View Full Version : Background Independence
Dear Forum,
As far as I can tell, the usual statement of background independence in string theory is that even though one chooses a specific metric (and other target space fields) in the string theory path integral, a particular choice of background field is exactly equivalent to any other choice - so long as one inserts appropriate vertex operators in the path integral. For example, the path integral with a curved metric is the same as the path integral with a flat target space AND a certain coherent state of gravitons inserted.
In this sense, two different background fields do not give two different theories, but can be thought of as two different states of the same string theory. Hopefully this is OK so far, and agrees with the usual interpretations.
If so, my question is as follows; suppose I consider a target space manifold with, say, a flat metric for simplicity, but with non-trivial homology groups. Can I think of the string theory path integral with this target space as just being the flat-space R^10 path integral but with an insertion of some appropriate string vertex operators?
In other words, is there an obvious way to think of two topologically-different target manifolds as corresponding to different states of the same string theory, just as there is for two different choices of background fields?
Sorry, I possibly haven't said this too clearly, but if anyone can put me straight it's just something that's been bugging me....
Urs Schreiber
Aug12-04, 12:44 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>"d70yxj" <redbull_j@yahoo.com> schrieb im Newsbeitrag\nnews:d70yxj.1aqwm7-100000@physicsforums.com...\n\n> If so, my question is as follows; suppose I consider a target space\n> manifold with, say, a flat metric for simplicity, but with non-trivial\n> homology groups. Can I think of the string theory path integral with\n> this target space as just being the flat-space R^10 path integral but\n> with an insertion of some appropriate string vertex operators?\n\nNo, if the metric is flat, no vertex operators are inserted.\n\n> In other words, is there an obvious way to think of two\n> topologically-different target manifolds as corresponding to different\n> states of the same string theory, just as there is for two different\n> choices of background fields?\n\nI believe this is best thought of in terms of covering spaces. work in the\ncovering space (e.g. flat space for toroidal compactifications) and take\ncare of the correct identifications.\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>"d70yxj" <redbull_j@yahoo.com> schrieb im Newsbeitrag
news:d70yxj.1aqwm7-100000@physicsforums.com...
> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space R^{10} path integral but
> with an insertion of some appropriate string vertex operators?
No, if the metric is flat, no vertex operators are inserted.
> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?
I believe this is best thought of in terms of covering spaces. work in the
covering space (e.g. flat space for toroidal compactifications) and take
care of the correct identifications.
> > In other words, is there an obvious way to think of two
> > topologically-different target manifolds as corresponding to different
> > states of the same string theory, just as there is for two different
> > choices of background fields?[/color]
> I believe this is best thought of in terms of covering spaces. work in the
> covering space (e.g. flat space for toroidal compactifications) and take
> care of the correct identifications.
So I do the path integral with a cover of my original manifold, and then do something (what?) at the end to get to the correct answer? This doesn't seem as obviously `stringy' as the way different background fields can be seen as string states sitting on a flat background.
Also, a covering space for a given manifold needn't be topologically trivial anyway. So it's not obvious to me that I can start with any target manifold and relate it back to R^10 using what you say.
Rufus Anton
Aug17-04, 12:00 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>( Redbull d70 wrote: )\n\n> In this sense, two different background fields do not give two\n> different theories, but can be thought of as two different states of\n> the same string theory. Hopefully this is OK so far, and agrees with\n> the usual interpretations.\n\nIt think that\'s fine so far.\n\n> If so, my question is as follows; suppose I consider a target space\n> manifold with, say, a flat metric for simplicity, but with non-trivial\n> homology groups. Can I think of the string theory path integral with\n> this target space as just being the flat-space R^10 path integral but\n> with an insertion of some appropriate string vertex operators?\n>\n> In other words, is there an obvious way to think of two\n> topologically-different target manifolds as corresponding to different\n> states of the same string theory, just as there is for two different\n> choices of background fields?\n\nWell, let\'s back off for a second. The Polyakov path integral\nprescription tells you to gauge fix as follows: Divide the space of\nall metrics that needs to be summed over in the path integral into\nequivalence classes, then (arbitrarily) choose a fiducial metric in\neach equivalence class and sum over those fiducial metrics instead.\nThis procedure mods out the diff x Weyl gauge group volumes (at the\nexpense of introducing ghosts).\nNow, if you need to include a topologically non-trivial sector into\nyour path integral, this will necessarily involve an equivalence class\ndifferent from the one that contains the topologically trivial R^10.\nHence you will have to choose a different fiducial metric in that\nsector.\nWithin each equivalence class different choices of the fiducial metric\ncorrespond to different insertions of vertex operators. But you can\nnever get from one equivalence class to another by any insertion of\nvertex operators. This should be clear as the vertex operators are\nlocal insertions (on the worldsheet) and as such cannot account for a\nglobal change of topology. The trick of thinking of a curved target\nspace metric in terms of a coherent states of gravitons is cute, but\nit should be pointed out that it only works as long as the *global*\ntopology of the curved metric is that of the flat R^10, that is, the\nhomology groups must be the same.\n\nThe statement of background independence is preserved as follows.\nBefore gauge fixing nothing depends on a specific metric because you\nsum over all of them. (An integral is always independent of the\nintegration variable...) After gauge fixing you have a sum over\nfiducial metrics, so superficially you might worry that this seems to\ndepend on the choice of fiducial metrics you had to make. But, by\nconstruction, the coice of fiducial metrics is irrelevant. The sum\nover them is equivalent to the initial sum over everything and hence\nbackground independent.\n\nHope that helps.\n\nBest,\nRufus\n\n[Moderator\'s note: have not you confused the worldsheet metric and the\ntarget manifold\'s metric, Rufus? In string theory, the target space\nmetric is fixed at the beginning, and one calculates the S-matrix on\nthis particular background. Physics at other backgrounds of the same\ntopology can be obtained by inserting vertex operators to the action,\nby deforming the worldsheet action with perturbative string states.\nIt\'s because an infinitesimal change of geometry corresponds to a\ncondensate of closed strings. However, if you want to obtain a manifold\nwith completely different homology groups, you must make a true\ntopology change transition, and switching from one topology to another\ntopology (branch) is represented by a condensation of non-perturbative\nstates, e.g. massless D3-branes (see chapter 13 of The Elegant Universe\nfor an elementary introduction). Massless D3-branes are not really\nlocal vertex operators on the worldsheet; in some sense, they can be\ndescribed by nonlocal vertex operators that add a boundary to the\nworldsheet. At any rate, the stringy perturbative expansion breaks\ndown once you change the geometry in such a way that some wrapped\nD3-branes become massless. For a more technical description of the\nconifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 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>( Redbull d70 wrote: )
> In this sense, two different background fields do not give two
> different theories, but can be thought of as two different states of
> the same string theory. Hopefully this is OK so far, and agrees with
> the usual interpretations.
It think that's fine so far.
> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space R^{10} path integral but
> with an insertion of some appropriate string vertex operators?
>
> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?
Well, let's back off for a second. The Polyakov path integral
prescription tells you to gauge fix as follows: Divide the space of
all metrics that needs to be summed over in the path integral into
equivalence classes, then (arbitrarily) choose a fiducial metric in
each equivalence class and sum over those fiducial metrics instead.
This procedure mods out the diff x Weyl gauge group volumes (at the
expense of introducing ghosts).
Now, if you need to include a topologically non-trivial sector into
your path integral, this will necessarily involve an equivalence class
different from the one that contains the topologically trivial R^{10}.
Hence you will have to choose a different fiducial metric in that
sector.
Within each equivalence class different choices of the fiducial metric
correspond to different insertions of vertex operators. But you can
never get from one equivalence class to another by any insertion of
vertex operators. This should be clear as the vertex operators are
local insertions (on the worldsheet) and as such cannot account for a
global change of topology. The trick of thinking of a curved target
space metric in terms of a coherent states of gravitons is cute, but
it should be pointed out that it only works as long as the *global*
topology of the curved metric is that of the flat R^{10}, that is, the
homology groups must be the same.
The statement of background independence is preserved as follows.
Before gauge fixing nothing depends on a specific metric because you
sum over all of them. (An integral is always independent of the
integration variable...) After gauge fixing you have a sum over
fiducial metrics, so superficially you might worry that this seems to
depend on the choice of fiducial metrics you had to make. But, by
construction, the coice of fiducial metrics is irrelevant. The sum
over them is equivalent to the initial sum over everything and hence
background independent.
Hope that helps.
Best,
Rufus
[Moderator's note: have not you confused the worldsheet metric and the
target manifold's metric, Rufus? In string theory, the target space
metric is fixed at the beginning, and one calculates the S-matrix on
this particular background. Physics at other backgrounds of the same
topology can be obtained by inserting vertex operators to the action,
by deforming the worldsheet action with perturbative string states.
It's because an infinitesimal change of geometry corresponds to a
condensate of closed strings. However, if you want to obtain a manifold
with completely different homology groups, you must make a true
topology change transition, and switching from one topology to another
topology (branch) is represented by a condensation of non-perturbative
states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
for an elementary introduction). Massless D3-branes are not really
local vertex operators on the worldsheet; in some sense, they can be
described by nonlocal vertex operators that add a boundary to the
worldsheet. At any rate, the stringy perturbative expansion breaks
down once you change the geometry in such a way that some wrapped
D3-branes become massless. For a more technical description of the
conifold transition, see e.g. http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9504145 LM]
> [Moderator's note: have not you confused the worldsheet metric and the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background. Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings.
> However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to another
> topology (branch) is represented by a condensation of non-perturbative
> states,
Thanks moderator, I think this is exactly the question I was asking.
> in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet.
Is there a good reference to see it in this way? The Greene et al paper doesn't seem to talk about it in quite this language.
> At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless.
So I specify infinite string coupling in order that the tension of the D3-brane goes to zero, is that right? Maybe I've misinterpreted that, but it somehow seems a bit odd; perturbative string theory on a target space with non-trivial homology groups can only be thought of as an infinite coupling string theory on flat space (with D3-branes added). Whereas perturbative string theory in manifold with just different geometry can always be thought of as perturbative string theory on flat space (with gravitons). Is that odd? Maybe it's what I should expect, I'm not sure. Thanks for the help, anyway.
I have a slightly different question, partly related to this, and also to the recent Hawking material. In his talk here he emphasised that in AdS/CFT one should sum over all supergravity configurations compatible with the appropriate boundary conditions. (I guess this is also emphasised in the Maldacena paper he references, and in 9803131 too). So at the level of partition functions you have
Z_cft = Z_s where Z_s is obtained by integrating or summing the sugra action over all appropriate metrics. In particular, in those above examples the interesting thing is that both AdS and AdS-Schwartzschild need to be included when comparing bulk results for correlation functions etc to results in the CFT.
So what happens when I move from just supergravity in the bulk, to string theory? Do I also need to sum over all appropriate target space metrics, or can I just choose one and string theory somehow does the rest for me?
(And is AdS-Schwartschild target space related to an AdS target space but with some kind of condensation of non-perturbative states?)
arivero
Aug17-04, 02:02 PM
Dear Forum,
As far as I can tell, the usual statement of background independence in string theory is that even though one chooses a specific metric (and other target space fields) in the string theory path integral, a particular choice of background field is exactly equivalent to any other choice ...
At the epoch of the T-duality revolution (about 1995, I mean) , there was around another concept of background independence; some lecturers presented the target space as a temporary concept, to be substituted in the future by some other calculation technique where only "spaces of strings" were to be consirered, and "spaces of points" were to be not needed anymore. I have not idea is such conception has been preserved in the publications.
Rufus Anton
Aug18-04, 12:17 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>Lubos Motl wrote:\n\n> Moderator\'s note: have not you confused the worldsheet metric and the\n> target manifold\'s metric, Rufus? In string theory, the target space\n> metric is fixed at the beginning, and one calculates the S-matrix on\n> this particular background.\n\nThanks for your remarks, Lubos. Apparently I didn\'t make this very\nclear, but the point is that the set of equivalence classes of\n*worldsheet metrics* depends in an essential way on the homology\ngroups of the *target space*. For every nontrivial cycle on the target\nspace there will be a set of equivalence classes of worldsheet metrics\ncoming from world sheets that wrap this cycle. Toroidal\ncompactification is possibly the simplest example for this: The\npartition function breaks into a sum of topologically distinct sectors\nlabeled by winding numbers.\n\n[Moderator\'s note: Yes, these configurations are called the worldsheet\ninstantons. LM]\n\n> Physics at other backgrounds of the same\n> topology can be obtained by inserting vertex operators to the action,\n> by deforming the worldsheet action with perturbative string states.\n> It\'s because an infinitesimal change of geometry corresponds to a\n> condensate of closed strings. However, if you want to obtain a manifold\n> with completely different homology groups, you must make a true\n> topology change transition, and switching from one topology to another\n> topology (branch) is represented by a condensation of non-perturbative\n> states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe\n> for an elementary introduction). Massless D3-branes are not really\n> local vertex operators on the worldsheet; in some sense, they can be\n> described by nonlocal vertex operators that add a boundary to the\n> worldsheet. At any rate, the stringy perturbative expansion breaks\n> down once you change the geometry in such a way that some wrapped\n> D3-branes become massless. For a more technical description of the\n> conifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 LM\n\nWhy do you think this is in conflict with what I said? If you insert\nsomething nonlocal into the path integral, so that you add another\nboundary to the worldsheet as you say, then you end up in a different\nequivalence class within the space of worldsheet metrics. But if you\nstarted with a *complete* set of such classes, as I suggested, it was\nalready there to begin with. This is the very definition of the\n(first-quantized) approach to string theory as we understand it and it\ncontains both perturbative and non-perturbative states. So all you do\nis pretending that you can transcend the classification of metrics\ninto equivalence classes by allowing for non-local vertex operator\ninsertions.\n\n[Moderator\'s note: I just don\'t quite understand the role that you want to\nassign to worldsheet instantons in answering the question. No doubt,\nworldsheet instantons DO play a very important role in topology change,\nas Witten showed in the example of the flop transition, see\nhttp://arxiv.org/abs/hep-th/9301042 - but the role seems to be different\nthan your comments. The worldsheet instantons contribute to various\nobservables such as the particle masses (or Yukawa couplings) and\nguarantee that the total value is continuous throughout the transition,\neven though the contribution of the classical Yukawa couplings has\na discontinuity. Nevertheless, different target space manifolds have\ndifferent spectrum of possible worldsheet instantons - they have\ndifferent second homology - and therefore the division of the worldsheet\nconfigurations into topological classes (the division into worldsheet\ninstantons) DOES depend on the target space manifold\'s topology. You\nseem to propose a way how to mask this difference and use a unified\ntreatment, but I just don\'t understand how it works and whether there\nis anything physical about such a description. 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 wrote:
> Moderator's note: have not you confused the worldsheet metric and the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background.
Thanks for your remarks, Lubos. Apparently I didn't make this very
clear, but the point is that the set of equivalence classes of
*worldsheet metrics* depends in an essential way on the homology
groups of the *target space*. For every nontrivial cycle on the target
space there will be a set of equivalence classes of worldsheet metrics
coming from world sheets that wrap this cycle. Toroidal
compactification is possibly the simplest example for this: The
partition function breaks into a sum of topologically distinct sectors
labeled by winding numbers.
[Moderator's note: Yes, these configurations are called the worldsheet
instantons. LM]
> Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings. However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to another
> topology (branch) is represented by a condensation of non-perturbative
> states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
> for an elementary introduction). Massless D3-branes are not really
> local vertex operators on the worldsheet; in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet. At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless. For a more technical description of the
> conifold transition, see e.g. http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9504145 LM
Why do you think this is in conflict with what I said? If you insert
something nonlocal into the path integral, so that you add another
boundary to the worldsheet as you say, then you end up in a different
equivalence class within the space of worldsheet metrics. But if you
started with a *complete* set of such classes, as I suggested, it was
already there to begin with. This is the very definition of the
(first-quantized) approach to string theory as we understand it and it
contains both perturbative and non-perturbative states. So all you do
is pretending that you can transcend the classification of metrics
into equivalence classes by allowing for non-local vertex operator
insertions.
