View Full Version : Re: quantizing gravity
Lubos Motl
May30-04, 02:49 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 30 May 2004, Squark wrote on sci.physics.research:\n\n> I\'m not sure this is fair to say. The "quantization" of certain asymptotically\n> Minkwosky and asymptotically AdS SUGRAs provided by matrix theory and\n> AdS/CFT appears to be "a quantization" due to superstring theory, i.e. due\n> to the fact we have serious evidence these models have a perturbative\n> expansion in terms of strings moving through Minkowski / AdS space and\n\nWell, the quantization of gravity in the 11D (and similar) flat space and\nin anti de Sitter space is "only" due to string theory because string\ntheory is the only known consistent way to quantize gravity.\n\n> there is evidence for the later reducing to the appropriate SUGRA in the low\n> energy limit, in some sense.\n\nThere is more than just evidence; we can simply calculate what does CFT in\nAdS/CFT predict about the low energy gravity, for example, and it agrees\nwith SUGRA and gives us a lot of information beyond it.\n\n> The trouble is, it is difficult to give an exact physical\n> explanation of this sense.\n\nIt is not so difficult. The exact sense is that string theory in the\nappropriate background is exactly equivalent to the conformal (gauge)\ntheory that describes it holographically. The low-energy spectrum of\nstring theory is exactly what SUGRA predicts, and the interactions between\nthese objects are exactly what you get from classical SUGRA plus\ncorrections that (relatively) go to zero at low energies. What are you\nexactly dissatisfied with?\n\n> In fact, the sole "gauge invariant" observable in stringy quantization\n> of asymptotically Minkowsky SUGRA is the S-matrix (though as far as I\n> understand there are certain complications in the way of producing it\n> in matrix theory which hasn\'t been completely resolved).\n\nThe reason why current string theory only allows to calculate the S-matrix\nis that the S-matrix is the only known gauge-invariant observable in flat\nspace.\n\n> The sole\n> "gauge invariant" observable in stringy quantization of asymptotically AdS\n> SUGRA is the set of n-point functions on the boundary of spacetime.\n\nAlso, the only reason why current string theory in AdS/CFT space can, as\nfar as dynamics goes, only calculate the boundary correlators - except for\nthe light-cone gauge treatment that can go beyond it - is that the\nboundary correlators (scattering from the AdS boundaries) are the only\nknown covariant gauge-invariant observables in this background.\n\nIf you know some other nice enough and generally neglected gauge-invariant\nobservable in gravity that should be calculable, I am eager to hear about\nit! ;-)\n\n> In the former case I don\'t completely understand the physical meaning\n> of this observable, since the S-matrix usually implies turning off the\n> coupling constant in past and future infinity.\n\nThe interactions are effectively turned off near the AdS boundary, too -\nsimply because the physical space becomes infinitely large due to the warp\nfactor and the gravitons etc. are "infinitely diluted". Your intuition\nthat it is only the usual coupling constant that can make things decoupled\nis correct in flat space, but it breaks in the curved e.g. AdS space. It\nis not hard to see by explicit calculations that all the correlators that\nyou are disputing have a well-defined interpretation on both sides, and\ntherefore your criticism is simply incorrect.\n\n> However, for instance in type IIA superstring theory the coupling\n> constant is the compactification radius of M-theory and I don\'t see\n> how it can be artificially enforced to become zero in future and past\n> infinity. At any rate, these observables are hardly sufficient to\n> describe what the processes going on around us, which happen both in\n> finite time and finite space. So while we have clues we are onto\n> something, it is far-fetched to claim we have found quantizations of\n> gravity (for spacetime dimension > 3).\n\nIt is difficult to define phenomena at finite time - and finite space - in\nsuch a way that these quantities are both exact as well as covariant and\ngauge-invariant (under general diffeomorphisms). If you want to talk about\nphenomena taking place in some "finite piece of spacetime", you must first\ndefine where the "finite piece of spacetime" ends. But coordinates are not\ngood enough - they are transformed to different coordinates by the gauge\ntransformations. Proper distances etc. would be better, but many\nquantities like that become divergent once quantum corrections are added.\n\nOnce we gauge-fix general covariance, for example by imposing the light\ncone gauge, we can calculate finite (light-cone) time evolution, too. All\nsuch ways to fix general covariance are necessarily non-covariant.\n______________________________________ ________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 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 Sun, 30 May 2004, Squark wrote on sci.physics.research:
> I'm not sure this is fair to say. The "quantization" of certain asymptotically
> Minkwosky and asymptotically AdS SUGRAs provided by matrix theory and
> AdS/CFT appears to be "a quantization" due to superstring theory, i.e. due
> to the fact we have serious evidence these models have a perturbative
> expansion in terms of strings moving through Minkowski / AdS space and
Well, the quantization of gravity in the 11D (and similar) flat space and
in anti de Sitter space is "only" due to string theory because string
theory is the only known consistent way to quantize gravity.
> there is evidence for the later reducing to the appropriate SUGRA in the low
> energy limit, in some sense.
There is more than just evidence; we can simply calculate what does CFT in
AdS/CFT predict about the low energy gravity, for example, and it agrees
with SUGRA and gives us a lot of information beyond it.
> The trouble is, it is difficult to give an exact physical
> explanation of this sense.
It is not so difficult. The exact sense is that string theory in the
appropriate background is exactly equivalent to the conformal (gauge)
theory that describes it holographically. The low-energy spectrum of
string theory is exactly what SUGRA predicts, and the interactions between
these objects are exactly what you get from classical SUGRA plus
corrections that (relatively) go to zero at low energies. What are you
exactly dissatisfied with?
> In fact, the sole "gauge invariant" observable in stringy quantization
> of asymptotically Minkowsky SUGRA is the S-matrix (though as far as I
> understand there are certain complications in the way of producing it
> in matrix theory which hasn't been completely resolved).
The reason why current string theory only allows to calculate the S-matrix
is that the S-matrix is the only known gauge-invariant observable in flat
space.
> The sole
> "gauge invariant" observable in stringy quantization of asymptotically AdS
> SUGRA is the set of n-point functions on the boundary of spacetime.
Also, the only reason why current string theory in AdS/CFT space can, as
far as dynamics goes, only calculate the boundary correlators - except for
the light-cone gauge treatment that can go beyond it - is that the
boundary correlators (scattering from the AdS boundaries) are the only
known covariant gauge-invariant observables in this background.
If you know some other nice enough and generally neglected gauge-invariant
observable in gravity that should be calculable, I am eager to hear about
it! ;-)
> In the former case I don't completely understand the physical meaning
> of this observable, since the S-matrix usually implies turning off the
> coupling constant in past and future infinity.
The interactions are effectively turned off near the AdS boundary, too -
simply because the physical space becomes infinitely large due to the warp
factor and the gravitons etc. are "infinitely diluted". Your intuition
that it is only the usual coupling constant that can make things decoupled
is correct in flat space, but it breaks in the curved e.g. AdS space. It
is not hard to see by explicit calculations that all the correlators that
you are disputing have a well-defined interpretation on both sides, and
therefore your criticism is simply incorrect.
> However, for instance in type IIA superstring theory the coupling
> constant is the compactification radius of M-theory and I don't see
> how it can be artificially enforced to become zero in future and past
> infinity. At any rate, these observables are hardly sufficient to
> describe what the processes going on around us, which happen both in
> finite time and finite space. So while we have clues we are onto
> something, it is far-fetched to claim we have found quantizations of
> gravity (for spacetime dimension > 3).
It is difficult to define phenomena at finite time - and finite space - in
such a way that these quantities are both exact as well as covariant and
gauge-invariant (under general diffeomorphisms). If you want to talk about
phenomena taking place in some "finite piece of spacetime", you must first
define where the "finite piece of spacetime" ends. But coordinates are not
good enough - they are transformed to different coordinates by the gauge
transformations. Proper distances etc. would be better, but many
quantities like that become divergent once quantum corrections are added.
Once we gauge-fix general covariance, for example by imposing the light
cone gauge, we can calculate finite (light-cone) time evolution, too. All
such ways to fix general covariance are necessarily non-covariant.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<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>This was posted a while ago (it appears on sci.physics.research,\nfor instance), but for some reason [moderator\'s note:continuing June\nproblems with the FAS newsserver, LM] haven\'t got through\n\n"Lubos Motl" <motl@feynman.harvard.edu> wrote in message\nnews:Pine.LNX.4.31.0405301432410.10458-100000@feynman.harvard.edu...\n\n> There is more than just evidence; we can simply calculate what does CFT in\n> AdS/CFT predict about the low energy gravity, for example, and it agrees\n> with SUGRA and gives us a lot of information beyond it.\n\nThe way you calculate is is via the AdS/CFT conjecture, via replacing the CFT\nby a string expansion. Or is there a way to compute things directly in the CFT?\nThis would have to be a method that works in the high coupling regime! At any\nrate the only entirely physical (gauge invariant) information we can compute\ntoday is the n-point functions on the spacetime boundary.\n\n> It is not so difficult. The exact sense is that string theory in the\n> appropriate background is exactly equivalent to the conformal (gauge)\n> theory that describes it holographically. The low-energy spectrum of\n> string theory is exactly what SUGRA predicts, and the interactions between\n> these objects are exactly what you get from classical SUGRA plus\n> corrections that (relatively) go to zero at low energies. What are you\n> exactly dissatisfied with?\n\nThat in SUGRA you can discuss processes occuring in a finite patch of\nspacetime whereas in string theory you cannot (or don\'t know how).\n\n> The reason why current string theory only allows to calculate the S-matrix\n> is that the S-matrix is the only known gauge-invariant observable in flat\n> space.\n> ...\n> Also, the only reason why current string theory in AdS/CFT space can, as\n> far as dynamics goes, only calculate the boundary correlators - except for\n> the light-cone gauge treatment that can go beyond it - is that the\n> boundary correlators (scattering from the AdS boundaries) are the only\n> known covariant gauge-invariant observables in this background.\n\nThis is clear to me. It also appears to me that Edward Witten (page 7,\n"Quantum Gravity in de Sitter Space", hep-th/0106109) believes this\nis the only observable that can or need to be defined in principle.\n\n> If you know some other nice enough and generally neglected gauge-invariant\n> observable in gravity that should be calculable, I am eager to hear about\n> it! ;-)\n\nI don\'t know the solution but I claim there is definitely a problem. Btw, after\nreading Lenny Susskind\'s "The World as a Hologram" (hep-th/9409089) it\nappears to me such an observable exists. Consider asymptotically AdS\ngravity for instance. Choose a point on the asymptotic boundary and a\npositive real number. For any classical solution one may construct a\nspacelike geodesic beginning from the chosen point on the boundary and\northogonal to it. A natural parameter exists on the geodesic (proper length\nstarting from the boundary)\n\n(More precisely, it is proper length with offset chosen to match a\ngiven choice on AdS space at the asymptotics. Proper length from\nthe boundary is of course infinite. The parameter is thus definable\nin an invariant fashion, and it is natural up to a constant shift.)\n\n- lets call it "depth". The observables are then\nvarious scalars constructed out of fields in the theory at the point on the\ngeodesic for which the depth is the chosen real number. One may construct\nmore observables using derivatives w.r.t. depth and the location of the\nbasepoint on the boundary. Did anyone try defining such observables in\nAdS/CFT? Or maybe it can be shown they are not "nice enough"?\n\n> > In the former case I don\'t completely understand the physical meaning\n> > of this observable, since the S-matrix usually implies turning off the\n> > coupling constant in past and future infinity.\n>\n> The interactions are effectively turned off near the AdS boundary, too -\n> simply because the physical space becomes infinitely large due to the warp\n> factor and the gravitons etc. are "infinitely diluted".\n\nPossibly I formulated myself badly. I have no problem with the AdS\nboundary n-point functions as observables. I have a problem with the\nS-matrix as an observable of superstring theory in asymptotically\nflat spacetime. The usual definition of the S-matrix in QFT includes\nturning off the coupling constant in past and future infinity but I don\'t\nunderstand how can it be consistently then in superstring theory:\n\n> > However, for instance in type IIA superstring theory the coupling\n> > constant is the compactification radius of M-theory and I don\'t see\n> > how it can be artificially enforced to become zero in future and past\n> > infinity.\n\n> It is difficult to define phenomena at finite time - and finite space - in\n> such a way that these quantities are both exact as well as covariant and\n> gauge-invariant (under general diffeomorphisms). If you want to talk about\n> phenomena taking place in some "finite piece of spacetime", you must first\n> define where the "finite piece of spacetime" ends. But coordinates are not\n> good enough - they are transformed to different coordinates by the gauge\n> transformations.\n\nI realize all that completely well. The thing is, these problems are solvable\nin classical gravity. It appears to me that to quantize gravity implies providing\na way to solve this problems for quantum gravity as well, within the setting of\nyour quantization.\n\n> Proper distances etc. would be better, but many\n> quantities like that become divergent once quantum corrections are added.\n\nDoes it include the thingies I mentioned above? How do you know they\nbecome divergent?\n\n\nBest regards,\nSquark\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>This was posted a while ago (it appears on sci.physics.research,
for instance), but for some reason [moderator's note:continuing June
problems with the FAS newsserver, LM] haven't got through
"Lubos Motl" <motl@feynman.harvard.edu> wrote in message
news:Pine.LNX.4.31.0405301432410.10458-100000@feynman.harvard.edu...
> There is more than just evidence; we can simply calculate what does CFT in
> AdS/CFT predict about the low energy gravity, for example, and it agrees
> with SUGRA and gives us a lot of information beyond it.
The way you calculate is is via the AdS/CFT conjecture, via replacing the CFT
by a string expansion. Or is there a way to compute things directly in the CFT?
This would have to be a method that works in the high coupling regime! At any
rate the only entirely physical (gauge invariant) information we can compute
today is the n-point functions on the spacetime boundary.
> It is not so difficult. The exact sense is that string theory in the
> appropriate background is exactly equivalent to the conformal (gauge)
> theory that describes it holographically. The low-energy spectrum of
> string theory is exactly what SUGRA predicts, and the interactions between
> these objects are exactly what you get from classical SUGRA plus
> corrections that (relatively) go to zero at low energies. What are you
> exactly dissatisfied with?
That in SUGRA you can discuss processes occuring in a finite patch of
spacetime whereas in string theory you cannot (or don't know how).
> The reason why current string theory only allows to calculate the S-matrix
> is that the S-matrix is the only known gauge-invariant observable in flat
> space.
> ...
> Also, the only reason why current string theory in AdS/CFT space can, as
> far as dynamics goes, only calculate the boundary correlators - except for
> the light-cone gauge treatment that can go beyond it - is that the
> boundary correlators (scattering from the AdS boundaries) are the only
> known covariant gauge-invariant observables in this background.
This is clear to me. It also appears to me that Edward Witten (page 7,
"Quantum Gravity in de Sitter Space", http://www.arxiv.org/abs/hep-th/0106109) believes this
is the only observable that can or need to be defined in principle.
> If you know some other nice enough and generally neglected gauge-invariant
> observable in gravity that should be calculable, I am eager to hear about
> it! ;-)
I don't know the solution but I claim there is definitely a problem. Btw, after
reading Lenny Susskind's "The World as a Hologram" (http://www.arxiv.org/abs/hep-th/9409089) it
appears to me such an observable exists. Consider asymptotically AdS
gravity for instance. Choose a point on the asymptotic boundary and a
positive real number. For any classical solution one may construct a
spacelike geodesic beginning from the chosen point on the boundary and
orthogonal to it. A natural parameter exists on the geodesic (proper length
starting from the boundary)
(More precisely, it is proper length with offset chosen to match a
given choice on AdS space at the asymptotics. Proper length from
the boundary is of course infinite. The parameter is thus definable
in an invariant fashion, and it is natural up to a constant shift.)
- lets call it "depth". The observables are then
various scalars constructed out of fields in the theory at the point on the
geodesic for which the depth is the chosen real number. One may construct
more observables using derivatives w.r.t. depth and the location of the
basepoint on the boundary. Did anyone try defining such observables in
AdS/CFT? Or maybe it can be shown they are not "nice enough"?
> > In the former case I don't completely understand the physical meaning
> > of this observable, since the S-matrix usually implies turning off the
> > coupling constant in past and future infinity.
>
> The interactions are effectively turned off near the AdS boundary, too -
> simply because the physical space becomes infinitely large due to the warp
> factor and the gravitons etc. are "infinitely diluted".
Possibly I formulated myself badly. I have no problem with the AdS
boundary n-point functions as observables. I have a problem with the
S-matrix as an observable of superstring theory in asymptotically
flat spacetime. The usual definition of the S-matrix in QFT includes
turning off the coupling constant in past and future infinity but I don't
understand how can it be consistently then in superstring theory:
> > However, for instance in type IIA superstring theory the coupling
> > constant is the compactification radius of M-theory and I don't see
> > how it can be artificially enforced to become zero in future and past
> > infinity.
> It is difficult to define phenomena at finite time - and finite space - in
> such a way that these quantities are both exact as well as covariant and
> gauge-invariant (under general diffeomorphisms). If you want to talk about
> phenomena taking place in some "finite piece of spacetime", you must first
> define where the "finite piece of spacetime" ends. But coordinates are not
> good enough - they are transformed to different coordinates by the gauge
> transformations.
I realize all that completely well. The thing is, these problems are solvable
in classical gravity. It appears to me that to quantize gravity implies providing
a way to solve this problems for quantum gravity as well, within the setting of
your quantization.
> Proper distances etc. would be better, but many
> quantities like that become divergent once quantum corrections are added.
Does it include the thingies I mentioned above? How do you know they
become divergent?
Best regards,
Squark
Lubos Motl
Jun6-04, 10:34 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 Wed, 2 Jun 2004, Squark wrote:\n\n> The way you calculate is is via the AdS/CFT conjecture, via replacing\n> the CFT by a string expansion. Or is there a way to compute things\n> directly in the CFT?\n\nYes, that\'s the whole point of the AdS/CFT checks - that you can compute\nmany things on both sides, and they agree for all quantities that are\nexactly calculable on both sides (and people have found many of them). The\ncalculations on the CFT side are being done without string theory - using\nold-fashioned Feynman diagrams in N=4 super Yang-Mills, for example.\nFollowing \'t Hooft, we can reorganize these calculations according to the\ngenus (and focus on the planar diagrams), but they are still\nfield-theoretical calculations.\n\nMost quantities (the more complicated ones) are however unprotected, and\nthey are difficult to calculate exactly on both sides. It makes the\ncontinuation in both directions difficult, of course, and the approximate\nresults then can\'t agree. For example, the entropy density of hot\nD3-branes is known to be renormalized by a factor of 3/4 as you change\nthe \'t Hooft coupling from 0 to infinity.\n\n> This would have to be a method that works in the high coupling regime!\n\nThe SUSY-protected quantities work out correctly in the whole range.\n\n> At any rate the only entirely physical (gauge\n> invariant) information we can compute today is the n-point functions\n> on the spacetime boundary.\n\nAnd according to old-fashioned physics, it is the only exact physical\ninformation that exists. It is a successful consistency check because we\nsee that no wrong and gauge-variant observables can be calculated.\n\n> That in SUGRA you can discuss processes occuring in a finite patch of\n> spacetime whereas in string theory you cannot (or don\'t know how).\n\nYou probably mean classical solutions of SUGRA (defined in a finite patch\nof spacetime) only? Classical SUGRA is a limit of superstring theory in\nthe same sense as it is a limit of any other quantum completion of\nclassical supergravity (well, it\'s because string/M-theory is the only\nconsistent quantum extension of classical SUGRA). You can use your\nfavorite local solutions of classical SUGRA for "finite patches", and you\ncan prove that these solutions are relevant even in string theory, in the\nlimit hbar=0 - at least if the red shifts are bounded and all distances\nyou study are much longer than l_{planck}. But these classical solutions\nare not quantum exact observables.\n\nIn a theory of quantum gravity, such as SUGRA, it is just impossible to\ndefine local gauge-invariant operators and compute their correlators or\nanything of this type that we are used to in other quantum theories.\n\nThe difference that you are trying to picture as a difference between\nSUGRA and superstring theory is actually something completely different:\nyou are talking about the things that can be done in classical physics,\nand you seem to complain that they are not meaningful in a quantum theory.\nYour criticism is actually a criticism of (the very framework of) quantum\ntheory, not a specific criticism of string theory. The only problem with\nthis criticism is that quantum theory is proved to be a feature of\nreality.\n\nIf one uses a non-stringy framework, one must be very careful to avoid all\nthese potential "quantities" that must be non-physical. String theory does\nit automatically, and it only allows us to calculate physically meaningful\nobservables such as the S-matrix or the boundary correlators.\n\n> > If you know some other nice enough and generally neglected gauge-invariant\n> > observable in gravity that should be calculable, I am eager to hear about\n> > it! ;-)\n>\n> I don\'t know the solution but I claim there is definitely a problem.\n\nHow can you claim that there is "definitely" a problem even if you\'re\nunable to formulate what the problem could be? ;-)\n\n> Btw, after reading Lenny Susskind\'s "The World as a Hologram"\n> (hep-th/9409089) it appears to me such an observable exists. Consider\n> asymptotically AdS gravity for instance. Choose a point on the\n> asymptotic boundary and a positive real number. For any classical\n> solution one may construct a spacelike geodesic beginning from the\n> chosen point on the boundary and orthogonal to it. A natural parameter\n> exists on the geodesic (proper length starting from the boundary) -\n> lets call it "depth".\n\nThe proper length of a line starting from the AdS boundary is infinite\neven in the classical theory. Even if you chose a different geometry where\nthis proper length is finite - or subtracted the divergent piece in some\nway, it would become infinite anyway once you took quantum fluctuations of\nthe geometry into account. In the full theory, geometry *does* fluctuate,\nand your intuition about the finite distances is only valid in the\nclassical, long-distance approximation.\n\n> The observables are then various scalars constructed out of fields in\n> the theory at the point on the geodesic for which the depth is the\n> chosen real number. One may construct more observables using\n> derivatives w.r.t. depth and the location of the basepoint on the\n> boundary. Did anyone try defining such observables in AdS/CFT? Or\n> maybe it can be shown they are not "nice enough"?\n\nTry it. I won\'t try it because these observables do not seem well\nmotivated and justified to me.\n\n> Possibly I formulated myself badly. I have no problem with the AdS\n> boundary n-point functions as observables. I have a problem with the\n> S-matrix as an observable of superstring theory in asymptotically\n> flat spacetime.\n\nThat\'s about an equally serious objection. ;-)\n\n> The usual definition of the S-matrix in QFT includes\n> turning off the coupling constant in past and future infinity but I don\'t\n> understand how can it be consistently then in superstring theory:\n\nTurning off the interactions is just a trick to define the asymptotic\nscattering states as the adiabatic continuation of the corresponding\nstates at g=0 (even though they exist, as slightly different states, for\nevery value of "g", as long as the particle is stable), and it is equally\nmeaningful in string theory like in a perturbative quantum field theory.\nWhere do you exactly see any difference? In the fact that the stringy\ncoupling constant is the expectation value of the dilaton (exponentiated)?\nWhy would you believe that this could invalidate perturbative expansions?\n\nOn the contrary, it adds more structure and knowledge - not less - because\nthe amplitudes with an extra dilaton in the process can be used to relate\ndifferent orders of "g" contributing to a general amplitude. But otherwise\nbe sure that as long as the dilaton is a modulus, you can tune its value\nto an arbitrarily small positive number. Did you say that you want zero?\nIn field theory, you don\'t need the exact zero to be achieved either; you\njust need the right limit.\n\n> I realize all that completely well. The thing is, these problems are\n> solvable in classical gravity. It appears to me that to quantize\n> gravity implies providing a way to solve this problems for quantum\n> gravity as well, within the setting of your quantization.\n\nIt appears to me that you want to reject basic principles in quantum\nmechanics. In classical physics, the proper length of a geodesic between\ntwo points is a sharply defined number. But in quantum physics, this\nobservable - just like any other observable - becomes an operator whose\nvalues fluctuate, whose values span a probability distribution and they\nhave different amplitudes to take different values. Quantum fluctuations\ncontribute and make these classical quantities - such as the proper length\n- cutoff-dependent, which will prevent you from defining an exact quantity\nas your cutoff approaches the Planck scale. You might hope that there will\nbe some sharply defined value of the "proper length" where the cutoff is\n"infinite" - which might be effectively equivalent to the Planck scale\ncutoff - but this logic is misguided because the classical geometry is\nsimply an incorrect description for processes that occur at the Planck\nscale (or shorter).\n\n> Does it include the thingies I mentioned above? How do you know they\n> become divergent?\n\nThe things you mentioned above were divergent even at the classical level.\nMore generally, we can calculate the cutoff-dependence of these quantities\nin effective field theory, and the fluctuations (as well as the mean\nvalue) increases with your cutoff energy. Intuitively this result is\nclear; if you imagine the spacetime geometry to be more foamy, the total\nlength will count all the local features of the foam, and it will grow.\nWell, in Minkowski geometry, you can on the contrary always find a\n*shorter* geodesic, but at any rate, the proper length computed from\n"quantized general relativity" won\'t be finite.\n\nIn string theory, we often compute meaningful "quantum volumes", but their\ndefinition is different. They are counted from masses of the branes\nwrapped on the corresponding cycles, and in a finite theory (with a lot of\nSUSY calculations), these masses (or energies) are finite, of course. But\nonce again, they are not being calculated in the naive classical way as\nthe integral of "sqrt{g_{mn}dx^m dx^n}" with an unlimited UV cutoff\nbecause this prescription is physically misled.\n_________________________________________ _____________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n[This post has been posted for the 4th time because of continuing problems with the\nFAS newsserver. LM]\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, 2 Jun 2004, Squark wrote:
> The way you calculate is is via the AdS/CFT conjecture, via replacing
> the CFT by a string expansion. Or is there a way to compute things
> directly in the CFT?
