Why strings? Why the Polyakov action? [Archive]

very_cryptic@hotmail.com

Dec1-04, 11:07 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 have been considering trying to learn a bit of string theory, but I\nhave to admit I don\'t find the motivations for it all that convincing.\nHere are a couple of questions I have.\n\n1-dimensional extended objects are the worldlines of particles which\nhave been well studied and 2-d objects are possibly strings. While I\nknow (from hearsay, I haven\'t managed to learn anywhere near enough to\nunderstand why) string theory predicts p-branes I wonder why string\ntheory starts with 2-d extended objects. The usual rationalizations\nmight go something like this: The Polyakov action is simplest in 2D and\nonly possesses conformal symmetry in 2D. I don\'t find either of the two\nreasons all that convincing. Simplicity is simply a matter of\nconvenience and a priori, conformal symmetry appears unnecessary.\nAnother closely related reason is that only in 1 and 2D is it always\npossible for the intrinsic metric to be isomorphic to flat 1 or 2D\nspacetime in some gauge. The latter is possible thanks to conformal\nsymmetry. In higher dimensions, this is not the case in general. But\nonce again, I feel this is merely a matter of convenience for\nquantization. While it is generally a good idea to study toy models\nbecause they are simpler, it would be mistaken to insist nature is\ndescribed by a toy model.\n\nBut even if we restrict ourselves to 2D objects, why do we have to\ninsist upon the Polyakov action? Why not add higher order terms? After\nall, most physical strings have stresses which depend upon their\n(extrinsic) curvature. Why does the action have to be proportional to\nthe area alone?\n\nOr for that matter, why do we have to model strings as a map from a 2D\nspacetime to a target space (pseudoRiemannian for bosonic strings and\nsuperspace for superstrings)? This way of modeling strings makes it\nhard to add interaction terms dealing with the self-intersection of a\nstring or the intersection of two strings, which is something we would\nexpect for physical strings. In the Polyakov model, strings simply pass\nthrough each other and themselves like ghosts.\n\nAnd why can\'t strings branch? Soap bubble surfaces do (when three soap\nbubbles are touching each other).\n\nWhy do strings have no other (position dependent) internal degrees of\nfreedom other than its position in the target space and the intrinsic\nmetric?\n\nI don\'t know too much about string theory yet (I\'m still learning), so\nI might not be able to understand explanations alluding to more\nadvanced concepts, but on the other hand, I don\'t wish to have\noversimplified answers either.\n\nThanks 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 been considering trying to learn a bit of string theory, but I
have to admit I don't find the motivations for it all that convincing.
Here are a couple of questions I have.

1-dimensional extended objects are the worldlines of particles which
have been well studied and 2-d objects are possibly strings. While I
know (from hearsay, I haven't managed to learn anywhere near enough to
understand why) string theory predicts p-branes I wonder why string
theory starts with 2-d extended objects. The usual rationalizations
might go something like this: The Polyakov action is simplest in 2D and
only possesses conformal symmetry in 2D. I don't find either of the two
reasons all that convincing. Simplicity is simply a matter of
convenience and a priori, conformal symmetry appears unnecessary.
Another closely related reason is that only in 1 and 2D is it always
possible for the intrinsic metric to be isomorphic to flat 1 or 2D
spacetime in some gauge. The latter is possible thanks to conformal
symmetry. In higher dimensions, this is not the case in general. But
once again, I feel this is merely a matter of convenience for
quantization. While it is generally a good idea to study toy models
because they are simpler, it would be mistaken to insist nature is
described by a toy model.

But even if we restrict ourselves to 2D objects, why do we have to
insist upon the Polyakov action? Why not add higher order terms? After
all, most physical strings have stresses which depend upon their
(extrinsic) curvature. Why does the action have to be proportional to
the area alone?

Or for that matter, why do we have to model strings as a map from a 2D
spacetime to a target space (pseudoRiemannian for bosonic strings and
superspace for superstrings)? This way of modeling strings makes it
hard to add interaction terms dealing with the self-intersection of a
string or the intersection of two strings, which is something we would
expect for physical strings. In the Polyakov model, strings simply pass
through each other and themselves like ghosts.