[Moderator's note: I just don't quite understand the role that you want to
assign to worldsheet instantons in answering the question. No doubt,
worldsheet instantons DO play a very important role in topology change,
as Witten showed in the example of the flop transition, see
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9301042 - but the role seems to be different
than your comments. The worldsheet instantons contribute to various
observables such as the particle masses (or Yukawa couplings) and
guarantee that the total value is continuous throughout the transition,
even though the contribution of the classical Yukawa couplings has
a discontinuity. Nevertheless, different target space manifolds have
different spectrum of possible worldsheet instantons - they have
different second homology - and therefore the division of the worldsheet
configurations into topological classes (the division into worldsheet
instantons) DOES depend on the target space manifold's topology. You
seem to propose a way how to mask this difference and use a unified
treatment, but I just don't understand how it works and whether there
is anything physical about such a description. LM]
Rufus, do you mean that by summing correctly over all equivalence classes of worldsheet metrics, one is actually summing over all possible topologies and geometries of the target space, almost without realizing it?
I don't think I've seen this expressed in quite this way in the usual sources, but perhaps I've missed something. Is there a good reference?
Rufus Anton
Sep9-04, 01:51 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>d70yxj <redbull_j@yahoo.com> wrote in message news:<d70yxj.1c8a8r-100000@physicsforums.com>...\n\n> Rufus, do you mean that by summing correctly over all equivalence\n> classes of worldsheet metrics, one is actually summing over all\n> possible topologies and geometries of the target space, almost without\n> realizing it?\n\nLet me first say that what I am saying is not half as outlandish as\nyou apparently think and it certainly is nothing new. It may just be a\nview point that is slightly different from the standard lore.\n\nD-branes were first discovered as soliton solutions of the low-energy\neffective equations of motion of (super-) string theory. Polchinski\'s\naccomplishment (in \'96) was to show that these solitons of the\neffective theory correspond to nonperturbative microscopic states in\nthe full-fledged string theory. Nevertheless, Polchinski\'s treatment\nstill pertains to a description in terms of classical target space\nproperties. While such a description provides intuitive pictures of\nthe low-energy description and is useful for a number of important\nissues (including the structure of moduli space), it is not quite\nsufficient. After all, the great thing about string theory is that it\nis a theory of quantum gravity and, hence, we should not expect to be\nable to describe it solely in terms of classical geometry. Maybe in\nthe future a more natural and/or complete understanding of the quantum\ngeometry of target space will emerge, but for now we can study the\nquantum regime by studying the CFT on the world sheet. The spectrum of\nthis CFT includes both perturbative and non-perturbative states and\nencodes the (quantum) geometry of target space. Surely if you\nformulate the CFT on the world-sheet general enough it will have\nexcitations in its spectrum that correspond to target spaces of\ndifferent topolgy in the low-energy effective description.\n\nHow does this relate to Lubos\' remarks concerning, e.g., Witten\'s\nwork? Well, if you insist on describing the target space in classical\nterms, some hard work is needed to show that physical properties\nremain continous even if the topology changes. The reason why one\nwould choose the classical description is, of course, that CFTs are\nnot sufficiently under control to calculate all the things we want to\nknow. It is simply too hard to solve them.\n\nHowever, even if we can\'t calculate many things explicitly, we can\nstill learn a great deal about the conceptual workings of string\ntheory. Despite the fact that the most general CFT that is consistent\nwith all classical string backgrounds hasn\'t been found yet, it\npresumably does exist. The original question in this thread was about\nbackground independence. This question has a particularly simple and\ngeneral answer in the abstract CFT language. That\'s what I tried to\nexplain using familiar terms.\n\nBest,\nRufus\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>d70yxj <redbull_j@yahoo.com> wrote in message news:<d70yxj.1c8a8r-100000@physicsforums.com>...
> Rufus, do you mean that by summing correctly over all equivalence
> classes of worldsheet metrics, one is actually summing over all
> possible topologies and geometries of the target space, almost without
> realizing it?
Let me first say that what I am saying is not half as outlandish as
you apparently think and it certainly is nothing new. It may just be a
view point that is slightly different from the standard lore.
D-branes were first discovered as soliton solutions of the low-energy
effective equations of motion of (super-) string theory. Polchinski's
accomplishment (in '96) was to show that these solitons of the
effective theory correspond to nonperturbative microscopic states in
the full-fledged string theory. Nevertheless, Polchinski's treatment
still pertains to a description in terms of classical target space
properties. While such a description provides intuitive pictures of
the low-energy description and is useful for a number of important
issues (including the structure of moduli space), it is not quite
sufficient. After all, the great thing about string theory is that it
is a theory of quantum gravity and, hence, we should not expect to be
able to describe it solely in terms of classical geometry. Maybe in
the future a more natural and/or complete understanding of the quantum
geometry of target space will emerge, but for now we can study the
quantum regime by studying the CFT on the world sheet. The spectrum of
this CFT includes both perturbative and non-perturbative states and
encodes the (quantum) geometry of target space. Surely if you
formulate the CFT on the world-sheet general enough it will have
excitations in its spectrum that correspond to target spaces of
different topolgy in the low-energy effective description.
How does this relate to Lubos' remarks concerning, e.g., Witten's
work? Well, if you insist on describing the target space in classical
terms, some hard work is needed to show that physical properties
remain continous even if the topology changes. The reason why one
would choose the classical description is, of course, that CFTs are
not sufficiently under control to calculate all the things we want to
know. It is simply too hard to solve them.
However, even if we can't calculate many things explicitly, we can
still learn a great deal about the conceptual workings of string
theory. Despite the fact that the most general CFT that is consistent
with all classical string backgrounds hasn't been found yet, it
presumably does exist. The original question in this thread was about
background independence. This question has a particularly simple and
general answer in the abstract CFT language. That's what I tried to
explain using familiar terms.
Best,
Rufus
Urs Schreiber
Sep9-04, 02:22 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>"Rufus Anton" <rufusanton@gmx.de> schrieb im Newsbeitrag\nnews:a1c70df8.0409091040.108b4b3d-100000@posting.google.com...\n> d70yxj <redbull_j@yahoo.com> wrote in message\nnews:<d70yxj.1c8a8r-100000@physicsforums.com>...\n\n> Surely if you\n> formulate the CFT on the world-sheet general enough it will have\n> excitations in its spectrum that correspond to target spaces of\n> different topolgy in the low-energy effective description.\n\n[...]\n\n>Despite the fact that the most general CFT that is consistent\n> with all classical string backgrounds hasn\'t been found yet, it\n> presumably does exist.\n\nI am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\nwith the correct central charge) corresponds to precisely one classical\nbackground of string theory. This indeed is precisely the general definition\nof "classical background" in string theory, isn\'t it?\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>"Rufus Anton" <rufusanton@gmx.de> schrieb im Newsbeitrag
news:a1c70df8.0409091040.108b4b3d-100000@posting.google.com...
> d70yxj <redbull_j@yahoo.com> wrote in message
news:<d70yxj.1c8a8r-100000@physicsforums.com>...
> Surely if you
> formulate the CFT on the world-sheet general enough it will have
> excitations in its spectrum that correspond to target spaces of
> different topolgy in the low-energy effective description.
[...]
>Despite the fact that the most general CFT that is consistent
> with all classical string backgrounds hasn't been found yet, it
> presumably does exist.
I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
with the correct central charge) corresponds to precisely one classical
background of string theory. This indeed is precisely the general definition
of "classical background" in string theory, isn't it?
Lubos Motl
Sep9-04, 03:36 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 Thu, 9 Sep 2004, Urs Schreiber wrote:\n\n> I am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\n> with the correct central charge) corresponds to precisely one classical\n> background of string theory. This indeed is precisely the general definition\n> of "classical background" in string theory, isn\'t it?\n\nMy guess is that Rufus might have meant a general (S)CFT with some\nparameters and other defining features whose choice correspond to a\nselection of a particular classical background - in other words, Rufus\nwanted to find a map of the whole perturbative beach of the landscape,\nwhich means the universal description of all backgrounds (at least all\nperturbative backgrounds); do I understand you well, Rufus?\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)\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 Thu, 9 Sep 2004, Urs Schreiber wrote:
> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?
My guess is that Rufus might have meant a general (S)CFT with some
parameters and other defining features whose choice correspond to a
selection of a particular classical background - in other words, Rufus
wanted to find a map of the whole perturbative beach of the landscape,
which means the universal description of all backgrounds (at least all
perturbative backgrounds); do I understand you well, Rufus?
__{_______________________________________________ _____________________________}
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Rufus Anton
Sep10-04, 03:08 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>> do I understand you well, Rufus?\n\nYes, you do. Thanks for clarifying. In the early 90s some people\nthought that finding the "master (S)CFT" would be the key to making\nprogress. But then spacetime rather worldsheet symmetries took over as\nthe most important tools for understanding more features of string\ntheory. Maybe the original plan should not be completely forgotten...\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>> do I understand you well, Rufus?
Yes, you do. Thanks for clarifying. In the early 90s some people
thought that finding the "master (S)CFT" would be the key to making
progress. But then spacetime rather worldsheet symmetries took over as
the most important tools for understanding more features of string
theory. Maybe the original plan should not be completely forgotten...
Urs Schreiber
Sep10-04, 03:28 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409091632270.1610 3-100000@feynman.harvard.edu...\n> On Thu, 9 Sep 2004, Urs Schreiber wrote:\n>\n> > I am wondering what you mean by that. I\'d say that every CFT (sort for:\nSCFT\n> > with the correct central charge) corresponds to precisely one classical\n> > background of string theory. This indeed is precisely the general\ndefinition\n> > of "classical background" in string theory, isn\'t it?\n>\n> My guess is that Rufus might have meant a general (S)CFT with some\n> parameters and other defining features whose choice correspond to a\n> selection of a particular classical background - in other words, Rufus\n> wanted to find a map of the whole perturbative beach of the landscape,\n> which means the universal description of all backgrounds (at least all\n> perturbative backgrounds); do I understand you well, Rufus?\n\nOk. And background *independence* (which is desireable as opposed to\nbackground *freedom* which may be problematic) means that each such SCFT can\nbe obtained from any other one by turning on some sort of coherent state in\nthe latter, roughly.\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 Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409091632270.16103-100000@feynman.harvard.edu...
> On Thu, 9 Sep 2004, Urs Schreiber wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> My guess is that Rufus might have meant a general (S)CFT with some
> parameters and other defining features whose choice correspond to a
> selection of a particular classical background - in other words, Rufus
> wanted to find a map of the whole perturbative beach of the landscape,
> which means the universal description of all backgrounds (at least all
> perturbative backgrounds); do I understand you well, Rufus?
Ok. And background *independence* (which is desireable as opposed to
background *freedom* which may be problematic) means that each such SCFT can
be obtained from any other one by turning on some sort of coherent state in
the latter, roughly.
Robert C. Helling
Sep10-04, 07:26 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, 9 Sep 2004 15:22:28 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> I am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\n> with the correct central charge) corresponds to precisely one classical\n> background of string theory. This indeed is precisely the general definition\n> of "classical background" in string theory, isn\'t it?\n\nWhat exactly do you mean by "classical"? And targets related by\nT-duality (or mirror symmetry if you like) have the same CFT but with\ntwo classical backgrounds described by it. Furthermore, I am not sure\nthat the converse is true: That would mean that each CFT has at least\none point in its moduli space ("large volume") where it is described\nby a sigma model.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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, 9 Sep 2004 15:22:28 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?
What exactly do you mean by "classical"? And targets related by
T-duality (or mirror symmetry if you like) have the same CFT but with
two classical backgrounds described by it. Furthermore, I am not sure
that the converse is true: That would mean that each CFT has at least
one point in its moduli space ("large volume") where it is described
by a \sigma model.
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Urs Schreiber
Sep10-04, 10:56 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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2qdasnFuimkpU1-100000@uni-berlin.de...\n> On Thu, 9 Sep 2004 15:22:28 -0400, Urs Schreiber\n<Urs.Schreiber@uni-essen.de> wrote:\n>\n> > I am wondering what you mean by that. I\'d say that every CFT (sort for:\nSCFT\n> > with the correct central charge) corresponds to precisely one classical\n> > background of string theory. This indeed is precisely the general\ndefinition\n> > of "classical background" in string theory, isn\'t it?\n>\n> What exactly do you mean by "classical"?\n\nFor instance that the (S)CFT is a classical solution of the corresponding\nstring field theory action.\n\nThat\'s what a background is, isn\'t it? A classical solution, i.e. a saddle\npoint of the full string field action that we\'d like to compute the path\nintegral of (if only we could) but which we can only perturb about.\n\n(From the classical solution Phi to the string field theory one gets a\ndeformed BRST operator Q_Phi which is the BRST operator of the new (S)CFT.)\n\nThere is some fine print here, of course, but up to that defining a (S)CFT\nmeans defining the (generalized) classical background that the string whose\nworldsheet dynamics is described by that CFT.\n\nI guess the reason why you feel uncomfortable with me saying "classical\nbackground" is that the most general such background is far from being a\n"classical spacetime" with smooth space and everything. It may be an exotic\nquantum gravitic thingy. But it is still the saddle point solution on which\nthe perturbative string propagates. Wouldn\'t you agree?\n\n> And targets related by\n> T-duality (or mirror symmetry if you like) have the same CFT but with\n> two classical backgrounds described by it.\n\nOk, right. I should have said that the SCFT describes the background up to\nsymmetries like gauge symmetries and dualities. But that\'s more a matter of\nlanguage, depending on if you consider two spacetime theories related by\nduality to be "different".\n\n> Furthermore, I am not sure\n> that the converse is true: That would mean that each CFT has at least\n> one point in its moduli space ("large volume") where it is described\n> by a sigma model.\n\nOh, no, that\'s not what I mean. By "classical background" I don\'t\nnecessarily mean one described by a CFT which is a sigma model.\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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2qdasnFuimkpU1-100000@uni-berlin.de...
> On Thu, 9 Sep 2004 15:22:28 -0400, Urs Schreiber
<Urs.Schreiber@uni-essen.de> wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> What exactly do you mean by "classical"?
For instance that the (S)CFT is a classical solution of the corresponding
string field theory action.
That's what a background is, isn't it? A classical solution, i.e. a saddle
point of the full string field action that we'd like to compute the path
integral of (if only we could) but which we can only perturb about.
(From the classical solution \Phi to the string field theory one gets a
deformed BRST operator Q_{Phi} which is the BRST operator of the new (S)CFT.)
There is some fine print here, of course, but up to that defining a (S)CFT
means defining the (generalized) classical background that the string whose
worldsheet dynamics is described by that CFT.
I guess the reason why you feel uncomfortable with me saying "classical
background" is that the most general such background is far from being a
"classical spacetime" with smooth space and everything. It may be an exotic
quantum gravitic thingy. But it is still the saddle point solution on which
the perturbative string propagates. Wouldn't you agree?
> And targets related by
> T-duality (or mirror symmetry if you like) have the same CFT but with
> two classical backgrounds described by it.
Ok, right. I should have said that the SCFT describes the background up to
symmetries like gauge symmetries and dualities. But that's more a matter of
language, depending on if you consider two spacetime theories related by
duality to be "different".
> Furthermore, I am not sure
> that the converse is true: That would mean that each CFT has at least
> one point in its moduli space ("large volume") where it is described
> by a \sigma model.
Oh, no, that's not what I mean. By "classical background" I don't
necessarily mean one described by a CFT which is a \sigma model.
<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>Rufus Anton <rufusanton@gmx.de> wrote in message news:<a1c70df8.0409091859.63fd8b9e-100000@posting.google.com>...\n\n> > do I understand you well, Rufus?\n>\n> Yes, you do. Thanks for clarifying. In the early 90s some people\n> thought that finding the "master (S)CFT" would be the key to making\n> progress. But then spacetime rather worldsheet symmetries took over as\n> the most important tools for understanding more features of string\n> theory. Maybe the original plan should not be completely forgotten...\n\nWorld-sheet concepts like CFT are pretty much useless for describing\nnon-perturbative backgrounds, such as eg F-theory compactifications\nwhich are non-perturbative backgrounds of the type IIB string\n(involving eg mutually non-local 7-branes; I wouldn\'t know of any\nCFT description of this situation). In the space of all consistent\nground states, it seems that only a small subset has a\nperturbative decription in terms of weakly coupled worldsheet\ntheories, like sigma models on gently curved "classical" manifolds.\nSuch backgrounds are fine as toy models, but if one wants to gain\na better understanding of the whole space of string backgrounds,\none definitely needs to go beyond on-shell worldsheet physics.\n\n[Moderator\'s note: I agree, of course, but my feeling was that the\ndiscussion was focussed on weakly-coupled perturbative backgrounds\nthat still do not have to be geometric sigma-models, such as various\nGepner-like models. One can still ask whether they can be always\nconnected with geometric backgrounds. Various islands, orbifolds\nby T-dualities and other CFTs with possibly frozen "size" modulus\nsuggest that the answer is "No", at least if we want them to connect\non-shell. 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>Rufus Anton <rufusanton@gmx.de> wrote in message news:<a1c70df8.0409091859.63fd8b9e-100000@posting.google.com>...