Yes, that's the whole point of the AdS/CFT checks - that you can compute
many things on both sides, and they agree for all quantities that are
exactly calculable on both sides (and people have found many of them). The
calculations on the CFT side are being done without string theory - using
old-fashioned Feynman diagrams in N=4 super Yang-Mills, for example.
Following 't Hooft, we can reorganize these calculations according to the
genus (and focus on the planar diagrams), but they are still
field-theoretical calculations.
Most quantities (the more complicated ones) are however unprotected, and
they are difficult to calculate exactly on both sides. It makes the
continuation in both directions difficult, of course, and the approximate
results then can't agree. For example, the entropy density of hot
D3-branes is known to be renormalized by a factor of 3/4 as you change
the 't Hooft coupling from to infinity.
> This would have to be a method that works in the high coupling regime!
The SUSY-protected quantities work out correctly in the whole range.
> At any rate the only entirely physical (gauge
> invariant) information we can compute today is the n-point functions
> on the spacetime boundary.
And according to old-fashioned physics, it is the only exact physical
information that exists. It is a successful consistency check because we
see that no wrong and gauge-variant observables can be calculated.
> That in SUGRA you can discuss processes occuring in a finite patch of
> spacetime whereas in string theory you cannot (or don't know how).
You probably mean classical solutions of SUGRA (defined in a finite patch
of spacetime) only? Classical SUGRA is a limit of superstring theory in
the same sense as it is a limit of any other quantum completion of
classical supergravity (well, it's because string/M-theory is the only
consistent quantum extension of classical SUGRA). You can use your
favorite local solutions of classical SUGRA for "finite patches", and you
can prove that these solutions are relevant even in string theory, in the
limit \hbar=0 - at least if the red shifts are bounded and all distances
you study are much longer than l_{planck}. But these classical solutions
are not quantum exact observables.
In a theory of quantum gravity, such as SUGRA, it is just impossible to
define local gauge-invariant operators and compute their correlators or
anything of this type that we are used to in other quantum theories.
The difference that you are trying to picture as a difference between
SUGRA and superstring theory is actually something completely different:
you are talking about the things that can be done in classical physics,
and you seem to complain that they are not meaningful in a quantum theory.
Your criticism is actually a criticism of (the very framework of) quantum
theory, not a specific criticism of string theory. The only problem with
this criticism is that quantum theory is proved to be a feature of
reality.
If one uses a non-stringy framework, one must be very careful to avoid all
these potential "quantities" that must be non-physical. String theory does
it automatically, and it only allows us to calculate physically meaningful
observables such as the S-matrix or the boundary correlators.
> > If you know some other nice enough and generally neglected gauge-invariant
> > observable in gravity that should be calculable, I am eager to hear about
> > it! ;-)
>
> I don't know the solution but I claim there is definitely a problem.
How can you claim that there is "definitely" a problem even if you're
unable to formulate what the problem could be? ;-)
> Btw, after reading Lenny Susskind's "The World as a Hologram"
> (http://www.arxiv.org/abs/hep-th/9409089) it appears to me such an observable exists. Consider
> asymptotically AdS gravity for instance. Choose a point on the
> asymptotic boundary and a positive real number. For any classical
> solution one may construct a spacelike geodesic beginning from the
> chosen point on the boundary and orthogonal to it. A natural parameter
> exists on the geodesic (proper length starting from the boundary) -
> lets call it "depth".
The proper length of a line starting from the AdS boundary is infinite
even in the classical theory. Even if you chose a different geometry where
this proper length is finite - or subtracted the divergent piece in some
way, it would become infinite anyway once you took quantum fluctuations of
the geometry into account. In the full theory, geometry *does* fluctuate,
and your intuition about the finite distances is only valid in the
classical, long-distance approximation.
> The observables are then various scalars constructed out of fields in
> the theory at the point on the geodesic for which the depth is the
> chosen real number. One may construct more observables using
> derivatives w.r.t. depth and the location of the basepoint on the
> boundary. Did anyone try defining such observables in AdS/CFT? Or
> maybe it can be shown they are not "nice enough"?
Try it. I won't try it because these observables do not seem well
motivated and justified to me.
> Possibly I formulated myself badly. I have no problem with the AdS
> boundary n-point functions as observables. I have a problem with the
> S-matrix as an observable of superstring theory in asymptotically
> flat spacetime.
That's about an equally serious objection. ;-)
> The usual definition of the S-matrix in QFT includes
> turning off the coupling constant in past and future infinity but I don't
> understand how can it be consistently then in superstring theory:
Turning off the interactions is just a trick to define the asymptotic
scattering states as the adiabatic continuation of the corresponding
states at g=0 (even though they exist, as slightly different states, for
every value of "g", as long as the particle is stable), and it is equally
meaningful in string theory like in a perturbative quantum field theory.
Where do you exactly see any difference? In the fact that the stringy
coupling constant is the expectation value of the dilaton (exponentiated)?
Why would you believe that this could invalidate perturbative expansions?
On the contrary, it adds more structure and knowledge - not less - because
the amplitudes with an extra dilaton in the process can be used to relate
different orders of "g" contributing to a general amplitude. But otherwise
be sure that as long as the dilaton is a modulus, you can tune its value
to an arbitrarily small positive number. Did you say that you want zero?
In field theory, you don't need the exact zero to be achieved either; you
just need the right limit.
> I realize all that completely well. The thing is, these problems are
> solvable in classical gravity. It appears to me that to quantize
> gravity implies providing a way to solve this problems for quantum
> gravity as well, within the setting of your quantization.
It appears to me that you want to reject basic principles in quantum
mechanics. In classical physics, the proper length of a geodesic between
two points is a sharply defined number. But in quantum physics, this
observable - just like any other observable - becomes an operator whose
values fluctuate, whose values span a probability distribution and they
have different amplitudes to take different values. Quantum fluctuations
contribute and make these classical quantities - such as the proper length
- cutoff-dependent, which will prevent you from defining an exact quantity
as your cutoff approaches the Planck scale. You might hope that there will
be some sharply defined value of the "proper length" where the cutoff is
"infinite" - which might be effectively equivalent to the Planck scale
cutoff - but this logic is misguided because the classical geometry is
simply an incorrect description for processes that occur at the Planck
scale (or shorter).
> Does it include the thingies I mentioned above? How do you know they
> become divergent?
The things you mentioned above were divergent even at the classical level.
More generally, we can calculate the cutoff-dependence of these quantities
in effective field theory, and the fluctuations (as well as the mean
value) increases with your cutoff energy. Intuitively this result is
clear; if you imagine the spacetime geometry to be more foamy, the total
length will count all the local features of the foam, and it will grow.
Well, in Minkowski geometry, you can on the contrary always find a
*shorter* geodesic, but at any rate, the proper length computed from
"quantized general relativity" won't be finite.
In string theory, we often compute meaningful "quantum volumes", but their
definition is different. They are counted from masses of the branes
wrapped on the corresponding cycles, and in a finite theory (with a lot of
SUSY calculations), these masses (or energies) are finite, of course. But
once again, they are not being calculated in the naive classical way as
the integral of "\sqrt{g_{mn}dx^m dx^n}" with an unlimited UV cutoff
because this prescription is physically misled.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
[This post has been posted for the 4th time because of continuing problems with the
FAS newsserver. LM]
Charlie Stromeyer Jr.
Jun9-04, 04:28 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>Squark <fiis5d@yahoo.com> wrote in message news:\n\n> Possibly I formulated myself badly. I have no problem with the AdS\n> boundary n-point functions as observables. I have a problem with the\n> S-matrix as an observable of superstring theory in asymptotically\n> flat spacetime.\n\nYou might be interested to know that S. Giddings once defined a\n"boundary S-matrix" for AdS/CFT [1]. I gained some important insights\nabout what would seem to be a minimal inherent amount of non-locality\nin string theory from reading three older papers either authored or\ncoauthored by S. Giddings within the AdS/CFT setting.\n\nI myself have never thought about how to translate Gidding\'s ideas\ninto a more realistic setting such as dS/CFT but A. Strominger has\nthought about this question and wrote an important paper about it [2].\n\n\n[1] http://arxiv.org/abs/hep-th/9903048\n\n[2] http://arxiv.org/abs/hep-th/0106113\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>Squark <fiis5d@yahoo.com> wrote in message news:
> Possibly I formulated myself badly. I have no problem with the AdS
> boundary n-point functions as observables. I have a problem with the
> S-matrix as an observable of superstring theory in asymptotically
> flat spacetime.
You might be interested to know that S. Giddings once defined a
"boundary S-matrix" for AdS/CFT [1]. I gained some important insights
about what would seem to be a minimal inherent amount of non-locality
in string theory from reading three older papers either authored or
coauthored by S. Giddings within the AdS/CFT setting.
I myself have never thought about how to translate Gidding's ideas
into a more realistic setting such as dS/CFT but A. Strominger has
thought about this question and wrote an important paper about it [2].
[1] http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9903048
[2] http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0106113
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message\nnews:<20040607043417.S2574@mail.kolej.mff .cuni.cz>...\n\n> Yes, that\'s the whole point of the AdS/CFT checks - that you can compute\n> many things on both sides, and they agree for all quantities that are\n> exactly calculable on both sides (and people have found many of them).\n\nThat is OK, however, it is still unclear to me in precisely in what sense\ndo you get SUGRA as the classical limit.\n\n> You can use your\n> favorite local solutions of classical SUGRA for "finite patches", and you\n> can prove that these solutions are relevant even in string theory, in the\n> limit hbar=0 - at least if the red shifts are bounded and all distances\n> you study are much longer than l_{planck}. But these classical solutions\n> are not quantum exact observables.\n\nWhat precisely is the meaning of the word "relevant" as you use it here?\n\n> The difference that you are trying to picture as a difference between\n> SUGRA and superstring theory is actually something completely different:\n> you are talking about the things that can be done in classical physics,\n> and you seem to complain that they are not meaningful in a quantum theory.\n> Your criticism is actually a criticism of (the very framework of) quantum\n> theory, not a specific criticism of string theory. The only problem with\n> this criticism is that quantum theory is proved to be a feature of\n> reality.\n\nYou are missing a very simple point though. My own existence, your\nexistence and all of our experiences are things occuring in finite\nspace and through finite times. In previous models of reality such\nas general relativity and QFT it is possible, at least in principle,\nto derive these experiences from the theory. However, it appears to\nme no way is known to do this (even in principle) in string theory.\nIf that is impossible string theory is meaningless as a model of\nreality. Therefore, the things I\'m not about are not things that\n"can be done in classical physics", they are things that have to be\npossible to do in any meaningful model of reality.\n\n> If one uses a non-stringy framework, one must be very careful to avoid all\n> these potential "quantities" that must be non-physical. String theory does\n> it automatically, and it only allows us to calculate physically meaningful\n> observables such as the S-matrix or the boundary correlators.\n\nYou call them "physically meaningful" but I don\'t know any experimentalist\nwho knows how to travel infinite distances and measure the boundary\ncorrelators, not speaking of living from time -infinity to time +infinity\nand controlling the coupling constant in the process.\n\n> How can you claim that there is "definitely" a problem even if you\'re\n> unable to formulate what the problem could be? ;-)\n\nI am able to formulate the _problem_, but not the _solution_.\n\n> The proper length of a line starting from the AdS boundary is infinite\n> even in the classical theory.\n\nI know, I indicated in a recent reposting of this post how to solve\nthis problem. You have to choose a certain offset for the proper\nlength in the corresponding geodesic for genuine AdS and then match\nyour parameter with the AdS one in the asymptotics.\n\n> Try it. I won\'t try it because these observables do not seem well\n> motivated and justified to me.\n\nI think I explained the motivation, though possibly\nmy suggested solution is not a good one.\n\n> Turning off the interactions is just a trick to define the asymptotic\n> scattering states as the adiabatic continuation of the corresponding\n> states at g=0 (even though they exist, as slightly different states, for\n> every value of "g", as long as the particle is stable), and it is equally\n> meaningful in string theory like in a perturbative quantum field theory.\n> Where do you exactly see any difference? In the fact that the stringy\n> coupling constant is the expectation value of the dilaton (exponentiated)?\n> Why would you believe that this could invalidate perturbative expansions?\n\nIn string theory the perturbative expansion is meaningful when\nthe spacetime you\'re expanding around is a solution of the\nclassical equations of motion (possibly with alpha\' corrections)\nand this makes perfect sense. However, I don\'t see how to\nconstruct a classical solution in which the dilaton goes to\n-infinity both in the past and the future but is finite at\nfinite times.\n\n> More generally, we can calculate the cutoff-dependence of these quantities\n> in effective field theory, and the fluctuations (as well as the mean\n> value) increases with your cutoff energy. Intuitively this result is\n> clear; if you imagine the spacetime geometry to be more foamy, the total\n> length will count all the local features of the foam, and it will grow.\n> Well, in Minkowski geometry, you can on the contrary always find a\n> *shorter* geodesic, but at any rate, the proper length computed from\n> "quantized general relativity" won\'t be finite.\n\nYet it is possible the e.v. of the scalar evaluated at the\nfluctuating point will be finite. Also, an alternative\nsuggestion: consider an n-dimensional spacelike subspace\nof the boundary. Define for any classical asymptotically\nAdS spacetime the minimal volume n+1-dimensional spacelike\nsubspace ending on the given one. You might also attempt\nto fix its toplogy. Now take the integral of some\ncovariantly defined n+1-form over this subspace.\n\nBest regards,\nSquark.\n\n{this post was submitted six times but\nfor some reason it still hasn\'t appeared. it is\nalso available on sci.physics.research}\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Lubos Motl <motl@feynman.harvard.edu> wrote in message
news:<20040607043417.S2574@mail.kolej.mff.cuni.cz>...
> Yes, that's the whole point of the AdS/CFT checks - that you can compute
> many things on both sides, and they agree for all quantities that are
> exactly calculable on both sides (and people have found many of them).
That is OK, however, it is still unclear to me in precisely in what sense
do you get SUGRA as the classical limit.
> You can use your
> favorite local solutions of classical SUGRA for "finite patches", and you
> can prove that these solutions are relevant even in string theory, in the
> limit \hbar=0 - at least if the red shifts are bounded and all distances
> you study are much longer than l_{planck}. But these classical solutions
> are not quantum exact observables.
What precisely is the meaning of the word "relevant" as you use it here?
> The difference that you are trying to picture as a difference between
> SUGRA and superstring theory is actually something completely different:
> you are talking about the things that can be done in classical physics,
> and you seem to complain that they are not meaningful in a quantum theory.
> Your criticism is actually a criticism of (the very framework of) quantum
> theory, not a specific criticism of string theory. The only problem with
> this criticism is that quantum theory is proved to be a feature of
> reality.
You are missing a very simple point though. My own existence, your
existence and all of our experiences are things occuring in finite
space and through finite times. In previous models of reality such
as general relativity and QFT it is possible, at least in principle,
to derive these experiences from the theory. However, it appears to
me no way is known to do this (even in principle) in string theory.
If that is impossible string theory is meaningless as a model of
reality. Therefore, the things I'm not about are not things that
"can be done in classical physics", they are things that have to be
possible to do in any meaningful model of reality.
> If one uses a non-stringy framework, one must be very careful to avoid all
> these potential "quantities" that must be non-physical. String theory does
> it automatically, and it only allows us to calculate physically meaningful
> observables such as the S-matrix or the boundary correlators.
You call them "physically meaningful" but I don't know any experimentalist
who knows how to travel infinite distances and measure the boundary
correlators, not speaking of living from time -infinity to time +infinity
and controlling the coupling constant in the process.
> How can you claim that there is "definitely" a problem even if you're
> unable to formulate what the problem could be? ;-)
I am able to formulate the _problem_, but not the _solution_.
> The proper length of a line starting from the AdS boundary is infinite
> even in the classical theory.
I know, I indicated in a recent reposting of this post how to solve
this problem. You have to choose a certain offset for the proper
length in the corresponding geodesic for genuine AdS and then match
your parameter with the AdS one in the asymptotics.
> Try it. I won't try it because these observables do not seem well
> motivated and justified to me.
I think I explained the motivation, though possibly
my suggested solution is not a good one.
> Turning off the interactions is just a trick to define the asymptotic
> scattering states as the adiabatic continuation of the corresponding
> states at g=0 (even though they exist, as slightly different states, for
> every value of "g", as long as the particle is stable), and it is equally
> meaningful in string theory like in a perturbative quantum field theory.
> Where do you exactly see any difference? In the fact that the stringy
> coupling constant is the expectation value of the dilaton (exponentiated)?
> Why would you believe that this could invalidate perturbative expansions?
In string theory the perturbative expansion is meaningful when
the spacetime you're expanding around is a solution of the
classical equations of motion (possibly with \alpha' corrections)
and this makes perfect sense. However, I don't see how to
construct a classical solution in which the dilaton goes to
-infinity both in the past and the future but is finite at
finite times.
> More generally, we can calculate the cutoff-dependence of these quantities
> in effective field theory, and the fluctuations (as well as the mean
> value) increases with your cutoff energy. Intuitively this result is
> clear; if you imagine the spacetime geometry to be more foamy, the total
> length will count all the local features of the foam, and it will grow.
> Well, in Minkowski geometry, you can on the contrary always find a
> *shorter* geodesic, but at any rate, the proper length computed from
> "quantized general relativity" won't be finite.
Yet it is possible the e.v. of the scalar evaluated at the
fluctuating point will be finite. Also, an alternative
suggestion: consider an n-dimensional spacelike subspace
of the boundary. Define for any classical asymptotically
AdS spacetime the minimal volume n+1-dimensional spacelike
subspace ending on the given one. You might also attempt
to fix its toplogy. Now take the integral of some
covariantly defined n+1-form over this subspace.
Best regards,
Squark.
{this post was submitted six times but
for some reason it still hasn't appeared. it is
also available on sci.physics.research}
Lubos Motl
Jun16-04, 07:07 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 7 Jun 2004, Squark wrote:\n\n> That is OK, however, it is still unclear to me in precisely in what sense\n> do you get SUGRA as the classical limit.\n\nIn the sense that the \\hbar\\to 0 limit of the stringy S-matrix (multiplied\nby a proper power of G_{Newton} etc.) is the same S-matrix that you would\nderive from a Taylor expansion of scattering of waves in classical\nsupergravity. This fact identifies the local classical limit of both\ntheories. Also, you can obtain the classical (10D) SUGRA equations as\nconsistency equations for the string theoretical background - from the\ncondition that the theory is conformal. (Well, this conformality is\nreplaced by something else nonperturbatively, and we don\'t know what this\nsomething exactly is; nevertheless we know a lot about physics.)\n\n> > You can use your\n> > favorite local solutions of classical SUGRA for "finite patches", and you\n> > can prove that these solutions are relevant even in string theory, ...\n>\n> What precisely is the meaning of the word "relevant" as you use it here?\n\nIt\'s just an informal usage of this English word - these classical\nsolutions are "worth looking at" because they describe, with the same\naccuracy/approximation, the classical backgrounds around which you may\nexpand string theory. Your questions are not specific to string theory.\nYou could ask the same question "how to you decide that a classical field\ntheory is the \\hbar=0 limit of a quantum field theory", and it is often an\ninteresting question (although for most known QFTs it\'s mostly trivial\nbecause we construct them from the classical starting point). The\ndifference is that string theory only agrees with the effective field\ntheories such as SUGRA at distances much greater than l_{string}.\n\n> > Your criticism is actually a criticism of (the very framework of) quantum\n> > theory, not a specific criticism of string theory. The only problem with\n> > this criticism is that quantum theory is proved to be a feature of\n> > reality.\n>\n> You are missing a very simple point though. My own existence, your\n> existence and all of our experiences are things occuring in finite\n> space and through finite times. In previous models of reality such\n> as general relativity and QFT it is possible, at least in principle,\n> to derive these experiences from the theory.\n\nI understand pretty well what you feel because it bothered me, too\n(actually not when I studied string theory, but already quantum field\ntheory, which already had a lot of "S-matrix is everything" ideology in\nit, too). It is possible to derive the experience in finite time from\nstring theory, too, but you must just live with the fact that such\nfinite-time calculations will always either require some gauge-fixing to\nchoose a priviliged frame (e.g. the light-cone gauge), or they will only\nbe approximate.\n\n> However, it appears to me no way is known to do this (even in\n> principle) in string theory.\n\nRight. This is a consequence of the general covariance (diffeomorphism\nlocal invariance) combined with the postulates of quantum theory. We can\nprobably rule out a candidate for a theory of quantum gravity that would\nbe able to define finite-volume gauge-invariant operators - such a theory\nwould be inconsistent.\n\n> If that is impossible string theory is meaningless as a model of\n> reality.\n\nNope. If that is impossible, and one can show that it is almost certainly\nimpossible, it just means that your way of thinking is incorrect which is\nnot as big deal as the far-reaching conclusions about string theory that\nyou would like to promote. In quantum physics, one computes the\nprobabilities of various histories - let me use the general decoherent\nhistories framework - and they are calculated from the matrix elements of\nproducts of various operators. It is simply incompatible with general\ndiffeomorphism (local) symmetry to construct a local or even\nfinite-volume (gauge-invariant) operator, in a quantum theory that\nincludes general covariance.\n\nGauge invariance (coordinate transformations) relate an operator at point\nP with operators at any other points, and therefore gauge-invariant\noperators necessarily depend on the basic operators in the whole space.\n\nAll microprocesses involving electrons in your body are perfectly well\napproximated by the Standard Model defined on a classical background\ngeometry. One can derive that the Standard Model on a classical\nbackground geometry is a valid approximation of string theory for these\npurposes (assuming that we pick a vacuum with the right Standard Model),\nbut one can also derive that in the full theory of quantum gravity, such\nan approximation cannot be exact, and the intuition that exact,\nfinite-volume gauge-invariant Lorentz-covariant quantities can be\ncalculated is misguided, too.\n\n> Therefore, the things I\'m not about are not things that\n> "can be done in classical physics", they are things that have to be\n> possible to do in any meaningful model of reality.\n\nIf someone wrote such things many decades ago, it would be perfectly fine.\nBut today such statements simply express that their author has not\nunderstood what quantum mechanics is all about. Talking about "phenomena\nat finite volume V" with a specific number "V" in a theory of quantum\ngravity is just a misunderstanding of the basic postulates of quantum\ntheory, simply because every physical observable - such as "V" - becomes\nan operator that does not commute with other operators.\n\nThe desire to calculate exact quantities of this sort has more or less\nexact counterparts in QED. Let me invent an example.\n\nIn QED, you could also "argue" that any meaningful model of reality allows\nyou to calculate the probability that an electron is in the region of\nspace where the electric field is smaller than X. Nevertheless, we know\nthat in QED such a question is meaningless at the quantum level.\nClassically, when we ignore the quantum fluctuations, it is a good\napproximation to imagine that that the electric field is a classical,\nc-number function of space and time.\n\nHowever once we take quantum mechanics into account - and we certainly\nwant to take quantum mechanics into account in a theory of *quantum*\ngravity, too - then it is just guaranteed that the electric field\nfluctuates. It is an operator whose values are not well-defined. If you\ntake the ultraviolet cutoff (distance) to zero, the fluctuations become\narbitrarily large (in both directions), and you will see that the\n(absolute value of the) electric field is greater than X almost\neverywhere.\n\nYour objection is an almost exact analogy of this example in the context\nof quantum gravity, there cannot exist any finite meaningful answer to\nyour questions either, and your lack of will to accept that your thinking\nis misguided just shows how strong some classical prejudices can be.\n\nYour self-confident statements like "Any meaningful theory of reality\nshould allow..." should be replaced by "A classical theory allows ...".\nBut classical theories are not correctly describing the real worlds, and\nyour conclusions derived from the classical intuition are incorrect, too.\n\n> You call them "physically meaningful" but I don\'t know any experimentalist\n> who knows how to travel infinite distances and measure the boundary\n> correlators, not speaking of living from time -infinity to time +infinity\n> and controlling the coupling constant in the process.\n\nThis is a very unreasonable statement because *all* experiments that\nhave ever been done on the Tevatron, for example, probed the S-matrix (or\nthings that can be derived from it) only. The distances how far the newly\ncreated electrons travel are infinite in any practical meaning of this\nword, all collissions that have ever been seen can be calculated from the\nS-matrix, and the corrections introduced by taking the "finiteness" of the\nTevatron into account (deviation from the S-matrix) are by tens of orders\nof magnitude smaller than any other real errors that affect their results.\n\nThe physicists take the finiteness of the Tevatron into account when they\nestimate where a muon or something else will decay - but this is already a\nsimple argument based on "gluing two infinite spaces" together.\n\nWhat you are trying to emphasize has virtually zero importance (it is\nessentially a vanishing contribution to the correct results), and\ntherefore it is morally wrong.\n\nIn condensed matter physics, for example, one often wants to study\nshort-time events and phenomena, and if you need it, you can use the\nlight-cone gauge to compute finite-time evolution, if you wish.\n\n> > How can you claim that there is "definitely" a problem even if you\'re\n> > unable to formulate what the problem could be? ;-)\n>\n> I am able to formulate the _problem_, but not the _solution_.\n\nI think it is just your invividual problem because you want others to\ncalculate an exact quantity that cannot exist.\n\n> > The proper length of a line starting from the AdS boundary is infinite\n> > even in the classical theory.\n>\n> I know, I indicated in a recent reposting of this post how to solve\n> this problem. You have to choose a certain offset for the proper\n> length in the corresponding geodesic for genuine AdS and then match\n> your parameter with the AdS one in the asymptotics.\n\nThis is only possible in the classical geometry approximation, I think.\n\n> In string theory the perturbative expansion is meaningful when\n> the spacetime you\'re expanding around is a solution of the\n> classical equations of motion (possibly with alpha\' corrections)\n> and this makes perfect sense. However, I don\'t see how to\n> construct a classical solution in which the dilaton goes to\n> -infinity both in the past and the future but is finite at\n> finite times.\n\nI don\'t know whether such solution exists or not - I see no reason why it\ncould not - but at any rate, it is certainly not necessary to have such a\nsolution in order to define the S-matrix. The framework in which the\ncoupling constant is turned off at both asymptotic limits is just a\npsychological trick, in no way it describes a configuration that must\nexist. (In field theory it violates the equations of motion, too, because\nthe "equations of motion" say that the coupling constant is a "constant".)\n\nThe real S-matrix is only obtained in the limit in which the coupling\nconstant changes infinitely slowly, and even if the solution you are\ncalling for does not exist, the S-matrix limit is precisely the limit in\nwhich such a configuration with a time-dependent dilaton becomes\narbitrarily close to a solution of equations of motion (the |LHS-RHS| of\nall equations of motion is bounded by a number epsilon that goes to zero).\n\n> Yet it is possible the e.v. of the scalar evaluated at the\n> fluctuating point will be finite.\n\nNo, it\'s not. If the UV (energy) cutoff is taken to infinity, the\nprobability that the scalar field at a given point P - without smearing it\nout or anything like that - belongs to a particular finite interval (a,b)\ngoes to zero. This sort of situation where you make an apparently\nincorrect conclusion is the reason why I humbly say that you might want to\nspend more time to learn quantum (field) theory in more detail.\n\n> Also, an alternative suggestion: consider an n-dimensional spacelike\n> subspace of the boundary. Define for any classical asymptotically AdS\n> spacetime the minimal volume n+1-dimensional spacelike subspace ending\n> on the given one. You might also attempt to fix its toplogy. Now take\n> the integral of some covariantly defined n+1-form over this subspace.\n\nInteresting context, but you should probably try to describe somewhat more\nexactly which forms in which theory you are talking about. If the\n(n+1)-form is exact, the integral will not depend on the bulk at all. If\nthe (n+1)-form is a field strength, the integral will be an integer more\nor less determined by the superselection sector (flux). Different forms\nhave differently behaving integrals and the explanation may differ between\nthe cases. Your proposal so far sounds too vague for me to say anything\nuseful about it, but if you clarify the details, it might be interesting\nto answer this question.\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n[reposted because the original was lost already three times]\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 Mon, 7 Jun 2004, Squark wrote:
> That is OK, however, it is still unclear to me in precisely in what sense
> do you get SUGRA as the classical limit.