And why can't strings branch? Soap bubble surfaces do (when three soap
bubbles are touching each other).

Why do strings have no other (position dependent) internal degrees of
freedom other than its position in the target space and the intrinsic
metric?

I don't know too much about string theory yet (I'm still learning), so
I might not be able to understand explanations alluding to more
advanced concepts, but on the other hand, I don't wish to have
oversimplified answers either.

Thanks in advance.

Aaron Bergman

Dec2-04, 06:13 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>In article <1101901431.971521.323450@c13g2000cwb.googlegroups .com>,\nvery_cryptic@hotmail.com wrote:\n\n> I have been considering trying to learn a bit of string theory, but I\n> have to admit I don\'t find the motivations for it all that convincing.\n> Here are a couple of questions I have.\n>\n> 1-dimensional extended objects are the worldlines of particles which\n> have been well studied and 2-d objects are possibly strings. While I\n> know (from hearsay, I haven\'t managed to learn anywhere near enough to\n> understand why) string theory predicts p-branes I wonder why string\n> theory starts with 2-d extended objects.\n\nIt\'s historical. String theory has its origins in trying to understand\nthe strong force. An amplitude was written down way back when by\nVeneziano that had some nice properties and it was realized by Susskind\nthat this amplitude could be derived from a theory of strings. Later, it\nwas realized that string theory had a graviton like particle and thus\nstrings as a theory of quantum gravity was born.\n\nAaron\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>In article <1101901431.971521.323450@c13g2000cwb.googlegroups. com>,
very_cryptic@hotmail.com wrote:

> I have been considering trying to learn a bit of string theory, but I
> have to admit I don't find the motivations for it all that convincing.
> Here are a couple of questions I have.
>
> 1-dimensional extended objects are the worldlines of particles which
> have been well studied and 2-d objects are possibly strings. While I
> know (from hearsay, I haven't managed to learn anywhere near enough to
> understand why) string theory predicts p-branes I wonder why string
> theory starts with 2-d extended objects.

It's historical. String theory has its origins in trying to understand
the strong force. An amplitude was written down way back when by
Veneziano that had some nice properties and it was realized by Susskind
that this amplitude could be derived from a theory of strings. Later, it
was realized that string theory had a graviton like particle and thus
strings as a theory of quantum gravity was born.