> > do I understand you well, Rufus?
>
> Yes, you do. Thanks for clarifying. In the early 90s some people
> thought that finding the "master (S)CFT" would be the key to making
> progress. But then spacetime rather worldsheet symmetries took over as
> the most important tools for understanding more features of string
> theory. Maybe the original plan should not be completely forgotten...
World-sheet concepts like CFT are pretty much useless for describing
non-perturbative backgrounds, such as eg F-theory compactifications
which are non-perturbative backgrounds of the type IIB string
(involving eg mutually non-local 7-branes; I wouldn't know of any
CFT description of this situation). In the space of all consistent
ground states, it seems that only a small subset has a
perturbative decription in terms of weakly coupled worldsheet
theories, like \sigma models on gently curved "classical" manifolds.
Such backgrounds are fine as toy models, but if one wants to gain
a better understanding of the whole space of string backgrounds,
one definitely needs to go beyond on-shell worldsheet physics.
[Moderator's note: I agree, of course, but my feeling was that the
discussion was focussed on weakly-coupled perturbative backgrounds
that still do not have to be geometric \sigma-models, such as various
Gepner-like models. One can still ask whether they can be always
connected with geometric backgrounds. Various islands, orbifolds
by T-dualities and other CFTs with possibly frozen "size" modulus
suggest that the answer is "No", at least if we want them to connect
on-shell. LM]
jgraber
Sep14-04, 02:45 AM
Urs Schreiber wrote (in part):
> Ok. And background *independence* (which is desireable as opposed to
> background *freedom* which may be problematic) means that each such SCFT can
>be obtained from any other one by turning on some sort of coherent state in
>the latter, roughly.
I find this very interesting. Could you take a minute to explain why background independence is desirable, but background freedom is (possibly) problematic.
Also it would help me understand if you defined the difference, or gave references to appropriate definitions for each term. Of course, I see an implied partial definition for background independence, at least in one particular context, in the post I am responding to. TIA. Jim Graber
Urs Schreiber
Sep14-04, 03:35 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 Tue, 14 Sep 2004, jgraber wrote:\n\n> Urs Schreiber wrote (in part):\n>\n> > Ok. And background *independence* (which is desireable as opposed to\n> > background *freedom* which may be problematic) means that each such\n> SCFT can\n> >be obtained from any other one by turning on some sort of coherent\n> state in\n> >the latter, roughly.\n>\n> I find this very interesting. Could you take a minute to explain why\n> background independence is desirable, but background freedom is\n> (possibly) problematic.\n> Also it would help me understand if you defined the difference, or gave\n> references to appropriate definitions for each term. Of course, I see\n> an implied partial definition for background independence, at least in\n> one particular context, in the post I am responding to. TIA. Jim\n> Graber\n\n\nFirst of all it is important to clarify what we mean by "background".\nYou\'ll sometimes hear people say things like:\n\n"The big lesson learned grom general relativity is that physics must be\nbackground free."\n\nThe next sentence will either advertize LQG as a background free theory\ntaking this lesson serious or criticize string theory for apparently not\ntaking this lesson serious.\n\nBut a little reflection shows that the meaning of "background" in the\nsecond sentence is not the same as in the first one.\n\nGR is "background free" in the sense that an object in its action which is\nnot varied (integrated over in the path integral) in other field\ntheories like pure Yang-Mills for instance, now is varied\n(integrated) - namely the metric tensor. Hence GR does not assume a fixed\nmetric background structure. (It still assumes a fixed topological\nmanifold, though for instance, which is not "varied".)\n\nBut if you look at the effective target space action of string theory\nyou\'ll note that it is precisely of the same "background free" form as\nthe Einstein-Hilbert action (no wonder, because it is just\nEinstein-Hilbert plus dilaton/axion and higher order terms) and that the\nmetric indeed is dynamical, just as in GR. That\'s pretty obvious.\n\nBut now when a theory is quantized there is a new notion of "background"\narising. Quantization means evaluating a path integral and perturbatively\nquantizing means approximating the path integral at a saddle point and\nthen computing corrections to that approximation.\n\nSo when we say "quantizing theory X on background Y" we mean "Pick the\nsaddle point of the path integral given by the classical solution Y of\ntheory X and then compute quantum fluctuations of it".\n\nThis notion of background may seem similar but is completely different\nfrom the one above. Every classical solution of a theory X which is\n"background free" in the first sense is a "background" in the second\nsense.\n\nAnd this already tells you why "background freedom" in the first\nsense is good (everything should be dynamical), while "background\nfreedom" in the second sense is bad - it would mean that there are no\nclassical solutions of your field theory!\n\nString theory is nicely "background free" in the first sense, even more so\nthan ordinary GR, for instance. Not only is the metric a dynamical\nquantity, but even the number of (macroscopic) dimensions, the coupling\nconstant, and to a large extend the entire field (particle) content of\nthe theory is not a fixed ingredient of the Lagrangian but is dynamical.\n\nThat\'s the very reason why one can even consider dynamics in the string\ntheory "landscape". String theory is so immensely "background free" in the\nfirst sense of the word that it is at present very hard to say anything\nabout which values all these dynamical quantities it contains actually\nwill obtain after some evolution. The landscape discussion is one of\ndealing with a theory which is highly background free, so that you first\nhave to solve equations of motion to even be able to say something about\nthe matter content of the theory. This is quite in contrast to other\napproaches to quantum gravity, which are often considered to be truly\n"background free", where all the matter content, the coupling constants,\nthe number of dimensions is fixed by hand.\n\nSo the problems with answering the "landscape question" in string theory\nshould be seen in light of the fact that they arise due to a difficult\nquestion which other theories can\'t even ask.\n\nThis directly leads to the next point: String theory is certainly not\n"background free" in the second sense. It does have classical solutions!\nAnd that\'s good. These classical solutions can be used for perturbative\nquantization by using them as a "background" about which to start a\nperturbative expansion. But it must be realized that this notion of\n"background" is one of how to do practical computations, not one of\nprinciple.\n\nIn principle you could use other tools to compute the quantum theory, like\nfor instance computing in non-perturbatively. And there are ways to do\nthat in string theory, too, but these are as yet not completely general.\n\nIt is this non-perturbartive quantization which people really have in mind\nwhen they say that, for instance, "LQG is a background free quantization\nof gravity". This just means that, indeed, LQG is an attempt to quantize\ngravity in one stroke, without perturbing about classical solutions of it\n(="backgrounds" in the second sense).\n\nBut currently it seems that LQG, which is certainly "background free" in\nthe first sense of the word (the metric is dynamical) is maybe even\n"background free" in the second sense of the word - which however\nwould mean that it admits no classical solutions! But that would be very\nundesirable, since the world we perceive is so obviously well described by\nclassical gravity to good approximation, that it is a great challange for\nexperimentalists to find any quantum fluctuations (of gravity).\n\n(Personally I feel that it is maybe not such a great surprise that the\nquantization method used in LQG has problems finding sensible solutions,\nsince we know that when applied to 1+1 dimensional gravity it does not\nreproduce the correct path integral quantization. In 1+1 dimensions there\nis a well known theorem that when you solve all the diffeomorphism\nconstraints the Hamiltonian constraint cannot have any solutions at all.\nThis seems to be precisely the problem encountered for years in 3+1d LQG,\ntoo.)\n\nNext, what is the difference between "background freedom" and "background\nindependence". Well, I guess when people use "background freedom" in the\nfirst sense (meaning that the metric is dynamical) it is pretty much\nsynonymous to "background independence" in this first sense.\n\nBut in the second sense of the word "background" (=classical solution\nused in quantum perturbation theory) the term "background indepence" is\nan important concept. It refers to the question if the results obtained by\nperturbing about one background will coincide with the results obtained by\nperturbing about another one. This is a nontrivial question and an\nimportant consistency check of perturbative string theory. And indeed, it\ncan be seen in various nice ways that perturbative string theory, while\nrequiring the specification of some background (=classical solution) is\nindependent of this choice. So it is a fixed but arbitrary background (in\nthe second sense) that is used in perturbative string theory.\n\nAnd that\'s a good thing.\n\n\n\n\nP.S.\nSee the recent discussion on SCFTs and their relation to "backgrounds" to\nsee more fine print to that last paragraph.\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, 14 Sep 2004, jgraber wrote:
> Urs Schreiber wrote (in part):
>
> > Ok. And background *independence* (which is desireable as opposed to
> > background *freedom* which may be problematic) means that each such
> SCFT can
> >be obtained from any other one by turning on some sort of coherent
> state in
> >the latter, roughly.
>
> I find this very interesting. Could you take a minute to explain why
> background independence is desirable, but background freedom is
> (possibly) problematic.
> Also it would help me understand if you defined the difference, or gave
> references to appropriate definitions for each term. Of course, I see
> an implied partial definition for background independence, at least in
> one particular context, in the post I am responding to. TIA. Jim
> Graber
First of all it is important to clarify what we mean by "background".
You'll sometimes hear people say things like:
"The big lesson learned grom general relativity is that physics must be
background free."
The next sentence will either advertize LQG as a background free theory
taking this lesson serious or criticize string theory for apparently not
taking this lesson serious.
But a little reflection shows that the meaning of "background" in the
second sentence is not the same as in the first one.
GR is "background free" in the sense that an object in its action which is
not varied (integrated over in the path integral) in other field
theories like pure Yang-Mills for instance, now is varied
(integrated) - namely the metric tensor. Hence GR does not assume a fixed
metric background structure. (It still assumes a fixed topological
manifold, though for instance, which is not "varied".)
But if you look at the effective target space action of string theory
you'll note that it is precisely of the same "background free" form as
the Einstein-Hilbert action (no wonder, because it is just
Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
metric indeed is dynamical, just as in GR. That's pretty obvious.
But now when a theory is quantized there is a new notion of "background"
arising. Quantization means evaluating a path integral and perturbatively
quantizing means approximating the path integral at a saddle point and
then computing corrections to that approximation.
So when we say "quantizing theory X on background Y" we mean "Pick the
saddle point of the path integral given by the classical solution Y of
theory X and then compute quantum fluctuations of it".
This notion of background may seem similar but is completely different
from the one above. Every classical solution of a theory X which is
"background free" in the first sense is a "background" in the second
sense.
And this already tells you why "background freedom" in the first
sense is good (everything should be dynamical), while "background
freedom" in the second sense is bad - it would mean that there are no
classical solutions of your field theory!
String theory is nicely "background free" in the first sense, even more so
than ordinary GR, for instance. Not only is the metric a dynamical
quantity, but even the number of (macroscopic) dimensions, the coupling
constant, and to a large extend the entire field (particle) content of
the theory is not a fixed ingredient of the Lagrangian but is dynamical.
That's the very reason why one can even consider dynamics in the string
theory "landscape". String theory is so immensely "background free" in the
first sense of the word that it is at present very hard to say anything
about which values all these dynamical quantities it contains actually
will obtain after some evolution. The landscape discussion is one of
dealing with a theory which is highly background free, so that you first
have to solve equations of motion to even be able to say something about
the matter content of the theory. This is quite in contrast to other
approaches to quantum gravity, which are often considered to be truly
"background free", where all the matter content, the coupling constants,
the number of dimensions is fixed by hand.
So the problems with answering the "landscape question" in string theory
should be seen in light of the fact that they arise due to a difficult
question which other theories can't even ask.
This directly leads to the next point: String theory is certainly not
"background free" in the second sense. It does have classical solutions!
And that's good. These classical solutions can be used for perturbative
quantization by using them as a "background" about which to start a
perturbative expansion. But it must be realized that this notion of
"background" is one of how to do practical computations, not one of
principle.
In principle you could use other tools to compute the quantum theory, like
for instance computing in non-perturbatively. And there are ways to do
that in string theory, too, but these are as yet not completely general.
It is this non-perturbartive quantization which people really have in mind
when they say that, for instance, "LQG is a background free quantization
of gravity". This just means that, indeed, LQG is an attempt to quantize
gravity in one stroke, without perturbing about classical solutions of it
(="backgrounds" in the second sense).
But currently it seems that LQG, which is certainly "background free" in
the first sense of the word (the metric is dynamical) is maybe even
"background free" in the second sense of the word - which however
would mean that it admits no classical solutions! But that would be very
undesirable, since the world we perceive is so obviously well described by
classical gravity to good approximation, that it is a great challange for
experimentalists to find any quantum fluctuations (of gravity).
(Personally I feel that it is maybe not such a great surprise that the
quantization method used in LQG has problems finding sensible solutions,
since we know that when applied to 1+1 dimensional gravity it does not
reproduce the correct path integral quantization. In 1+1 dimensions there
is a well known theorem that when you solve all the diffeomorphism
constraints the Hamiltonian constraint cannot have any solutions at all.
This seems to be precisely the problem encountered for years in 3+1d LQG,
too.)
Next, what is the difference between "background freedom" and "background
independence". Well, I guess when people use "background freedom" in the
first sense (meaning that the metric is dynamical) it is pretty much
synonymous to "background independence" in this first sense.
But in the second sense of the word "background" (=classical solution
used in quantum perturbation theory) the term "background indepence" is
an important concept. It refers to the question if the results obtained by
perturbing about one background will coincide with the results obtained by
perturbing about another one. This is a nontrivial question and an
important consistency check of perturbative string theory. And indeed, it
can be seen in various nice ways that perturbative string theory, while
requiring the specification of some background (=classical solution) is
independent of this choice. So it is a fixed but arbitrary background (in
the second sense) that is used in perturbative string theory.
And that's a good thing.
P.S.
See the recent discussion on SCFTs and their relation to "backgrounds" to
see more fine print to that last paragraph.
Robert C. Helling
Sep14-04, 06:29 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 Tue, 14 Sep 2004 04:35:19 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> GR is "background free" in the sense that an object in its action which is\n> not varied (integrated over in the path integral) in other field\n> theories like pure Yang-Mills for instance, now is varied\n> (integrated) - namely the metric tensor. Hence GR does not assume a fixed\n> metric background structure. (It still assumes a fixed topological\n> manifold, though for instance, which is not "varied".)\n\nYesterday, my office mate and me spend nearly two hours discussing\nthis (or a closely related) question, we tried to understand the\nstatement that GR is diffeomorphism invariant or that diffeomorphisms\nare a gauge symmetry in GR. I think we solved it in the end and sorted\nout that there are at least two different meanings to these words but\nlet me get you thinking in this direction: There is an obvious tension\nbetween the following statement that each individually seem to be\ntrue:\n\n1) GR is diffeomorphism invariant.\n\n2) Diffeomorphisms are a gauge symmetry in GR in the sense that before\nmodding out the description contains redundant degrees of freedom.\n\n3) Gauge symmetries cannot be broken spontaneously (as the degrees of\nfreedom are not really there). [This might sound surprising but it is\nnot: In the Higgs effect it is not the local gauge freedom that is\nbroken it is the global one]\n\n4) The ground state of (quantum) GR should be diffeomorphism invariant\nas otherwise diffeomorphisms would be spontaneously broken.\n\n5) Minkowski space is not diffeomorphism invariant.\n\n6) In fact no classical pseudo Riemannian manifold is diffeomorphism\ninvariant.\n\nThis appears to prove that classical manifolds can never be ground\nstates of quantum gravity (for some definition of "ground" state, not\nnecessarily referring to some energy).\n\nOnce you manage to sort this out for yourselves you learned a great\ndeal about LQG, I promise you.\n\n> But if you look at the effective target space action of string theory\n> you\'ll note that it is precisely of the same "background free" form as\n> the Einstein-Hilbert action (no wonder, because it is just\n> Einstein-Hilbert plus dilaton/axion and higher order terms) and that the\n> metric indeed is dynamical, just as in GR. That\'s pretty obvious.\n\nWhat exactly are you talking about, sugra or string field theory?\nSugra definitelty is a classical theory (and non-renormalizable as a\nquantum theory) and my impression from vague looks at the literature\nis that string field theory (of Witten type and its possible\ngeneralizations to super and closed) is only used to compute tree\ndiagrams (although in light cone sft there are loop calculations but\nother problems). Is this correct?\n\nMy point is, that what you describe is at best theoretical because for\nproper quantum (ie loop) processes you have methods available only for\na very limited number of backgrounds (flat space, group manifolds,\norbifolds etc). And you don\'t really know if it is possible to\ngeneralize these methods that are the only ones so far to reliably\ncompute scattering data to general backgrounds even in principle. It\nseems to me it is not impossible but this is a question of faith. Of\ncourse I have in mind the example of strings in AdS5 which is still a\nsymmetric space so sufficiently simple but turns out very hard to\ntreat with world sheet methods.\n\nNow you can come and say that in the past 10 years we have learned so\nmuch about non-perturbative string theory. True. But this is mostly\nabout states and moduli spaces etc and not yet a full theory that\ncontains all the dynamics.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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, 14 Sep 2004 04:35:19 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> GR is "background free" in the sense that an object in its action which is
> not varied (integrated over in the path integral) in other field
> theories like pure Yang-Mills for instance, now is varied
> (integrated) - namely the metric tensor. Hence GR does not assume a fixed
> metric background structure. (It still assumes a fixed topological
> manifold, though for instance, which is not "varied".)
