Jack Tremarco
Dec3-04, 04:49 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>Hi very_cryptic --\n\nConsidering the geometric action, which is proportional to its area,\nis motivated by analogy to the point particle where this gives you the\nquantum field theories we know and love. In string theory the simplest\ngeometric action is the Nambu Goto action, which is however highly\nnon-linear (square rott). One therefore considers an equivalent\naction, which is only quadratic in the fields at the expense of some\nextra redundancy. This is the Polyakov action. The requirement that a\ngiven world sheet embedding does not depend on how you choose to\nparameterize it (something obviously necessary) translates to the\nrequirement of diff times Weyl invariance in the Polyakov fromalism.\n\nNone of your ideas for extending this concept holds any water. You\nwant the action to depend on "extrinsic curvature"? There is no such\nthing for a 2d surface: By Gauss\' "egregious theorm" curvature in 2d\ndepends only on the 2d metric and is entirely independent on how you\nembed this surface into some target space. It is also a fact that\nevery world-sheet is conformally flat, so don\'t try to replace the\nworld extrinsic by intrinsic. There is no freedom here.\n\nWhat ever else you might suggest, it has to be checked for\nconsistency. It turns out that the amount of symmetry a 2d CFT has is\nso huge that the above described is all we can do. It is unique.\n\nYour ideas for more direct string interactions are also not new and\nhave been ruled out. The point here is essentially that if you try to\nadd a local interaction between world sheets, you do something that\'s\nhighly non-local in momentum space. String theory is a theory of\ngravity, so you cannot uniquely desribe points in target space, since\nsuch an attempt will always depend on the coordinates you choose on\nyour curved target space. Nonlocal interactions will immediately break\nLorentz invariance and lead to all sorts of inconsistencies.\n\nThe only string interactions that are mathematically consistent are\nthe ones that are already present in the sum over embeddings. They are\nunique, too!\n\nThe fact that strings define maps from the world sheet into target\nspace is just the mathematical version of the statement that we want\nthe strings to live in spacetime. What do you want to replace this\nwith? I am not sure I see your point here.\n\nIf you need motivation to study string theory look at the beautiful\nresults derived from it. A serious introduction like Polchinski will\ngive you better answers than a forum like this one.\n\nBest, Jack\n\nvery_cryptic@hotmail.com wrote in message news:<cokunb\\$1so8\\$1@fiasco.xenopsyche.net>...\ 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. The usual rationalizations\n> might go something like this: The Polyakov action is simplest in 2D and\n> only possesses conformal symmetry in 2D. I don\'t find either of the two\n> reasons all that convincing. Simplicity is simply a matter of\n> convenience and a priori, conformal symmetry appears unnecessary.\n> Another closely related reason is that only in 1 and 2D is it always\n> possible for the intrinsic metric to be isomorphic to flat 1 or 2D\n> spacetime in some gauge. The latter is possible thanks to conformal\n> symmetry. In higher dimensions, this is not the case in general. But\n> once again, I feel this is merely a matter of convenience for\n> quantization. While it is generally a good idea to study toy models\n> because they are simpler, it would be mistaken to insist nature is\n> described by a toy model.\n>\n> 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>\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>\n> And why can\'t strings branch? Soap bubble surfaces do (when three soap\n> bubbles are touching each other).\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>\n> I don\'t know too much about string theory yet (I\'m still learning), so\n> I might not be able to understand explanations alluding to more\n> advanced concepts, but on the other hand, I don\'t wish to have\n> oversimplified answers either.\n>\n> Thanks 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>Hi very_cryptic --
Considering the geometric action, which is proportional to its area,
is motivated by analogy to the point particle where this gives you the
quantum field theories we know and love. In string theory the simplest
geometric action is the Nambu Goto action, which is however highly
non-linear (square rott). One therefore considers an equivalent
action, which is only quadratic in the fields at the expense of some
extra redundancy. This is the Polyakov action. The requirement that a
given world sheet embedding does not depend on how you choose to
parameterize it (something obviously necessary) translates to the
requirement of diff times Weyl invariance in the Polyakov fromalism.
None of your ideas for extending this concept holds any water. You
want the action to depend on "extrinsic curvature"? There is no such
thing for a 2d surface: By Gauss' "egregious theorm" curvature in 2d
depends only on the 2d metric and is entirely independent on how you
embed this surface into some target space. It is also a fact that
every world-sheet is conformally flat, so don't try to replace the
world extrinsic by intrinsic. There is no freedom here.
What ever else you might suggest, it has to be checked for
consistency. It turns out that the amount of symmetry a 2d CFT has is
so huge that the above described is all we can do. It is unique.
Your ideas for more direct string interactions are also not new and
have been ruled out. The point here is essentially that if you try to
add a local interaction between world sheets, you do something that's
highly non-local in momentum space. String theory is a theory of
gravity, so you cannot uniquely desribe points in target space, since
such an attempt will always depend on the coordinates you choose on
your curved target space. Nonlocal interactions will immediately break
Lorentz invariance and lead to all sorts of inconsistencies.
The only string interactions that are mathematically consistent are
the ones that are already present in the sum over embeddings. They are
unique, too!
The fact that strings define maps from the world sheet into target
space is just the mathematical version of the statement that we want
the strings to live in spacetime. What do you want to replace this
with? I am not sure I see your point here.
If you need motivation to study string theory look at the beautiful
results derived from it. A serious introduction like Polchinski will
give you better answers than a forum like this one.
Best, Jack
very_cryptic@hotmail.com wrote in message news:<cokunb$1so8$1@fiasco.xenopsyche.net>...
> 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.
Considering the geometric action, which is proportional to its area,
is motivated by analogy to the point particle where this gives you the
quantum field theories we know and love. In string theory the simplest
geometric action is the Nambu Goto action, which is however highly
non-linear (square rott). One therefore considers an equivalent
action, which is only quadratic in the fields at the expense of some
extra redundancy. This is the Polyakov action. The requirement that a
given world sheet embedding does not depend on how you choose to
parameterize it (something obviously necessary) translates to the
requirement of diff times Weyl invariance in the Polyakov fromalism.
None of your ideas for extending this concept holds any water. You
want the action to depend on "extrinsic curvature"? There is no such
thing for a 2d surface: By Gauss' "egregious theorm" curvature in 2d
depends only on the 2d metric and is entirely independent on how you
embed this surface into some target space. It is also a fact that
every world-sheet is conformally flat, so don't try to replace the
world extrinsic by intrinsic. There is no freedom here.
What ever else you might suggest, it has to be checked for
consistency. It turns out that the amount of symmetry a 2d CFT has is
so huge that the above described is all we can do. It is unique.
Your ideas for more direct string interactions are also not new and
have been ruled out. The point here is essentially that if you try to
add a local interaction between world sheets, you do something that's
highly non-local in momentum space. String theory is a theory of
gravity, so you cannot uniquely desribe points in target space, since
such an attempt will always depend on the coordinates you choose on
your curved target space. Nonlocal interactions will immediately break
Lorentz invariance and lead to all sorts of inconsistencies.
The only string interactions that are mathematically consistent are
the ones that are already present in the sum over embeddings. They are
unique, too!
The fact that strings define maps from the world sheet into target
space is just the mathematical version of the statement that we want
the strings to live in spacetime. What do you want to replace this
with? I am not sure I see your point here.
If you need motivation to study string theory look at the beautiful
results derived from it. A serious introduction like Polchinski will
give you better answers than a forum like this one.
Best, Jack
very_cryptic@hotmail.com wrote in message news:<cokunb$1so8$1@fiasco.xenopsyche.net>...
> 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.