Aaron

Lubos Motl

Dec4-04, 08:00 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>On Thu, 2 Dec 2004, Aaron Bergman wrote:\n\n> very_cryptic@hotmail.com wrote:\n\n> > ... I haven\'t managed to learn anywhere near enough to\n> > understand why - string theory predicts p-branes I wonder why string\n> > theory starts with 2-d extended objects.\n>\n> It\'s historical. ...\n\nWell, it\'s not quite historical, is it? Dualities give us some sort of\n"Brane democracy" (the word comes from Paul Townsend), but on the other\nhand, (the fundamental) strings remain special.\n\nWhen we investigate dynamics and properties of other objects, such as\nD-branes of any dimension, we always reduce the calculation to a\ncomputation involving fundamental strings.\n\nFor example, all fields and particles that propagate inside a D-brane are\ndescribed by different vibration modes of open strings that can be\nattached to such a D-brane with their endpoints.\n\nThe fundamental strings are the only objects whose internal dynamics is\ndescribed by ordinary field theory - namely a two-dimensional theory of\ngravity with some extra matter fields in it. Two is the worldsheet\ndimension (one time, one spatial direction along the string).\n\nTwo-dimensional gravity is special. The metric tensor has three\nindependent components, g_{11}, g_{12}, g_{22}. Also, there are three\nparameters of a local symmetry at each point: two parameters of a\ncoordinate redefinition (diffeomorphism), namely delta sigma^1 and delta\nsigma^2, and also delta omega. The latter determines a Weyl symmetry, a\nrescaling of the metric tensor by a numerical factor.\n\nWe are allowed to require that the new, transformed g_{ij} equals\ndelta_{ij}, for example, and this set of 3 conditions can be solved for\nthe 3 parameters of the symmetry: each equivalence class of configurations\n(related by the local symmetry) contains a representative that has\ng_{ij}=delta_{ij}, at least in some finite neighborhood of a point. In\nother words, the metric tensor in two dimensions, if we have the Weyl\nsymmetry, can always (locally) be set to a predetermined non-singular\ntensor.\n\nThis implies that the two-dimensional gravity has no propagating degrees\nof freedom, and the dynamics of the string reduces to the dynamics of the\nscalar (and fermionic) fields on the pre-determined (flat) worldsheet.\nThis can be written as a nice, free theory.\n\nSomething like that is not possible for objects whose dimension is higher\nthan 1+1, simply because the metric tensor on the worldvolume has too many\ncomponents, and there are not enough symmetries to get rid of this metric\ntensor. Consequently, we deal with a nonlinear theory in dimension greater\nthan 2 that includes gravity, and it turns out that this specific type of\ntheory is as divergent (non-renormalizable) as much as four-dimensional\ngeneral relativity, for example. It\'s not a good starting point for a\nwell-behaved theory.\n\nStrings are the only compromise of objects that are not pointlike - so\nthat they are able to regularize the divergences in spacetime - but whose\ndimension is small enough so that we don\'t introduce new lethal\ndivergences in their worldvolume. It\'s not just about the dimension of\nthese strings: their other properties are also heavily constrained by\nconsistency. In the "maximal" dimension 10, there only exist 5 types of\nperturbative superstring theory. Using methods that go beyond perturbative\nstring theory, one can show that all of them are connected.\n\nEventually one can show that string theory predicts the existence of\np-branes for virtually any p, but their precise properties are calculated\nfrom the strings themselves. We would be happy to have a formulation in\nwhich the strings were not fundamental, but rather appeared as a solution\nthat would be on par with other objects - but we don\'t have a universal\ndefinition of string theory where it would be possible.\n\nBest\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</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, 2 Dec 2004, Aaron Bergman wrote:

> very_cryptic@hotmail.com wrote:

> > ... I haven't managed to learn anywhere near enough to
> > understand why - string theory predicts p-branes I wonder why string
> > theory starts with 2-d extended objects.
>
> It's historical. ...

Well, it's not quite historical, is it? Dualities give us some sort of
"Brane democracy" (the word comes from Paul Townsend), but on the other
hand, (the fundamental) strings remain special.

When we investigate dynamics and properties of other objects, such as
D-branes of any dimension, we always reduce the calculation to a
computation involving fundamental strings.

For example, all fields and particles that propagate inside a D-brane are
described by different vibration modes of open strings that can be
attached to such a D-brane with their endpoints.

The fundamental strings are the only objects whose internal dynamics is
described by ordinary field theory - namely a two-dimensional theory of
gravity with some extra matter fields in it. Two is the worldsheet
dimension (one time, one spatial direction along the string).

Two-dimensional gravity is special. The metric tensor has three
independent components, g_{11}, g_{12}, g_{22}. Also, there are three
parameters of a local symmetry at each point: two parameters of a
coordinate redefinition (diffeomorphism), namely \delta \sigma^1 and \delta\sigma^2, and also \delta \omega. The latter determines a Weyl symmetry, a
rescaling of the metric tensor by a numerical factor.

We are allowed to require that the new, transformed g_{ij} equals
\delta_{ij}, for example, and this set of 3 conditions can be solved for
the 3 parameters of the symmetry: each equivalence class of configurations
(related by the local symmetry) contains a representative that has
g_{ij}=\delta_{ij}, at least in some finite neighborhood of a point. In
other words, the metric tensor in two dimensions, if we have the Weyl
symmetry, can always (locally) be set to a predetermined non-singular
tensor.