Yesterday, my office mate and me spend nearly two hours discussing
this (or a closely related) question, we tried to understand the
statement that GR is diffeomorphism invariant or that diffeomorphisms
are a gauge symmetry in GR. I think we solved it in the end and sorted
out that there are at least two different meanings to these words but
let me get you thinking in this direction: There is an obvious tension
between the following statement that each individually seem to be
true:
1) GR is diffeomorphism invariant.
2) Diffeomorphisms are a gauge symmetry in GR in the sense that before
modding out the description contains redundant degrees of freedom.
3) Gauge symmetries cannot be broken spontaneously (as the degrees of
freedom are not really there). [This might sound surprising but it is
not: In the Higgs effect it is not the local gauge freedom that is
broken it is the global one]
4) The ground state of (quantum) GR should be diffeomorphism invariant
as otherwise diffeomorphisms would be spontaneously broken.
5) Minkowski space is not diffeomorphism invariant.
6) In fact no classical pseudo Riemannian manifold is diffeomorphism
invariant.
This appears to prove that classical manifolds can never be ground
states of quantum gravity (for some definition of "ground" state, not
necessarily referring to some energy).
Once you manage to sort this out for yourselves you learned a great
deal about LQG, I promise you.
> But if you look at the effective target space action of string theory
> you'll note that it is precisely of the same "background free" form as
> the Einstein-Hilbert action (no wonder, because it is just
> Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
> metric indeed is dynamical, just as in GR. That's pretty obvious.
What exactly are you talking about, sugra or string field theory?
Sugra definitelty is a classical theory (and non-renormalizable as a
quantum theory) and my impression from vague looks at the literature
is that string field theory (of Witten type and its possible
generalizations to super and closed) is only used to compute tree
diagrams (although in light cone sft there are loop calculations but
other problems). Is this correct?
My point is, that what you describe is at best theoretical because for
proper quantum (ie loop) processes you have methods available only for
a very limited number of backgrounds (flat space, group manifolds,
orbifolds etc). And you don't really know if it is possible to
generalize these methods that are the only ones so far to reliably
compute scattering data to general backgrounds even in principle. It
seems to me it is not impossible but this is a question of faith. Of
course I have in mind the example of strings in AdS5 which is still a
symmetric space so sufficiently simple but turns out very hard to
treat with world sheet methods.
Now you can come and say that in the past 10 years we have learned so
much about non-perturbative string theory. True. But this is mostly
about states and moduli spaces etc and not yet a full theory that
contains all the dynamics.
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
Sep14-04, 08:36 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 Tue, 14 Sep 2004, Robert C. Helling wrote:\n\n> 1) GR is diffeomorphism invariant.\n\nVery good.\n\n> 2) Diffeomorphisms are a gauge symmetry in GR in the sense that before\n> modding out the description contains redundant degrees of freedom.\n\nAgreed.\n\n> 3) Gauge symmetries cannot be broken spontaneously (as the degrees of\n> freedom are not really there). [This might sound surprising but it is\n> not: In the Higgs effect it is not the local gauge freedom that is\n> broken it is the global one]\n\nFirst of all - yes, this is a correct interpretation if you do it\ncarefully. Second: it is not quite fair - the global symmetry is a\nsubgroup of the local symmetry group, so you can\'t really break the former\nwithout breaking the latter (at least a little bit). What you probably\nwanted to say is that the state must be invariant under "normalizable"\nlocal transformations - those that converge to the identity at infinity\nquickly enough. The "asymptotically changing" transformations can be\nbroken - this is probably what you mean by the "global ones"?.\n\nThe electroweak symmetry breaking can be phrased as spontaneous symmetry\nbreaking. In this ordinary approach, the vacuum is not invariant under the\nglobal transformations of the gauge group (not even under those that only\naffect a small piece of space) - which means that it partially breaks the\ngauge group (spontaneously), too. But I agree, the more correct way to\ndescribe the situation is to "dress" all operators with a function of the\nHiggs, and take the vacuum which is the usual symmetry-breaking vacuum,\nbut averaged over all gauge copies of itself.\n\nThe situation in GR is analogous. You can take a background metric tensor,\nand it breaks most of the coordinate transformations, except for the\nisometries of the background. If you were picky, you could also work with\na general state invariant under *all* coordinate transformations, and\ndress all operators with some (very nonlocal) functions of the metric.\nThis would completely obscure the geometric interpretation. I don\'t know\nhow exactly this procedure should be done, and I am not sure whether it is\ntoo important, interesting, or useful.\n\nI think that this way of thinking - with the completely non-geometric\nmess forming your Hilbert space - is the first thing that one must avoid\nif she wants to escape from the trap of pseudo-physics where everything is\nobscure. I don\'t know if you agree, but the standard particle physics\ntreatment - the expansion around a chosen value of the vacuum expectation\nvalue (of the Higgs, in the electroweak case, or the metric, in the GR\ncase) is a perfectly legitimate approach that preserves the general\ndiffeomorphism symmetry - it just reinterprets them in terms of new\nformulae. It is the way to go. It may be derivable from something deeper\nand more obscure, but this derived formalism (expansions around a\nbackground) will remain correct.\n\n> 4) The ground state of (quantum) GR should be diffeomorphism invariant\n> as otherwise diffeomorphisms would be spontaneously broken.\n\nFirst of all, the very notion of "background independence" or the rules of\nGR are meant to emphasize the fact that GR does not have a unique ground\nstate - it has many solutions, certainly if you allow the asymptotic\nbehavior of the metric tensor to be whatever you want.\n\nIn the previous paragraphs I tried to explain that the "non-normalizable"\ndiffeomorphisms *can* be spontaneously broken, using the first formalism,\nand they allow you to fix the asymptotic behavior of the metric tensor at\ninfinity. You may want to require such a state to be invariant under the\n"normalizable" local transformations, which simply means that you would\naverage it over all coherent states containing the null (pure gauge)\nunphysical polarizations of the graviton. We usually don\'t do it - not\neven in gauge theories - and I don\'t know what you precisely want to gain.\nWe simply define our physical Hilbert space as excitations of a given\nbackground, and remove the unphysical polarizations corresponding to the\ngauge symmetry.\n\nThere is nothing wrong to take a geometrically interpretable state which\nis not gauge-invariant and work with it, imagining that the allowed state\nis the average of your state over the gauge group.\n\n> 5) Minkowski space is not diffeomorphism invariant.\n\nRight. The Minkowski space is Poincare invariant. The Poincare group is a\nsubgroup of general diffeomorphisms (of any manifold with the trivial\ntopology); it is the isometry group of the Minkowski space. There is\nnothing wrong with it. Minkowski space is an example of a state that\nspontaneously breaks most of the general coordinate symmetries. Those\ntransformations that change the asymptotics are allowed to be broken\nspontaneously because they mix different superselection sectors of the\ntheory - this is why the (ADM) energy can be nonzero even though the\nenergy is superficially a generator of coordinate transformations that\nshould annihilate all physical states. It is completely analogous to the\nfact that the total electric charge can be nonzero in QED, even though it\nnaively generates a gauge transformation. But it is a gauge transformation\nthat changes the asymptotic behavior (of the fields) at infinity, and\ntherefore it maps the states to states in a different superselection\nsector. Therefore it is allowed to break the symmetry under this operation\nspontaneously.\n\n> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism\n> invariant.\n\nRight.\n\n> This appears to prove that classical manifolds can never be ground\n> states of quantum gravity (for some definition of "ground" state, not\n> necessarily referring to some energy).\n\nI hope that you agree that if there is any sense in which the manifolds\nare "not allowed", it is just an error or a very awkward artifact of a\nparticular formalism - not anything that should be labeled "a physical\ninsight". The universe certainly looks much like a pseudo-Riemannian\nmanifold, and classical GR models the Cosmos in the same way. A theory\nthat seriously predicts that there is no physics of manifolds implied by\nquantum GR is ruled out both theoretically as well as experimentally - it\nis just a mere stupidity.\n\n> Once you manage to sort this out for yourselves you learned a great\n> deal about LQG, I promise you.\n\nOnce again.\n\nIf you\'re seriously studying a theory that does not imply (or even does\nnot want to imply) physics in spacetime that looks like a\npseudo-Riemannian manifold, not even in the classical limit, then you are\nnot doing any natural science, because you are contradicting the very\nbasic property of our Universe - that it *does* behave like a manifold.\nAll experiments in all sciences that have been done so far support this\nclaim, and any meaningful theory of physics must be able to reconstruct\nit. Moreover, if you derived that your "quantum theory of gravity" cannot\nreproduce any physics of its classical counterpart - which certainly\n*does* describe the spacetime as a manifold - then you have certainly made\na serious error in your quantization.\n\nIf you want to learn something about loop quantum gravity, see\nhttp://en.wikipedia.org/wiki/Loop_gravity\n\n> What exactly are you talking about, sugra or string field theory?\n> Sugra definitelty is a classical theory (and non-renormalizable as a\n> quantum theory) and my impression from vague looks at the literature\n> is that string field theory (of Witten type and its possible\n> generalizations to super and closed) is only used to compute tree\n> diagrams (although in light cone sft there are loop calculations but\n> other problems). Is this correct?\n\nThat\'s definitely not correct. Witten\'s open string field theory was\nconstructed to reproduce the whole perturbative expansions, and if you ask\nBarton Zwiebach, for example, he will certainly be able to show you the\nproof that the Feynman calculations in Witten\'s cubic SFT cover all\nRiemann surfaces with at least one boundary. So SFT has certainly\nsucceeded to reproduce the whole perturbative quantum theory, otherwise it\nwould be in a very bad shape. SFT was even meant to be a\nnon-perturbatively complete formulation. Well, it can construct D-branes\nas classical solutions (lumps), but otherwise I think that SFT has failed\nin this more ambitious task. But it can certainly give you loop\namplitudes!\n\nThe same holds for closed string field theory - the only new major\nproblem is that the action of closed SFT is itself an infinite expansion\nin the coupling constant, and the local symmetries must be dealt with the\nBV formalism, the annoying complicated generalization of the BRST methods.\nFields, antifields, ghosts for ghosts etc.\n\n> My point is, that what you describe is at best theoretical because for\n> proper quantum (ie loop) processes you have methods available only for\n> a very limited number of backgrounds (flat space, group manifolds,\n> orbifolds etc). And you don\'t really know if it is possible to\n> generalize these methods that are the only ones so far to reliably\n> compute scattering data to general backgrounds even in principle.\n\nI have no idea what you mean. Have you heard the phrase "non-linear sigma\nmodel"? It is a description of perturbative string theory in any\nacceptable geometrical background. Are you complaining that the sigma\nmodels do not exist, or do you criticize the fact that a general enough\nsigma model does not allow us to write compact formulae for all quantities\n- that it is not integrable? Well, most things in the world, including\nclassical physics of 3 interacting bodies, are not integrable. Or are you\ncomplaining that we do not have nonperturbative definition for general\nbackgrounds? Yes, I agree - but we do not even have any non-perturbative\ndefinitions for flat space, e.g. type II string theory on T^6 at a general\nvalue of the coupling.\n\n> It seems to me it is not impossible but this is a question of faith.\n> Of course I have in mind the example of strings in AdS5 which is still\n> a symmetric space so sufficiently simple but turns out very hard to\n> treat with world sheet methods.\n\nThe fully curved AdS5 is not well-formulated because we don\'t know how to\ntreat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know\neverything (perturbatively). But you seem to have doubts that a nonlinear\nsigma model, that can be proved to be a conformal theory, is a correct\ntheory of physics at the appropriate background. I have no idea how would\nyou want to justify these doubts.\n\nBest wishes\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)\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, 14 Sep 2004, Robert C. Helling wrote:
> 1) GR is diffeomorphism invariant.
Very good.
> 2) Diffeomorphisms are a gauge symmetry in GR in the sense that before
> modding out the description contains redundant degrees of freedom.
Agreed.
> 3) Gauge symmetries cannot be broken spontaneously (as the degrees of
> freedom are not really there). [This might sound surprising but it is
> not: In the Higgs effect it is not the local gauge freedom that is
> broken it is the global one]
First of all - yes, this is a correct interpretation if you do it
carefully. Second: it is not quite fair - the global symmetry is a
subgroup of the local symmetry group, so you can't really break the former
without breaking the latter (at least a little bit). What you probably
wanted to say is that the state must be invariant under "normalizable"
local transformations - those that converge to the identity at infinity
quickly enough. The "asymptotically changing" transformations can be
broken - this is probably what you mean by the "global ones"?.
The electroweak symmetry breaking can be phrased as spontaneous symmetry
breaking. In this ordinary approach, the vacuum is not invariant under the
global transformations of the gauge group (not even under those that only
affect a small piece of space) - which means that it partially breaks the
gauge group (spontaneously), too. But I agree, the more correct way to
describe the situation is to "dress" all operators with a function of the
Higgs, and take the vacuum which is the usual symmetry-breaking vacuum,
but averaged over all gauge copies of itself.
The situation in GR is analogous. You can take a background metric tensor,
and it breaks most of the coordinate transformations, except for the
isometries of the background. If you were picky, you could also work with
a general state invariant under *all* coordinate transformations, and
dress all operators with some (very nonlocal) functions of the metric.
This would completely obscure the geometric interpretation. I don't know
how exactly this procedure should be done, and I am not sure whether it is
too important, interesting, or useful.
I think that this way of thinking - with the completely non-geometric
mess forming your Hilbert space - is the first thing that one must avoid
if she wants to escape from the trap of pseudo-physics where everything is
obscure. I don't know if you agree, but the standard particle physics
treatment - the expansion around a chosen value of the vacuum expectation
value (of the Higgs, in the electroweak case, or the metric, in the GR
case) is a perfectly legitimate approach that preserves the general
diffeomorphism symmetry - it just reinterprets them in terms of new
formulae. It is the way to go. It may be derivable from something deeper
and more obscure, but this derived formalism (expansions around a
background) will remain correct.
> 4) The ground state of (quantum) GR should be diffeomorphism invariant
> as otherwise diffeomorphisms would be spontaneously broken.
First of all, the very notion of "background independence" or the rules of
GR are meant to emphasize the fact that GR does not have a unique ground
state - it has many solutions, certainly if you allow the asymptotic
behavior of the metric tensor to be whatever you want.
In the previous paragraphs I tried to explain that the "non-normalizable"
diffeomorphisms *can* be spontaneously broken, using the first formalism,
and they allow you to fix the asymptotic behavior of the metric tensor at
infinity. You may want to require such a state to be invariant under the
"normalizable" local transformations, which simply means that you would
average it over all coherent states containing the null (pure gauge)
unphysical polarizations of the graviton. We usually don't do it - not
even in gauge theories - and I don't know what you precisely want to gain.
We simply define our physical Hilbert space as excitations of a given
background, and remove the unphysical polarizations corresponding to the
gauge symmetry.
There is nothing wrong to take a geometrically interpretable state which
is not gauge-invariant and work with it, imagining that the allowed state
is the average of your state over the gauge group.
> 5) Minkowski space is not diffeomorphism invariant.