In the sense that the \hbar\to limit of the stringy S-matrix (multiplied
by a proper power of G_{Newton} etc.) is the same S-matrix that you would
derive from a Taylor expansion of scattering of waves in classical
supergravity. This fact identifies the local classical limit of both
theories. Also, you can obtain the classical (10D) SUGRA equations as
consistency equations for the string theoretical background - from the
condition that the theory is conformal. (Well, this conformality is
replaced by something else nonperturbatively, and we don't know what this
something exactly is; nevertheless we know a lot about physics.)
> > You can use your
> > favorite local solutions of classical SUGRA for "finite patches", and you
> > can prove that these solutions are relevant even in string theory, ...
>
> What precisely is the meaning of the word "relevant" as you use it here?
It's just an informal usage of this English word - these classical
solutions are "worth looking at" because they describe, with the same
accuracy/approximation, the classical backgrounds around which you may
expand string theory. Your questions are not specific to string theory.
You could ask the same question "how to you decide that a classical field
theory is the \hbar=0 limit of a quantum field theory", and it is often an
interesting question (although for most known QFTs it's mostly trivial
because we construct them from the classical starting point). The
difference is that string theory only agrees with the effective field
theories such as SUGRA at distances much greater than l_{string}.
> > Your criticism is actually a criticism of (the very framework of) quantum
> > theory, not a specific criticism of string theory. The only problem with
> > this criticism is that quantum theory is proved to be a feature of
> > reality.
>
> You are missing a very simple point though. My own existence, your
> existence and all of our experiences are things occuring in finite
> space and through finite times. In previous models of reality such
> as general relativity and QFT it is possible, at least in principle,
> to derive these experiences from the theory.
I understand pretty well what you feel because it bothered me, too
(actually not when I studied string theory, but already quantum field
theory, which already had a lot of "S-matrix is everything" ideology in
it, too). It is possible to derive the experience in finite time from
string theory, too, but you must just live with the fact that such
finite-time calculations will always either require some gauge-fixing to
choose a priviliged frame (e.g. the light-cone gauge), or they will only
be approximate.
> However, it appears to me no way is known to do this (even in
> principle) in string theory.
Right. This is a consequence of the general covariance (diffeomorphism
local invariance) combined with the postulates of quantum theory. We can
probably rule out a candidate for a theory of quantum gravity that would
be able to define finite-volume gauge-invariant operators - such a theory
would be inconsistent.
> If that is impossible string theory is meaningless as a model of
> reality.
Nope. If that is impossible, and one can show that it is almost certainly
impossible, it just means that your way of thinking is incorrect which is
not as big deal as the far-reaching conclusions about string theory that
you would like to promote. In quantum physics, one computes the
probabilities of various histories - let me use the general decoherent
histories framework - and they are calculated from the matrix elements of
products of various operators. It is simply incompatible with general
diffeomorphism (local) symmetry to construct a local or even
finite-volume (gauge-invariant) operator, in a quantum theory that
includes general covariance.
Gauge invariance (coordinate transformations) relate an operator at point
P with operators at any other points, and therefore gauge-invariant
operators necessarily depend on the basic operators in the whole space.
All microprocesses involving electrons in your body are perfectly well
approximated by the Standard Model defined on a classical background
geometry. One can derive that the Standard Model on a classical
background geometry is a valid approximation of string theory for these
purposes (assuming that we pick a vacuum with the right Standard Model),
but one can also derive that in the full theory of quantum gravity, such
an approximation cannot be exact, and the intuition that exact,
finite-volume gauge-invariant Lorentz-covariant quantities can be
calculated is misguided, too.
> Therefore, the things I'm not about are not things that
> "can be done in classical physics", they are things that have to be
> possible to do in any meaningful model of reality.
If someone wrote such things many decades ago, it would be perfectly fine.
But today such statements simply express that their author has not
understood what quantum mechanics is all about. Talking about "phenomena
at finite volume V" with a specific number "V" in a theory of quantum
gravity is just a misunderstanding of the basic postulates of quantum
theory, simply because every physical observable - such as "V" - becomes
an operator that does not commute with other operators.
The desire to calculate exact quantities of this sort has more or less
exact counterparts in QED. Let me invent an example.
In QED, you could also "argue" that any meaningful model of reality allows
you to calculate the probability that an electron is in the region of
space where the electric field is smaller than X. Nevertheless, we know
that in QED such a question is meaningless at the quantum level.
Classically, when we ignore the quantum fluctuations, it is a good
approximation to imagine that that the electric field is a classical,
c-number function of space and time.
However once we take quantum mechanics into account - and we certainly
want to take quantum mechanics into account in a theory of *quantum*
gravity, too - then it is just guaranteed that the electric field
fluctuates. It is an operator whose values are not well-defined. If you
take the ultraviolet cutoff (distance) to zero, the fluctuations become
arbitrarily large (in both directions), and you will see that the
(absolute value of the) electric field is greater than X almost
everywhere.
Your objection is an almost exact analogy of this example in the context
of quantum gravity, there cannot exist any finite meaningful answer to
your questions either, and your lack of will to accept that your thinking
is misguided just shows how strong some classical prejudices can be.
Your self-confident statements like "Any meaningful theory of reality
should allow..." should be replaced by "A classical theory allows ...".
But classical theories are not correctly describing the real worlds, and
your conclusions derived from the classical intuition are incorrect, too.
> You call them "physically meaningful" but I don't know any experimentalist
> who knows how to travel infinite distances and measure the boundary
> correlators, not speaking of living from time -infinity to time +infinity
> and controlling the coupling constant in the process.
This is a very unreasonable statement because *all* experiments that
have ever been done on the Tevatron, for example, probed the S-matrix (or
things that can be derived from it) only. The distances how far the newly
created electrons travel are infinite in any practical meaning of this
word, all collissions that have ever been seen can be calculated from the
S-matrix, and the corrections introduced by taking the "finiteness" of the
Tevatron into account (deviation from the S-matrix) are by tens of orders
of magnitude smaller than any other real errors that affect their results.
The physicists take the finiteness of the Tevatron into account when they
estimate where a muon or something else will decay - but this is already a
simple argument based on "gluing two infinite spaces" together.
What you are trying to emphasize has virtually zero importance (it is
essentially a vanishing contribution to the correct results), and
therefore it is morally wrong.
In condensed matter physics, for example, one often wants to study
short-time events and phenomena, and if you need it, you can use the
light-cone gauge to compute finite-time evolution, if you wish.
> > How can you claim that there is "definitely" a problem even if you're
> > unable to formulate what the problem could be? ;-)
>
> I am able to formulate the _problem_, but not the _solution_.
I think it is just your invividual problem because you want others to
calculate an exact quantity that cannot exist.
> > The proper length of a line starting from the AdS boundary is infinite
> > even in the classical theory.
>
> I know, I indicated in a recent reposting of this post how to solve
> this problem. You have to choose a certain offset for the proper
> length in the corresponding geodesic for genuine AdS and then match
> your parameter with the AdS one in the asymptotics.
This is only possible in the classical geometry approximation, I think.
> In string theory the perturbative expansion is meaningful when
> the spacetime you're expanding around is a solution of the
> classical equations of motion (possibly with \alpha' corrections)
> and this makes perfect sense. However, I don't see how to
> construct a classical solution in which the dilaton goes to
> -infinity both in the past and the future but is finite at
> finite times.
I don't know whether such solution exists or not - I see no reason why it
could not - but at any rate, it is certainly not necessary to have such a
solution in order to define the S-matrix. The framework in which the
coupling constant is turned off at both asymptotic limits is just a
psychological trick, in no way it describes a configuration that must
exist. (In field theory it violates the equations of motion, too, because
the "equations of motion" say that the coupling constant is a "constant".)
The real S-matrix is only obtained in the limit in which the coupling
constant changes infinitely slowly, and even if the solution you are
calling for does not exist, the S-matrix limit is precisely the limit in
which such a configuration with a time-dependent dilaton becomes
arbitrarily close to a solution of equations of motion (the |LHS-RHS| of
all equations of motion is bounded by a number \epsilon that goes to zero).
> Yet it is possible the e.v. of the scalar evaluated at the
> fluctuating point will be finite.
No, it's not. If the UV (energy) cutoff is taken to infinity, the
probability that the scalar field at a given point P - without smearing it
out or anything like that - belongs to a particular finite interval (a,b)
goes to zero. This sort of situation where you make an apparently
incorrect conclusion is the reason why I humbly say that you might want to
spend more time to learn quantum (field) theory in more detail.
> Also, an alternative suggestion: consider an n-dimensional spacelike
> subspace of the boundary. Define for any classical asymptotically AdS
> spacetime the minimal volume n+1-dimensional spacelike subspace ending
> on the given one. You might also attempt to fix its toplogy. Now take
> the integral of some covariantly defined n+1-form over this subspace.
Interesting context, but you should probably try to describe somewhat more
exactly which forms in which theory you are talking about. If the
(n+1)-form is exact, the integral will not depend on the bulk at all. If
the (n+1)-form is a field strength, the integral will be an integer more
or less determined by the superselection sector (flux). Different forms
have differently behaving integrals and the explanation may differ between
the cases. Your proposal so far sounds too vague for me to say anything
useful about it, but if you clarify the details, it might be interesting
to answer this question.
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
[reposted because the original was lost already three times]
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<20040616130507.S41977@mail.kolej.mff.cuni.cz >...\n\n> In the sense that the \\hbar\\to 0 limit of the stringy S-matrix (multiplied\n> by a proper power of G_{Newton} etc.) is the same S-matrix that you would\n> derive from a Taylor expansion of scattering of waves in classical\n> supergravity.\n\nIt is not clear to me how you define the S-matrix in classical\nsupergravity. Classical supergravity has no Fock space or\nanything of the sort. Maybe there are particular amplitudes\n(such as gravitational scattering of a pair of particles) for\nwhich classical cross-sections may be defined, but it\'s\nunclear how to define proccesses like pair production and\nannihilation classically. Also, it doesn\'t quite show the\nclassical limit is SUGRA since SUGRA contains observables\nother than the S-matrix.\n\n> This fact identifies the local classical limit of both\n> theories.\n\nThis sentence implies you are thinking about the "naive"\nQFT quantization SUGRA as the "second theory", is it so?\n\n> Also, you can obtain the classical (10D) SUGRA equations as\n> consistency equations for the string theoretical background - from the\n> condition that the theory is conformal. (Well, this conformality is\n> replaced by something else nonperturbatively, and we don\'t know what this\n> something exactly is; nevertheless we know a lot about physics.)\n\nOK, but what meaning do you assign to this? You are merely\ntalking about pertubation theory here. Possibly you can do\nbetter if you assing a state in the non-perturbative theory\n(using AdS/CFT or matrix theory) to the background (that\nwould be some kind of a coherent state, I guess). However,\nI\'m not quite sure what meaning (if any) does the state\nspace of matrix theory have. The only observable is\nsupposed to be the S-matrix and the S-matrix is defined on\nthe free theory state space (Fock space).\n\n> It\'s just an informal usage of this English word - these classical\n> solutions are "worth looking at" because they describe, with the same\n> accuracy/approximation, the classical backgrounds around which you may\n> expand string theory.\n\nSo what? The observables that we use in classical and\nsemiclassical (QFT on fixed backgroudn) theory might\nbe approximate but they are tightly related to real\nobservables which are the things we measure when we\nthink we measure the approximate observables. Therefore\nyou have to show how does string theory give rise to\nthese real observables. And it\'s far from obvious how\nto show this using the fact the classical backgrounds\ncan be used to build string theory expansions.\n\n> I understand pretty well what you feel because it bothered me, too\n> (actually not when I studied string theory, but already quantum field\n> theory, which already had a lot of "S-matrix is everything" ideology in\n> it, too).\n\nHowever, QFT contains observables beyond the S-matrix.\n\n> It is possible to derive the experience in finite time from\n> string theory, too, but you must just live with the fact that such\n> finite-time calculations will always either require some gauge-fixing to\n> choose a priviliged frame (e.g. the light-cone gauge), or they will only\n> be approximate.\n\nWhat exactly do you mean by gauge fixing? If you have a way\nto gauge-fix diffeomorphism invariance and define observables\nlocalized with respect to the coordinates after gauge fixing\nthen\n\nA) It\'s very interesting and I\'d like to know how you do that.\n\nB) It contradicts your claim local gauge invariant observables\ncannot be defined.\n\nC) It contradicts the claim the S-matrix is the only observable.\n\n> Right. This is a consequence of the general covariance (diffeomorphism\n> local invariance) combined with the postulates of quantum theory. We can\n> probably rule out a candidate for a theory of quantum gravity that would\n> be able to define finite-volume gauge-invariant operators - such a theory\n> would be inconsistent.\n\nI\'m not so sure. If the finite-volume operators are localized\nusing a state-dependent procedure ("the trace of the energy\nmomentum tensor near the guy in the brown coat") it\'s not\nobvious it can\'t be defined.\n\n> All microprocesses involving electrons in your body are perfectly well\n> approximated by the Standard Model defined on a classical background\n> geometry. One can derive that the Standard Model on a classical\n> background geometry is a valid approximation of string theory for these\n> purposes (assuming that we pick a vacuum with the right Standard Model),\n\nIt is again unclear in what sense it is an approximation. Let me\ndemonstrate. Assume I want to compute what my neighbour will eat\nfrom breakfast tomorrow. If the world is described by a QFT, I\ncan in principle imagine how to do this.\n\nA) Idenitify a sufficient amount of (local) observables.\n\nB) Find the state of the universe (locally) using\nmeasurement of the observables in a sufficient volume\naround yourself.\n\nC) Define "neighbour", "breakfast" etc. in terms of your\nobservables.\n\nD) Compute the state evolution until tomorrow morning\n(probably using some assumptions about the state of the\nentire universe).\n\nE) Decompose your state in terms of "breakfast" eigenvalues\nand find the probabilities for the various breakfasts.\n\nHow would you do that in string theory?\n\n> The desire to calculate exact quantities of this sort has more or less\n> exact counterparts in QED. Let me invent an example.\n>\n> In QED, you could also "argue" that any meaningful model of reality allows\n> you to calculate the probability that an electron is in the region of\n> space where the electric field is smaller than X. Nevertheless, we know\n> that in QED such a question is meaningless at the quantum level.\n> Classically, when we ignore the quantum fluctuations, it is a good\n> approximation to imagine that that the electric field is a classical,\n> c-number function of space and time.\n\nFirst of all your observable is problematic already on the classical\nlevel since you have to specify which electron you are intersted in.\nSecondly, if we ignore thar problem it probably makes sense in QED\nif you "smooth it" well enough (for instance take the convolution\nfo the electric field with some smooth kernel).\n\n> However once we take quantum mechanics into account - and we certainly\n> want to take quantum mechanics into account in a theory of *quantum*\n> gravity, too - then it is just guaranteed that the electric field\n> fluctuates. It is an operator whose values are not well-defined. If you\n> take the ultraviolet cutoff (distance) to zero, the fluctuations become\n> arbitrarily large (in both directions), and you will see that the\n> (absolute value of the) electric field is greater than X almost\n> everywhere.\n\nYes but "smoothened out" objects constructed out of the electric\nfield are meaningful and that\'s the thingies we measure in reality.\nWhat kind of "smoothing out" can you suggest in the quantum gravity\ncase?\n\n> This is a very unreasonable statement because *all* experiments that\n> have ever been done on the Tevatron, for example, probed the S-matrix (or\n> things that can be derived from it) only.\n\nNo, they probed the finite approximation to the S-matrix.\nIn QFT you can define it and show the exact S-matrix really\napproximates it extremely well. However you can\'t do the\nsame for string theory (as far as I know).\n\n> In condensed matter physics, for example, one often wants to study\n> short-time events and phenomena, and if you need it, you can use the\n> light-cone gauge to compute finite-time evolution, if you wish.\n\nSee my comments regarding the light-cone gauge above.\n\n> > I know, I indicated in a recent reposting of this post how to solve\n> > this problem. You have to choose a certain offset for the proper\n> > length in the corresponding geodesic for genuine AdS and then match\n> > your parameter with the AdS one in the asymptotics.\n>\n> This is only possible in the classical geometry approximation, I think.\n\nWhy are you so sure?\n\n> ...The framework in which the\n> coupling constant is turned off at both asymptotic limits is just a\n> psychological trick, in no way it describes a configuration that must\n> exist. (In field theory it violates the equations of motion, too, because\n> the "equations of motion" say that the coupling constant is a "constant".)\n\nNo, in QFT you can take the coupling constant to be any\nreasonable function of spacetime (as far as I know) while\nin string theory it\'s a field, and as you indicated\nyourself, the string expansion only makes sense around\nsolutions of the field equations of motion.\n\n> The real S-matrix is only obtained in the limit in which the coupling\n> constant changes infinitely slowly, and even if the solution you are\n> calling for does not exist, the S-matrix limit is precisely the limit in\n> which such a configuration with a time-dependent dilaton becomes\n> arbitrarily close to a solution of equations of motion (the |LHS-RHS| of\n> all equations of motion is bounded by a number epsilon that goes to zero).\n\nThis is already a much better argument. Still the S-matrix\nis defined as the limit of something that is not defined.\nThe fact it becomes "almost defined" in the limit we\'re taking\nis nice but has to be worked on to build a precise definition.\n\n> No, it\'s not. If the UV (energy) cutoff is taken to infinity, the\n> probability that the scalar field at a given point P - without smearing it\n> out or anything like that - belongs to a particular finite interval (a,b)\n> goes to zero.\n\nThis is nitpicking. You can smear it out using the\nsame "gauge fixed coordinates" I used to construct it.\n\n> This sort of situation where you make an apparently\n> incorrect conclusion is the reason why I humbly say that you might want to\n> spend more time to learn quantum (field) theory in more detail.\n\nThis sort of situation where you indicate a flaw in my\nargument the correction of which is obvious to both of\nus (so obvious I was too lazy to make the argument\nprecise in the first place) is the reason why I humbly\nsay this is hardly an honest method to avoid tough\nquestions ;-)\n\n> Interesting context, but you should probably try to describe somewhat more\n> exactly which forms in which theory you are talking about.\n\n1) Take your favorite RR or NS-NS form of field strength\nand mutliply it by your favorite scalar constructed out\nof the fields.\n\n2) Take the form with unit norm with respect to the induced\nmetric on the brane (pick a sign using the orientation) and\nmutliply it by your favorite scalar constructed out of the\nfields.\n\nBest regards,\nSquark.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<20040616130507.S41977@mail.kolej.mff.cuni.cz>...
> In the sense that the \hbar\to limit of the stringy S-matrix (multiplied
> by a proper power of G_{Newton} etc.) is the same S-matrix that you would
> derive from a Taylor expansion of scattering of waves in classical
> supergravity.
It is not clear to me how you define the S-matrix in classical
supergravity. Classical supergravity has no Fock space or
anything of the sort. Maybe there are particular amplitudes
(such as gravitational scattering of a pair of particles) for
which classical cross-sections may be defined, but it's
unclear how to define proccesses like pair production and
annihilation classically. Also, it doesn't quite show the
classical limit is SUGRA since SUGRA contains observables
other than the S-matrix.
> This fact identifies the local classical limit of both
> theories.
This sentence implies you are thinking about the "naive"
QFT quantization SUGRA as the "second theory", is it so?
> Also, you can obtain the classical (10D) SUGRA equations as
> consistency equations for the string theoretical background - from the
> condition that the theory is conformal. (Well, this conformality is
> replaced by something else nonperturbatively, and we don't know what this
> something exactly is; nevertheless we know a lot about physics.)