This implies that the two-dimensional gravity has no propagating degrees
of freedom, and the dynamics of the string reduces to the dynamics of the
scalar (and fermionic) fields on the pre-determined (flat) worldsheet.
This can be written as a nice, free theory.

Something like that is not possible for objects whose dimension is higher
than 1+1, simply because the metric tensor on the worldvolume has too many
components, and there are not enough symmetries to get rid of this metric
tensor. Consequently, we deal with a nonlinear theory in dimension greater
than 2 that includes gravity, and it turns out that this specific type of
theory is as divergent (non-renormalizable) as much as four-dimensional
general relativity, for example. It's not a good starting point for a
well-behaved theory.

Strings are the only compromise of objects that are not pointlike - so
that they are able to regularize the divergences in spacetime - but whose
dimension is small enough so that we don't introduce new lethal
divergences in their worldvolume. It's not just about the dimension of
these strings: their other properties are also heavily constrained by
consistency. In the "maximal" dimension 10, there only exist 5 types of
perturbative superstring theory. Using methods that go beyond perturbative
string theory, one can show that all of them are connected.

Eventually one can show that string theory predicts the existence of
p-branes for virtually any p, but their precise properties are calculated
from the strings themselves. We would be happy to have a formulation in
which the strings were not fundamental, but rather appeared as a solution
that would be on par with other objects - but we don't have a universal
definition of string theory where it would be possible.

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/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Aaron Bergman

Dec4-04, 12:16 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>Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> Well, it\'s not quite historical, is it? Dualities give us some sort of\n> "Brane democracy" (the word comes from Paul Townsend), but on the other\n> hand, (the fundamental) strings remain special.\n\nAt zero coupling. As you move the coupling away from zero, as I know you\nknow, your strings can grow an extra dimension or become difficult to\ndistinguish from D-strings.\n\nAaron\n\n[Moderator\'s note: sure, no doubt about that. But the dynamics of these\nD-branes or strings-turned-D-branes is still calculated from strings,\nis not it? LM]\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 Motl <motl@feynman.harvard.edu> wrote:

> Well, it's not quite historical, is it? Dualities give us some sort of
> "Brane democracy" (the word comes from Paul Townsend), but on the other
> hand, (the fundamental) strings remain special.

At zero coupling. As you move the coupling away from zero, as I know you
know, your strings can grow an extra dimension or become difficult to
distinguish from D-strings.

Aaron

[Moderator's note: sure, no doubt about that. But the dynamics of these
D-branes or strings-turned-D-branes is still calculated from strings,
is not it? LM]

Jack Tremarco

Dec6-04, 07:16 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>> But even if we restrict ourselves to 2D objects, why do we have to\n> insist upon the Polyakov action? Why not add higher order terms? After\n> all, most physical strings have stresses which depend upon their\n> (extrinsic) curvature. Why does the action have to be proportional to\n> the area alone?\n\nWhat do you mean by extrinsic curvature? Gauss\' "egregious theorem"\ntells us that the curvature of a 2d space is independent of its\nembedding into a higher dimensional (target) space. It only depends on\nthe 2d metric. This in turn can be taken care of by the fact the every\n2d space is conformally flat. So you would have to explain what kind\nof terms you want to add. There simply are no known candidates that\ncan consistently be added and, in fact, one can formulate far reaching\nno-go theorems that such terms almost certainly do not exist.\n\n> Or for that matter, why do we have to model strings as a map from a 2D\n> spacetime to a target space (pseudoRiemannian for bosonic strings and\n> superspace for superstrings)? This way of modeling strings makes it\n> hard to add interaction terms dealing with the self-intersection of a\n> string or the intersection of two strings, which is something we would\n> expect for physical strings. In the Polyakov model, strings simply pass\n> through each other and themselves like ghosts.\n\nThe fact that strings define maps from the world sheet into target\nspace is just the mathematical version of the statement that strings\nlive in spacetime. What do you want to replace this with? The fact\nthat the only allowed interactions are the ones already implicit in\nthe Polyakov path integral is one of the biggest virtues of string\ntheory and makes it so unique. You cannot add interactions by hand,\nbecause they would be nonlocal and spoil essential symmetries like\nLorentz invariance.\n\n> And why can\'t strings branch? Soap bubble surfaces do (when three soap\n> bubbles are touching each other).\n\nSure they can. Look at Polchinski chapter 3 for a comprehensible\nexplanation of which string interactions are and aren\'t allowed and\nwhy.\n\n> Why do strings have no other (position dependent) internal degrees of\n> freedom other than its position in the target space and the intrinsic\n> metric?\n\nAgain, it is not clear that any such degrees of freedom exist that can\nbe added without causing obvious inconsistencies.\n\nBest,\nJack\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>> But even if we restrict ourselves to 2D objects, why do we have to
> insist upon the Polyakov action? Why not add higher order terms? After
> all, most physical strings have stresses which depend upon their
> (extrinsic) curvature. Why does the action have to be proportional to
> the area alone?