Right. The Minkowski space is Poincare invariant. The Poincare group is a
subgroup of general diffeomorphisms (of any manifold with the trivial
topology); it is the isometry group of the Minkowski space. There is
nothing wrong with it. Minkowski space is an example of a state that
spontaneously breaks most of the general coordinate symmetries. Those
transformations that change the asymptotics are allowed to be broken
spontaneously because they mix different superselection sectors of the
theory - this is why the (ADM) energy can be nonzero even though the
energy is superficially a generator of coordinate transformations that
should annihilate all physical states. It is completely analogous to the
fact that the total electric charge can be nonzero in QED, even though it
naively generates a gauge transformation. But it is a gauge transformation
that changes the asymptotic behavior (of the fields) at infinity, and
therefore it maps the states to states in a different superselection
sector. Therefore it is allowed to break the symmetry under this operation
spontaneously.
> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> invariant.
Right.
> This appears to prove that classical manifolds can never be ground
> states of quantum gravity (for some definition of "ground" state, not
> necessarily referring to some energy).
I hope that you agree that if there is any sense in which the manifolds
are "not allowed", it is just an error or a very awkward artifact of a
particular formalism - not anything that should be labeled "a physical
insight". The universe certainly looks much like a pseudo-Riemannian
manifold, and classical GR models the Cosmos in the same way. A theory
that seriously predicts that there is no physics of manifolds implied by
quantum GR is ruled out both theoretically as well as experimentally - it
is just a mere stupidity.
> Once you manage to sort this out for yourselves you learned a great
> deal about LQG, I promise you.
Once again.
If you're seriously studying a theory that does not imply (or even does
not want to imply) physics in spacetime that looks like a
pseudo-Riemannian manifold, not even in the classical limit, then you are
not doing any natural science, because you are contradicting the very
basic property of our Universe - that it *does* behave like a manifold.
All experiments in all sciences that have been done so far support this
claim, and any meaningful theory of physics must be able to reconstruct
it. Moreover, if you derived that your "quantum theory of gravity" cannot
reproduce any physics of its classical counterpart - which certainly
*does* describe the spacetime as a manifold - then you have certainly made
a serious error in your quantization.
If you want to learn something about loop quantum gravity, see
http://en.wikipedia.org/wiki/Loop_gravity
> What exactly are you talking about, sugra or string field theory?
> Sugra definitelty is a classical theory (and non-renormalizable as a
> quantum theory) and my impression from vague looks at the literature
> is that string field theory (of Witten type and its possible
> generalizations to super and closed) is only used to compute tree
> diagrams (although in light cone sft there are loop calculations but
> other problems). Is this correct?
That's definitely not correct. Witten's open string field theory was
constructed to reproduce the whole perturbative expansions, and if you ask
Barton Zwiebach, for example, he will certainly be able to show you the
proof that the Feynman calculations in Witten's cubic SFT cover all
Riemann surfaces with at least one boundary. So SFT has certainly
succeeded to reproduce the whole perturbative quantum theory, otherwise it
would be in a very bad shape. SFT was even meant to be a
non-perturbatively complete formulation. Well, it can construct D-branes
as classical solutions (lumps), but otherwise I think that SFT has failed
in this more ambitious task. But it can certainly give you loop
amplitudes!
The same holds for closed string field theory - the only new major
problem is that the action of closed SFT is itself an infinite expansion
in the coupling constant, and the local symmetries must be dealt with the
BV formalism, the annoying complicated generalization of the BRST methods.
Fields, antifields, ghosts for ghosts etc.
> My point is, that what you describe is at best theoretical because for
> proper quantum (ie loop) processes you have methods available only for
> a very limited number of backgrounds (flat space, group manifolds,
> orbifolds etc). And you don't really know if it is possible to
> generalize these methods that are the only ones so far to reliably
> compute scattering data to general backgrounds even in principle.
I have no idea what you mean. Have you heard the phrase "non-linear \sigma
model"? It is a description of perturbative string theory in any
acceptable geometrical background. Are you complaining that the \sigma
models do not exist, or do you criticize the fact that a general enough
\sigma model does not allow us to write compact formulae for all quantities
- that it is not integrable? Well, most things in the world, including
classical physics of 3 interacting bodies, are not integrable. Or are you
complaining that we do not have nonperturbative definition for general
backgrounds? Yes, I agree - but we do not even have any non-perturbative
definitions for flat space, e.g. type II string theory on T^6 at a general
value of the coupling.
> It seems to me it is not impossible but this is a question of faith.
> Of course I have in mind the example of strings in AdS5 which is still
> a symmetric space so sufficiently simple but turns out very hard to
> treat with world sheet methods.
The fully curved AdS5 is not well-formulated because we don't know how to
treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
everything (perturbatively). But you seem to have doubts that a nonlinear
\sigma model, that can be proved to be a conformal theory, is a correct
theory of physics at the appropriate background. I have no idea how would
you want to justify these doubts.
Best wishes
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Sep14-04, 08:47 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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2qntssF10f45tU1-100000@uni-berlin.de...\n\n> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism\n> invariant.\n\nI don\'t see what you mean by that. Could you explain?\n\nSay my manifold is flat R^3. To me this statement is quite independent of\nany diffeomorpisms R^3 -> R^3. So I don\'t even know in which sense a\nRiemannian manifold could be "diffeomorphism invariant" or not.\n\n> This appears to prove that classical manifolds can never be ground\n> states of quantum gravity (for some definition of "ground" state, not\n> necessarily referring to some energy).\n>\n> Once you manage to sort this out for yourselves you learned a great\n> deal about LQG, I promise you.\n\nMaybe, but I don\'t see yet what you are getting at, sorry.\n\nYou may have noticed that I have developed a weakness for first thinking\nabout this stuff in 2-dimensional gravity. So let me suggest we discuss this\npoint you have in mind (which I don\'t see yet) for 2d gravity on S^1 x R.\n\nThe spatial diffeomorphism constrains are L_n - \\bar L_{-n} for all n.\nThey act, roughly, on the space of fuctionals of maps from the circle into\ntarget space. A state is diffeo invariant, i.e. annihilated by all these\nL_n - \\bar L_{-n}, precisely if takes the same value on any one of these\nmaps that it takes on a map that differs from the first only by a\nreparameterization.\n\nSuch states includes for instance traced Wilson lines along the loop with\nrespect to some (auxiliary) connection on target space. One can show that we\ncan deal with the quantum operator ordering issues in these (for instance by\nrelating them to DDF states).\n\nSo such reparameterization invariant states do exist in the quantum theory\nof 2d gravity. Could you point out in terms of these what you have in mind\nconcerning more general quantum gravity?\n\n(BTW, as I have mentioned before, these states in principle cannot be in the\nkernel of the Hamiltonian constraint operators.)\n\n\n> > But if you look at the effective target space action of string theory\n> > you\'ll note that it is precisely of the same "background free" form as\n> > the Einstein-Hilbert action (no wonder, because it is just\n> > Einstein-Hilbert plus dilaton/axion and higher order terms) and that the\n> > metric indeed is dynamical, just as in GR. That\'s pretty obvious.\n>\n> What exactly are you talking about, sugra or string field theory?\n> Sugra definitelty is a classical theory (and non-renormalizable as a\n> quantum theory) and my impression from vague looks at the literature\n> is that string field theory (of Witten type and its possible\n> generalizations to super and closed) is only used to compute tree\n> diagrams (although in light cone sft there are loop calculations but\n> other problems). Is this correct?\n\n\nIt may be correct that in practice it is mostly used for tree level\ncalculations, but as far as I know it is also true that SFT is known to\nreproduce all higher loop diagrams.\n\nSo I was thinking about string field theory all along, because it makes the\ncomparison of string theory to other approaches to quantum gravity most\nconvenient. What we really should be doing is computing the path integral\nfor the string field action. That statement can be found explicitly for\ninstance in discussions of AdS-CFT. where the conjecture really is that the\nCFT on the boundary is computed by that SFT integral in the bult. Of course\none mostly discusses the limit where the SFT action becomes just the\nordinary sugra action. It\'s a limit.\n\n\n> My point is, that what you describe is at best theoretical because for\n> proper quantum (ie loop) processes you have methods available only for\n> a very limited number of backgrounds (flat space, group manifolds,\n> orbifolds etc). And you don\'t really know if it is possible to\n> generalize these methods that are the only ones so far to reliably\n> compute scattering data to general backgrounds even in principle.\n\nIn principle given any SCFT with c=15 one can compute its correlators on\nparameter spaces of arbitrary genus. Are this not the loop processes that\nyou are talking about?\n\n> Now you can come and say that in the past 10 years we have learned so\n> much about non-perturbative string theory. True. But this is mostly\n> about states and moduli spaces etc and not yet a full theory that\n> contains all the dynamics.\n\nI\'d say the most important problem of the known non-perturbative methods is\nthat they rely on asymptotic boundary conditions, like asymptotic Minkowski\nor asymptotic AdS/CFT, which are not quite those that we would like to\nconsider.\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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2qntssF10f45tU1-100000@uni-berlin.de...
> 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> invariant.
I don't see what you mean by that. Could you explain?
Say my manifold is flat R^3. To me this statement is quite independent of
any diffeomorpisms R^3 -> R^3. So I don't even know in which sense a
Riemannian manifold could be "diffeomorphism invariant" or not.
> This appears to prove that classical manifolds can never be ground
> states of quantum gravity (for some definition of "ground" state, not
> necessarily referring to some energy).
>
> Once you manage to sort this out for yourselves you learned a great
> deal about LQG, I promise you.
Maybe, but I don't see yet what you are getting at, sorry.
You may have noticed that I have developed a weakness for first thinking
about this stuff in 2-dimensional gravity. So let me suggest we discuss this
point you have in mind (which I don't see yet) for 2d gravity on S^1 x R.
The spatial diffeomorphism constrains are L_n - \bar L_{-n} for all n.
They act, roughly, on the space of fuctionals of maps from the circle into
target space. A state is diffeo invariant, i.e. annihilated by all these
L_n - \bar L_{-n}, precisely if takes the same value on any one of these
maps that it takes on a map that differs from the first only by a
reparameterization.
Such states includes for instance traced Wilson lines along the loop with
respect to some (auxiliary) connection on target space. One can show that we
can deal with the quantum operator ordering issues in these (for instance by
relating them to DDF states).
So such reparameterization invariant states do exist in the quantum theory
of 2d gravity. Could you point out in terms of these what you have in mind
concerning more general quantum gravity?
(BTW, as I have mentioned before, these states in principle cannot be in the
kernel of the Hamiltonian constraint operators.)
> > But if you look at the effective target space action of string theory
> > you'll note that it is precisely of the same "background free" form as
> > the Einstein-Hilbert action (no wonder, because it is just
> > Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
> > metric indeed is dynamical, just as in GR. That's pretty obvious.
>
> What exactly are you talking about, sugra or string field theory?
> Sugra definitelty is a classical theory (and non-renormalizable as a
> quantum theory) and my impression from vague looks at the literature
> is that string field theory (of Witten type and its possible
> generalizations to super and closed) is only used to compute tree
> diagrams (although in light cone sft there are loop calculations but
> other problems). Is this correct?
It may be correct that in practice it is mostly used for tree level
calculations, but as far as I know it is also true that SFT is known to
reproduce all higher loop diagrams.
So I was thinking about string field theory all along, because it makes the
comparison of string theory to other approaches to quantum gravity most
convenient. What we really should be doing is computing the path integral
for the string field action. That statement can be found explicitly for
instance in discussions of AdS-CFT. where the conjecture really is that the
CFT on the boundary is computed by that SFT integral in the bult. Of course
one mostly discusses the limit where the SFT action becomes just the
ordinary sugra action. It's a limit.
> My point is, that what you describe is at best theoretical because for
> proper quantum (ie loop) processes you have methods available only for
> a very limited number of backgrounds (flat space, group manifolds,
> orbifolds etc). And you don't really know if it is possible to
> generalize these methods that are the only ones so far to reliably
> compute scattering data to general backgrounds even in principle.
In principle given any SCFT with c=15 one can compute its correlators on
parameter spaces of arbitrary genus. Are this not the loop processes that
you are talking about?
> Now you can come and say that in the past 10 years we have learned so
> much about non-perturbative string theory. True. But this is mostly
> about states and moduli spaces etc and not yet a full theory that
> contains all the dynamics.
I'd say the most important problem of the known non-perturbative methods is
that they rely on asymptotic boundary conditions, like asymptotic Minkowski
or asymptotic AdS/CFT, which are not quite those that we would like to
consider.
Urs Schreiber
Sep14-04, 09:00 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 Tue, 14 Sep 2004, Urs Schreiber wrote:\n\n> "Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\n> Newsbeitrag news:2qntssF10f45tU1-100000@uni-berlin.de...\n>\n> > 6) In fact no classical pseudo Riemannian manifold is diffeomorphism\n> > invariant.\n>\n> I don\'t see what you mean by that. Could you explain?\n\nI just see Lubos\' reply to this and see that I should clarify: Of course\nany given diffeomorphism is not an in general an isometry. Lubos is, I\nthink, referring to isometries, while I was thinking of general diffeos.\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, 14 Sep 2004, Urs Schreiber wrote:
> "Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
> Newsbeitrag news:2qntssF10f45tU1-100000@uni-berlin.de...
>
> > 6) In fact no classical pseudo Riemannian manifold is diffeomorphism
> > invariant.
>
> I don't see what you mean by that. Could you explain?
I just see Lubos' reply to this and see that I should clarify: Of course
any given diffeomorphism is not an in general an isometry. Lubos is, I
think, referring to isometries, while I was thinking of general diffeos.
Lubos Motl
Sep14-04, 02:56 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, 14 Sep 2004, Urs Schreiber wrote:\n\n> I just see Lubos\' reply to this and see that I should clarify: Of course\n> any given diffeomorphism is not an in general an isometry. Lubos is, I\n> think, referring to isometries, while I was thinking of general diffeos.\n\nWe are talking about all these concepts simultaneously. Let me say once\nagain the statements that I find important: the isometry group of a\nmanifold is the only unbroken part of the diffeomorphism symmetry group;\nthe rest of the diffeomorphism symmetry is spontaneously broken by the vev\nof the metric, i.e. by the geometry.\n\nSomeone may argue that local symmetries should not really be broken, and\nall physical states should be invariant under all diffeomorphisms. But\nthis condition can be given up for all diffeomorphisms that are "large",\nnon-normalizable - those that move the points even at infinity. Such\ndiffeomorphisms are typically mapping states from a superselection sector\nA into states in a different superselection sector B.\n\nIn other words, consistency only requires that we demand the physical\nstates to be annihilated by the generators of the normalizable\ndiffeomorphisms - those that move the points "in the middle of the\nspacetime". Such invariant states are easily obtained from a completely\ngeometric state, just by demanding that we average it over the gauge\ngroup. Averaging over the "normalizable" part of the gauge group is\nequivalent to adding the mixture of coherent states of the pure gauge\nunphysically polarized gravitons; averaging over the "non-normalizable"\npart of the gauge symmetry combines the state with its copies in other\nsuperselection sectors, and it is totally legitimate to work with a single\nrepresentative within a single sector, and we obtain the correct results.\nIf we recall that the full state may be thought of as a mixture of states\nfrom different sectors, we also realize that the isometries of the\nsuperselection sectors don\'t have to annihilate the physical states.\n\nThe goal of these two messages is to convince the reader that the gauge\nsymmetries are subtle, there are various types and they can be treated in\nmany ways - and the idea that a state in quantum gravity must always be a\nnon-geometrical obscure mixture of everything in the world, because it\nmust be invariant under all transformations, and that geometrical way of\nthinking is impossible if we deal with quantum gravity properly, is naive\nand would not lead anywhere. Gauge symmetries are finally not as\nfundamental as Robert tries to picture - they represent a nice tool to get\nrid of some degrees of freedom, especially some polarizations of vectors\nand tensors that would otherwise have a negative norm. The theories with\ngauge symmetries can be equivalently rewritten in other formalisms, for\nexample the BRST formalism or a gauge-fixed form such as the light-cone\ngauge. The physics is unchanged, and if Robert were working with the BRST\ninvariance (or, perhaps, a BV-type formalism, if necessary), he would not\nencounter his restrictions and the geometrically clear states would be\nclearly possible.\n\nI realize that many people in loop quantum gravity often and explicitly\nsay that there is no Lorentz or other symmetry in GR, and I just don\'t\nunderstand what meaningful can they ever mean. GR has a symmetry under\n*all* coordinate transformations. For a trivial topology, there are\ninfinitely many ways to embed the Lorentz group within the group of all\ndiffeomorphisms, and the theory is invariant under all these\ntransformations. The symmetry under a diffeomorphism is only broken\nspontaneously, by the values of the metric (which we call the "vev"s in\nthe quantum theory).\n\nBut even in this case, if we only consider local physics of classical GR\nin a sufficiently small piece of spacetime - and let us pick the locally\ninertial frame - then physics of GR always reduces to special relativity,\nby the princple of equivalence, and curvature etc. can be neglected. It is\nonly the combination of the locally valid Lorentz symmetry *with* the\n"background independence" - or the ability to curve spacetime - that makes\nGR nontrivial and constrained. A theory whose local physics does not\nrespect the rules of special relativity cannot be called general\nrelativity (or its ramification).\n\nNote that the Lorentz symmetry enters GR in many different ways. It is an\nunbroken symmetry of local physics, but it is also a symmetry applicable\nglobally in a topologically trivial spacetime - the latter copy of the\nLorentz symmetry is usually spontaneously broken by most backgrounds\n(except the Minkowski space).\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)\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, 14 Sep 2004, Urs Schreiber wrote:
> I just see Lubos' reply to this and see that I should clarify: Of course
> any given diffeomorphism is not an in general an isometry. Lubos is, I
> think, referring to isometries, while I was thinking of general diffeos.