OK, but what meaning do you assign to this? You are merely
talking about pertubation theory here. Possibly you can do
better if you assing a state in the non-perturbative theory
(using AdS/CFT or matrix theory) to the background (that
would be some kind of a coherent state, I guess). However,
I'm not quite sure what meaning (if any) does the state
space of matrix theory have. The only observable is
supposed to be the S-matrix and the S-matrix is defined on
the free theory state space (Fock space).
> It's just an informal usage of this English word - these classical
> solutions are "worth looking at" because they describe, with the same
> accuracy/approximation, the classical backgrounds around which you may
> expand string theory.
So what? The observables that we use in classical and
semiclassical (QFT on fixed backgroudn) theory might
be approximate but they are tightly related to real
observables which are the things we measure when we
think we measure the approximate observables. Therefore
you have to show how does string theory give rise to
these real observables. And it's far from obvious how
to show this using the fact the classical backgrounds
can be used to build string theory expansions.
> I understand pretty well what you feel because it bothered me, too
> (actually not when I studied string theory, but already quantum field
> theory, which already had a lot of "S-matrix is everything" ideology in
> it, too).
However, QFT contains observables beyond the S-matrix.
> It is possible to derive the experience in finite time from
> string theory, too, but you must just live with the fact that such
> finite-time calculations will always either require some gauge-fixing to
> choose a priviliged frame (e.g. the light-cone gauge), or they will only
> be approximate.
What exactly do you mean by gauge fixing? If you have a way
to gauge-fix diffeomorphism invariance and define observables
localized with respect to the coordinates after gauge fixing
then
A) It's very interesting and I'd like to know how you do that.
B) It contradicts your claim local gauge invariant observables
cannot be defined.
C) It contradicts the claim the S-matrix is the only observable.
> Right. This is a consequence of the general covariance (diffeomorphism
> local invariance) combined with the postulates of quantum theory. We can
> probably rule out a candidate for a theory of quantum gravity that would
> be able to define finite-volume gauge-invariant operators - such a theory
> would be inconsistent.
I'm not so sure. If the finite-volume operators are localized
using a state-dependent procedure ("the trace of the energy
momentum tensor near the guy in the brown coat") it's not
obvious it can't be defined.
> All microprocesses involving electrons in your body are perfectly well
> approximated by the Standard Model defined on a classical background
> geometry. One can derive that the Standard Model on a classical
> background geometry is a valid approximation of string theory for these
> purposes (assuming that we pick a vacuum with the right Standard Model),
It is again unclear in what sense it is an approximation. Let me
demonstrate. Assume I want to compute what my neighbour will eat
from breakfast tomorrow. If the world is described by a QFT, I
can in principle imagine how to do this.
A) Idenitify a sufficient amount of (local) observables.
B) Find the state of the universe (locally) using
measurement of the observables in a sufficient volume
around yourself.
C) Define "neighbour", "breakfast" etc. in terms of your
observables.
D) Compute the state evolution until tomorrow morning
(probably using some assumptions about the state of the
entire universe).
E) Decompose your state in terms of "breakfast" eigenvalues
and find the probabilities for the various breakfasts.
How would you do that in string theory?
> The desire to calculate exact quantities of this sort has more or less
> exact counterparts in QED. Let me invent an example.
>
> In QED, you could also "argue" that any meaningful model of reality allows
> you to calculate the probability that an electron is in the region of
> space where the electric field is smaller than X. Nevertheless, we know
> that in QED such a question is meaningless at the quantum level.
> Classically, when we ignore the quantum fluctuations, it is a good
> approximation to imagine that that the electric field is a classical,
> c-number function of space and time.
First of all your observable is problematic already on the classical
level since you have to specify which electron you are intersted in.
Secondly, if we ignore thar problem it probably makes sense in QED
if you "smooth it" well enough (for instance take the convolution
fo the electric field with some smooth kernel).
> However once we take quantum mechanics into account - and we certainly
> want to take quantum mechanics into account in a theory of *quantum*
> gravity, too - then it is just guaranteed that the electric field
> fluctuates. It is an operator whose values are not well-defined. If you
> take the ultraviolet cutoff (distance) to zero, the fluctuations become
> arbitrarily large (in both directions), and you will see that the
> (absolute value of the) electric field is greater than X almost
> everywhere.
Yes but "smoothened out" objects constructed out of the electric
field are meaningful and that's the thingies we measure in reality.
What kind of "smoothing out" can you suggest in the quantum gravity
case?
> This is a very unreasonable statement because *all* experiments that
> have ever been done on the Tevatron, for example, probed the S-matrix (or
> things that can be derived from it) only.
No, they probed the finite approximation to the S-matrix.
In QFT you can define it and show the exact S-matrix really
approximates it extremely well. However you can't do the
same for string theory (as far as I know).
> In condensed matter physics, for example, one often wants to study
> short-time events and phenomena, and if you need it, you can use the
> light-cone gauge to compute finite-time evolution, if you wish.
See my comments regarding the light-cone gauge above.
> > I know, I indicated in a recent reposting of this post how to solve
> > this problem. You have to choose a certain offset for the proper
> > length in the corresponding geodesic for genuine AdS and then match
> > your parameter with the AdS one in the asymptotics.
>
> This is only possible in the classical geometry approximation, I think.
Why are you so sure?
> ...The framework in which the
> coupling constant is turned off at both asymptotic limits is just a
> psychological trick, in no way it describes a configuration that must
> exist. (In field theory it violates the equations of motion, too, because
> the "equations of motion" say that the coupling constant is a "constant".)
No, in QFT you can take the coupling constant to be any
reasonable function of spacetime (as far as I know) while
in string theory it's a field, and as you indicated
yourself, the string expansion only makes sense around
solutions of the field equations of motion.
> The real S-matrix is only obtained in the limit in which the coupling
> constant changes infinitely slowly, and even if the solution you are
> calling for does not exist, the S-matrix limit is precisely the limit in
> which such a configuration with a time-dependent dilaton becomes
> arbitrarily close to a solution of equations of motion (the |LHS-RHS| of
> all equations of motion is bounded by a number \epsilon that goes to zero).
This is already a much better argument. Still the S-matrix
is defined as the limit of something that is not defined.
The fact it becomes "almost defined" in the limit we're taking
is nice but has to be worked on to build a precise definition.
> No, it's not. If the UV (energy) cutoff is taken to infinity, the
> probability that the scalar field at a given point P - without smearing it
> out or anything like that - belongs to a particular finite interval (a,b)
> goes to zero.
This is nitpicking. You can smear it out using the
same "gauge fixed coordinates" I used to construct it.
> This sort of situation where you make an apparently
> incorrect conclusion is the reason why I humbly say that you might want to
> spend more time to learn quantum (field) theory in more detail.
This sort of situation where you indicate a flaw in my
argument the correction of which is obvious to both of
us (so obvious I was too lazy to make the argument
precise in the first place) is the reason why I humbly
say this is hardly an honest method to avoid tough
questions ;-)
> Interesting context, but you should probably try to describe somewhat more
> exactly which forms in which theory you are talking about.
1) Take your favorite RR or NS-NS form of field strength
and mutliply it by your favorite scalar constructed out
of the fields.
2) Take the form with unit norm with respect to the induced
metric on the brane (pick a sign using the orientation) and
mutliply it by your favorite scalar constructed out of the
fields.
Best regards,
Squark.
Lubos Motl
Jun17-04, 04:43 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, 17 Jun 2004, Squark wrote:\n\n> It is not clear to me how you define the S-matrix in classical\n> supergravity. Classical supergravity has no Fock space or\n> anything of the sort.\n\nWhile it\'s true that the interpretation of the S-matrix in terms of the\nHilbert space and quantum amplitudes is not relevant for a classical\ntheory, it is also true that all information encoded in the tree level\nS-matrix is completely isomorphic to all facts that you can derive about\nthe scattering of classical waves in the classical theory.\n\nThis is why the tree-level S-matrix "is" the classical theory, much like\nthe one-loop level "is" semiclassical physics.\n\nThe tree level S-matrix is nothing else than the Taylor expansion of the\nsolutions describing the scattering of classical waves - an expansion\naround the vanishing strength of these waves (around the background).\nThe classical, tree-level approximation of the S-matrix carries the same\ninformation as the space of classical solutions of the classical theory -\nclassical supergravity in this case. The fact that I am only talking about\nthe solutions - as opposed to general configurations - is what we label by\nthe term "on-shell".\n\n> Maybe there are particular amplitudes\n> (such as gravitational scattering of a pair of particles) for\n> which classical cross-sections may be defined, but it\'s\n> unclear how to define proccesses like pair production and\n> annihilation classically.\n\nThere are subtleties associated with the classical interpretation of the\nfermion fields - the fermion fields should really classically be set equal\nto zero (and they carry, in some sense, a square root of hbar) - and\ntherefore it is easier to look at the case of bosons. Be sure that the\npair production of W+ W- (from two photons) in the Standard Model is at\nthe tree-level approximation described by the same math as the classical\nfield theory involving the photon and W+- classical fields.\n\n> Also, it doesn\'t quite show the\n> classical limit is SUGRA since SUGRA contains observables\n> other than the S-matrix.\n\nI am not sure what you exactly mean by the phrase "contains observables".\nIt may contain some observables - like the metric at point with\ncoordinates (x,y,z,t) - but it does not allow you to calculate anything\nabout them except for the classical solutions and other data implied by\nthe classical solutions. In a classical theory, every configuration that\nis not a solution of the equations of motion is sort of unphysical. This\njustifies us to focus on the on-shell information only in the "analogous"\nquantum theory, too, and all such information is encoded in the S-matrix.\n\nIt is only *quantum* local field theory where we can consider off-shell\ninformation, for example the local Green\'s functions. It is only\nmeaningful to compute them because we can couple the local field theory to\narbitrary local sources - and measure their correlations and response\nfunctions etc. A necessary assumption to be able to consider such local\ncorrelators is that the theory *admits* fully localized probes. Quantum\ngravity - meaning string theory - does not allow arbitrarily small probes\nto be included consistently to the theory.\n\n> > This fact identifies the local classical limit of both\n> > theories.\n>\n> This sentence implies you are thinking about the "naive"\n> QFT quantization SUGRA as the "second theory", is it so?\n\nBased on the fact that I don\'t know what you mean by the "second theory"\nsuggests that the person among the two of us who has a naive viewpoint\nbased on the "second theory" is not me.\n\nYou probably meant "second quantized theory", did not you? Well, yes, if\nyou did, then what you\'re pretending that I am saying is true indeed, in\nthe limit that we discuss. Yes, in the same classical or semiclassical\nlimit, M-theory reduces to SUGRA which is nothing else than the\n"second-quantized" (formerly) classical supergravity.\n\nThe problem with your sentence is that you are directly talking about\nSUGRA, not M-theory. Sure, (quantum) SUGRA is nothing else than the\nprocedure of second quantization applied to the classical SUGRA theory. If\nsomeone thinks that it is something else, then she or he is wrong, and if\nshe or he also says that the correct answer is naive, then she or he is\nnot only wrong but also arrogant. ;-)\n\nM-theory is more than just SUGRA, but this "more" is not seen in the\nclassical limit of SUGRA. This "more" are purely quantum, or equivanently\nshort-distance, phenomena.\n\n> > Also, you can obtain the classical (10D) SUGRA equations as\n> > consistency equations for the string theoretical background - from the\n> > condition that the theory is conformal. (Well, this conformality is\n> > replaced by something else nonperturbatively, and we don\'t know what this\n> > something exactly is; nevertheless we know a lot about physics.)\n>\n> OK, but what meaning do you assign to this? You are merely\n> talking about pertubation theory here.\n\nIt\'s because "solving the equations of motion" and "being described by a\nconformal field theory" *is* only true at the level of perturbative string\ntheory.\n\n> Possibly you can do better if you assing a state in the\n> non-perturbative theory (using AdS/CFT or matrix theory) to the\n> background (that would be some kind of a coherent state, I guess).\n> However, I\'m not quite sure what meaning (if any) does the state space\n> of matrix theory have.\n\nM(atrix) theory is a quantum mechanical model, and therefore the Hilbert\nspace of states is nothing else than the space of possible complex-valued\nfunctions of the (bosonic and fermionic) elements of the matrices. It is\nconceptually as easy as quantum mechanics of the Hydrogen atom.\n\n> The only observable is supposed to be the S-matrix and the S-matrix is\n> defined on the free theory state space (Fock space).\n\nM(atrix) theory is a (discrete) light-cone gauge approach to SUGRA, and\ntherefore the general covariance (local symmetry) is fixed in a specific\nway - a way that breaks the Lorentz symmetry. Indeed, several boost\ngenerators are not manifest in M(atrix) theory. Consequently, M(atrix)\ntheory allows us to calculate not only the S-matrix, but also the\nevolution over finite intervals in the light-cone time x^+.\n\n> So what? The observables that we use in classical and\n> semiclassical (QFT on fixed backgroudn) theory might\n> be approximate but they are tightly related to real\n> observables which are the things we measure when we\n> think we measure the approximate observables. Therefore\n> you have to show how does string theory give rise to\n> these real observables. And it\'s far from obvious how\n> to show this using the fact the classical backgrounds\n> can be used to build string theory expansions.\n\nI\'ve explained both why string theory reduces to the appropriate classical\ndescriptions in the classical limit, and therefore why it agrees with\neverything that can be correctly derived from the classical theory; as\nwell as the reasons why the approximate questions present in a classical\ntheory have no exact generalization in the full quantum (string/M) theory;\nand it would be a waste of time to repeat it again, I think. You did not\nget it - no problem; I would guess that others did.\n\n> However, QFT contains observables beyond the S-matrix.\n\nLocal QFTs contain observables beyond the S-matrix because the local\nfields can be coupled to point-like, infinitely small, local probes. It is\nnot the case in quantum gravity. In quantum gravity - as exemplified in\nstring theory - it is not possible to consistently couple gravity to some\nnew completely local objects or fields, and therefore one cannot assume\nthat it will be possible to compute local Green\'s functions.\n\n> What exactly do you mean by gauge fixing?\n\nBy gauge fixing I mean the same thing as everyone else who knows what\ngauge fixing means. Your question should have been "what does it mean\ngauge fixing?". The answer is that gauge fixing means imposing additional\nconstraints on the fields (or other degrees of freedom) of your theory\nsuch that each class of configurations that are related by gauge\ntransformations contains at least one representative that satisfies the\nadditional constraints. If it contains exactly one such representative, we\ndeal with "complete gauge fixing".\n\n> If you have a way to gauge-fix diffeomorphism invariance and define\n> observables localized with respect to the coordinates after gauge\n> fixing then\n\nYes, for example it is the light cone gauge, as shown in M(atrix) theory.\n\n> A) It\'s very interesting and I\'d like to know how you do that.\n\nOn the string, light-cone gauge just means that X^+ is set to tau (the\nworldsheet coordinate) and X^- (another light-like direction) is\ncalculated in terms of the remaining D-2 "purely transverse" bosons\nX^i(sigma,tau). In spacetime it means that only the transverse components\nof various fields - such as the metric tensor fluctuation h_{ij} -\nappear, and the light-like Hamiltonian is written as a functional of\nthese components of the fields.\n\n> B) It contradicts your claim local gauge invariant observables\n> cannot be defined.\n\nMost of the times, I was careful to say that local gauge-invariant\n*covariant* observables cannot be defined. The local fields derived from\nthe light-cone gauge are not covariant (meaning Lorentz covariant).\nDepending on the exact definitions of the words, many people would argue\nthat the spacetime fields as seen in the light-cone gauge are not\ngauge-invariant either because they are expressed as functions of a\nspecific set of coordinates, including X^+ and X^-, and these coordinates\nare not gauge-invariant (invariant under coordinate transformations). Of\ncourse, one could say that we can construct "priviliged" coordinates X^+\nand X^- for any asymptotically-Minkowski background, and these coordinates\nX^+ and X^- are then therefore "gauge-invariant" observables, but it is a\nstretched interpretation. Of course, dogmatically speaking, gauge symmetry\nis *never* broken, and therefore any observable we express is "deeply"\ngauge-invariant.\n\nNevertheless, the common sense usage of the words "gauge-invariant" leads\nto the conclusion that a special choice of coordinates just *cannot* be\ngauge-invariant (under coordinate redefinitions).\n\n> C) It contradicts the claim the S-matrix is the only observable.\n\nOnce again, the S-matrix is the only Lorentz-covariant gauge-invariant\nobservable. Incidentally, it is the only information about the gravitons\nand quantum gravity that people were interested to compute - even in the\nlight cone gauge. Although the light-cone gauge allows us to evolve a\nstring field by a finite light-cone time X^+, the evolution from minus\ninfinity to infinity, which defines the S-matrix, is the only thing that\npeople really want (and can) compute well.\n\nThe idea that there is a large amount of interesting particle physics\ninformation in quantum gravity beyond the S-matrix is largely an illusion.\nIn condensed matter physics, one usually wants to compute properties of\nnon-relativistic, nearly static, configurations, and this implies other\nrules. But high-energy physics *is* about the existing states and their\nS-matrix.\n\n> > We can\n> > probably rule out a candidate for a theory of quantum gravity that would\n> > be able to define finite-volume gauge-invariant operators - such a theory\n> > would be inconsistent.\n>\n> I\'m not so sure. If the finite-volume operators are localized\n> using a state-dependent procedure ("the trace of the energy\n> momentum tensor near the guy in the brown coat") it\'s not\n> obvious it can\'t be defined.\n\nIt is naive to think that these sentences can make any exact sense in a\nquantum theory, and the fact that they make perfect sense to you is caused\nby your thinking merely in terms of the classical approximation. Yes, at\nthe macroscopic level, the laws of physics conspire in such a way that we\ncan see a cat and the tip of its tail has a pretty well-defined classical\nposition in a classical geometry - but at the quantum level, such\ninformation becomes meaningless or heavily modified.\n\n> It is again unclear in what sense it is an approximation.\n\nIt is approximation in the sense as every other approximation, something\nthat is not true exactly, that is only true (or equal to something else)\nif you put the operator "lim hbar -> 0" in this case on both sides of\nthis equation.\n\n> Let me demonstrate. Assume I want to compute what my neighbour will\n> eat from breakfast tomorrow. If the world is described by a QFT, I can\n> in principle imagine how to do this.\n>\n> A) Idenitify a sufficient amount of (local) observables.\n>\n> B) Find the state of the universe (locally) using\n> measurement of the observables in a sufficient volume\n> around yourself.\n\nIn a theory of gravity, it is meaningless to say "the Universe looks XY at\ntime t" because there is no universal definition of "t". Any well-behaved\nscalar function "t" on spacetime can be used as a time coordinate. OK, let\nme assume that you gauge-fix general covariance, e.g. by the light-cone\ngauge (or another fixing that effectively allows you to consider the\nspacetime geometry as fixed object). Then you can do both A and B, in a\nsense, regardless whether you are in string theory or another\n(hypothetical) theory of quantum gravity.\n\n> C) Define "neighbour", "breakfast" etc. in terms of your\n> observables.\n\nLight-cone gauge - or another gauge-fixed description - makes the quantum\ngravitational theory conceptually analogous to local quantum field theory,\nand you can in principle define the observables "the number of breakfasts\ncloser than 5 miles" analogously like in QFT without gravity. Once again,\nif you want to preserve the diffeomorphism invariance, you won\'t be able\nto localize breakfasts to a finite volume.\n\n> D) Compute the state evolution until tomorrow morning\n> (probably using some assumptions about the state of the\n> entire universe).\n\nWith diffeomorphism invariance unbroken, again, it makes no sense to talk\nabout the "evolution of a state by time t". Such a phrase can only make\nsense once you define a priviliged time coordinate - for example the\nlight-cone gauge X^+ - but then your description is not gauge-invariant.\nUnless the definition of "t" is derived from some properties of\n"important" objects, and you declare that you are allowed to use these\n"priviliged" objects, and still derive "gauge-invariant" quantities.\n\n> E) Decompose your state in terms of "breakfast" eigenvalues\n> and find the probabilities for the various breakfasts.\n\nRight.\n\n> How would you do that in string theory?\n\nI don\'t understand why you think that your pretty vague description has\nanything to do with having string theory or not. The problem of your\nconstruction is that there is no priviliged coordinate "t" that you could\nuse to measure time, no coordinates "x,y,z" to define your neighborhood,\nno "T" to evolve over a fixed time interval T, and all other things that\nyou mentioned. The fact that there is no priviliged "T" is a consequence\nof general covariance, and any theory that respects the basic rules of GR\n- not just string theory - will prevent you from choosing such coordinates\ncovariantly. On the other hand, once you allow the gauge freedom to be\nfixed, these questions and operators etc. will make essentially the same\nsense as in non-gravitational QFT - and again, it does not matter whether\nyou have string theory or something else.\n\nIn practice, what you said is something that we compute neither in string\ntheory nor in quantum field theory.\n\n> > The desire to calculate exact quantities of this sort has more or less\n> > exact counterparts in QED. Let me invent an example.\n> >\n> > In QED, you could also "argue" that any meaningful model of reality allows\n> > you to calculate the probability that an electron is in the region of\n> > space where the electric field is smaller than X. Nevertheless, we know\n> > that in QED such a question is meaningless at the quantum level.\n> > Classically, when we ignore the quantum fluctuations, it is a good\n> > approximation to imagine that that the electric field is a classical,\n> > c-number function of space and time.\n>\n> First of all your observable is problematic already on the classical\n> level since you have to specify which electron you are intersted in.\n\nOK, but this is an artificially invented new problem that has nothing to\ndo with the core of our discussion. I can consider a state where we know\nthat the number of electrons is one - and consider physics at distances\nlonger than the Compton wavelength, so that the virtual positron-electron\npairs are irrelevant. Nevertheless, the scheme will still be ill-defined\nbecause of the fluctuations of the electromagnetic field that appear at\n*any* scale.\n\nAt any rate, you are right that it is problematic to choose quantum\nobservables that fit some classical counterparts. Be sure that it becomes\nmuch much more problematic in a theory of quantum *gravity*, i.e. in a\ntheory with the diffeomorphism local symmetry.\n\n> Secondly, if we ignore thar problem it probably makes sense in QED\n> if you "smooth it" well enough (for instance take the convolution\n> fo the electric field with some smooth kernel).\n\nRight. This smoothing is again more or less equivalent to taking the\nclassical limit. Once again, string theory gives the correct classical\nlimit with everything that works, and physics beyond the classical limit\nis different. Your objections based on the statement that something\n(described by classical language) is missing are misguided and vacuous.\n\n> Yes but "smoothened out" objects constructed out of the electric\n> field are meaningful and that\'s the thingies we measure in reality.\n> What kind of "smoothing out" can you suggest in the quantum gravity\n> case?\n\nThis is exactly another part of the story that can never work in quantum\ngravity. In a non-gravitational theory, you can smear a field over a\nregion of size V, because you have a fixed background geometry and you\nknow what such an average means. In a theory of quantum gravity, the\nmetric tensor defining the background geometry is a quantum observable\nitself, and there is no way how to define the size of the region over\nwhich you should smear the field out. The fluctuations of the metric at\ndistances of order 1000 Planck lengths are always roughly identical - a\nsort of critical behavior - and the only invariant information about the\nmetric in a given region is the totally classical limit in which you\nassume that the metric is uniquely well-defined - something that is\npossible without errors only assuming hbar=0.\n\n> > This is a very unreasonable statement because *all* experiments that\n> > have ever been done on the Tevatron, for example, probed the S-matrix (or\n> > things that can be derived from it) only.\n>\n> No, they probed the finite approximation to the S-matrix.\n\nWhich experiments at the Tevatron that measure non-S-matrix (finite-time)\neffects do you exactly mean? Is it correct to conjecture that the reason\nwhy you wrote no example is that you also realize that what you are saying\nis pure nonsense? Well, people use a lot of non-S-matrix thinking to\ndevelop effective theories of the interior of the baryons, for example,\nbut at any rate, what you finally measure at the Tevatron is the S-matrix\nfor quarks and gluons.\n\nThe QCD effects take place at distances of order 10^{-15} meters, and the\nmaximal accuracy how can we deal with the jets etc. is never much smaller\nthan a millimeter. You would have to measure all these things with\nincredible accuracy in order for the available time to look "finite". The\nanomalous magnetic moment of an electron is measured pretty well, but it\nis again extractable from the S-matrix element coupling two electron\nfields and a (nearly) zero-momentum photon. We usually extract it from\nthe 3-point function, but because the photon in the relevant limit is very\nsoft, it becomes also arbitrarily close to being on-shell, and therefore\nit is encoded in the S-matrix, too.\n\n> In QFT you can define it and show the exact S-matrix really\n> approximates it extremely well. However you can\'t do the\n> same for string theory (as far as I know).\n\nAnd I\'ve explained many times why you cannot do such things in a\nconsistent quantum theory of gravity.\n\n> > > I know, I indicated in a recent reposting of this post how to solve\n> > > this problem. You have to choose a certain offset for the proper\n> > > length in the corresponding geodesic for genuine AdS and then match\n> > > your parameter with the AdS one in the asymptotics.\n> >\n> > This is only possible in the classical geometry approximation, I think.\n>\n> Why are you so sure?\n\nBecause I know that quantum mechanically, the metric fluctuates, and in\nfact, we can calculate how *much* it fluctuates. A calculation based on\nthese metric tensor degrees of freedom becomes increasingly ill-behaved as\nyou go to shorter distances, and at very short distances, the metric is\nnot the whole story and the idea that it is a good degree of freedom,\nisolated from others, breaks itself.\n\nAdS/CFT however gives you again a "preferred" choice of coordinates - but\nthe best way to extract the observables that depend on them is to use the\nconformal field theory side. Such calculations work, but they are never\ntrue "quantum gravity" calculations that would take the quantum mechanics\nfor the metric into account. It\'s another feature of string theory that\nwhile we can calculate the gravitons\' S-matrix at each order in\nperturbative expansion in g, none of them really probes the Planck scale.\nWe can only probe the string length distances - which is much longer than\nthe Planck length.\n\nOne needs the full non-perturbative string theory to probe the Planck\nscale.\n\nThe increasing oscillations at short distance scales is how quantum theory\nand quantum gravity works, and the remaining conclusions can be seen in\nstring theory, and the reason why I don\'t care about other approaches with\nfundamentally different conclusions about these basic questions is that no\nsuch convincing alternatives exist - at least no one has shown any.\n\n> > ...The framework in which the\n> > coupling constant is turned off at both asymptotic limits is just a\n> > psychological trick, in no way it describes a configuration that must\n> > exist. (In field theory it violates the equations of motion, too, because\n> > the "equations of motion" say that the coupling constant is a "constant".)\n>\n> No, in QFT you can take the coupling constant to be any\n> reasonable function of spacetime (as far as I know) while\n> in string theory it\'s a field, and as you indicated\n> yourself, the string expansion only makes sense around\n> solutions of the field equations of motion.\n\nThat\'s right - and it is one of the biggest advantages of string theory\nthat all of its "coupling constants" are subjects to dynamics - and\ntherefore one can either derive that they are exact moduli, which lead to\nnew long-range forces and variation - or they have potentially calculable\npotentials that uniquely determine their values.\n\nBut once again, treating the coupling as a general spacetime-dependent\nfunction is not what we usually want in most QFT calculations. We only\n*imagine* that the coupling goes (very slowly) to zero at |t|=infinity in\norder to calculate the perturbative S-matrix meaningfully (to define the\nasymptotic states by adiabatic continuations of the states at g=0) - but\nwhat we\'re really calculating is the scattering of real objects in a\nspacetime where the coupling constant is a positive constant in the whole\nspacetime.\n\nThis calculation (and the interpretation) of the real S-matrix goes\nconceptually unchanged in string theory, and one can also imagine that the\ncoupling constant goes slowly to zero at t=|infinity|. We want this\nevolution of the coupling be arbitrarily slow, because the S-matrix is\ntaken to count the scattering around *constant* values of the couplings,\nwhich is the same like saying that this evolution is arbitrarily close to\nsolve the stringy equations of motion.\n\n> This is already a much better argument. Still the S-matrix\n> is defined as the limit of something that is not defined.\n\nThat\'s correct. It\'s exactly the same statement as saying the the S-matrix\nis the only thing that makes sense, and I feel that you are slowly\nstarting to understand the origin of this statement. You may dislike\nvarious principles of physics, but it is the only thing you can do against\nthem.\n\n> The fact it becomes "almost defined" in the limit we\'re taking\n> is nice but has to be worked on to build a precise definition.\n\nWell, your concrete proposed procedure how the S-matrix should be defined\nis simply wrong in string theory. We certainly don\'t need any such wrong\nprocedures - involving backgrounds that don\'t solve the equations of\nmotion - neither to compute the stringy S-matrix, nor to interpret it or\ncheck it. It is just your proposal, and it is an incorrect proposal.\n\n> > No, it\'s not. If the UV (energy) cutoff is taken to infinity, the\n> > probability that the scalar field at a given point P - without smearing it\n> > out or anything like that - belongs to a particular finite interval (a,b)\n> > goes to zero.\n>\n> This is nitpicking. You can smear it out using the\n> same "gauge fixed coordinates" I used to construct it.\n\nYou cannot smear over "region of size V" in quantum gravity because the\n"region of size V" depends on the metric which is dynamical and its\nquantum fluctuations are really growing all the time as you increase the\nresolution. Yes, all such effects become unimportant in the classical\nlimit, but they are enough to invalidate your (and similar) procedures in\nthe full quantum theory.\n\n> 1) Take your favorite RR or NS-NS form of field strength\n> and mutliply it by your favorite scalar constructed out\n> of the fields.\n>\n> 2) Take the form with unit norm with respect to the induced\n> metric on the brane (pick a sign using the orientation) and\n> mutliply it by your favorite scalar constructed out of the\n> fields.\n\nIf you send the UV cutoff to infinity and includ the quantum effect, the\nobservable you are proposing - and define as an integral over a minimum\ncurve - will be zero because for any configuration that contributes to the\npath integral, you will find a spacelike subspace whose volume is\n(arbitrarily close to) zero, and the integrals of all things in your\ndefinitions will vanish.\n\nLet me say something more general. You might try to define various\nquantum observables as quantum generalizations of various classical\nobservables - but in most cases, you will fail. You will get either zero,\ninfinity, nonsense, or a gauge-variant object. There is no reason why this\nprocedure will succeed. The constraint is that the classical limit of a\nquantum theory should be what we want it to be - but it certainly does not\nmean that all notions that we are used to use in the classical theory will\nbe meaningful in the full quantum theory! Most of them are not, and the\nS-matrix is certainly a key, consistent and interesting observable in\nquantum gravity. There are not too many things like the S-matrix.\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: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\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, 17 Jun 2004, Squark wrote:
> It is not clear to me how you define the S-matrix in classical
> supergravity. Classical supergravity has no Fock space or
> anything of the sort.
