. by functions X^i(\sigma) that only depend on one
coordinate. The image of this map in spacetime is an infinitesimally thin
string.
When you turn on the interactions, the strings start to split and join,
and in some sense it must mean that the conformal field theory based on
X^i(\sigma) is just an approximation, something that becomes invalid if the
string coupling is nonzero.
If you imagine that the strings are literally 3-dimensional tubes, for
example, you will not be able to derive a meaningful theory - Heisenberg
and Dirac tried this, and it did not work. The only loophole of their
no-go theorems seem to be the relativistic string, which is 1-dimensional.
Nevertheless, the strings become "different" if the coupling is finite.
In some cases, we know what the strings become. In "type IIA" or "E8 x E8
heterotic" string theory, the strings become two-dimensional membranes in
M-theory, wrapped on a circle (or stretched between the two domain walls -
the boundaries of spacetime). The membranes are still infinitely thin, in
a sense, but they are two-dimensional - they are extended in 2 dimensions
instead of one, unlike the strings.
Well, the description purely in terms of infinitely thin membranes does
not really work too well, and one must discretize it to get Matrix theory,
and so forth. This makes the local geometry "fuzzy", in a sense.
It must be morally true that the concept of the infinitesimal string is
just an approximation valid in perturbative string theory where "g" is
infinitesimally small: not only spacetime geometry, but also the existence
of a string is a derived approximate notion. Well, we know that the
previous sentence is definitely true - the whole perturbative string
theory is just an approximation. But by saying the sentence in this way, I
also implicitly propose that we should try to look for the framework that
"generalizes" the description of string theory at a finite value of the
coupling, in which the strings become "M", something more general than a
string, a fattened fuzzy modified generalized string whose dimension can
oscillate, but whose theory is algebraically analogous to the
two-dimensional CFTs that describe the infinitesimally thin strings.
Look at the formula for the S-matrix amplitudes calculated from a CFT, and
imagine that you replace the detailed components by something else. The
path integral will not be over 2-dimensional CFTs, but some general fields
- imagine the configuration space of large matrices. There will be a new
action, and there will be vertex operators that won't be 2-dimensional
integrals anymore, but rather traces or something more general, and
\int d^2\sigma \exp(i.k.X(z)) can become \Tr \exp(i.k.X) where X is a
matrix.
Do you know how to generalize the argument that could show that the
S-matrix calculated along these lines will be unitary? The conformal
symmetry may be generalized to something else that can remove the
unphysical polarizations - i.e. the N-scaling symmetry. Do we know how to
construct the analogue of the stress energy tensor for the symmetry under
scaling of N? Is there some replacement for the central charge, and some
counting of the critical dimension?
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<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>Lubos says:\n\n"If you imagine that the strings are literally 3-dimensional tubes, for\nexample, you will not be able to derive a meaningful theory - Heisenberg\nand Dirac tried this"\n\nYou mean to say that Heisenberg and Dirac\nactually took a 4-dimensional manifold and\nmade a theory of 4 quantum scalar fields on\nthis 4 dimensional manifold and made the fields\nsatisfy all the lorenz symmetries on the target\nspace of the fields and found this to be\ninconsistent?\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>Lubos says:
"If you imagine that the strings are literally 3-dimensional tubes, for
example, you will not be able to derive a meaningful theory - Heisenberg
and Dirac tried this"
You mean to say that Heisenberg and Dirac
actually took a 4-dimensional manifold and
made a theory of 4 quantum scalar fields on
this 4 dimensional manifold and made the fields
satisfy all the lorenz symmetries on the target
space of the fields and found this to be
inconsistent?