We are talking about all these concepts simultaneously. Let me say once
again the statements that I find important: the isometry group of a
manifold is the only unbroken part of the diffeomorphism symmetry group;
the rest of the diffeomorphism symmetry is spontaneously broken by the vev
of the metric, i.e. by the geometry.
Someone may argue that local symmetries should not really be broken, and
all physical states should be invariant under all diffeomorphisms. But
this condition can be given up for all diffeomorphisms that are "large",
non-normalizable - those that move the points even at infinity. Such
diffeomorphisms are typically mapping states from a superselection sector
A into states in a different superselection sector B.
In other words, consistency only requires that we demand the physical
states to be annihilated by the generators of the normalizable
diffeomorphisms - those that move the points "in the middle of the
spacetime". Such invariant states are easily obtained from a completely
geometric state, just by demanding that we average it over the gauge
group. Averaging over the "normalizable" part of the gauge group is
equivalent to adding the mixture of coherent states of the pure gauge
unphysically polarized gravitons; averaging over the "non-normalizable"
part of the gauge symmetry combines the state with its copies in other
superselection sectors, and it is totally legitimate to work with a single
representative within a single sector, and we obtain the correct results.
If we recall that the full state may be thought of as a mixture of states
from different sectors, we also realize that the isometries of the
superselection sectors don't have to annihilate the physical states.
The goal of these two messages is to convince the reader that the gauge
symmetries are subtle, there are various types and they can be treated in
many ways - and the idea that a state in quantum gravity must always be a
non-geometrical obscure mixture of everything in the world, because it
must be invariant under all transformations, and that geometrical way of
thinking is impossible if we deal with quantum gravity properly, is naive
and would not lead anywhere. Gauge symmetries are finally not as
fundamental as Robert tries to picture - they represent a nice tool to get
rid of some degrees of freedom, especially some polarizations of vectors
and tensors that would otherwise have a negative norm. The theories with
gauge symmetries can be equivalently rewritten in other formalisms, for
example the BRST formalism or a gauge-fixed form such as the light-cone
gauge. The physics is unchanged, and if Robert were working with the BRST
invariance (or, perhaps, a BV-type formalism, if necessary), he would not
encounter his restrictions and the geometrically clear states would be
clearly possible.
I realize that many people in loop quantum gravity often and explicitly
say that there is no Lorentz or other symmetry in GR, and I just don't
understand what meaningful can they ever mean. GR has a symmetry under
*all* coordinate transformations. For a trivial topology, there are
infinitely many ways to embed the Lorentz group within the group of all
diffeomorphisms, and the theory is invariant under all these
transformations. The symmetry under a diffeomorphism is only broken
spontaneously, by the values of the metric (which we call the "vev"s in
the quantum theory).
But even in this case, if we only consider local physics of classical GR
in a sufficiently small piece of spacetime - and let us pick the locally
inertial frame - then physics of GR always reduces to special relativity,
by the princple of equivalence, and curvature etc. can be neglected. It is
only the combination of the locally valid Lorentz symmetry *with* the
"background independence" - or the ability to curve spacetime - that makes
GR nontrivial and constrained. A theory whose local physics does not
respect the rules of special relativity cannot be called general
relativity (or its ramification).
Note that the Lorentz symmetry enters GR in many different ways. It is an
unbroken symmetry of local physics, but it is also a symmetry applicable
globally in a topologically trivial spacetime - the latter copy of the
Lorentz symmetry is usually spontaneously broken by most backgrounds
(except the Minkowski space).
__{_______________________________________________ _____________________________}
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Robert C. Helling
Sep15-04, 06:17 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 Tue, 14 Sep 2004 09:36:52 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> First of all - yes, this is a correct interpretation if you do it\n> carefully. Second: it is not quite fair - the global symmetry is a\n> subgroup of the local symmetry group, so you can\'t really break the former\n> without breaking the latter (at least a little bit). What you probably\n> wanted to say is that the state must be invariant under "normalizable"\n> local transformations - those that converge to the identity at infinity\n> quickly enough. The "asymptotically changing" transformations can be\n> broken - this is probably what you mean by the "global ones"?.\n\nIndeed. When I say local gauge transformations I mean those that tend\nto the identity at infinity (in some precise sense that you call\n"normalizable". The global ones are then in all gauge trafos mod local\ngauge trafos and that coset should be a copy of the gauge group (at\nleast if "infinity" is simple enough).\n\n>> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.\n>\n> Right.\n\nActually, this is where I would probably say no. All this \'confusion\'\n(which I claim is wide spread in certain circles) comes from the\nmixing of two meanings of "diffeomorphism invariant": It always means\n"invariant under general changes of coordinates" but the difference\nbetween the two meanings is whether you transform your fields as well:\nYou can describe Minkowski space in many different coordinate systems,\noften for example radial coordinates are quite usefui. But of course\nyou have to change the metric as well: The metric in radial\ncoordinates is different from diag(-1,1,1,1). But this change of\ncoordinates is a diffeomorphism. (There are isometries as well, where\nyou don\'t have to pull back the metric but those are only special\ndiffeos). And it is these changes of coordinates that GR is invarant\nunder. As is any other theories that I can formulate in terms of\ntensors (that I pull back when applying diffeomorphisms). It is just\nthat in "field theory on flat space" one of these tensors (eta) is not\ndynamical and thus background. Even Newtonian gravity is diffeo\ninvariant (see MTW), there there is background one form dt as well.\n\nWhat one should really require is that the result of the quantization\nprocedure does not depend on the choice of coordinates. And therefore\nif one uses coordinates at some point in the quantization, one has to\nfind the unitary operators that do the pull back on the metric and\nimpose the constraints that under changes of coordinates also the\nquantum analogue of the metric is pulled back correctly.\n\n>> This appears to prove that classical manifolds can never be ground\n>> states of quantum gravity (for some definition of "ground" state, not\n>> necessarily referring to some energy).\n\nAgain this mixes the two meanings and would correctly be translated to\n"The proper quantum state should have all diffeos as isometries" which\nis of course nonsense.\n\n> I hope that you agree that if there is any sense in which the manifolds\n> are "not allowed", it is just an error or a very awkward artifact of a\n> particular formalism\n\n100% ACK. That was what I wanted to point out.\n\n>> My point is, that what you describe is at best theoretical because for\n>> proper quantum (ie loop) processes you have methods available only for\n>> a very limited number of backgrounds (flat space, group manifolds,\n>> orbifolds etc). And you don\'t really know if it is possible to\n>> generalize these methods that are the only ones so far to reliably\n>> compute scattering data to general backgrounds even in principle.\n>\n> I have no idea what you mean. Have you heard the phrase "non-linear sigma\n> model"? It is a description of perturbative string theory in any\n> acceptable geometrical background. Are you complaining that the sigma\n> models do not exist, or do you criticize the fact that a general enough\n> sigma model does not allow us to write compact formulae for all quantities\n> - that it is not integrable?\n\nNeither. I complain that for most non-linear sigmar models you don\'t\nknow what the asymptotic states are, so you cannot go an compute\nscattering amplitudes. You don\'t even know the particle spectrum.\n\n> The fully curved AdS5 is not well-formulated because we don\'t know how to\n> treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know\n> everything (perturbatively). But you seem to have doubts that a nonlinear\n> sigma model, that can be proved to be a conformal theory, is a correct\n> theory of physics at the appropriate background. I have no idea how would\n> you want to justify these doubts.\n\nCorrect yes. But it does not tell you much about observed physics.\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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, 14 Sep 2004 09:36:52 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:
> First of all - yes, this is a correct interpretation if you do it
> carefully. Second: it is not quite fair - the global symmetry is a
> subgroup of the local symmetry group, so you can't really break the former
> without breaking the latter (at least a little bit). What you probably
> wanted to say is that the state must be invariant under "normalizable"
> local transformations - those that converge to the identity at infinity
> quickly enough. The "asymptotically changing" transformations can be
> broken - this is probably what you mean by the "global ones"?.
Indeed. When I say local gauge transformations I mean those that tend
to the identity at infinity (in some precise sense that you call
"normalizable". The global ones are then in all gauge trafos mod local
gauge trafos and that coset should be a copy of the gauge group (at
least if "infinity" is simple enough).
>> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.
>
> Right.
Actually, this is where I would probably say no. All this 'confusion'
(which I claim is wide spread in certain circles) comes from the
mixing of two meanings of "diffeomorphism invariant": It always means
"invariant under general changes of coordinates" but the difference
between the two meanings is whether you transform your fields as well:
You can describe Minkowski space in many different coordinate systems,
often for example radial coordinates are quite usefui. But of course
you have to change the metric as well: The metric in radial
coordinates is different from diag(-1,1,1,1). But this change of
coordinates is a diffeomorphism. (There are isometries as well, where
you don't have to pull back the metric but those are only special
diffeos). And it is these changes of coordinates that GR is invarant
under. As is any other theories that I can formulate in terms of
tensors (that I pull back when applying diffeomorphisms). It is just
that in "field theory on flat space" one of these tensors (\eta) is not
dynamical and thus background. Even Newtonian gravity is diffeo
invariant (see MTW), there there is background one form dt as well.
What one should really require is that the result of the quantization
procedure does not depend on the choice of coordinates. And therefore
if one uses coordinates at some point in the quantization, one has to
find the unitary operators that do the pull back on the metric and
impose the constraints that under changes of coordinates also the
quantum analogue of the metric is pulled back correctly.
>> This appears to prove that classical manifolds can never be ground
>> states of quantum gravity (for some definition of "ground" state, not
>> necessarily referring to some energy).
Again this mixes the two meanings and would correctly be translated to
"The proper quantum state should have all diffeos as isometries" which
is of course nonsense.
> I hope that you agree that if there is any sense in which the manifolds
> are "not allowed", it is just an error or a very awkward artifact of a
> particular formalism
100% ACK. That was what I wanted to point out.
>> My point is, that what you describe is at best theoretical because for
>> proper quantum (ie loop) processes you have methods available only for
>> a very limited number of backgrounds (flat space, group manifolds,
>> orbifolds etc). And you don't really know if it is possible to
>> generalize these methods that are the only ones so far to reliably
>> compute scattering data to general backgrounds even in principle.
>
> I have no idea what you mean. Have you heard the phrase "non-linear \sigma
> model"? It is a description of perturbative string theory in any
> acceptable geometrical background. Are you complaining that the \sigma
> models do not exist, or do you criticize the fact that a general enough
> \sigma model does not allow us to write compact formulae for all quantities
> - that it is not integrable?
Neither. I complain that for most non-linear sigmar models you don't
know what the asymptotic states are, so you cannot go an compute
scattering amplitudes. You don't even know the particle spectrum.
> The fully curved AdS5 is not well-formulated because we don't know how to
> treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
> everything (perturbatively). But you seem to have doubts that a nonlinear
> \sigma model, that can be proved to be a conformal theory, is a correct
> theory of physics at the appropriate background. I have no idea how would
> you want to justify these doubts.
Correct yes. But it does not tell you much about observed physics.
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
Sep15-04, 07:09 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 Wed, 15 Sep 2004, Robert C. Helling wrote:\n\n> >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.\n> >\n> > Right.\n>\n> Actually, this is where I would probably say no. All this \'confusion\'\n> (which I claim is wide spread in certain circles) comes from the\n> mixing of two meanings of "diffeomorphism invariant":\n\nIf you now say that the Minkowski space preserves *all* diffeomorphisms,\nthen you should consider the option that it is you who is very confused.\n\n> It always means "invariant under general changes of coordinates" but\n> the difference between the two meanings is whether you transform your\n> fields as well:\n\nIf you want to do the transformation twice and undo it at the same moment,\nthen one cannot be surprised that you get the same thing back. You don\'t\nreally mean "invariant under gauge transformations". You rather mean\n"invariant under gauge transformations up to a gauge transformation".\nHowever, this is a tautology. If you used your definition of "being\ninvariant under something", then everything would always be invariant\nunder everything.\n\nIncidentally, you want to mask this trick by making the transformation\nactive, and undo it in a passive way.\n\nIn any theory of fields - that includes GR - we do not really mean "change\nof coordinates" but usually the associated transformation of the fields.\nBecause the diffeomorphisms act equally on all fields, it is sort of\nenough to describe the action on the coordinates. But this statement is a\nspecial property of diffeomorphisms, and if we want to use the standard\nmachinery of field theory rules and methods, we *always* describe any\ntransformation as a transformation acting on the fields because the fields\nare the only dynamical degrees of freedom. The coordinates are not\ndynamical degrees of freedom in GR, because GR is a theory of fields, and\nit is conceptually misleading to transform them.\n\nFor example, a scalar field transforms as\n\n\\phi(x^\\mu) -> \\phi\'(x^\\mu) := \\phi(f^\\mu(\\{x^\\nu\\}))\n\nwhile the non-scalar tensor fields also include one or more copies of the\nJacobi matrix, and so on. This is the only way how to treat\ndiffeomorphisms uniformly with all other transformations that one can find\nin a theory of fields. If you paid attention to supersymmetry, for\nexample, you would also realize that the SUSic theories without a\nsuperspace formulation simply force you to study the transformation of the\nfields, not coordinates, because this is the only way how SUSY is\nrealized in this context.\n\n> You can describe Minkowski space in many different coordinate systems,\n> often for example radial coordinates are quite usefui. But of course\n> you have to change the metric as well: The metric in radial\n> coordinates is different from diag(-1,1,1,1).\n\nThis is why the flat space is *not* invariant under the diffeomorphism\nthat moves the point (x,y) to (x\',y\') = (r,\\phi) where r\\exp(i\\phi) =\nx+iy. If you transform the fields (!) according to the correct rule, which\nis my rule, not yours, you get different fields - a different form of\nmetric, as you observed, too. The metric tensor describing a manifold is\ninvariant under the isometries only; others are spontaneously broken. The\nword "different" means that the configuration is *not* invariant under the\noperation. Do you think that "different" means that it *is* invariant, or\nwhat do you exactly think that the word "invariance" means?\n\n> But this change of coordinates is a diffeomorphism.\n\nYes, it is a diffeomorphism, but because it does not preserve the\nnumerical values of the components of the metric, the metric is not\ninvariant under this diffeomorphism. By the "metric" or the "metric\ntensor" we do not mean the "asthetic impression that we get by visualizing\nit as a manifold" which is probably what you think it is; instead, we are\ndoing quantitative science and we mean the collection of 10 functions of 4\ncoordinates (in the d=4 case). These functions change, and therefore they\n(the metric) is *not* invariant under the operation.\n\n> (There are isometries as well, where\n> you don\'t have to pull back the metric but those are only special\n> diffeos). And it is these changes of coordinates that GR is invarant\n> under.\n\nIf you say that the only transformations under which GR is invariant is\nyour operation in which you transform the coordinates, which is equivalent\nto transforming all fields, but then you undo this transformation by\nanother action on the fields, then you are very confused. GR is diff.\ninvariant only because it is invariant under the transformation of the\nfields only, those described above. The laws (and the action, as the\ninvariant scalar) of GR is invariant under those; the particular\nconfigurations themselves are not, unless you consider an isometry of the\nbackground.\n\n> As is any other theories that I can formulate in terms of\n> tensors (that I pull back when applying diffeomorphisms). It is just\n> that in "field theory on flat space" one of these tensors (eta) is not\n> dynamical and thus background. Even Newtonian gravity is diffeo\n> invariant (see MTW), there there is background one form dt as well.\n\nIf you add new fields, you can make virtually any theory be invariant\nunder virtually any group. The fields that you must add to Newton\'s theory\nto make it generally covariant are non-dynamical and they can easily be\nerased, together with these symmetries, and therefore Newton\'s theory did\nnot morally satisfy Einstein\'s requirement of a diff. invariant theory.\nWhen we talk about a gauge symmetry of a theory, we must say in which\ndegrees of freedom we formulate the theory because different choices can\nhave different gauge symmetries. Of course, in the case of GR, I meant the\nmetric tensor (plus matter fields, if you wish) that are functions of the\nspacetime coordinates. If we know the degrees of freedom and the action,\nwe can decide what the symmetry group is.\n\n> What one should really require is that the result of the quantization\n> procedure does not depend on the choice of coordinates.\n\nThis is not the field theory approach. In field theory, GR is invariant\nunder the transformations of the fields associated with a transformation\nof the coordinates. Once again, the fields - including the metric - are\nthe only real degrees of freedom, which means that they are the only\nobjects used in constructing any questions, and the only objects that\nshould be transformed.\n\n> And therefore if one uses coordinates at some point in the\n> quantization,\n\nIf we o quantitative physics whose purpose is to be compared with\nexperiments, we almost always use coordinates and fields that are\nfunctions of them. You might dislike it, but it is the only way to get\nquantitative results as opposed to vague hand-waving and unusable abstract\nformulae.\n\n> one has to find the unitary operators that do the pull back on the\n> metric and impose the constraints that under changes of coordinates\n> also the quantum analogue of the metric is pulled back correctly.\n\nWhat is important is that the operators (the metric, for example) are\n*not* invariant under the conjugation by the unitary operator associated\nwith a generic diffeomorphism. I say "conjugation" because this is how\noperators transform under the operation encoded in a unitary operator.\nIt would be totally wrong to *require* the operators to be strongly\ninvariant under the conjugation indicated above. For example, for scalar\nfields such a requirement would mean that the scalar field is a constant\nfunction of spacetime - a clear nonsense because we shuold not really call\nit a field in this case.\n\nLet me tell you a trivial example in gauge theory. The fields are by no\nmeans required to be invariant under gauge transformations, not even the\nglobal ones. If they were, it would mean that all charged fields are\nprohibited, for example, because charged fields *do* transform under\nU!)_{elmg}, for example. In the same way, I obtained the nonsensical\nresult above that the fields in GR would have zero momentum (constant\nfunctions of spacetime) if you required *them* to be invariant. If we\nrequire something like that, it is the invariance of the *states* - or,\nequivalently, the BRST-invariance of the states.\n\nAlso, the "operators" of the metric tensor only exist in the low energy\nlimit and it is by no means guaranteed that the full theory has these\noperators exactly. Well, string theory probably does not allow these\noperators to be defined in the short-distance regime, yet it is a\nconsistent general covariant theory; the unphysical polarizations of the\ngravitons decouple, which is the perturbative incarnation of the local\nsymmetry.\n\n> Neither. I complain that for most non-linear sigmar models you don\'t\n> know what the asymptotic states are, so you cannot go an compute\n> scattering amplitudes. You don\'t even know the particle spectrum.\n\nThis is one of the options I gave you. So you *are* complaining that a\ngeneric model is not solvable. Well, even the three-body Newtonian system\nis not solvable; if you don\'t like that most models in this Universe are\nnot solvable and one cannot calculate the particle spectrum analytically,\nyou should consider moving into a different Universe.\n\n> > treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know\n> > everything (perturbatively). But you seem to have doubts that a nonlinear\n> ...\n> Correct yes. But it does not tell you much about observed physics.\n\nWhat does not tell me much about observed physics? The complete solution\nof the strings on the pp-wave? It tells me everything I want to know about\nobserved physics (in that Universe).\n\nCheers,\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)\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 Wed, 15 Sep 2004, Robert C. Helling wrote:
> >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism invariant.