While it's true that the interpretation of the S-matrix in terms of the
Hilbert space and quantum amplitudes is not relevant for a classical
theory, it is also true that all information encoded in the tree level
S-matrix is completely isomorphic to all facts that you can derive about
the scattering of classical waves in the classical theory.
This is why the tree-level S-matrix "is" the classical theory, much like
the one-loop level "is" semiclassical physics.
The tree level S-matrix is nothing else than the Taylor expansion of the
solutions describing the scattering of classical waves - an expansion
around the vanishing strength of these waves (around the background).
The classical, tree-level approximation of the S-matrix carries the same
information as the space of classical solutions of the classical theory -
classical supergravity in this case. The fact that I am only talking about
the solutions - as opposed to general configurations - is what we label by
the term "on-shell".
> Maybe there are particular amplitudes
> (such as gravitational scattering of a pair of particles) for
> which classical cross-sections may be defined, but it's
> unclear how to define proccesses like pair production and
> annihilation classically.
There are subtleties associated with the classical interpretation of the
fermion fields - the fermion fields should really classically be set equal
to zero (and they carry, in some sense, a square root of \hbar) - and
therefore it is easier to look at the case of bosons. Be sure that the
pair production of W+ W- (from two photons) in the Standard Model is at
the tree-level approximation described by the same math as the classical
field theory involving the photon and W+- classical fields.
> Also, it doesn't quite show the
> classical limit is SUGRA since SUGRA contains observables
> other than the S-matrix.
I am not sure what you exactly mean by the phrase "contains observables".
It may contain some observables - like the metric at point with
coordinates (x,y,z,t) - but it does not allow you to calculate anything
about them except for the classical solutions and other data implied by
the classical solutions. In a classical theory, every configuration that
is not a solution of the equations of motion is sort of unphysical. This
justifies us to focus on the on-shell information only in the "analogous"
quantum theory, too, and all such information is encoded in the S-matrix.
It is only *quantum* local field theory where we can consider off-shell
information, for example the local Green's functions. It is only
meaningful to compute them because we can couple the local field theory to
arbitrary local sources - and measure their correlations and response
functions etc. A necessary assumption to be able to consider such local
correlators is that the theory *admits* fully localized probes. Quantum
gravity - meaning string theory - does not allow arbitrarily small probes
to be included consistently to the theory.
> > This fact identifies the local classical limit of both
> > theories.
>
> This sentence implies you are thinking about the "naive"
> QFT quantization SUGRA as the "second theory", is it so?
Based on the fact that I don't know what you mean by the "second theory"
suggests that the person among the two of us who has a naive viewpoint
based on the "second theory" is not me.
You probably meant "second quantized theory", did not you? Well, yes, if
you did, then what you're pretending that I am saying is true indeed, in
the limit that we discuss. Yes, in the same classical or semiclassical
limit, M-theory reduces to SUGRA which is nothing else than the
"second-quantized" (formerly) classical supergravity.
The problem with your sentence is that you are directly talking about
SUGRA, not M-theory. Sure, (quantum) SUGRA is nothing else than the
procedure of second quantization applied to the classical SUGRA theory. If
someone thinks that it is something else, then she or he is wrong, and if
she or he also says that the correct answer is naive, then she or he is
not only wrong but also arrogant. ;-)
M-theory is more than just SUGRA, but this "more" is not seen in the
classical limit of SUGRA. This "more" are purely quantum, or equivanently
short-distance, phenomena.
> > Also, you can obtain the classical (10D) SUGRA equations as
> > consistency equations for the string theoretical background - from the
> > condition that the theory is conformal. (Well, this conformality is
> > replaced by something else nonperturbatively, and we don't know what this
> > something exactly is; nevertheless we know a lot about physics.)
>
> OK, but what meaning do you assign to this? You are merely
> talking about pertubation theory here.
It's because "solving the equations of motion" and "being described by a
conformal field theory" *is* only true at the level of perturbative string
theory.
> Possibly you can do better if you assing a state in the
> non-perturbative theory (using AdS/CFT or matrix theory) to the
> background (that would be some kind of a coherent state, I guess).
> However, I'm not quite sure what meaning (if any) does the state space
> of matrix theory have.
M(atrix) theory is a quantum mechanical model, and therefore the Hilbert
space of states is nothing else than the space of possible complex-valued
functions of the (bosonic and fermionic) elements of the matrices. It is
conceptually as easy as quantum mechanics of the Hydrogen atom.
> The only observable is supposed to be the S-matrix and the S-matrix is
> defined on the free theory state space (Fock space).
M(atrix) theory is a (discrete) light-cone gauge approach to SUGRA, and
therefore the general covariance (local symmetry) is fixed in a specific
way - a way that breaks the Lorentz symmetry. Indeed, several boost
generators are not manifest in M(atrix) theory. Consequently, M(atrix)
theory allows us to calculate not only the S-matrix, but also the
evolution over finite intervals in the light-cone time x^+.
> So what? The observables that we use in classical and
> semiclassical (QFT on fixed backgroudn) theory might
> be approximate but they are tightly related to real
> observables which are the things we measure when we
> think we measure the approximate observables. Therefore
> you have to show how does string theory give rise to
> these real observables. And it's far from obvious how
> to show this using the fact the classical backgrounds
> can be used to build string theory expansions.
I've explained both why string theory reduces to the appropriate classical
descriptions in the classical limit, and therefore why it agrees with
everything that can be correctly derived from the classical theory; as
well as the reasons why the approximate questions present in a classical
theory have no exact generalization in the full quantum (string/M) theory;
and it would be a waste of time to repeat it again, I think. You did not
get it - no problem; I would guess that others did.
> However, QFT contains observables beyond the S-matrix.
Local QFTs contain observables beyond the S-matrix because the local
fields can be coupled to point-like, infinitely small, local probes. It is
not the case in quantum gravity. In quantum gravity - as exemplified in
string theory - it is not possible to consistently couple gravity to some
new completely local objects or fields, and therefore one cannot assume
that it will be possible to compute local Green's functions.
> What exactly do you mean by gauge fixing?
By gauge fixing I mean the same thing as everyone else who knows what
gauge fixing means. Your question should have been "what does it mean
gauge fixing?". The answer is that gauge fixing means imposing additional
constraints on the fields (or other degrees of freedom) of your theory
such that each class of configurations that are related by gauge
transformations contains at least one representative that satisfies the
additional constraints. If it contains exactly one such representative, we
deal with "complete gauge fixing".
> If you have a way to gauge-fix diffeomorphism invariance and define
> observables localized with respect to the coordinates after gauge
> fixing then
Yes, for example it is the light cone gauge, as shown in M(atrix) theory.
> A) It's very interesting and I'd like to know how you do that.
On the string, light-cone gauge just means that X^+ is set to \tau (the
worldsheet coordinate) and X^- (another light-like direction) is
calculated in terms of the remaining D-2 "purely transverse" bosons
X^i(\sigma,\tau). In spacetime it means that only the transverse components
of various fields - such as the metric tensor fluctuation h_{ij} -
appear, and the light-like Hamiltonian is written as a functional of
these components of the fields.
> B) It contradicts your claim local gauge invariant observables
> cannot be defined.
Most of the times, I was careful to say that local gauge-invariant
*covariant* observables cannot be defined. The local fields derived from
the light-cone gauge are not covariant (meaning Lorentz covariant).
Depending on the exact definitions of the words, many people would argue
that the spacetime fields as seen in the light-cone gauge are not
gauge-invariant either because they are expressed as functions of a
specific set of coordinates, including X^+ and X^-, and these coordinates
are not gauge-invariant (invariant under coordinate transformations). Of
course, one could say that we can construct "priviliged" coordinates X^+
and X^- for any asymptotically-Minkowski background, and these coordinates
X^+ and X^- are then therefore "gauge-invariant" observables, but it is a
stretched interpretation. Of course, dogmatically speaking, gauge symmetry
is *never* broken, and therefore any observable we express is "deeply"
gauge-invariant.
Nevertheless, the common sense usage of the words "gauge-invariant" leads
to the conclusion that a special choice of coordinates just *cannot* be
gauge-invariant (under coordinate redefinitions).
> C) It contradicts the claim the S-matrix is the only observable.
Once again, the S-matrix is the only Lorentz-covariant gauge-invariant
observable. Incidentally, it is the only information about the gravitons
and quantum gravity that people were interested to compute - even in the
light cone gauge. Although the light-cone gauge allows us to evolve a
string field by a finite light-cone time X^+, the evolution from minus
infinity to infinity, which defines the S-matrix, is the only thing that
people really want (and can) compute well.
The idea that there is a large amount of interesting particle physics
information in quantum gravity beyond the S-matrix is largely an illusion.
In condensed matter physics, one usually wants to compute properties of
non-relativistic, nearly static, configurations, and this implies other
rules. But high-energy physics *is* about the existing states and their
S-matrix.
> > We can
> > probably rule out a candidate for a theory of quantum gravity that would
> > be able to define finite-volume gauge-invariant operators - such a theory
> > would be inconsistent.
>
> I'm not so sure. If the finite-volume operators are localized
> using a state-dependent procedure ("the trace of the energy
> momentum tensor near the guy in the brown coat") it's not
> obvious it can't be defined.
It is naive to think that these sentences can make any exact sense in a
quantum theory, and the fact that they make perfect sense to you is caused
by your thinking merely in terms of the classical approximation. Yes, at
the macroscopic level, the laws of physics conspire in such a way that we
can see a cat and the tip of its tail has a pretty well-defined classical
position in a classical geometry - but at the quantum level, such
information becomes meaningless or heavily modified.
> It is again unclear in what sense it is an approximation.
It is approximation in the sense as every other approximation, something
that is not true exactly, that is only true (or equal to something else)
if you put the operator "lim \hbar -> " in this case on both sides of
this equation.
> Let me demonstrate. Assume I want to compute what my neighbour will
> eat from breakfast tomorrow. If the world is described by a QFT, I can
> in principle imagine how to do this.
>
> A) Idenitify a sufficient amount of (local) observables.
>
> B) Find the state of the universe (locally) using
> measurement of the observables in a sufficient volume
> around yourself.
In a theory of gravity, it is meaningless to say "the Universe looks XY at
time t" because there is no universal definition of "t". Any well-behaved
scalar function "t" on spacetime can be used as a time coordinate. OK, let
me assume that you gauge-fix general covariance, e.g. by the light-cone
gauge (or another fixing that effectively allows you to consider the
spacetime geometry as fixed object). Then you can do both A and B, in a
sense, regardless whether you are in string theory or another
(hypothetical) theory of quantum gravity.
> C) Define "neighbour", "breakfast" etc. in terms of your
> observables.
Light-cone gauge - or another gauge-fixed description - makes the quantum
gravitational theory conceptually analogous to local quantum field theory,
and you can in principle define the observables "the number of breakfasts
closer than 5 miles" analogously like in QFT without gravity. Once again,
if you want to preserve the diffeomorphism invariance, you won't be able
to localize breakfasts to a finite volume.
> D) Compute the state evolution until tomorrow morning
> (probably using some assumptions about the state of the
> entire universe).
With diffeomorphism invariance unbroken, again, it makes no sense to talk
about the "evolution of a state by time t". Such a phrase can only make
sense once you define a priviliged time coordinate - for example the
light-cone gauge X^+ - but then your description is not gauge-invariant.
Unless the definition of "t" is derived from some properties of
"important" objects, and you declare that you are allowed to use these
"priviliged" objects, and still derive "gauge-invariant" quantities.
> E) Decompose your state in terms of "breakfast" eigenvalues
> and find the probabilities for the various breakfasts.
Right.
> How would you do that in string theory?
I don't understand why you think that your pretty vague description has
anything to do with having string theory or not. The problem of your
construction is that there is no priviliged coordinate "t" that you could
use to measure time, no coordinates "x,y,z" to define your neighborhood,
no "T" to evolve over a fixed time interval T, and all other things that
you mentioned. The fact that there is no priviliged "T" is a consequence
of general covariance, and any theory that respects the basic rules of GR
- not just string theory - will prevent you from choosing such coordinates
covariantly. On the other hand, once you allow the gauge freedom to be
fixed, these questions and operators etc. will make essentially the same
sense as in non-gravitational QFT - and again, it does not matter whether
you have string theory or something else.
In practice, what you said is something that we compute neither in string
theory nor in quantum field theory.
> > The desire to calculate exact quantities of this sort has more or less
> > exact counterparts in QED. Let me invent an example.
> >
> > In QED, you could also "argue" that any meaningful model of reality allows
> > you to calculate the probability that an electron is in the region of
> > space where the electric field is smaller than X. Nevertheless, we know
> > that in QED such a question is meaningless at the quantum level.
> > Classically, when we ignore the quantum fluctuations, it is a good
> > approximation to imagine that that the electric field is a classical,
> > c-number function of space and time.
>
> First of all your observable is problematic already on the classical
> level since you have to specify which electron you are intersted in.
OK, but this is an artificially invented new problem that has nothing to
do with the core of our discussion. I can consider a state where we know
that the number of electrons is one - and consider physics at distances
longer than the Compton wavelength, so that the virtual positron-electron
pairs are irrelevant. Nevertheless, the scheme will still be ill-defined
because of the fluctuations of the electromagnetic field that appear at
*any* scale.
At any rate, you are right that it is problematic to choose quantum
observables that fit some classical counterparts. Be sure that it becomes
much much more problematic in a theory of quantum *gravity*, i.e. in a
theory with the diffeomorphism local symmetry.
> Secondly, if we ignore thar problem it probably makes sense in QED
> if you "smooth it" well enough (for instance take the convolution
> fo the electric field with some smooth kernel).
Right. This smoothing is again more or less equivalent to taking the
classical limit. Once again, string theory gives the correct classical
limit with everything that works, and physics beyond the classical limit
is different. Your objections based on the statement that something
(described by classical language) is missing are misguided and vacuous.
> Yes but "smoothened out" objects constructed out of the electric
> field are meaningful and that's the thingies we measure in reality.
> What kind of "smoothing out" can you suggest in the quantum gravity
> case?
This is exactly another part of the story that can never work in quantum
gravity. In a non-gravitational theory, you can smear a field over a
region of size V, because you have a fixed background geometry and you
know what such an average means. In a theory of quantum gravity, the
metric tensor defining the background geometry is a quantum observable
itself, and there is no way how to define the size of the region over
which you should smear the field out. The fluctuations of the metric at
distances of order 1000 Planck lengths are always roughly identical - a
sort of critical behavior - and the only invariant information about the
metric in a given region is the totally classical limit in which you
assume that the metric is uniquely well-defined - something that is
possible without errors only assuming \hbar=0.
> > This is a very unreasonable statement because *all* experiments that
> > have ever been done on the Tevatron, for example, probed the S-matrix (or
> > things that can be derived from it) only.
>
> No, they probed the finite approximation to the S-matrix.
Which experiments at the Tevatron that measure non-S-matrix (finite-time)
effects do you exactly mean? Is it correct to conjecture that the reason
why you wrote no example is that you also realize that what you are saying
is pure nonsense? Well, people use a lot of non-S-matrix thinking to
develop effective theories of the interior of the baryons, for example,
but at any rate, what you finally measure at the Tevatron is the S-matrix
for quarks and gluons.
The QCD effects take place at distances of order 10^{-15} meters, and the
maximal accuracy how can we deal with the jets etc. is never much smaller
than a millimeter. You would have to measure all these things with
incredible accuracy in order for the available time to look "finite". The
anomalous magnetic moment of an electron is measured pretty well, but it
is again extractable from the S-matrix element coupling two electron
fields and a (nearly) zero-momentum photon. We usually extract it from
the 3-point function, but because the photon in the relevant limit is very
soft, it becomes also arbitrarily close to being on-shell, and therefore
it is encoded in the S-matrix, too.
> In QFT you can define it and show the exact S-matrix really
> approximates it extremely well. However you can't do the
> same for string theory (as far as I know).
And I've explained many times why you cannot do such things in a
consistent quantum theory of gravity.
> > > I know, I indicated in a recent reposting of this post how to solve
> > > this problem. You have to choose a certain offset for the proper
> > > length in the corresponding geodesic for genuine AdS and then match
> > > your parameter with the AdS one in the asymptotics.
> >
> > This is only possible in the classical geometry approximation, I think.
>
> Why are you so sure?
Because I know that quantum mechanically, the metric fluctuates, and in
fact, we can calculate how *much* it fluctuates. A calculation based on
these metric tensor degrees of freedom becomes increasingly ill-behaved as
you go to shorter distances, and at very short distances, the metric is
not the whole story and the idea that it is a good degree of freedom,
isolated from others, breaks itself.
AdS/CFT however gives you again a "preferred" choice of coordinates - but
the best way to extract the observables that depend on them is to use the
conformal field theory side. Such calculations work, but they are never
true "quantum gravity" calculations that would take the quantum mechanics
for the metric into account. It's another feature of string theory that
while we can calculate the gravitons' S-matrix at each order in
perturbative expansion in g, none of them really probes the Planck scale.
We can only probe the string length distances - which is much longer than
the Planck length.
One needs the full non-perturbative string theory to probe the Planck
scale.
The increasing oscillations at short distance scales is how quantum theory
and quantum gravity works, and the remaining conclusions can be seen in
string theory, and the reason why I don't care about other approaches with
fundamentally different conclusions about these basic questions is that no
such convincing alternatives exist - at least no one has shown any.
> > ...The framework in which the
> > coupling constant is turned off at both asymptotic limits is just a
> > psychological trick, in no way it describes a configuration that must
> > exist. (In field theory it violates the equations of motion, too, because
> > the "equations of motion" say that the coupling constant is a "constant".)
>
> No, in QFT you can take the coupling constant to be any
> reasonable function of spacetime (as far as I know) while
> in string theory it's a field, and as you indicated
> yourself, the string expansion only makes sense around
> solutions of the field equations of motion.
That's right - and it is one of the biggest advantages of string theory
that all of its "coupling constants" are subjects to dynamics - and
therefore one can either derive that they are exact moduli, which lead to
new long-range forces and variation - or they have potentially calculable
potentials that uniquely determine their values.