Matti Pitkanen
Nov5-04, 02:41 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>mandro <ultraman2002@hotmail.com> wrote in message news:<dec722c5.0411030649.23d7fcff-100000@posting.google.com>...\n> Lubos says:\n>\n> "If you imagine that the strings are literally 3-dimensional tubes, for\n> example, you will not be able to derive a meaningful theory - Heisenberg\n> and Dirac tried this"\n>\n> You mean to say that Heisenberg and Dirac\n> actually took a 4-dimensional manifold and\n> made a theory of 4 quantum scalar fields on\n> this 4 dimensional manifold and made the fields\n> satisfy all the lorenz symmetries on the target\n> space of the fields and found this to be\n> inconsistent?\n\n\nA apologize for my inability to resist the temptation to interfere\nwith a comment.\n\nThe extreme non-linearity of any general coordinate invariant action\nprinciple is a serious obstacle for constructing theories of\nhigher-dimensional objects: in string theories conformal invariance\nallows to overcome the difficulties.\n\n>From my own experience during 10 first years of TGD (I began around\n1978) I can tell that both canonical quantization and functional\nintegral approach failed completely for 4-D objects in H=M^4xCP_2 for\nthe physically realistic choice of action (generalization of Maxwell\naction but with Maxwell field defined by the induced Kahler form). The\nreason was that gauge degeneracy corresponds geometrically to a\ngigantic vacuum degeneracy: any 4-surface with CP_2 projection Y^2\nwhich is Lagrange manifold defines a vacuum extremal.\n\nFor instance, for small perturbations around the canonical imbedding\nof M^4 to H the first terms in the perturbative expansion are fourth\norder. Since the kinetic term is absent, the failure of the\nfunctional integral approach could not be more dramatic. The same\napplies to the canonical quantization: the time derivatives of the\nimbedding space coordinates cannot be solved in terms of canonical\nmomentum densities as single valued functions.\n\nThis forced to replace the functional integral approach with\ngeometrization of the configuration space of 3-surfaces. The dream is\nthat the infinite-dimensional (necessarily) Kahler geometry is unique\nby the existence of Riemann connection as found in the case of of loop\nspaces by Dan Freed. Configuration space would be a union of\ninfinite-dimensional symmetric spaces labelled by zero modes of the\nmetric. Symmetric space property is also required by calculability\n(all points of symmetric space are metrically equivalent) and zero\nmode degeneracy by the fact that a 3-surface with a size electron\ncannot be metrically equivalent with a 3-surface with a size of\ngalaxy.\n\n\n\nThe loss of conformal invariance and Kac-Moody algebra in dimension 3\nwould certainly be good sounding modern arguments for giving up the\nidea of 3-dimensional fundamental objects. These arguments are however\non a shaky ground.\n\na) Let us accept quantum holography and general coordinate invariance\nin the sense that any 3-dimensional section X^3 of 4-D space-time\nsurface X^4 can serve as a "causal determinant". What this means that\nonce you know 3-surface X^3 you know X^4 by classical field equations:\nX^4(X^3) is an analog of Bohr orbit. This argument involves an\nover-simplicifaction but this is not essential for what follows.\n\nb) Suppose that you can select this 3-surface X^3 to be a light like\nsub-manifold analogous to a light front created by a light source of\narbitrary shape in the curved space-time (curvature implies that the\nfront need not look expanding when viewed from imbedding space). This\nallows a unique gauge fixing allowing to circumvent technical problems\nproduced by the 4-D general coordinate invariance acting as a gauge\nsymmetry.\n\nc) By its lighlikeness X^3 is metrically 2-dimensional and allows\ngeneralized conformal symmetries. X^3 possesses also degenerate\nKaehler and symplectic structures. One of the most fascinating\ngeometric properties is that the group of isometries is\ninfinite-dimensional and in 1-1 correspondence with conformal\nsymmetries. Indeed, taking lightcone boundary as a simple example and\nwriting it as V= S^2xR_+, one realizes that the Weyl scaling induced\nby the conformal transformation of S^2 can be compensated by an\nS^2-local scaling of the light-like radial coordinate so that an\nisometry results. These properties makes space-time dimension D=4\ncompletely unique.\n\n\nIt requires a lot of work to work out the physical interpretation and\ndetails (it took 25 years for me). The outcome is however a theory\nwhich in some aspects resembles membrane theory.\n\na) There are actually 2 kinds of causal determinants. The\n7-dimensional lightlike surfaces X^7= V^3xCP_2 of imbedding space, V^3\nfuture or past light cone boundary, act as causal determinants at\nimbedding space-level playing a role analogous to that of "big bang".\nThey are forced by the loss of complete determinism due to the\nabove-mentioned vacuum degeneracy of geometrized Maxwell action. They\nare somewhat analogous to 6-branes in M-theory in the sense that pairs\nof space-time sheets with opposite time orientations are created on\nthem from vacuum and carrying opposite inertial energies. There are\n2 conformal invariances corresponding to imbedding space conformal\ninvariance associated with V^3 and space-time conformal invariance\nassociated with surfaces X^3. Hence there is much more conformla\nstructure than in string theory.\n\nb) As far as state construction is considered, the intersections of\nX^3 and V^7 defining 2-D "parton" surfaces X^2 are in a key role (as\none might guess from the fact that lightlike dimension of X^3 is not\ndynamical). These intersections are somewhat analogous to the\njunctures of strings and branes. One can speak of effective\n2-dimensionality so that there is definite resemblance with\nM(embrane) theories.\n\n\nI have explained these aspects in earlier discussion thread\n"Alternative to strings and Kaluza-Klein theories". Material related\nto the theory can be found at\n\nhttp://www.physics.helsinki.fi/~matpitka/tgd.html\n\nand\n\nhttp://www.physics.helsinki.fi/~matpitka/padtgd.html .\n\nMatti Pitkanen\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>mandro <ultraman2002@hotmail.com> wrote in message news:<dec722c5.0411030649.23d7fcff-100000@posting.google.com>...
> Lubos says:
>
> "If you imagine that the strings are literally 3-dimensional tubes, for
> example, you will not be able to derive a meaningful theory - Heisenberg
> and Dirac tried this"
>
> You mean to say that Heisenberg and Dirac
> actually took a 4-dimensional manifold and
> made a theory of 4 quantum scalar fields on
> this 4 dimensional manifold and made the fields
> satisfy all the lorenz symmetries on the target
> space of the fields and found this to be
> inconsistent?
A apologize for my inability to resist the temptation to interfere
with a comment.