> >
> > Right.
>
> Actually, this is where I would probably say no. All this 'confusion'
> (which I claim is wide spread in certain circles) comes from the
> mixing of two meanings of "diffeomorphism invariant":
If you now say that the Minkowski space preserves *all* diffeomorphisms,
then you should consider the option that it is you who is very confused.
> It always means "invariant under general changes of coordinates" but
> the difference between the two meanings is whether you transform your
> fields as well:
If you want to do the transformation twice and undo it at the same moment,
then one cannot be surprised that you get the same thing back. You don't
really mean "invariant under gauge transformations". You rather mean
"invariant under gauge transformations up to a gauge transformation".
However, this is a tautology. If you used your definition of "being
invariant under something", then everything would always be invariant
under everything.
Incidentally, you want to mask this trick by making the transformation
active, and undo it in a passive way.
In any theory of fields - that includes GR - we do not really mean "change
of coordinates" but usually the associated transformation of the fields.
Because the diffeomorphisms act equally on all fields, it is sort of
enough to describe the action on the coordinates. But this statement is a
special property of diffeomorphisms, and if we want to use the standard
machinery of field theory rules and methods, we *always* describe any
transformation as a transformation acting on the fields because the fields
are the only dynamical degrees of freedom. The coordinates are not
dynamical degrees of freedom in GR, because GR is a theory of fields, and
it is conceptually misleading to transform them.
For example, a scalar field transforms as
\phi(x^\mu) -> \phi'(x^\mu) := \phi(f^\mu(\{x^\nu\}))
while the non-scalar tensor fields also include one or more copies of the
Jacobi matrix, and so on. This is the only way how to treat
diffeomorphisms uniformly with all other transformations that one can find
in a theory of fields. If you paid attention to supersymmetry, for
example, you would also realize that the SUSic theories without a
superspace formulation simply force you to study the transformation of the
fields, not coordinates, because this is the only way how SUSY is
realized in this context.
> You can describe Minkowski space in many different coordinate systems,
> often for example radial coordinates are quite usefui. But of course
> you have to change the metric as well: The metric in radial
> coordinates is different from diag(-1,1,1,1).
This is why the flat space is *not* invariant under the diffeomorphism
that moves the point (x,y) to (x',y') = (r,\phi) where r\exp(i\phi) =x+iy. If you transform the fields (!) according to the correct rule, which
is my rule, not yours, you get different fields - a different form of
metric, as you observed, too. The metric tensor describing a manifold is
invariant under the isometries only; others are spontaneously broken. The
word "different" means that the configuration is *not* invariant under the
operation. Do you think that "different" means that it *is* invariant, or
what do you exactly think that the word "invariance" means?
> But this change of coordinates is a diffeomorphism.
Yes, it is a diffeomorphism, but because it does not preserve the
numerical values of the components of the metric, the metric is not
invariant under this diffeomorphism. By the "metric" or the "metric
tensor" we do not mean the "asthetic impression that we get by visualizing
it as a manifold" which is probably what you think it is; instead, we are
doing quantitative science and we mean the collection of 10 functions of 4
coordinates (in the d=4 case). These functions change, and therefore they
(the metric) is *not* invariant under the operation.
> (There are isometries as well, where
> you don't have to pull back the metric but those are only special
> diffeos). And it is these changes of coordinates that GR is invarant
> under.
If you say that the only transformations under which GR is invariant is
your operation in which you transform the coordinates, which is equivalent
to transforming all fields, but then you undo this transformation by
another action on the fields, then you are very confused. GR is diff.
invariant only because it is invariant under the transformation of the
fields only, those described above. The laws (and the action, as the
invariant scalar) of GR is invariant under those; the particular
configurations themselves are not, unless you consider an isometry of the
background.
> As is any other theories that I can formulate in terms of
> tensors (that I pull back when applying diffeomorphisms). It is just
> that in "field theory on flat space" one of these tensors (\eta) is not
> dynamical and thus background. Even Newtonian gravity is diffeo
> invariant (see MTW), there there is background one form dt as well.
If you add new fields, you can make virtually any theory be invariant
under virtually any group. The fields that you must add to Newton's theory
to make it generally covariant are non-dynamical and they can easily be
erased, together with these symmetries, and therefore Newton's theory did
not morally satisfy Einstein's requirement of a diff. invariant theory.
When we talk about a gauge symmetry of a theory, we must say in which
degrees of freedom we formulate the theory because different choices can
have different gauge symmetries. Of course, in the case of GR, I meant the
metric tensor (plus matter fields, if you wish) that are functions of the
spacetime coordinates. If we know the degrees of freedom and the action,
we can decide what the symmetry group is.
> What one should really require is that the result of the quantization
> procedure does not depend on the choice of coordinates.
This is not the field theory approach. In field theory, GR is invariant
under the transformations of the fields associated with a transformation
of the coordinates. Once again, the fields - including the metric - are
the only real degrees of freedom, which means that they are the only
objects used in constructing any questions, and the only objects that
should be transformed.
> And therefore if one uses coordinates at some point in the
> quantization,
If we o quantitative physics whose purpose is to be compared with
experiments, we almost always use coordinates and fields that are
functions of them. You might dislike it, but it is the only way to get
quantitative results as opposed to vague hand-waving and unusable abstract
formulae.
> one has to find the unitary operators that do the pull back on the
> metric and impose the constraints that under changes of coordinates
> also the quantum analogue of the metric is pulled back correctly.
What is important is that the operators (the metric, for example) are
*not* invariant under the conjugation by the unitary operator associated
with a generic diffeomorphism. I say "conjugation" because this is how
operators transform under the operation encoded in a unitary operator.
It would be totally wrong to *require* the operators to be strongly
invariant under the conjugation indicated above. For example, for scalar
fields such a requirement would mean that the scalar field is a constant
function of spacetime - a clear nonsense because we shuold not really call
it a field in this case.
Let me tell you a trivial example in gauge theory. The fields are by no
means required to be invariant under gauge transformations, not even the
global ones. If they were, it would mean that all charged fields are
prohibited, for example, because charged fields *do* transform under
U!)_{elmg}, for example. In the same way, I obtained the nonsensical
result above that the fields in GR would have zero momentum (constant
functions of spacetime) if you required *them* to be invariant. If we
require something like that, it is the invariance of the *states* - or,
equivalently, the BRST-invariance of the states.
Also, the "operators" of the metric tensor only exist in the low energy
limit and it is by no means guaranteed that the full theory has these
operators exactly. Well, string theory probably does not allow these
operators to be defined in the short-distance regime, yet it is a
consistent general covariant theory; the unphysical polarizations of the
gravitons decouple, which is the perturbative incarnation of the local
symmetry.
> Neither. I complain that for most non-linear sigmar models you don't
> know what the asymptotic states are, so you cannot go an compute
> scattering amplitudes. You don't even know the particle spectrum.
This is one of the options I gave you. So you *are* complaining that a
generic model is not solvable. Well, even the three-body Newtonian system
is not solvable; if you don't like that most models in this Universe are
not solvable and one cannot calculate the particle spectrum analytically,
you should consider moving into a different Universe.
> > treat general RR backgrounds. In the pp-wave limit of AdS5 x S5, we know
> > everything (perturbatively). But you seem to have doubts that a nonlinear
> ...
> Correct yes. But it does not tell you much about observed physics.
What does not tell me much about observed physics? The complete solution
of the strings on the pp-wave? It tells me everything I want to know about
observed physics (in that Universe).
Cheers,
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Sep15-04, 07:09 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>\n"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2qqk3sF12042nU1-100000@uni-berlin.de...\n\n> What one should really require is that the result of the quantization\n> procedure does not depend on the choice of coordinates. And therefore\n> if one uses coordinates at some point in the quantization, one has to\n> find the unitary operators that do the pull back on the metric and\n> impose the constraints that under changes of coordinates also the\n> quantum analogue of the metric is pulled back correctly.\n\nBut here one has to be very careful what one really means. The standard\nquantizations of 2d gravity don\'t depend on coordinates, yet physical states\nare *not* annihilated by the generator L_n - \\bar L_{-n} of spatial\ncoordinate transformations. They cannot. Coordinate invariance is restored\nonly for the expectation values and amplitudes. The states themselves are\nnot rep invariant.\n\nIf you wish to impose spatial rep invariance for conformal 1+1d gravity on\nthe cylinder you can do so, but you end up with "boundary states" (off-shell\nstates) not physical states. They can never be annihilated by any mode of\nthe Hamiltonian constraint.\n\nI was recently hoping (in vain) that Lee Smolin might reply to that\nobservation and its obvious implication for non-perturbative quantization of\nhigher dimensional gravity:\n\nhttp://golem.ph.utexas.edu/string/archives/000420.html\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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2qqk3sF12042nU1-100000@uni-berlin.de...
> What one should really require is that the result of the quantization
> procedure does not depend on the choice of coordinates. And therefore
> if one uses coordinates at some point in the quantization, one has to
> find the unitary operators that do the pull back on the metric and
> impose the constraints that under changes of coordinates also the
> quantum analogue of the metric is pulled back correctly.
But here one has to be very careful what one really means. The standard
quantizations of 2d gravity don't depend on coordinates, yet physical states
are *not* annihilated by the generator L_n - \bar L_{-n} of spatial
coordinate transformations. They cannot. Coordinate invariance is restored
only for the expectation values and amplitudes. The states themselves are
not rep invariant.
If you wish to impose spatial rep invariance for conformal 1+1d gravity on
the cylinder you can do so, but you end up with "boundary states" (off-shell
states) not physical states. They can never be annihilated by any mode of
the Hamiltonian constraint.
I was recently hoping (in vain) that Lee Smolin might reply to that
observation and its obvious implication for non-perturbative quantization of
higher dimensional gravity:
http://golem.ph.utexas.edu/string/archives/000420.html
Urs Schreiber
Sep15-04, 07:52 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409150720370.1903-100000@feynman.harvard.edu...\n> On Wed, 15 Sep 2004, Robert C. Helling wrote:\n>\n> > >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism\ninvariant.\n> > >\n> > > Right.\n> >\n> > Actually, this is where I would probably say no. All this \'confusion\'\n> > (which I claim is wide spread in certain circles) comes from the\n> > mixing of two meanings of "diffeomorphism invariant":\n>\n> If you now say that the Minkowski space preserves *all* diffeomorphisms,\n> then you should consider the option that it is you who is very confused.\n\nI think you clarify this further below. When saying "Minkowski space" you\nare thinking of a manifold with coordinates such that the metric is in the\nform diag(-1,1,...,1) and this is indeed not preserved by all\ndiffeomorphisms, but only by the isometries (which form the Poincare group\nin this case).\n\nWhen I say "Minkowski space" I usually mean "a flat pseudo-Riemannian\nmanifold with one timelike direction", no matter which coordinates I put on\nit and hence no matter if the metric has the diag(-1,1,...,1) form or that\nof polar coordinates or whatever. Apparently this is also what Robert has in\nmind.\n\nI believe this allows us to understand each other\'s use of terminology.\n\nWith that out of the way I would still like to understand what Robert had in\nmind in his original comment on non-perturbative quantization of gravity. I\nstill don\'t quite see what he is getting at.\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409150720370.1903-100000@feynman.harvard.edu...
> On Wed, 15 Sep 2004, Robert C. Helling wrote:
>
> > >> (R. Helling wrote:) 5) Minkowski space is not diffeomorphism
invariant.
> > >
> > > Right.
> >
> > Actually, this is where I would probably say no. All this 'confusion'
> > (which I claim is wide spread in certain circles) comes from the
> > mixing of two meanings of "diffeomorphism invariant":
>
> If you now say that the Minkowski space preserves *all* diffeomorphisms,
> then you should consider the option that it is you who is very confused.
I think you clarify this further below. When saying "Minkowski space" you
are thinking of a manifold with coordinates such that the metric is in the
form diag(-1,1,...,1) and this is indeed not preserved by all
diffeomorphisms, but only by the isometries (which form the Poincare group
in this case).
When I say "Minkowski space" I usually mean "a flat pseudo-Riemannian
manifold with one timelike direction", no matter which coordinates I put on
it and hence no matter if the metric has the diag(-1,1,...,1) form or that
of polar coordinates or whatever. Apparently this is also what Robert has in
mind.
I believe this allows us to understand each other's use of terminology.
With that out of the way I would still like to understand what Robert had in
mind in his original comment on non-perturbative quantization of gravity. I
still don't quite see what he is getting at.