What do you mean by extrinsic curvature? Gauss' "egregious theorem"
tells us that the curvature of a 2d space is independent of its
embedding into a higher dimensional (target) space. It only depends on
the 2d metric. This in turn can be taken care of by the fact the every
2d space is conformally flat. So you would have to explain what kind
of terms you want to add. There simply are no known candidates that
can consistently be added and, in fact, one can formulate far reaching
no-go theorems that such terms almost certainly do not exist.

> Or for that matter, why do we have to model strings as a map from a 2D
> spacetime to a target space (pseudoRiemannian for bosonic strings and
> superspace for superstrings)? This way of modeling strings makes it
> hard to add interaction terms dealing with the self-intersection of a
> string or the intersection of two strings, which is something we would
> expect for physical strings. In the Polyakov model, strings simply pass
> through each other and themselves like ghosts.

The fact that strings define maps from the world sheet into target
space is just the mathematical version of the statement that strings
live in spacetime. What do you want to replace this with? The fact
that the only allowed interactions are the ones already implicit in
the Polyakov path integral is one of the biggest virtues of string
theory and makes it so unique. You cannot add interactions by hand,
because they would be nonlocal and spoil essential symmetries like
Lorentz invariance.

> And why can't strings branch? Soap bubble surfaces do (when three soap
> bubbles are touching each other).

Sure they can. Look at Polchinski chapter 3 for a comprehensible
explanation of which string interactions are and aren't allowed and
why.

> Why do strings have no other (position dependent) internal degrees of
> freedom other than its position in the target space and the intrinsic
> metric?

Again, it is not clear that any such degrees of freedom exist that can
be added without causing obvious inconsistencies.

Best,
Jack

The Undergraduate

Feb1-05, 03:50 AM

Someone may have already asked this, but... can you (theoretially) cut a string in half? It seems like this would at least be mathematically possible. I mean, isn't it mathematically impossible to find a fundamental unit, since you can take any number, any mass, any volume, and cut it into smaller pieces? It seems to me that there might be a fundamental substance, but not a fundamental particle. Am I making any sense?