The extreme non-linearity of any general coordinate invariant action
principle is a serious obstacle for constructing theories of
higher-dimensional objects: in string theories conformal invariance
allows to overcome the difficulties.
>From my own experience during 10 first years of TGD (I began around
1978) I can tell that both canonical quantization and functional
integral approach failed completely for 4-D objects in H=M^{4xCP_2} for
the physically realistic choice of action (generalization of Maxwell
action but with Maxwell field defined by the induced Kahler form). The
reason was that gauge degeneracy corresponds geometrically to a
gigantic vacuum degeneracy: any 4-surface with CP_2 projection Y^2
which is Lagrange manifold defines a vacuum extremal.
For instance, for small perturbations around the canonical imbedding
of M^4 to H the first terms in the perturbative expansion are fourth
order. Since the kinetic term is absent, the failure of the
functional integral approach could not be more dramatic. The same
applies to the canonical quantization: the time derivatives of the
imbedding space coordinates cannot be solved in terms of canonical
momentum densities as single valued functions.
This forced to replace the functional integral approach with
geometrization of the configuration space of 3-surfaces. The dream is
that the infinite-dimensional (necessarily) Kahler geometry is unique
by the existence of Riemann connection as found in the case of of loop
spaces by Dan Freed. Configuration space would be a union of
infinite-dimensional symmetric spaces labelled by zero modes of the
metric. Symmetric space property is also required by calculability
(all points of symmetric space are metrically equivalent) and zero
mode degeneracy by the fact that a 3-surface with a size electron
cannot be metrically equivalent with a 3-surface with a size of
galaxy.
The loss of conformal invariance and Kac-Moody algebra in dimension 3
would certainly be good sounding modern arguments for giving up the
idea of 3-dimensional fundamental objects. These arguments are however
on a shaky ground.
a) Let us accept quantum holography and general coordinate invariance
in the sense that any 3-dimensional section X^3 of 4-D space-time
surface X^4 can serve as a "causal determinant". What this means that
once you know 3-surface X^3 you know X^4 by classical field equations:
X^4(X^3) is an analog of Bohr orbit. This argument involves an
over-simplicifaction but this is not essential for what follows.
b) Suppose that you can select this 3-surface X^3 to be a light like
sub-manifold analogous to a light front created by a light source of
arbitrary shape in the curved space-time (curvature implies that the
front need not look expanding when viewed from imbedding space). This
allows a unique gauge fixing allowing to circumvent technical problems
produced by the 4-D general coordinate invariance acting as a gauge
symmetry.
c) By its lighlikeness X^3 is metrically 2-dimensional and allows
generalized conformal symmetries. X^3 possesses also degenerate
Kaehler and symplectic structures. One of the most fascinating
geometric properties is that the group of isometries is
infinite-dimensional and in 1-1 correspondence with conformal
symmetries. Indeed, taking lightcone boundary as a simple example and
writing it as V= S^{2xR_}+, one realizes that the Weyl scaling induced
by the conformal transformation of S^2 can be compensated by an
S^2-local scaling of the light-like radial coordinate so that an
isometry results. These properties makes space-time dimension D=4
completely unique.
It requires a lot of work to work out the physical interpretation and
details (it took 25 years for me). The outcome is however a theory
which in some aspects resembles membrane theory.
a) There are actually 2 kinds of causal determinants. The
7-dimensional lightlike surfaces X^7= V^{3xCP_2} of imbedding space, V^3
future or past light cone boundary, act as causal determinants at
imbedding space-level playing a role analogous to that of "big bang".
They are forced by the loss of complete determinism due to the
above-mentioned vacuum degeneracy of geometrized Maxwell action. They
are somewhat analogous to 6-branes in M-theory in the sense that pairs
of space-time sheets with opposite time orientations are created on
them from vacuum and carrying opposite inertial energies. There are
2 conformal invariances corresponding to imbedding space conformal
invariance associated with V^3 and space-time conformal invariance
associated with surfaces X^3. Hence there is much more conformla
structure than in string theory.
b) As far as state construction is considered, the intersections of
X^3 and V^7 defining 2-D "parton" surfaces X^2 are in a key role (as
one might guess from the fact that lightlike dimension of X^3 is not
dynamical). These intersections are somewhat analogous to the
junctures of strings and branes. One can speak of effective
2-dimensionality so that there is definite resemblance with
M(embrane) theories.
I have explained these aspects in earlier discussion thread
"Alternative to strings and Kaluza-Klein theories". Material related
to the theory can be found at
http://www.physics.helsinki.fi/~matpitka/tgd.html
and
http://www.physics.helsinki.fi/~matpitka/padtgd.html .
Matti Pitkanen
DivineNathicana
Nov25-04, 02:28 AM
Alright, here's a stupid question:
Do the strings actually physically exist? Or are they assumed to physically exist due to the duality thing that says that it doesn't matter whether we consider the phenomenon as spacetime moving or an actual string vibrating?
How can anything having only one dimension exist? Lines per se don't exist. They are purely mathematical concepts. Please guys do not bring up any ant and garden hose univesre analogies lol.
- Alisa
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