Lubos Motl
Sep15-04, 09:48 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 Wed, 15 Sep 2004, Urs Schreiber wrote:\n\n> I think you clarify this further below. When saying "Minkowski space" you\n> are thinking of a manifold with coordinates such that the metric is in the\n> form diag(-1,1,...,1) and this is indeed not preserved by all\n> diffeomorphisms, but only by the isometries (which form the Poincare group\n> in this case).\n\nRight.\n\n> When I say "Minkowski space" I usually mean "a flat pseudo-Riemannian\n> manifold with one timelike direction", no matter which coordinates I put on\n> it and hence no matter if the metric has the diag(-1,1,...,1) form or that\n> of polar coordinates or whatever. Apparently this is also what Robert has in\n> mind.\n\nI mean the same thing.\n\n> I believe this allows us to understand each other\'s use of terminology.\n\nNot really. Regardless of your choice of the coordinates for the Minkowski\nspace - or any other manifold, for that matter - its metric is only\ninvariant under a group that is isomorphic to the isometry group. What\nRobert seems to mean is that the metric is invariant under *any*\ntransformation (of coordinates), up to a transformation (of coordinates).\nwhich is a vacuous tautology. Everything would be symmetric under\neverything if this notion of "invariance" were adopted, and I think that\nmy definition is the only definition of "being invariant" that makes any\nsense. So I don\'t really have understanding for any other terminology. Am\nI missing anything?\n\nCheers,\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)\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 Wed, 15 Sep 2004, Urs Schreiber wrote:
> I think you clarify this further below. When saying "Minkowski space" you
> are thinking of a manifold with coordinates such that the metric is in the
> form diag(-1,1,...,1) and this is indeed not preserved by all
> diffeomorphisms, but only by the isometries (which form the Poincare group
> in this case).
Right.
> When I say "Minkowski space" I usually mean "a flat pseudo-Riemannian
> manifold with one timelike direction", no matter which coordinates I put on
> it and hence no matter if the metric has the diag(-1,1,...,1) form or that
> of polar coordinates or whatever. Apparently this is also what Robert has in
> mind.
I mean the same thing.
> I believe this allows us to understand each other's use of terminology.
Not really. Regardless of your choice of the coordinates for the Minkowski
space - or any other manifold, for that matter - its metric is only
invariant under a group that is isomorphic to the isometry group. What
Robert seems to mean is that the metric is invariant under *any*
transformation (of coordinates), up to a transformation (of coordinates).
which is a vacuous tautology. Everything would be symmetric under
everything if this notion of "invariance" were adopted, and I think that
my definition is the only definition of "being invariant" that makes any
sense. So I don't really have understanding for any other terminology. Am
I missing anything?
Cheers,
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Sep15-04, 10:35 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409151043490.2026-100000@feynman.harvard.edu...\n\n> Not really. Regardless of your choice of the coordinates for the Minkowski\n> space - or any other manifold, for that matter - its metric is only\n> invariant under a group that is isomorphic to the isometry group. What\n> Robert seems to mean is that the metric is invariant under *any*\n> transformation (of coordinates), up to a transformation (of coordinates).\n> which is a vacuous tautology. Everything would be symmetric under\n> everything if this notion of "invariance" were adopted, and I think that\n> my definition is the only definition of "being invariant" that makes any\n> sense. So I don\'t really have understanding for any other terminology. Am\n> I missing anything?\n\nAgreed. For this reason I said before that I don\'t understand what Robert\nmeans when saying "a manifold is diff invariant". Either it is meant in "my"\nsense, that the abstract geometry does not care about the coordinates (in\nwhich case it is pretty tautologous) or it is meant in "your" sense saying\nthat the metric tensor is invariant under all diffeos, in which case it is\nnot correct.\n\nBut for something different: You mentioned recently the failure of SFT to\ncapture certain non-perturbative degrees of freedom. Is it conceivable that\none can somehow "augment" SFT in a nice way to include these?\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 Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409151043490.2026-100000@feynman.harvard.edu...
> Not really. Regardless of your choice of the coordinates for the Minkowski
> space - or any other manifold, for that matter - its metric is only
> invariant under a group that is isomorphic to the isometry group. What
> Robert seems to mean is that the metric is invariant under *any*
> transformation (of coordinates), up to a transformation (of coordinates).
> which is a vacuous tautology. Everything would be symmetric under
> everything if this notion of "invariance" were adopted, and I think that
> my definition is the only definition of "being invariant" that makes any
> sense. So I don't really have understanding for any other terminology. Am
> I missing anything?
Agreed. For this reason I said before that I don't understand what Robert
means when saying "a manifold is diff invariant". Either it is meant in "my"
sense, that the abstract geometry does not care about the coordinates (in
which case it is pretty tautologous) or it is meant in "your" sense saying
that the metric tensor is invariant under all diffeos, in which case it is
not correct.
But for something different: You mentioned recently the failure of SFT to
capture certain non-perturbative degrees of freedom. Is it conceivable that
one can somehow "augment" SFT in a nice way to include these?
Ilja Schmelzer
Sep16-04, 07:02 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im\n\n> If you add new fields, you can make virtually any theory be invariant\n> under virtually any group. The fields that you must add to Newton\'s theory\n> to make it generally covariant are non-dynamical and they can easily be\n> erased, together with these symmetries, and therefore Newton\'s theory did\n> not morally satisfy Einstein\'s requirement of a diff. invariant theory.\n\nI would be interested to understand how distinguishing "non-dynamical"\nfrom "dynamical" fields is supposed to work.\n\nTo describe a flat metric eta_mn, I can write down equations like R_ijkl=0\nwhich look quite dynamical. I don\'t see how identify R_ijkl=0 as\nnon-dynamical but R_ij=0 for GR in the vacuum as dynamical.\n\n[Moderator\'s note: \\eta_{mn} constrained to satisfy R_{ijkl}=0 is a\ntextbook example of non-dynamical degrees of freedom because once you\nfix the general covariance in the well-known way, the constraint forces\nyou to pick \\eta_{mn}=diag(-1,1,1,1), for example. Henceforth, there\nare no wave solutions of these constraints. Once again, the only allowed\nsolution is - up to symmetry transformations - static, and "static" means\n"completely non-dynamical". In fact, it is completely constant here,\nnot just static.\n\nIn the quantized version\nof the theory, there are no physical particle-like excitations coming\nfrom your constrained version of \\eta_{mn}, which is why it is called\nan auxilliary, non-dynamical degree of freedom. The situation is\ncompletely analogous to a gauge field A_\\mu whose fields strength is\nstrongly required to vanish - F_{\\mu\\nu}=0. In this case, one can also\nfix the gauge so that A_\\mu=0 to see that this gauge field has no\nphysical polarizations - it would become non-dynamical. (If you set\nthe coupling constant in usual gauge theories strictly to zero, you get\na decoupled gauge field - although the equations are not quite F=0 -\nwhich is still equivalent to its becoming unphysical.)\n\nOn the other hand, the Yang-Mills equations or the Einstein equations\nR_{ij}=0 (or G_{ij}=T_{ij}) are dynamical because they lead to\npropagating wave-like solutions for the gauge bosons and/or gravitons.\nAlthough it may be unclear to you a priori, a simple procedure of solving\nthe constraints shows that R_{ijkl}=0 is completely non-dynamical, while\nR_{ij}=0 is about dynamical metric if the spacetime dimension is greater\nthan two. Is it clear now? Adding a metric tensor and requiring\nthe complete flatness R_{ijkl}=0 is adding *nothing* to the Euclidean\ngeometry - it is about allowing different coordinates which people\nhad done for centuries - while adding the metric tensor with\nEinstein\'s equations is equivalent to discovery of GR, with all the\nnew dynamics and curvature of space(time) that guarantees the\ngravitational force. If you think that these are equally dynamical\nand physical situations, you must be doing something wrong. ;-) LM]\n\nI also don\'t see what means "easily erased". Of course we can simplify the\ntheory using a non-covariant formulation. But this is true in GR too,\nthe Einstein equations are much simpler in harmonic coordinates.\n\nIlja\n\n[Moderator\'s note: The procedure above showed that in the theory\nR_{ijkl}=0 you can always set the metric to the usual flat metric - all\nother solutions are gauge transformations of this one. Once you know\nthat the constraint is physically (up to symmetries) equivalent to requiring\n\\eta_{mn}=diag(-1,1,1,1), there is nothing that can be varied in\nin \\eta_{mn} anymore, and therefore its degrees of freedom have been\nerased. Of course, the whole point of GR is that such an elimination\nis impossible in GR - there are solutions of R_{ij}=0, for example\nthe gravitational waves, that are *not* equivalent to the flat space.\nThe number of degrees of freedom in the metric is larger than the amount\nof symmetry, and therefore some propagating degrees of freedom survive.\nThe same holds for gauge theories in d>2 dimensions.\n\nIncidentally, your flawed thinking, combined with Mach\'s principle,\nwas one of the reason why the relativists could not understand that\nthere exist gravitational waves for such a long time - they spent\nliterally years trying to prove that these solutions are pure gauge.\nWell, they are not! Every excitation of the R_{ijkl}=0 constraint -\nwhich by the way does not follow from a nice action, except for\nan action with a Lagrange multiplier - is easily seen to be pure gauge. 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> schrieb im
> If you add new fields, you can make virtually any theory be invariant
> under virtually any group. The fields that you must add to Newton's theory
> to make it generally covariant are non-dynamical and they can easily be
> erased, together with these symmetries, and therefore Newton's theory did
> not morally satisfy Einstein's requirement of a diff. invariant theory.
I would be interested to understand how distinguishing "non-dynamical"
from "dynamical" fields is supposed to work.
To describe a flat metric \eta_mn, I can write down equations like R_{ijkl}=0
which look quite dynamical. I don't see how identify R_{ijkl}=0 as
non-dynamical but R_{ij}=0 for GR in the vacuum as dynamical.
[Moderator's note: \eta_{mn} constrained to satisfy R_{ijkl}=0 is a
textbook example of non-dynamical degrees of freedom because once you
fix the general covariance in the well-known way, the constraint forces
you to pick \eta_{mn}=diag(-1,1,1,1), for example. Henceforth, there
are no wave solutions of these constraints. Once again, the only allowed
solution is - up to symmetry transformations - static, and "static" means
"completely non-dynamical". In fact, it is completely constant here,
not just static.
In the quantized version
of the theory, there are no physical particle-like excitations coming
from your constrained version of \eta_{mn}, which is why it is called
an auxilliary, non-dynamical degree of freedom. The situation is
completely analogous to a gauge field A_\mu whose fields strength is
strongly required to vanish - F_{\mu\nu}=0. In this case, one can also
fix the gauge so that A_\mu=0 to see that this gauge field has no
physical polarizations - it would become non-dynamical. (If you set
the coupling constant in usual gauge theories strictly to zero, you get
a decoupled gauge field - although the equations are not quite F=0 -
which is still equivalent to its becoming unphysical.)
On the other hand, the Yang-Mills equations or the Einstein equations
R_{ij}=0 (or G_{ij}=T_{ij}) are dynamical because they lead to
propagating wave-like solutions for the gauge bosons and/or gravitons.
Although it may be unclear to you a priori, a simple procedure of solving
the constraints shows that R_{ijkl}=0 is completely non-dynamical, while
R_{ij}=0 is about dynamical metric if the spacetime dimension is greater
than two. Is it clear now? Adding a metric tensor and requiring
the complete flatness R_{ijkl}=0 is adding *nothing* to the Euclidean
geometry - it is about allowing different coordinates which people
had done for centuries - while adding the metric tensor with
Einstein's equations is equivalent to discovery of GR, with all the
new dynamics and curvature of space(time) that guarantees the
gravitational force. If you think that these are equally dynamical
and physical situations, you must be doing something wrong. ;-) LM]
I also don't see what means "easily erased". Of course we can simplify the
theory using a non-covariant formulation. But this is true in GR too,
the Einstein equations are much simpler in harmonic coordinates.
Ilja
[Moderator's note: The procedure above showed that in the theory
R_{ijkl}=0 you can always set the metric to the usual flat metric - all
other solutions are gauge transformations of this one. Once you know
that the constraint is physically (up to symmetries) equivalent to requiring
\eta_{mn}=diag(-1,1,1,1), there is nothing that can be varied in
in \eta_{mn} anymore, and therefore its degrees of freedom have been
erased. Of course, the whole point of GR is that such an elimination
is impossible in GR - there are solutions of R_{ij}=0, for example
the gravitational waves, that are *not* equivalent to the flat space.
The number of degrees of freedom in the metric is larger than the amount
of symmetry, and therefore some propagating degrees of freedom survive.
The same holds for gauge theories in d>2 dimensions.
Incidentally, your flawed thinking, combined with Mach's principle,
was one of the reason why the relativists could not understand that
there exist gravitational waves for such a long time - they spent
literally years trying to prove that these solutions are pure gauge.
Well, they are not! Every excitation of the R_{ijkl}=0 constraint -
which by the way does not follow from a nice action, except for
an action with a Lagrange multiplier - is easily seen to be pure gauge. LM]
Robert C. Helling
Sep16-04, 08:50 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 Wed, 15 Sep 2004 11:35:28 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> Agreed. For this reason I said before that I don\'t understand what Robert\n> means when saying "a manifold is diff invariant". Either it is meant in "my"\n> sense, that the abstract geometry does not care about the coordinates (in\n> which case it is pretty tautologous) or it is meant in "your" sense saying\n> that the metric tensor is invariant under all diffeos, in which case it is\n> not correct.\n\nI never claimed to say anything deep. Of course I ment the first\nversion of the statement. You change coordinates and then have to\ndress tensors with the appropriate powers of the Jacobian. Lubos calls\nthis undoing the diffeomorphism. The difference between GR and say\nNewtonian physics is that in GR only dynamical fields get these powers\nof Jacobians whereas in theories with background, there are tensors\nthat get these Jacobians that are not dynamical (eta or dt in the\nexample).\n\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Wed, 15 Sep 2004 11:35:28 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> Agreed. For this reason I said before that I don't understand what Robert
> means when saying "a manifold is diff invariant". Either it is meant in "my"
> sense, that the abstract geometry does not care about the coordinates (in
> which case it is pretty tautologous) or it is meant in "your" sense saying
> that the metric tensor is invariant under all diffeos, in which case it is
> not correct.
I never claimed to say anything deep. Of course I ment the first
version of the statement. You change coordinates and then have to
dress tensors with the appropriate powers of the Jacobian. Lubos calls
this undoing the diffeomorphism. The difference between GR and say
Newtonian physics is that in GR only dynamical fields get these powers
of Jacobians whereas in theories with background, there are tensors
that get these Jacobians that are not dynamical (\eta or dt in the
example).
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
we can study the
quantum regime by studying the CFT on the world sheet. The spectrum of
this CFT includes both perturbative and non-perturbative states and
encodes the (quantum) geometry of target space. Surely if you
formulate the CFT on the world-sheet general enough it will have
excitations in its spectrum that correspond to target spaces of
different topolgy in the low-energy effective description.
Hi Rufus
So I think this is what I had thought. I presume you mean above that if the CFT on the worldsheet is defined appropriately, then in addition to graviton excitations corresponding to perturbative changes in the geometry of the target space, there will also be some CFT states corresponding to changes in the homology groups of the target space. (For example, the massless D3 branes that Lubos discussed above, I guess).
Am still not sure what this has to do with equivalence classes of the worldsheet metric, though, as you implied much earlier on in this thread. Possibly I am missing something obvious. Do you mean that choosing a different equivalence class of worldsheet metric is equivalent (apologies for mixed usage) to choosing a target space with different homology groups?
Robert, do you really not expect it to be possible for a classical manifold to be a ground state of (quantum) gravity? Or were you just saying that such a conclusion reflects sloppy definitions of diffeomorphism invariance and gauge invariance?
[Moderator's note: Well, if there are massless D3-branes, then the
CFT breaks down. The CFT can perturbatively treat the perturbative
string states only, and they are too "light" and unable to change
the topology too much. Perturbative string theory is only OK if the
states that it neglects - such as D-branes - are heavy or otherwise
decoupled. It's not just a matter of calculational
complexity: it is also difficult "physically" to change the topology
of space. LM]
I'm not sure if you're disagreeing with Rufus as I've quoted him in the post above, or just with my interpretation of what he was saying....
Also, I thought you said the D3-branes in the target space can be described by adding a boundary to the world sheet CFT. Is this what you mean by break down?
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