Urs Schreiber

Feb1-05, 05:30 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>"The Undergraduate" <amalphru@student.umass.edu> schrieb im Newsbeitrag\nnews:The.Undergraduate.1jqt44-100000@physicsforums.com...\n\n> Someone may have already asked this, but... can you (theoretially) cut a\n> string in half? It seems like this would at least be mathematically\n> possible. I mean, isn\'t it mathematically impossible to find a\n> fundamental unit, since you can take any number, any mass, any volume,\n> and cut it into smaller pieces? It seems to me that there might be a\n> fundamental substance, but not a fundamental particle. Am I making any\n> sense?\n\nRecall how a fundamental discrete unit appears as you go from classical to\nquantum mechanics:\n\nConsider a free particle smeared over some region of diameter d. Since the\nwave function is continuous you can just as well have a particle smeared\nover a region of diameter d/2 only. Or over d/4. And so on.\n\nBut a particle has more properties than just a position. It also has\nmomentum. And it turns out that while you reduce the uncertainty in the\nposition the uncertainty in the momentum will inevitably grow. And in such a\nway that the product of both is bigger than a fixed unit of action.\n\nNothing in the physics here is discrete, everything is continuous. And yet\nthere can be a fundamental unit.\n\nNow consider not a free particle but one in an oscillator potential. Its\nenergy is the sum of the kinetic energy due to its momentum plus the\npotential energy due to its position in the parabolic potential. Due to\nuncertainty the particle has minimum energy not when it sits precisely at\nthe bottom of the potential (as it would be classically) but when it is\nsmeared a bit in position space and a bit in momentum space. By a similar\nlogic it follows that the osciallor can pick up energy only in discrete\nunits - quanta.\n\nNow consider a free string. It has a tension which means that apart from the\nkinetic energy of each of its "points" there is potential energy coming from\nthe extension of the string along these points. Hence every single point of\nthe string is a little oscillator, in a sense. As you may know from the\nstudy of a chain of masses connected by springs, such a linear collection of\ncoupled oscillators can be conveniently reexpressed in terms of normal\nmodes, i.e. collective excitiations of these oscillators. These normal modes\nbehave like ordinary oscillators themselves, but uncoupled ones.\n\nIn any case, from the above it follows that there is a minimum fundamental\nunit of energy in all these modes and that all higher excitations come with\ndiscrete amounts of additional energy - even though the physical\nconfiguration is continuous!\n\nFor the string this means that even in its lowest energy state it has a\nfundamental unit of being smeared out over spacetime. And that its\nexcitations come in discrete packages of energy. For the string these quanta\nof excitation energy appear as quanta of mass of the center-of-mass motion\nof the string.\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>"The Undergraduate" <amalphru@student.umass.edu> schrieb im Newsbeitrag
news:The.Undergraduate.1jqt44-100000@physicsforums.com...

> Someone may have already asked this, but... can you (theoretially) cut a
> string in half? It seems like this would at least be mathematically
> possible. I mean, isn't it mathematically impossible to find a
> fundamental unit, since you can take any number, any mass, any volume,
> and cut it into smaller pieces? It seems to me that there might be a
> fundamental substance, but not a fundamental particle. Am I making any
> sense?

Recall how a fundamental discrete unit appears as you go from classical to
quantum mechanics:

Consider a free particle smeared over some region of diameter d. Since the
wave function is continuous you can just as well have a particle smeared
over a region of diameter d/2 only. Or over d/4. And so on.

But a particle has more properties than just a position. It also has
momentum. And it turns out that while you reduce the uncertainty in the
position the uncertainty in the momentum will inevitably grow. And in such a
way that the product of both is bigger than a fixed unit of action.

Nothing in the physics here is discrete, everything is continuous. And yet
there can be a fundamental unit.

Now consider not a free particle but one in an oscillator potential. Its
energy is the sum of the kinetic energy due to its momentum plus the
potential energy due to its position in the parabolic potential. Due to
uncertainty the particle has minimum energy not when it sits precisely at
the bottom of the potential (as it would be classically) but when it is
smeared a bit in position space and a bit in momentum space. By a similar
logic it follows that the osciallor can pick up energy only in discrete
units - quanta.

Now consider a free string. It has a tension which means that apart from the
kinetic energy of each of its "points" there is potential energy coming from
the extension of the string along these points. Hence every single point of
the string is a little oscillator, in a sense. As you may know from the
study of a chain of masses connected by springs, such a linear collection of
coupled oscillators can be conveniently reexpressed in terms of normal
modes, i.e. collective excitiations of these oscillators. These normal modes
behave like ordinary oscillators themselves, but uncoupled ones.

In any case, from the above it follows that there is a minimum fundamental
unit of energy in all these modes and that all higher excitations come with
discrete amounts of additional energy - even though the physical
configuration is continuous!

For the string this means that even in its lowest energy state it has a
fundamental unit of being smeared out over spacetime. And that its
excitations come in discrete packages of energy. For the string these quanta
of excitation energy appear as quanta of mass of the center-of-mass motion
of the string.