But once again, treating the coupling as a general spacetime-dependent
function is not what we usually want in most QFT calculations. We only
*imagine* that the coupling goes (very slowly) to zero at |t|=infinity in
order to calculate the perturbative S-matrix meaningfully (to define the
asymptotic states by adiabatic continuations of the states at g=0) - but
what we're really calculating is the scattering of real objects in a
spacetime where the coupling constant is a positive constant in the whole
spacetime.
This calculation (and the interpretation) of the real S-matrix goes
conceptually unchanged in string theory, and one can also imagine that the
coupling constant goes slowly to zero at t=|infinity|. We want this
evolution of the coupling be arbitrarily slow, because the S-matrix is
taken to count the scattering around *constant* values of the couplings,
which is the same like saying that this evolution is arbitrarily close to
solve the stringy equations of motion.
> This is already a much better argument. Still the S-matrix
> is defined as the limit of something that is not defined.
That's correct. It's exactly the same statement as saying the the S-matrix
is the only thing that makes sense, and I feel that you are slowly
starting to understand the origin of this statement. You may dislike
various principles of physics, but it is the only thing you can do against
them.
> The fact it becomes "almost defined" in the limit we're taking
> is nice but has to be worked on to build a precise definition.
Well, your concrete proposed procedure how the S-matrix should be defined
is simply wrong in string theory. We certainly don't need any such wrong
procedures - involving backgrounds that don't solve the equations of
motion - neither to compute the stringy S-matrix, nor to interpret it or
check it. It is just your proposal, and it is an incorrect proposal.
> > No, it's not. If the UV (energy) cutoff is taken to infinity, the
> > probability that the scalar field at a given point P - without smearing it
> > out or anything like that - belongs to a particular finite interval (a,b)
> > goes to zero.
>
> This is nitpicking. You can smear it out using the
> same "gauge fixed coordinates" I used to construct it.
You cannot smear over "region of size V" in quantum gravity because the
"region of size V" depends on the metric which is dynamical and its
quantum fluctuations are really growing all the time as you increase the
resolution. Yes, all such effects become unimportant in the classical
limit, but they are enough to invalidate your (and similar) procedures in
the full quantum theory.
> 1) Take your favorite RR or NS-NS form of field strength
> and mutliply it by your favorite scalar constructed out
> of the fields.
>
> 2) Take the form with unit norm with respect to the induced
> metric on the brane (pick a sign using the orientation) and
> mutliply it by your favorite scalar constructed out of the
> fields.
If you send the UV cutoff to infinity and includ the quantum effect, the
observable you are proposing - and define as an integral over a minimum
curve - will be zero because for any configuration that contributes to the
path integral, you will find a spacelike subspace whose volume is
(arbitrarily close to) zero, and the integrals of all things in your
definitions will vanish.
Let me say something more general. You might try to define various
quantum observables as quantum generalizations of various classical
observables - but in most cases, you will fail. You will get either zero,
infinity, nonsense, or a gauge-variant object. There is no reason why this
procedure will succeed. The constraint is that the classical limit of a
quantum theory should be what we want it to be - but it certainly does not
mean that all notions that we are used to use in the classical theory will
be meaningful in the full quantum theory! Most of them are not, and the
S-matrix is certainly a key, consistent and interesting observable in
quantum gravity. There are not too many things like the S-matrix.
Cheers,
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0406171443330.4202-100000@feynman.harvard.edu>...\n\n> The tree level S-matrix is nothing else than the Taylor expansion of the\n> solutions describing the scattering of classical waves - an expansion\n> around the vanishing strength of these waves (around the background).\n\nSo, you\'re saying that is I want to compute the scattering\nof classical waves, I have the imagine them as coherent\nstates in Fock space and act with the tree-level S-matrix?\nIs the result guaranteed to be a coherent state?\n\n> > Also, it doesn\'t quite show the\n> > classical limit is SUGRA since SUGRA contains observables\n> > other than the S-matrix.\n>\n> I am not sure what you exactly mean by the phrase "contains observables".\n> It may contain some observables - like the metric at point with\n> coordinates (x,y,z,t)\n\nThis is not an observable since it is not diffeomorphism\ninvariant. In classical SUGRA one can specify the initial\nconditions on a spacelike slice of finite volume and\npredict the patch inside the resulting spacetime which\nonly depends on that slice (i.e. such that any\nworld-point in it is connected to the slice with a casual\ncurve).\n\n> It is only *quantum* local field theory where we can consider off-shell\n> information, for example the local Green\'s functions. It is only\n> meaningful to compute them because we can couple the local field theory to\n> arbitrary local sources - and measure their correlations and response\n> functions etc.\n\nI.e. you\'re saying that the reason QFT contains\nobservables beyond the S-matrix whereas string\ntheory does not is that we can insert external\ncurrents into the QFT action and make the\nS-matrix depend on them?\n\n> > > This fact identifies the local classical limit of both\n> > > theories.\n> >\n> > This sentence implies you are thinking about the "naive"\n> > QFT quantization SUGRA as the "second theory", is it so?\n>\n> Based on the fact that I don\'t know what you mean by the "second theory"\n> suggests that the person among the two of us who has a naive viewpoint\n> based on the "second theory" is not me.\n\nThe phrase "second theory" is related to your phrase "both\ntheories". I assumed the first theory is superstring theory\nand tried to speculate on the nature of the second theory.\nThe purpose of the term "naive" is not the demonstration of\nsome arrogant standpoint, like you immediately concluded,\nbut the refence to the (divergent) QFT-fashion quantized\nSUGRA as opposed to the correct way to quantize gravity\n(which is conjectured to be superstring theory).\n\n> It\'s because "solving the equations of motion" and "being described by a\n> conformal field theory" *is* only true at the level of perturbative string\n> theory.\n\nAnd therefore it doesn\'t demonstrate superstring theory\nhas a SUGRA approximation since it doesn\'t show how\nto define SUGRA-like observables in the full\nsuperstring theory.\n\n> M(atrix) theory is a quantum mechanical model, and therefore the Hilbert\n> space of states is nothing else than the space of possible complex-valued\n> functions of the (bosonic and fermionic) elements of the matrices. It is\n> conceptually as easy as quantum mechanics of the Hydrogen atom.\n\nOr, if we compactify some dimensions, it is a QFT.\nThat much is obvious, however...\n\n> M(atrix) theory is a (discrete) light-cone gauge approach to SUGRA, and\n> therefore the general covariance (local symmetry) is fixed in a specific\n> way - a way that breaks the Lorentz symmetry. Indeed, several boost\n> generators are not manifest in M(atrix) theory. Consequently, M(atrix)\n> theory allows us to calculate not only the S-matrix, but also the\n> evolution over finite intervals in the light-cone time x^+.\n\nSo is the S-matrix the only observable or not?\n\n> I\'ve explained both why string theory reduces to the appropriate classical\n> descriptions in the classical limit, and therefore why it agrees with\n> everything that can be correctly derived from the classical theory; as\n> well as the reasons why the approximate questions present in a classical\n> theory have no exact generalization in the full quantum (string/M) theory;\n> and it would be a waste of time to repeat it again, I think. You did not\n> get it - no problem; I would guess that others did.\n\nThe problem is that you didn\'t get something else. If, as you claim,\nthese observables have no generalization in any sense in the full\ntheory it\'s not clear how can the full theory be a meaningful model\nof reality. The reason I\'m saying this is as follows: my perceptions,\nas a person, at a given moment of my life have to be an exactly (not\napproximately) defined observable of any meaningful model of reality.\nI hope you agree with me on that?! If the real world was classical\nSUGRA it would be conceivable that this observable could be defined\nin it. It is however not clear whether the same is true about\nsuperstring theory.\n\n> Local QFTs contain observables beyond the S-matrix because the local\n> fields can be coupled to point-like, infinitely small, local probes. It is\n> not the case in quantum gravity.\n\nHowever Matrix theory is a local QFT (just not local in spacetime),\ntherefore it should contain observables beyond the S-matrix. Or\nshouldn\'t it?\n\n> > If you have a way to gauge-fix diffeomorphism invariance and define\n> > observables localized with respect to the coordinates after gauge\n> > fixing then\n>\n> Yes, for example it is the light cone gauge, as shown in M(atrix) theory.\n>\n> > A) It\'s very interesting and I\'d like to know how you do that.\n>\n> On the string, light-cone gauge just means that X^+ is set to tau (the\n> worldsheet coordinate) and X^- (another light-like direction) is\n> calculated in terms of the remaining D-2 "purely transverse" bosons\n> X^i(sigma,tau). In spacetime it means that only the transverse components\n> of various fields - such as the metric tensor fluctuation h_{ij} -\n> appear, and the light-like Hamiltonian is written as a functional of\n> these components of the fields.\n\nAll of these things you said can be done in usual QFT\nand have nothing to do with gauge fixing diffeomorphism\ninvariance.\n\n> > B) It contradicts your claim local gauge invariant observables\n> > cannot be defined.\n>\n> Most of the times, I was careful to say that local gauge-invariant\n> *covariant* observables cannot be defined. The local fields derived from\n> the light-cone gauge are not covariant (meaning Lorentz covariant).\n\nAny scalar constructed out of these fields is Lorentz\ninvariant and therefore Lorentz covariant.\n\n> Depending on the exact definitions of the words, many people would argue\n> that the spacetime fields as seen in the light-cone gauge are not\n> gauge-invariant either because they are expressed as functions of a\n> specific set of coordinates, including X^+ and X^-, and these coordinates\n> are not gauge-invariant (invariant under coordinate transformations).\n\nIndeed I would argue that.\n\n> Of\n> course, one could say that we can construct "priviliged" coordinates X^+\n> and X^- for any asymptotically-Minkowski background, and these coordinates\n> X^+ and X^- are then therefore "gauge-invariant" observables,\n\nHow do you do that?\n\n> but it is a stretched interpretation.\n\nWhy is it stretched?\n\n> Nevertheless, the common sense usage of the words "gauge-invariant" leads\n> to the conclusion that a special choice of coordinates just *cannot* be\n> gauge-invariant (under coordinate redefinitions).\n\nIt can if it is "priviliged" as you say.\n\n> The idea that there is a large amount of interesting particle physics\n> information in quantum gravity beyond the S-matrix is largely an illusion.\n> In condensed matter physics, one usually wants to compute properties of\n> non-relativistic, nearly static, configurations, and this implies other\n> rules. But high-energy physics *is* about the existing states and their\n> S-matrix.\n\nHowever, superstring theory is supposed to be the ultimate\ntheory of everything. As such, it should contain not only\nthe results of particle accelerator experiments (to some\nreasonably good approximation) but everything that can\never be said about the universe!\n\n> > I\'m not so sure. If the finite-volume operators are localized\n> > using a state-dependent procedure ("the trace of the energy\n> > momentum tensor near the guy in the brown coat") it\'s not\n> > obvious it can\'t be defined.\n>\n> It is naive to think that these sentences can make any exact sense in a\n> quantum theory, and the fact that they make perfect sense to you is caused\n> by your thinking merely in terms of the classical approximation.\n\nBut you just said they do make exact sense by claiming there\nis a way to construct privileged coordinate and compute\neverything in terms of these coordinates in the full\nnonperturbative quantum theory!!\n\n> > Let me demonstrate. Assume I want to compute what my neighbour will\n> > eat from breakfast tomorrow. If the world is described by a QFT, I can\n> > in principle imagine how to do this.\n> >\n> > A) Idenitify a sufficient amount of (local) observables.\n> >\n> > B) Find the state of the universe (locally) using\n> > measurement of the observables in a sufficient volume\n> > around yourself.\n>\n> In a theory of gravity, it is meaningless to say "the Universe looks XY at\n> time t" because there is no universal definition of "t".\n\nYes, that\'s why I said "if the world is described by a QFT".\nOf course in reality it isn\'t (so I should really have said\n"if the world were described by a QFT).\n\n> In practice, what you said is something that we compute neither in string\n> theory nor in quantum field theory.\n\nWhy not in QFT?\n\n> > Yes but "smoothened out" objects constructed out of the electric\n> > field are meaningful and that\'s the thingies we measure in reality.\n> > What kind of "smoothing out" can you suggest in the quantum gravity\n> > case?\n>\n> This is exactly another part of the story that can never work in quantum\n> gravity.\n\nSo, your claim that my complaints can be addressed to QFT\nin the same fashion they can be adrressed to superstring\ntheory is plainly wrong.\n\n> Which experiments at the Tevatron that measure non-S-matrix (finite-time)\n> effects do you exactly mean?\n\nAll of them! Simply the finite-time effects are virtually\nzero in any practical situation. However, there\'s a\ndifference between saying that my theory can produce the\nexact result of the experiment but it can always be\napproximated by something called the S-matrix and saying\nthe exact result of the experiment is undefined in my\ntheory but the S-matrix is defined.\n\n> > > > I know, I indicated in a recent reposting of this post how to solve\n> > > > this problem. You have to choose a certain offset for the proper\n> > > > length in the corresponding geodesic for genuine AdS and then match\n> > > > your parameter with the AdS one in the asymptotics.\n> > >\n> > > This is only possible in the classical geometry approximation, I think.\n> >\n> > Why are you so sure?\n>\n> Because I know that quantum mechanically, the metric fluctuates, and in\n> fact, we can calculate how *much* it fluctuates.\n\nSo what?\n\n> A calculation based on\n> these metric tensor degrees of freedom becomes increasingly ill-behaved as\n> you go to shorter distances, and at very short distances, the metric is\n> not the whole story and the idea that it is a good degree of freedom,\n> isolated from others, breaks itself.\n\nOK, so give something else instead of the metric that works.\n\n> AdS/CFT however gives you again a "preferred" choice of coordinates\n\nHow does it do that?!\n\n> But once again, treating the coupling as a general spacetime-dependent\n> function is not what we usually want in most QFT calculations. We only\n> *imagine* that the coupling goes (very slowly) to zero at |t|=infinity in\n> order to calculate the perturbative S-matrix meaningfully (to define the\n> asymptotic states by adiabatic continuations of the states at g=0) - but\n> what we\'re really calculating is the scattering of real objects in a\n> spacetime where the coupling constant is a positive constant in the whole\n> spacetime.\n\nWhether we imagine that or not is a semantical game. The fact\nremains that we can\'t define the adiabatic continuation\nwithout allowing for the coupling constant to change. The\nfact the resulting object (the S-matrix) is a good\napproximation of real-world proccesses with constant coupling\nconstant is another matter. Even here though there are\nsubtleties. In the real world, the spectrum is different than\nin the free theory. We can consider scattering of bound\nstates and fundumental particles can be impossible to produce\ndue to confinement. Possibly we can compute real world\nscattering by plugging appropriate wavefunctions into the\nS-matrix, however, the computation of the wavefunctions\nthemselves goes beyond the S-matrix.\n\n> > This is already a much better argument. Still the S-matrix\n> > is defined as the limit of something that is not defined.\n>\n> That\'s correct.\n\nSo it\'s correct we define the only concept that leads to\ncomputable quantities in our theory through concepts that\nare meaningless?!\n\n> Well, your concrete proposed procedure how the S-matrix should be defined\n> is simply wrong in string theory. We certainly don\'t need any such wrong\n> procedures - involving backgrounds that don\'t solve the equations of\n> motion - neither to compute the stringy S-matrix, nor to interpret it or\n> check it. It is just your proposal, and it is an incorrect proposal.\n\nThen what is the correct proposal? Without the\ngradual changing of the coupling constant\nyou can\'t relate the real states to the free\ntheory states and the S-matrix has no relevance\nfor real world scattering.\n\n> You cannot smear over "region of size V" in quantum gravity because the\n> "region of size V" depends on the metric which is dynamical and its\n> quantum fluctuations are really growing all the time as you increase the\n> resolution.\n\nI don\'t see how the statement after "because" implies\nthe one before it.\n\n> If you send the UV cutoff to infinity and includ the quantum effect, the\n> observable you are proposing - and define as an integral over a minimum\n> curve - will be zero because for any configuration that contributes to the\n> path integral, you will find a spacelike subspace whose volume is\n> (arbitrarily close to) zero, and the integrals of all things in your\n> definitions will vanish.\n\nWhy is that?\n\nBest regards,\nSquark.\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> wrote in message news:<Pine.LNX.4.31.0406171443330.4202-100000@feynman.harvard.edu>...
> The tree level S-matrix is nothing else than the Taylor expansion of the
> solutions describing the scattering of classical waves - an expansion
> around the vanishing strength of these waves (around the background).
So, you're saying that is I want to compute the scattering
of classical waves, I have the imagine them as coherent
states in Fock space and act with the tree-level S-matrix?
Is the result guaranteed to be a coherent state?
> > Also, it doesn't quite show the
> > classical limit is SUGRA since SUGRA contains observables
> > other than the S-matrix.
>
> I am not sure what you exactly mean by the phrase "contains observables".
> It may contain some observables - like the metric at point with
> coordinates (x,y,z,t)
This is not an observable since it is not diffeomorphism
invariant. In classical SUGRA one can specify the initial
conditions on a spacelike slice of finite volume and
predict the patch inside the resulting spacetime which
only depends on that slice (i.e. such that any
world-point in it is connected to the slice with a casual
curve).
> It is only *quantum* local field theory where we can consider off-shell
> information, for example the local Green's functions. It is only
> meaningful to compute them because we can couple the local field theory to
> arbitrary local sources - and measure their correlations and response
> functions etc.
I.e. you're saying that the reason QFT contains
observables beyond the S-matrix whereas string
theory does not is that we can insert external
currents into the QFT action and make the
S-matrix depend on them?
> > > This fact identifies the local classical limit of both
> > > theories.
> >
> > This sentence implies you are thinking about the "naive"
> > QFT quantization SUGRA as the "second theory", is it so?
>
> Based on the fact that I don't know what you mean by the "second theory"
> suggests that the person among the two of us who has a naive viewpoint
> based on the "second theory" is not me.
The phrase "second theory" is related to your phrase "both
theories". I assumed the first theory is superstring theory
and tried to speculate on the nature of the second theory.
The purpose of the term "naive" is not the demonstration of
some arrogant standpoint, like you immediately concluded,
but the refence to the (divergent) QFT-fashion quantized
SUGRA as opposed to the correct way to quantize gravity
(which is conjectured to be superstring theory).
> It's because "solving the equations of motion" and "being described by a
> conformal field theory" *is* only true at the level of perturbative string
> theory.
And therefore it doesn't demonstrate superstring theory
has a SUGRA approximation since it doesn't show how
to define SUGRA-like observables in the full
superstring theory.
> M(atrix) theory is a quantum mechanical model, and therefore the Hilbert
> space of states is nothing else than the space of possible complex-valued
> functions of the (bosonic and fermionic) elements of the matrices. It is
> conceptually as easy as quantum mechanics of the Hydrogen atom.
Or, if we compactify some dimensions, it is a QFT.
That much is obvious, however...
> M(atrix) theory is a (discrete) light-cone gauge approach to SUGRA, and
> therefore the general covariance (local symmetry) is fixed in a specific
> way - a way that breaks the Lorentz symmetry. Indeed, several boost
> generators are not manifest in M(atrix) theory. Consequently, M(atrix)
> theory allows us to calculate not only the S-matrix, but also the
> evolution over finite intervals in the light-cone time x^+.
So is the S-matrix the only observable or not?
> I've explained both why string theory reduces to the appropriate classical
> descriptions in the classical limit, and therefore why it agrees with
> everything that can be correctly derived from the classical theory; as
> well as the reasons why the approximate questions present in a classical
> theory have no exact generalization in the full quantum (string/M) theory;
> and it would be a waste of time to repeat it again, I think. You did not
> get it - no problem; I would guess that others did.
The problem is that you didn't get something else. If, as you claim,
these observables have no generalization in any sense in the full
theory it's not clear how can the full theory be a meaningful model
of reality. The reason I'm saying this is as follows: my perceptions,
as a person, at a given moment of my life have to be an exactly (not
approximately) defined observable of any meaningful model of reality.
I hope you agree with me on that?! If the real world was classical
SUGRA it would be conceivable that this observable could be defined
in it. It is however not clear whether the same is true about
superstring theory.
> Local QFTs contain observables beyond the S-matrix because the local
> fields can be coupled to point-like, infinitely small, local probes. It is
> not the case in quantum gravity.
However Matrix theory is a local QFT (just not local in spacetime),
therefore it should contain observables beyond the S-matrix. Or
shouldn't it?
> > If you have a way to gauge-fix diffeomorphism invariance and define
> > observables localized with respect to the coordinates after gauge
> > fixing then
>
> Yes, for example it is the light cone gauge, as shown in M(atrix) theory.
>
> > A) It's very interesting and I'd like to know how you do that.
>
> On the string, light-cone gauge just means that X^+ is set to \tau (the
> worldsheet coordinate) and X^- (another light-like direction) is
> calculated in terms of the remaining D-2 "purely transverse" bosons
> X^i(\sigma,\tau). In spacetime it means that only the transverse components
> of various fields - such as the metric tensor fluctuation h_{ij} -
> appear, and the light-like Hamiltonian is written as a functional of
> these components of the fields.
All of these things you said can be done in usual QFT
and have nothing to do with gauge fixing diffeomorphism
invariance.
> > B) It contradicts your claim local gauge invariant observables
> > cannot be defined.
>
> Most of the times, I was careful to say that local gauge-invariant
> *covariant* observables cannot be defined. The local fields derived from
> the light-cone gauge are not covariant (meaning Lorentz covariant).
Any scalar constructed out of these fields is Lorentz
invariant and therefore Lorentz covariant.
> Depending on the exact definitions of the words, many people would argue
> that the spacetime fields as seen in the light-cone gauge are not
> gauge-invariant either because they are expressed as functions of a
> specific set of coordinates, including X^+ and X^-, and these coordinates
> are not gauge-invariant (invariant under coordinate transformations).
Indeed I would argue that.
> Of
> course, one could say that we can construct "priviliged" coordinates X^+
> and X^- for any asymptotically-Minkowski background, and these coordinates
> X^+ and X^- are then therefore "gauge-invariant" observables,
How do you do that?
> but it is a stretched interpretation.
Why is it stretched?
> Nevertheless, the common sense usage of the words "gauge-invariant" leads
> to the conclusion that a special choice of coordinates just *cannot* be
> gauge-invariant (under coordinate redefinitions).
It can if it is "priviliged" as you say.
> The idea that there is a large amount of interesting particle physics
> information in quantum gravity beyond the S-matrix is largely an illusion.
> In condensed matter physics, one usually wants to compute properties of
> non-relativistic, nearly static, configurations, and this implies other
> rules. But high-energy physics *is* about the existing states and their
> S-matrix.
However, superstring theory is supposed to be the ultimate
theory of everything. As such, it should contain not only
the results of particle accelerator experiments (to some
reasonably good approximation) but everything that can
ever be said about the universe!
> > I'm not so sure. If the finite-volume operators are localized
> > using a state-dependent procedure ("the trace of the energy
> > momentum tensor near the guy in the brown coat") it's not
> > obvious it can't be defined.
>
> It is naive to think that these sentences can make any exact sense in a
> quantum theory, and the fact that they make perfect sense to you is caused
> by your thinking merely in terms of the classical approximation.
But you just said they do make exact sense by claiming there
is a way to construct privileged coordinate and compute
everything in terms of these coordinates in the full
nonperturbative quantum theory!!
> > Let me demonstrate. Assume I want to compute what my neighbour will
> > eat from breakfast tomorrow. If the world is described by a QFT, I can
> > in principle imagine how to do this.
> >
> > A) Idenitify a sufficient amount of (local) observables.
> >
> > B) Find the state of the universe (locally) using
> > measurement of the observables in a sufficient volume
> > around yourself.
>
> In a theory of gravity, it is meaningless to say "the Universe looks XY at
> time t" because there is no universal definition of "t".
Yes, that's why I said "if the world is described by a QFT".
Of course in reality it isn't (so I should really have said
"if the world were described by a QFT).
> In practice, what you said is something that we compute neither in string
> theory nor in quantum field theory.
Why not in QFT?
> > Yes but "smoothened out" objects constructed out of the electric
> > field are meaningful and that's the thingies we measure in reality.
> > What kind of "smoothing out" can you suggest in the quantum gravity
> > case?
>
> This is exactly another part of the story that can never work in quantum
> gravity.
So, your claim that my complaints can be addressed to QFT
in the same fashion they can be adrressed to superstring
theory is plainly wrong.
> Which experiments at the Tevatron that measure non-S-matrix (finite-time)
> effects do you exactly mean?
All of them! Simply the finite-time effects are virtually
zero in any practical situation. However, there's a
difference between saying that my theory can produce the
exact result of the experiment but it can always be
approximated by something called the S-matrix and saying
the exact result of the experiment is undefined in my
theory but the S-matrix is defined.
> > > > I know, I indicated in a recent reposting of this post how to solve
> > > > this problem. You have to choose a certain offset for the proper
> > > > length in the corresponding geodesic for genuine AdS and then match
> > > > your parameter with the AdS one in the asymptotics.
> > >
> > > This is only possible in the classical geometry approximation, I think.
> >
> > Why are you so sure?
>
> Because I know that quantum mechanically, the metric fluctuates, and in
> fact, we can calculate how *much* it fluctuates.
So what?
> A calculation based on
> these metric tensor degrees of freedom becomes increasingly ill-behaved as
> you go to shorter distances, and at very short distances, the metric is
> not the whole story and the idea that it is a good degree of freedom,
> isolated from others, breaks itself.
OK, so give something else instead of the metric that works.
> AdS/CFT however gives you again a "preferred" choice of coordinates
How does it do that?!
> But once again, treating the coupling as a general spacetime-dependent
> function is not what we usually want in most QFT calculations. We only
> *imagine* that the coupling goes (very slowly) to zero at |t|=infinity in
> order to calculate the perturbative S-matrix meaningfully (to define the
> asymptotic states by adiabatic continuations of the states at g=0) - but
> what we're really calculating is the scattering of real objects in a
> spacetime where the coupling constant is a positive constant in the whole
> spacetime.
Whether we imagine that or not is a semantical game. The fact
remains that we can't define the adiabatic continuation
without allowing for the coupling constant to change. The
fact the resulting object (the S-matrix) is a good
approximation of real-world proccesses with constant coupling
constant is another matter. Even here though there are
subtleties. In the real world, the spectrum is different than
in the free theory. We can consider scattering of bound
states and fundumental particles can be impossible to produce
due to confinement. Possibly we can compute real world
scattering by plugging appropriate wavefunctions into the
S-matrix, however, the computation of the wavefunctions
themselves goes beyond the S-matrix.
> > This is already a much better argument. Still the S-matrix
> > is defined as the limit of something that is not defined.
>
> That's correct.
So it's correct we define the only concept that leads to
computable quantities in our theory through concepts that
are meaningless?!
> Well, your concrete proposed procedure how the S-matrix should be defined
> is simply wrong in string theory. We certainly don't need any such wrong
> procedures - involving backgrounds that don't solve the equations of
> motion - neither to compute the stringy S-matrix, nor to interpret it or
> check it. It is just your proposal, and it is an incorrect proposal.
Then what is the correct proposal? Without the
gradual changing of the coupling constant
you can't relate the real states to the free
theory states and the S-matrix has no relevance
for real world scattering.
> You cannot smear over "region of size V" in quantum gravity because the
> "region of size V" depends on the metric which is dynamical and its
> quantum fluctuations are really growing all the time as you increase the
> resolution.
I don't see how the statement after "because" implies
the one before it.
> If you send the UV cutoff to infinity and includ the quantum effect, the
> observable you are proposing - and define as an integral over a minimum
> curve - will be zero because for any configuration that contributes to the
> path integral, you will find a spacelike subspace whose volume is
> (arbitrarily close to) zero, and the integrals of all things in your
> definitions will vanish.
Why is that?
Best regards,
Squark.
Lubos Motl
Jun18-04, 12:17 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 Fri, 18 Jun 2004, Squark wrote:\n\n> > LM: The tree level S-matrix is nothing else than the Taylor expansion of the\n> > solutions describing the scattering of classical waves - an expansion\n> > around the vanishing strength of these waves (around the background).\n>\n> So, you\'re saying that is I want to compute the scattering\n> of classical waves, I have the imagine them as coherent\n> states in Fock space and act with the tree-level S-matrix?\n> Is the result guaranteed to be a coherent state?\n\n(Let me start with the comment that the amount of new information in this\nthread seems very small, and I will personally be rejecting similar posts\nas repetitive.)\n\nBack to your coherent states.\n\nThe participants who can read can easily check that I did not use the\nwords "coherent state" in my posting at all, and therefore I was obviously\nnot saying what you are claiming now that I was saying - but if you ask\nwhether the classical waves are coherent states, and whether their\nclassical scattering is encoded in the tree level S-matrix, the answer to\nboth questions is yes.\n\nConcerning the last question.\n\nNo, a general interaction process does not produce exact coherent states\nout of coherent states, and again, if it is a problem, it is a problem of\nyour understanding, not a problem of the fact that the classical\nscattering is encoded in the tree level S-matrix. Classical scattering\ndoes not care whether a "packet" is an exact coherent state (of course\nthat essentially any non-linear evolution makes it non-coherent) - it only\ncares about the position of the packet in the configuration space, and\nthis information can be extracted from the tree level approximation of the\nS-matrix.\n\nYou tend to impose a large number of assumptions "XYZ should be possible".\nMost of these assumptions are incorrect. Instead of correcting your\nunderstanding of the issue, and understanding why the assumption was not\nright, you permanently want to derive that something is wrong with string\ntheory. I don\'t find this dogmatic approach to be a particularly good\napproach to learn physics.\n\n> > I am not sure what you exactly mean by the phrase "contains observables".\n> > It may contain some observables - like the metric at point with\n> > coordinates (x,y,z,t)\n>\n> This is not an observable since it is not diffeomorphism\n> invariant.\n\nIt depends whether you define observables only as those that are\ngauge-invariant, or all of them. A terminological issue. But otherwise -\nyes, it was exactly my goal to argue that there are no nice (e.g. local)\ngauge-invariant observables in quantum gravity beyond the S-matrix.\n\n> I.e. you\'re saying that the reason QFT contains\n> observables beyond the S-matrix whereas string\n> theory does not is that we can insert external\n> currents into the QFT action and make the\n> S-matrix depend on them?\n\nYes, it\'s one of the things and interpretations I am saying. In quantum\ngravity (string theory), you cannot put (couple) external point-like\nsources consistently which is another reason why we should not expect\noff-shell local correlators.\n\n> The phrase "second theory" is related to your phrase "both\n> theories".\n\nIf someone uses the phrase "both theories", it does not make the phrase\n"second theory" well-defined, does it?\n\n> I assumed the first theory is superstring theory\n> and tried to speculate on the nature of the second theory.\n\nHow could anyone else besides you know what you\'re assuming to be the\n"first" and what is the "second" theory?\n\nBy the way, even in this new posting, you have not quite explained what\nyou mean by the "second theory" - probably (nonrenormalizable) SUGRA?\nWhichever interpretation I choose, your question\n\n> This sentence implies you are thinking about the "naive"\n> QFT quantization SUGRA as the "second theory", is it so?\n\nlooks either meaningless, or tautologic. It is you who wanted to show that\nstring/M-theory reproduces SUGRA (a theory that is only complete\nclassically), so I don\'t quite understand why you call the "second theory"\n(let\'s assume that it means SUGRA) "naive" if it is the ultimate goal of\nyour question. If we call it "naive", then we also must call your question\n"naive", don\'t we? Is not it better to call SUGRA simply and neutrally the\n"classical limit" of the full (string) theory?\n\n> > It\'s because "solving the equations of motion" and "being described by a\n> > conformal field theory" *is* only true at the level of perturbative string\n> > theory.\n>\n> And therefore it doesn\'t demonstrate superstring theory\n> has a SUGRA approximation since it doesn\'t show how\n> to define SUGRA-like observables in the full\n> superstring theory.\n\nWell, yes: perturbative calculations only show perturbative statements.\nBut it does not mean that we can\'t demonstrate various similar claims\nnon-perturbatively. For example, graviton scattering can be calculated in\nM(atrix) theory or AdS/CFT, and it agrees with SUGRA in the classical\nlimit.\n\n> > M(atrix) theory is a quantum mechanical model, and therefore the Hilbert\n> > space of states is nothing else than the space of possible complex-valued\n> > functions of the (bosonic and fermionic) elements of the matrices. It is\n> > conceptually as easy as quantum mechanics of the Hydrogen atom.\n>\n> Or, if we compactify some dimensions, it is a QFT.\n> That much is obvious, however...\n\nEven if we don\'t compactify BFSS, quantum mechanics is an example of a\n(0+1)-dimensional quantum field theory. Yes, I agree that it is "obvious"\n- but it was you who asked the obvious question, and therefore I answered\nit. Don\'t ask me why you asked your question, I don\'t know.\n\n> So is the S-matrix the only observable or not?\n\nIt\'s the only "nice" observable - the statement that it is the only\ngauge-invariant observable usually means that the off-shell correlators\n(Green\'s functions) of local operators - the correlators we like to use in\nQFT - can\'t be defined in quantum gravity - because there are no local\ngauge-invariant operators in quantum gravity.\n\n> The problem is that you didn\'t get something else. If, as you claim,\n> these observables have no generalization in any sense in the full\n> theory it\'s not clear how can the full theory be a meaningful model\n> of reality.\n\nMost observables that you think about classically don\'t lead to any nice\ngauge-invariant observables in quantum gravity. You *think* that this fact\nproves that quantum gravity is a meaningless theory. You are not right.\nThe real reason is that your classical intuition is oversimplified, much\nlike the observables that you think about. You apparently have problems to\nunderstand this statement - let me say it in different words. Your\nopinions what should be possible in quantum gravity are not qualified, and\nyour attempts to derive far-reaching conclusions about the validity of\nquantum gravity theories is an example that limited knowledge can still\ncombine with aggressive behavior. You should first try to understand some\ngeneral lessons of quantum mechanics before you try to argue that all the\nphysicists are doing something totally wrong.\n\n> The reason I\'m saying this is as follows: my perceptions,\n> as a person, at a given moment of my life have to be an exactly (not\n> approximately) defined observable of any meaningful model of reality.\n\nThat\'s a good summary of the naive classical thinking. Perceptions at a\ngiven moment of a life are totally vague, approximate notions, not only\nbecause the observables related to perceptions are notoriously inaccurate\nand hard to define even in QFT, but especially because even a "given\nmoment" cannot be defined in quantum gravity.\n\n> I hope you agree with me on that?!\n\nNo, I think it is centuries (or millenia) away from the truth in quantum\ngravity. Already quantum mechanics has showed us that many things that we\nbelieved had exact and canonical and universal meaning (position and\nvelocities of an electron) were not well-defined, and quantum gravity goes\neven further in this process. You seem to be directed in the opposite\ndirection - going from the 17th century to the 15th century.\n\nOne can formulate a lot of sentences that satisfy the rules of grammar,\nbut most of them won\'t have any exact (not even semi-exact) meaning in\nscience, especially not in quantum gravity where it is very difficult and\nconstrained to define exact objects.\n\n> If the real world was classical SUGRA it would be conceivable that\n> this observable could be defined in it.\n\nIf the world were classical SUGRA, one could not define any quantum\nobservables at all, because classical theories are incompatible with\nquantum mechanics. If the world were quantized SUGRA, as defined by the\nLagrangian, then it would be logically inconsistent, and the\ninconsistency could be seen near the Planck scale.\n\nNeglecting the UV problems, one can use quantized SUGRA, but the only\nmeaningfully extractable results from this theory are those that are\nencoded in the classical limit.\n\n> It is however not clear whether the same is true about\n> superstring theory.\n\nThe fact that there are no local gauge-invariant observables in quantum\ngravity does not rely on superstring theory. It relies on general\ncovariance.\n\n> However Matrix theory is a local QFT (just not local in spacetime),\n> therefore it should contain observables beyond the S-matrix. Or\n> shouldn\'t it?\n\nYes, and it does. The finite-time evolution in Matrix theory is - at least\nformally - related to a finite X^+ evolution in spacetime.\n\n> All of these things you said can be done in usual QFT\n> and have nothing to do with gauge fixing diffeomorphism\n> invariance.\n\nThe most important goal of what I am explaining you is that *less* things\ncan be defined and computed in quantum gravity, not more! The only way to\nget as much data in quantum gravity as in local QFT is to gauge-fix (or\nbreak) the diffeomorphism invariance. For example, in the light-cone\ngauge, superstring theory resembles non-gravitational local QF theories.\n\n> > Most of the times, I was careful to say that local gauge-invariant\n> > *covariant* observables cannot be defined. The local fields derived from\n> > the light-cone gauge are not covariant (meaning Lorentz covariant).\n>\n> Any scalar constructed out of these fields is Lorentz\n> invariant and therefore Lorentz covariant.\n\nNo, be sure that X^+ is not Lorentz-covariant. If you define a scalar to\nbe Lorentz invariant, then your sentence is a vacuous tautology, but\nagain, the subtlety is that there exist no Lorentz invariant local\ngauge-invariant observables in a theory of quantum gravity.\n\nThe Lorentz generators in the light cone gauge contain "compensating"\ndiffeomorphism generators that restore the light cone gauge, but\neffectively move the operators into different points.\n\n> > Of\n> > course, one could say that we can construct "priviliged" coordinates X^+\n> > and X^- for any asymptotically-Minkowski background, and these coordinates\n> > X^+ and X^- are then therefore "gauge-invariant" observables,\n>\n> How do you do that?\n\nAny asymptotically Minkowski state in the physical Hilbert space can be -\nat least in the perturbative expansion - obtained via the action of the\nlight cone gauge creation operators in a unique way - there exists no\ndiffeomorphism freedom (one that would preserve the spacetime at\ninfinity), and therefore the assignment of the spacetime coordinate\nbecomes unique. But it is a very Lorentz-non-invariant procedure.\n\n> > but it is a stretched interpretation.\n>\n> Why is it stretched?\n\nBecause many sets of coordinates can be obtained according to some\n"gauge-invariant, physical procedures", but it is unfair to pick one such\nprocedure and call it "canonical" or "invariant". The Schwarzschild\ngeometry can be written in the Schwarzschild coordinates where "r" is\ndefined ("covariantly") from the area of the spheres (orbits of the SO(3)\nsymmetry) - but nevertheless we would not say that the precise form of\nthe metric in the Schwarzschild coordinates is something diffeomorphism\ninvariant. Simply because there exist infinitely many other choices of\ncoordinates. Many of them can be described physically (distances from 3\ntrees), but it does not make them fundamental or universally priviliged.\n\n> > Nevertheless, the common sense usage of the words "gauge-invariant" leads\n> > to the conclusion that a special choice of coordinates just *cannot* be\n> > gauge-invariant (under coordinate redefinitions).\n>\n> It can if it is "priviliged" as you say.\n\nSome coordinates may be "priviliged" or "useful" for a particular purpose,\nbut there are no "generally priviliged" coordinates. This principle is\ncalled "general covariance", and it is one of the key insights underlying\ngeneral relativity.\n\n> However, superstring theory is supposed to be the ultimate\n> theory of everything. As such, it should contain not only\n> the results of particle accelerator experiments (to some\n> reasonably good approximation) but everything that can\n> ever be said about the universe!\n\nNope. By the words "theory of everything" we only mean "a theory of\neverything that really exists in the Universe" - and therefore string\ntheory should *not* contain nonsense. It should not work with nonsensical\nobservables that someone on the street invents when she is bored, and\nindeed, it does not operate with them.\n\nIt is just a wrong assumption of yours that any nonsense that someone\ninvents must be confirmed by string theory. String theory only contains\nall good ideas in the world, not *all* ideas. ;-)\n\n> But you just said they do make exact sense by claiming there\n> is a way to construct privileged coordinate and compute\n> everything in terms of these coordinates in the full\n> nonperturbative quantum theory!!\n\nIf you find e.g. the light-cone gauge treatment natural and useful for\nanswering finite-time questions, feel free to use it. It will work much\nlike local quantum field theory, but the underlying symmetries (coordinate\ntransformations) will be largerly obscured.\n\n> > In a theory of gravity, it is meaningless to say "the Universe looks XY at\n> > time t" because there is no universal definition of "t".\n>\n> Yes, that\'s why I said "if the world is described by a QFT".\n> Of course in reality it isn\'t (so I should really have said\n> "if the world were described by a QFT).\n\nSo why are you talking about it if you know that it is not true? The\ndiscussion is very different and simpler in QFT than in QG.\n\n> > In practice, what you said is something that we compute neither in string\n> > theory nor in quantum field theory.\n>\n> Why not in QFT?\n\nBecause in QFT we compute the spectrum of states and operators, their\nenergies, dimensions, correlators, scattering amplitudes, and things that\nfollow from this list. Your "observables" are not in this list; they are\neasy to be said in English, but it would be extremely difficult and\nambiguous if one tried to convert them into exact science in a quantum\ntheory. In reality, all questions and statements of the sort that you\ndescribe are answered by using various classical approximations, and most\nof the objects in such sentences make no sense outside the region of\nvalidity of the classical approximation.\n\n> > > Yes but "smoothened out" objects constructed out of the electric\n> > > field are meaningful and that\'s the thingies we measure in reality.\n> > > What kind of "smoothing out" can you suggest in the quantum gravity\n> > > case?\n> >\n> > This is exactly another part of the story that can never work in quantum\n> > gravity.\n>\n> So, your claim that my complaints can be addressed to QFT\n> in the same fashion they can be adrressed to superstring\n> theory is plainly wrong.\n\nMy statement was just the opposite. Many things that can work in QFT\ncannot, in principle, work in quantum gravity - and the smearing out of\nthe operators over a volume-V region of space is an example because the\nvolume (and geometry) itself becomes a quantum, "jittery" observable.\n\n> > Which experiments at the Tevatron that measure non-S-matrix (finite-time)\n> > effects do you exactly mean?\n>\n> All of them! Simply the finite-time effects are virtually\n> zero in any practical situation.\n\nThis is what I am trying to explain you. The extra stuff that you want to\nbe meaningful cannot be meaningful in quantum gravity, even in principle,\nand in practice, it is meaningless even in particle physics / QFT.\n\n> However, there\'s a difference between saying that my theory can\n> produce the exact result of the experiment but it can always be\n> approximated by something called the S-matrix and saying the exact\n> result of the experiment is undefined in my theory but the S-matrix is\n> defined.\n\nThe only problem is that "your theory" does not exist. There exists no\n(consistent) theory of quantum gravity where the local gauge-invariant\nobservables could be defined and predicted. All "loopholes" described\nabove, such as the light-cone gauge, are (and must be) fixing the\nsymmetries in a non-local fashion if they want to define gauge-invariant\nquantities, and such quantities never transform naturally under the\nLorentz transformations.\n\n> > Because I know that quantum mechanically, the metric fluctuates, and in\n> > fact, we can calculate how *much* it fluctuates.\n>\n> So what?\n\nI was just pointing out that you were using incorrect classical intuition\nfor the "proper length", assuming that the "geodesics" and other stuff\nmake sense even if quantum effects are taken into account.\n\n> > A calculation based on\n> > these metric tensor degrees of freedom becomes increasingly ill-behaved as\n> > you go to shorter distances, and at very short distances, the metric is\n> > not the whole story and the idea that it is a good degree of freedom,\n> > isolated from others, breaks itself.\n>\n> OK, so give something else instead of the metric that works.\n\nThese objects are not just "instead" of the metric. The extra fields and\nother degrees of freedom implied by string/M-theory *complete* the metric\ntensor into a complete picture where *everything* matters. You cannot\nremove pieces because the structure would become inconsistent.\n\nIf you want me to say some non-metric fields predicted by string theory,\nfor example the massive tensor\n\n\\alpha_{-2}^\\mu \\tilde\\alpha_{-2}^\\nu |0\\rangle.\n\n> > AdS/CFT however gives you again a "preferred" choice of coordinates\n>\n> How does it do that?!\n\nIt is much like in the light cone gauge or elsewhere. The asymptotic AdS\nregion at infinity - where the relative fluctuations go to zero - is\nmatched to its standard form, the rest of geometry is extrapolated to the\ninterior, and the physics of the bulk can be calculated from a CFT\n(although, of course, it becomes more subtle as you try to go deeper to\nthe bulk, away from the boundaries) - a CFT that has *no* diffeomorphism\ninvariance left. Only the physical states of the graviton are seen in the\nCFT, and therefore the bulk general covariance (diffeomorphism invariance)\nis clearly gauge-fixed much like in the case of the light cone gauge.\n\nBoth M(atrix) theory (and other light cone approaches) as well as the\nAdS/CFT are descriptions where the diffeomorphism invariance in spacetime\nhas been fixed, without our having done any work.\n\n> Whether we imagine that or not is a semantical game. The fact\n> remains that we can\'t define the adiabatic continuation\n> without allowing for the coupling constant to change.\n\nIt\'s not true. If you tried to study a textbook on string theory, you\nwould see that it allows you to calculate the S-matrix between the\nscattering states - which themselves are adiabatic continuations of the\nwell-known free states at g=0, much like in QFT - without considering any\ninconsistent backgrounds which would violate the equations of motion.\n\nI think that you must know that what you are saying most of the time is\nsimply wrong - because everyone (even some readers of the newspapers)\nknow(s) that string theory allows us to calculate a meaningful S-matrix\nthat follows the standard QFT interpretation in a more complex context,\nand it is more, not less, consistent than the QFT S-matrices.\n\nEveryone also knows that string theory expanded around "wrong" backgrounds\nis inconsistent. Nevertheless, you are saying that at least one of these\ntwo well-known facts must be wrong. I wonder why you say such\nunjustifiable statements that are known to be wrong. The way how *you*\nwant to define (or calculate?) the S-matrix is just incorrect, but *you*\nbeing wrong and *string theory* being wrong are two absolutely different\nsituations. Also, why don\'t you try to isolate the problem? If you think\nthat something must be wrong about the stringy S-matrix, why don\'t you try\nto isolate what the hypothetical problem should be? If you think that one\nshould be considering wrong backgrounds, why don\'t you try to invent an\nargument?\n\nString theory does not do any calculations in a meaningless way, and it\nnever requires us (it even does not allow us) to consider inconsistent\nbackgrounds that would violate the equations of motion. Nevertheless, it\nhas clear rules to calculate the S-matrix that directly generalize the\nrules in QFT and can be interpreted identically. The topic that you touch\nis very important - but unfortunately you are systematically saying just\nthe opposite of the correct answers.\n\n> Possibly we can compute real world\n> scattering by plugging appropriate wavefunctions into the\n> S-matrix, however, the computation of the wavefunctions\n> themselves goes beyond the S-matrix.\n\nYes, one needs to insert some knowledge about the internal structure of\nthe strongly interacting particles - form-factors etc. - if she wants to\ncalculate the baryon scattering, for example, in terms of QCD. This is\njust a fact; baryons are complicated, strongly coupled systems, that are\nfar from the perturbative scattering states (quarks and gluons). But these\nsituations or facts are again either true or not in a given context\nregardless whether we use string theory or QCD. In string theory, a proton\nis also a complicated composite of strings and perhaps other things (well,\nmesons are - in some sense - just plain open strings, but one must work a\nlot to see "which exactly strings"), and we would have to make some extra\nwork besides the simple perturbative graphs in order to calculate their\nscattering.\n\n> > > This is already a much better argument. Still the S-matrix\n> > > is defined as the limit of something that is not defined.\n> >\n> > That\'s correct.\n>\n> So it\'s correct we define the only concept that leads to\n> computable quantities in our theory through concepts that\n> are meaningless?!\n\nWe don\'t need any meaningless concepts - it is just you who apparently\ncannot live without them. I am telling you that the S-matrix is only\nmeaningful as a way to calculate scattering of excitations around a\nbackground that solves the equations of motion. You seem to consider some\nparadoxical objects, unacceptable backgrounds, and so on, and I am telling\nyou that if you really can\'t live without them, you can always consider a\nmeaningful background as a limit of meaningless backgrounds where the\n"error" goes to zero.\n\nBut I am only telling you this because *you* want to consider meaningless\nbackgrounds. String theory (and string theorists) never need any\nmeaningless backgrounds or contradictory obervables to be included in\ntheir calculations. Could you please try to understand the difference\nbetween YOU and between STRING THEORY?\n\nIt is very important that string theory cannot calculate physics on the\nbackgrounds that violate the equations of motion; it is very important\nthat string theory cannot (uniquely) construct local off-shell correlators\nof closed string fields. If anything like that were possible, the theory\nwould be inconsistent.\n\n> Then what is the correct proposal? Without the gradual changing of the\n> coupling constant you can\'t relate the real states to the free theory\n> states and the S-matrix has no relevance for real world scattering.\n\nI think that you only want to create problems, but in reality you don\'t\nbelieve that we can\'t define the scattering states. The graviton\nscattering states in superstring/M-theory are, for example, uniquely\ndetermined at any value of the coupling constant by their being\nsupersymmetric. We don\'t need to gradually change the coupling to identify\nthe states - nevertheless, we can still do it. We know how to solve the\nequations of motion for a dilaton that increases with time very slowly\netc. What problem do you exactly have in mind? All these facts about\nthe stringy S-matrix are totally analogous to QFT, and they are equally\ninterpreted. We just replace the Feynman diagrams by the worldsheets.\n\n> > You cannot smear over "region of size V" in quantum gravity because the\n> > "region of size V" depends on the metric which is dynamical and its\n> > quantum fluctuations are really growing all the time as you increase the\n> > resolution.\n>\n> I don\'t see how the statement after "because" implies\n> the one before it.\n\nIf you define the volume as the integral of the volume form (think about\n\\sqrt{g}), then this integral over any finite region in the coordinate\nspace blows up if you remove the UV cutoff. Quantum mechanically, volume\n"V" is one point (infinitesimally small neighborhood in the coordinate\nspace), and if you average over that, you won\'t cure any divergences of\nthe other operators you wanted to regulate.\n\n> Why is that?\n\nJust calculate it. If you expand the Einstein-Hilbert action, and write\nthe metric as g_{mn} = \\eta_{mn} + \\sqrt{G} h_{mn}, then the\nEinstein-Hilbert action will contain the kinetic term for h_{mn},\nschematically (dh)^2, with coefficient one. The dimension of "h" is of\nmass by dimensional analysis. There is no dimensionful parameter in the\naction (dh)^2, and therefore the average fluctuation of "h" must be equal\nto the only dimensionful parameter with the right dimension that you have\n- namely the cutoff \\Lambda - and therefore it goes to infinity if you try\nto send \\Lambda to infinity.\n\nThese infinities are sort of stopped when \\Lambda approaches the Planck\nscale - but according to string theory, the whole concept of geometry\nbreaks down at the Planck scale, too.\n\nThe space just fluctuates ever more violently as you look at it with a\nbetter resolution, and as the cutoff approaches the Planck scale, even the\ntopology is oscillating. String theory stops this chaos and replaces the\nnaive geometry by a more complete, "non-commutative" physical picture at\nthe Planck scale - well, perturbative string theory already at the string\nscale which is much longer.\n\nBest\nLubos\n__________________________ __________________________________________________ __\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 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 Fri, 18 Jun 2004, Squark wrote:
> > LM: The tree level S-matrix is nothing else than the Taylor expansion of the
> > solutions describing the scattering of classical waves - an expansion
> > around the vanishing strength of these waves (around the background).
>
> So, you're saying that is I want to compute the scattering
> of classical waves, I have the imagine them as coherent
> states in Fock space and act with the tree-level S-matrix?
> Is the result guaranteed to be a coherent state?
(Let me start with the comment that the amount of new information in this
thread seems very small, and I will personally be rejecting similar posts
as repetitive.)
Back to your coherent states.
The participants who can read can easily check that I did not use the
words "coherent state" in my posting at all, and therefore I was obviously
not saying what you are claiming now that I was saying - but if you ask
whether the classical waves are coherent states, and whether their
classical scattering is encoded in the tree level S-matrix, the answer to
both questions is yes.
Concerning the last question.
No, a general interaction process does not produce exact coherent states
out of coherent states, and again, if it is a problem, it is a problem of
your understanding, not a problem of the fact that the classical
scattering is encoded in the tree level S-matrix. Classical scattering
does not care whether a "packet" is an exact coherent state (of course
that essentially any non-linear evolution makes it non-coherent) - it only
cares about the position of the packet in the configuration space, and
this information can be extracted from the tree level approximation of the
S-matrix.
You tend to impose a large number of assumptions "XYZ should be possible".
Most of these assumptions are incorrect. Instead of correcting your
understanding of the issue, and understanding why the assumption was not
right, you permanently want to derive that something is wrong with string
theory. I don't find this dogmatic approach to be a particularly good
approach to learn physics.
> > I am not sure what you exactly mean by the phrase "contains observables".
> > It may contain some observables - like the metric at point with
> > coordinates (x,y,z,t)
>
> This is not an observable since it is not diffeomorphism
> invariant.
It depends whether you define observables only as those that are
gauge-invariant, or all of them. A terminological issue. But otherwise -
yes, it was exactly my goal to argue that there are no nice (e.g. local)
gauge-invariant observables in quantum gravity beyond the S-matrix.
> I.e. you're saying that the reason QFT contains
> observables beyond the S-matrix whereas string
> theory does not is that we can insert external
> currents into the QFT action and make the
> S-matrix depend on them?
Yes, it's one of the things and interpretations I am saying. In quantum
gravity (string theory), you cannot put (couple) external point-like
sources consistently which is another reason why we should not expect
off-shell local correlators.
> The phrase "second theory" is related to your phrase "both
> theories".
If someone uses the phrase "both theories", it does not make the phrase
"second theory" well-defined, does it?
> I assumed the first theory is superstring theory
> and tried to speculate on the nature of the second theory.
How could anyone else besides you know what you're assuming to be the
"first" and what is the "second" theory?
By the way, even in this new posting, you have not quite explained what
you mean by the "second theory" - probably (nonrenormalizable) SUGRA?
Whichever interpretation I choose, your question
> This sentence implies you are thinking about the "naive"
> QFT quantization SUGRA as the "second theory", is it so?
looks either meaningless, or tautologic. It is you who wanted to show that
string/M-theory reproduces SUGRA (a theory that is only complete
classically), so I don't quite understand why you call the "second theory"
(let's assume that it means SUGRA) "naive" if it is the ultimate goal of
your question. If we call it "naive", then we also must call your question
"naive", don't we? Is not it better to call SUGRA simply and neutrally the
"classical limit" of the full (string) theory?
> > It's because "solving the equations of motion" and "being described by a
> > conformal field theory" *is* only true at the level of perturbative string
> > theory.
>
> And therefore it doesn't demonstrate superstring theory
> has a SUGRA approximation since it doesn't show how
> to define SUGRA-like observables in the full
> superstring theory.
Well, yes: perturbative calculations only show perturbative statements.
But it does not mean that we can't demonstrate various similar claims
non-perturbatively. For example, graviton scattering can be calculated in
M(atrix) theory or AdS/CFT, and it agrees with SUGRA in the classical
limit.
> > M(atrix) theory is a quantum mechanical model, and therefore the Hilbert
> > space of states is nothing else than the space of possible complex-valued
> > functions of the (bosonic and fermionic) elements of the matrices. It is
> > conceptually as easy as quantum mechanics of the Hydrogen atom.
>
> Or, if we compactify some dimensions, it is a QFT.
> That much is obvious, however...
Even if we don't compactify BFSS, quantum mechanics is an example of a
(0+1)-dimensional quantum field theory. Yes, I agree that it is "obvious"
- but it was you who asked the obvious question, and therefore I answered
it. Don't ask me why you asked your question, I don't know.
> So is the S-matrix the only observable or not?
It's the only "nice" observable - the statement that it is the only
gauge-invariant observable usually means that the off-shell correlators
(Green's functions) of local operators - the correlators we like to use in
QFT - can't be defined in quantum gravity - because there are no local
gauge-invariant operators in quantum gravity.
> The problem is that you didn't get something else. If, as you claim,
> these observables have no generalization in any sense in the full
> theory it's not clear how can the full theory be a meaningful model
> of reality.
Most observables that you think about classically don't lead to any nice
gauge-invariant observables in quantum gravity. You *think* that this fact
proves that quantum gravity is a meaningless theory. You are not right.
The real reason is that your classical intuition is oversimplified, much
like the observables that you think about. You apparently have problems to
understand this statement - let me say it in different words. Your
opinions what should be possible in quantum gravity are not qualified, and
your attempts to derive far-reaching conclusions about the validity of
quantum gravity theories is an example that limited knowledge can still
combine with aggressive behavior. You should first try to understand some
general lessons of quantum mechanics before you try to argue that all the
physicists are doing something totally wrong.
> The reason I'm saying this is as follows: my perceptions,
> as a person, at a given moment of my life have to be an exactly (not
> approximately) defined observable of any meaningful model of reality.
That's a good summary of the naive classical thinking. Perceptions at a
given moment of a life are totally vague, approximate notions, not only
because the observables related to perceptions are notoriously inaccurate
and hard to define even in QFT, but especially because even a "given
moment" cannot be defined in quantum gravity.
> I hope you agree with me on that?!
No, I think it is centuries (or millenia) away from the truth in quantum
gravity. Already quantum mechanics has showed us that many things that we
believed had exact and canonical and universal meaning (position and
velocities of an electron) were not well-defined, and quantum gravity goes
even further in this process. You seem to be directed in the opposite
direction - going from the 17th century to the 15th century.
One can formulate a lot of sentences that satisfy the rules of grammar,
but most of them won't have any exact (not even semi-exact) meaning in
science, especially not in quantum gravity where it is very difficult and
constrained to define exact objects.
> If the real world was classical SUGRA it would be conceivable that
> this observable could be defined in it.
If the world were classical SUGRA, one could not define any quantum
observables at all, because classical theories are incompatible with
quantum mechanics. If the world were quantized SUGRA, as defined by the
Lagrangian, then it would be logically inconsistent, and the
inconsistency could be seen near the Planck scale.
Neglecting the UV problems, one can use quantized SUGRA, but the only
meaningfully extractable results from this theory are those that are
encoded in the classical limit.
> It is however not clear whether the same is true about
> superstring theory.
The fact that there are no local gauge-invariant observables in quantum
gravity does not rely on superstring theory. It relies on general
covariance.
> However Matrix theory is a local QFT (just not local in spacetime),
> therefore it should contain observables beyond the S-matrix. Or
> shouldn't it?
Yes, and it does. The finite-time evolution in Matrix theory is - at least
formally - related to a finite X^+ evolution in spacetime.
> All of these things you said can be done in usual QFT
> and have nothing to do with gauge fixing diffeomorphism
> invariance.
The most important goal of what I am explaining you is that *less* things
can be defined and computed in quantum gravity, not more! The only way to
get as much data in quantum gravity as in local QFT is to gauge-fix (or
break) the diffeomorphism invariance. For example, in the light-cone
gauge, superstring theory resembles non-gravitational local QF theories.
> > Most of the times, I was careful to say that local gauge-invariant
> > *covariant* observables cannot be defined. The local fields derived from
> > the light-cone gauge are not covariant (meaning Lorentz covariant).
>
> Any scalar constructed out of these fields is Lorentz
> invariant and therefore Lorentz covariant.
No, be sure that X^+ is not Lorentz-covariant. If you define a scalar to
be Lorentz invariant, then your sentence is a vacuous tautology, but
again, the subtlety is that there exist no Lorentz invariant local
gauge-invariant observables in a theory of quantum gravity.
The Lorentz generators in the light cone gauge contain "compensating"
diffeomorphism generators that restore the light cone gauge, but
effectively move the operators into different points.
> > Of
> > course, one could say that we can construct "priviliged" coordinates X^+
> > and X^- for any asymptotically-Minkowski background, and these coordinates
> > X^+ and X^- are then therefore "gauge-invariant" observables,
>
> How do you do that?
Any asymptotically Minkowski state in the physical Hilbert space can be -
at least in the perturbative expansion - obtained via the action of the
light cone gauge creation operators in a unique way - there exists no
diffeomorphism freedom (one that would preserve the spacetime at
infinity), and therefore the assignment of the spacetime coordinate
becomes unique. But it is a very Lorentz-non-invariant procedure.
> > but it is a stretched interpretation.
>
> Why is it stretched?
Because many sets of coordinates can be obtained according to some
"gauge-invariant, physical procedures", but it is unfair to pick one such
procedure and call it "canonical" or "invariant". The Schwarzschild
geometry can be written in the Schwarzschild coordinates where "r" is
defined ("covariantly") from the area of the spheres (orbits of the SO(3)
symmetry) - but nevertheless we would not say that the precise form of
the metric in the Schwarzschild coordinates is something diffeomorphism
invariant. Simply because there exist infinitely many other choices of
coordinates. Many of them can be described physically (distances from 3
trees), but it does not make them fundamental or universally priviliged.
> > Nevertheless, the common sense usage of the words "gauge-invariant" leads
> > to the conclusion that a special choice of coordinates just *cannot* be
> > gauge-invariant (under coordinate redefinitions).
>
> It can if it is "priviliged" as you say.
Some coordinates may be "priviliged" or "useful" for a particular purpose,
but there are no "generally priviliged" coordinates. This principle is
called "general covariance", and it is one of the key insights underlying
general relativity.
> However, superstring theory is supposed to be the ultimate
> theory of everything. As such, it should contain not only
> the results of particle accelerator experiments (to some
> reasonably good approximation) but everything that can
> ever be said about the universe!
Nope. By the words "theory of everything" we only mean "a theory of
everything that really exists in the Universe" - and therefore string
theory should *not* contain nonsense. It should not work with nonsensical
observables that someone on the street invents when she is bored, and
indeed, it does not operate with them.
It is just a wrong assumption of yours that any nonsense that someone
invents must be confirmed by string theory. String theory only contains
all good ideas in the world, not *all* ideas. ;-)
> But you just said they do make exact sense by claiming there
> is a way to construct privileged coordinate and compute
> everything in terms of these coordinates in the full
> nonperturbative quantum theory!!
If you find e.g. the light-cone gauge treatment natural and useful for
answering finite-time questions, feel free to use it. It will work much
like local quantum field theory, but the underlying symmetries (coordinate
transformations) will be largerly obscured.
> > In a theory of gravity, it is meaningless to say "the Universe looks XY at
> > time t" because there is no universal definition of "t".
>
> Yes, that's why I said "if the world is described by a QFT".
> Of course in reality it isn't (so I should really have said
> "if the world were described by a QFT).
So why are you talking about it if you know that it is not true? The
discussion is very different and simpler in QFT than in QG.
> > In practice, what you said is something that we compute neither in string
> > theory nor in quantum field theory.
>
> Why not in QFT?
Because in QFT we compute the spectrum of states and operators, their
energies, dimensions, correlators, scattering amplitudes, and things that
follow from this list. Your "observables" are not in this list; they are
easy to be said in English, but it would be extremely difficult and
ambiguous if one tried to convert them into exact science in a quantum
theory. In reality, all questions and statements of the sort that you
describe are answered by using various classical approximations, and most
of the objects in such sentences make no sense outside the region of
validity of the classical approximation.
> > > Yes but "smoothened out" objects constructed out of the electric
> > > field are meaningful and that's the thingies we measure in reality.
> > > What kind of "smoothing out" can you suggest in the quantum gravity
> > > case?
> >
> > This is exactly another part of the story that can never work in quantum
> > gravity.
>
> So, your claim that my complaints can be addressed to QFT
> in the same fashion they can be adrressed to superstring
> theory is plainly wrong.
My statement was just the opposite. Many things that can work in QFT
cannot, in principle, work in quantum gravity - and the smearing out of
the operators over a volume-V region of space is an example because the
volume (and geometry) itself becomes a quantum, "jittery" observable.
> > Which experiments at the Tevatron that measure non-S-matrix (finite-time)
> > effects do you exactly mean?
>
> All of them! Simply the finite-time effects are virtually
> zero in any practical situation.
This is what I am trying to explain you. The extra stuff that you want to
be meaningful cannot be meaningful in quantum gravity, even in principle,
and in practice, it is meaningless even in particle physics / QFT.
> However, there's a difference between saying that my theory can
> produce the exact result of the experiment but it can always be
> approximated by something called the S-matrix and saying the exact
> result of the experiment is undefined in my theory but the S-matrix is
> defined.
The only problem is that "your theory" does not exist. There exists no
(consistent) theory of quantum gravity where the local gauge-invariant
observables could be defined and predicted. All "loopholes" described
above, such as the light-cone gauge, are (and must be) fixing the
symmetries in a non-local fashion if they want to define gauge-invariant
quantities, and such quantities never transform naturally under the
Lorentz transformations.
> > Because I know that quantum mechanically, the metric fluctuates, and in
> > fact, we can calculate how *much* it fluctuates.
>
> So what?
I was just pointing out that you were using incorrect classical intuition
for the "proper length", assuming that the "geodesics" and other stuff
make sense even if quantum effects are taken into account.
> > A calculation based on
> > these metric tensor degrees of freedom becomes increasingly ill-behaved as
> > you go to shorter distances, and at very short distances, the metric is
> > not the whole story and the idea that it is a good degree of freedom,
> > isolated from others, breaks itself.
>
> OK, so give something else instead of the metric that works.
These objects are not just "instead" of the metric. The extra fields and
other degrees of freedom implied by string/M-theory *complete* the metric
tensor into a complete picture where *everything* matters. You cannot
remove pieces because the structure would become inconsistent.
If you want me to say some non-metric fields predicted by string theory,
for example the massive tensor
\alpha_{-2}^\mu \tilde\alpha_{-2}^\nu |0\rangle.
> > AdS/CFT however gives you again a "preferred" choice of coordinates
>
> How does it do that?!
It is much like in the light cone gauge or elsewhere. The asymptotic AdS
region at infinity - where the relative fluctuations go to zero - is
matched to its standard form, the rest of geometry is extrapolated to the
interior, and the physics of the bulk can be calculated from a CFT
(although, of course, it becomes more subtle as you try to go deeper to
the bulk, away from the boundaries) - a CFT that has *no* diffeomorphism
invariance left. Only the physical states of the graviton are seen in the
CFT, and therefore the bulk general covariance (diffeomorphism invariance)
is clearly gauge-fixed much like in the case of the light cone gauge.
Both M(atrix) theory (and other light cone approaches) as well as the
AdS/CFT are descriptions where the diffeomorphism invariance in spacetime
has been fixed, without our having done any work.
> Whether we imagine that or not is a semantical game. The fact
> remains that we can't define the adiabatic continuation
> without allowing for the coupling constant to change.
It's not true. If you tried to study a textbook on string theory, you
would see that it allows you to calculate the S-matrix between the
scattering states - which themselves are adiabatic continuations of the
well-known free states at g=0, much like in QFT - without considering any
inconsistent backgrounds which would violate the equations of motion.
I think that you must know that what you are saying most of the time is
simply wrong - because everyone (even some readers of the newspapers)
know(s) that string theory allows us to calculate a meaningful S-matrix
that follows the standard QFT interpretation in a more complex context,
and it is more, not less, consistent than the QFT S-matrices.
Everyone also knows that string theory expanded around "wrong" backgrounds
is inconsistent. Nevertheless, you are saying that at least one of these
two well-known facts must be wrong. I wonder why you say such
unjustifiable statements that are known to be wrong. The way how *you*
want to define (or calculate?) the S-matrix is just incorrect, but *you*
being wrong and *string theory* being wrong are two absolutely different
situations. Also, why don't you try to isolate the problem? If you think
that something must be wrong about the stringy S-matrix, why don't you try
to isolate what the hypothetical problem should be? If you think that one
should be considering wrong backgrounds, why don't you try to invent an
argument?
String theory does not do any calculations in a meaningless way, and it
never requires us (it even does not allow us) to consider inconsistent
backgrounds that would violate the equations of motion. Nevertheless, it
has clear rules to calculate the S-matrix that directly generalize the
rules in QFT and can be interpreted identically. The topic that you touch
is very important - but unfortunately you are systematically saying just
the opposite of the correct answers.
> Possibly we can compute real world
> scattering by plugging appropriate wavefunctions into the
> S-matrix, however, the computation of the wavefunctions
> themselves goes beyond the S-matrix.
Yes, one needs to insert some knowledge about the internal structure of
the strongly interacting particles - form-factors etc. - if she wants to
calculate the baryon scattering, for example, in terms of QCD. This is
just a fact; baryons are complicated, strongly coupled systems, that are
far from the perturbative scattering states (quarks and gluons). But these
situations or facts are again either true or not in a given context
regardless whether we use string theory or QCD. In string theory, a proton
is also a complicated composite of strings and perhaps other things (well,
mesons are - in some sense - just plain open strings, but one must work a
lot to see "which exactly strings"), and we would have to make some extra
work besides the simple perturbative graphs in order to calculate their
scattering.
> > > This is already a much better argument. Still the S-matrix
> > > is defined as the limit of something that is not defined.
> >
> > That's correct.
>
> So it's correct we define the only concept that leads to
> computable quantities in our theory through concepts that
> are meaningless?!
We don't need any meaningless concepts - it is just you who apparently
cannot live without them. I am telling you that the S-matrix is only
meaningful as a way to calculate scattering of excitations around a
background that solves the equations of motion. You seem to consider some
paradoxical objects, unacceptable backgrounds, and so on, and I am telling
you that if you really can't live without them, you can always consider a
meaningful background as a limit of meaningless backgrounds where the
"error" goes to zero.
But I am only telling you this because *you* want to consider meaningless
backgrounds. String theory (and string theorists) never need any
meaningless backgrounds or contradictory obervables to be included in
their calculations. Could you please try to understand the difference
between YOU and between STRING THEORY?
It is very important that string theory cannot calculate physics on the
backgrounds that violate the equations of motion; it is very important
that string theory cannot (uniquely) construct local off-shell correlators
of closed string fields. If anything like that were possible, the theory
would be inconsistent.
> Then what is the correct proposal? Without the gradual changing of the
> coupling constant you can't relate the real states to the free theory
> states and the S-matrix has no relevance for real world scattering.
I think that you only want to create problems, but in reality you don't
believe that we can't define the scattering states. The graviton
scattering states in superstring/M-theory are, for example, uniquely
determined at any value of the coupling constant by their being
supersymmetric. We don't need to gradually change the coupling to identify
the states - nevertheless, we can still do it. We know how to solve the
equations of motion for a dilaton that increases with time very slowly
etc. What problem do you exactly have in mind? All these facts about
the stringy S-matrix are totally analogous to QFT, and they are equally
interpreted. We just replace the Feynman diagrams by the worldsheets.
> > You cannot smear over "region of size V" in quantum gravity because the
> > "region of size V" depends on the metric which is dynamical and its
> > quantum fluctuations are really growing all the time as you increase the
> > resolution.
>
> I don't see how the statement after "because" implies
> the one before it.
If you define the volume as the integral of the volume form (think about
\sqrt{g}), then this integral over any finite region in the coordinate
space blows up if you remove the UV cutoff. Quantum mechanically, volume
"V" is one point (infinitesimally small neighborhood in the coordinate
space), and if you average over that, you won't cure any divergences of
the other operators you wanted to regulate.
> Why is that?
Just calculate it. If you expand the Einstein-Hilbert action, and write
the metric as g_{mn} = \eta_{mn} + \sqrt{G} h_{mn}, then the
Einstein-Hilbert action will contain the kinetic term for h_{mn},
schematically (dh)^2, with coefficient one. The dimension of "h" is of
mass by dimensional analysis. There is no dimensionful parameter in the
action (dh)^2, and therefore the average fluctuation of "h" must be equal
to the only dimensionful parameter with the right dimension that you have
- namely the cutoff \Lambda - and therefore it goes to infinity if you try
to send \Lambda to infinity.
These infinities are sort of stopped when \Lambda approaches the Planck
scale - but according to string theory, the whole concept of geometry
breaks down at the Planck scale, too.
The space just fluctuates ever more violently as you look at it with a
better resolution, and as the cutoff approaches the Planck scale, even the
topology is oscillating. String theory stops this chaos and replaces the
naive geometry by a more complete, "non-commutative" physical picture at
the Planck scale - well, perturbative string theory already at the string
scale which is much longer.
Best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0406181010160.5331-100000@feynman.harvard.edu>...\n\n> On Fri, 18 Jun 2004, Squark wrote:\n> > So, you\'re saying that is I want to compute the scattering\n> > of classical waves, I have the imagine them as coherent\n> > states in Fock space and act with the tree-level S-matrix?\n> > Is the result guaranteed to be a coherent state?\n> ...\n> The participants who can read can easily check that I did not use the\n> words "coherent state" in my posting at all, and therefore I was obviously\n> not saying what you are claiming now that I was saying - but if you ask\n> whether the classical waves are coherent states, and whether their\n> classical scattering is encoded in the tree level S-matrix, the answer to\n> both questions is yes.\n\nI was merely trying to understand the detailed mathematical\nrelation between the tree-level S-matrix and the classical\ntheory.\n\n> No, a general interaction process does not produce exact coherent states\n> out of coherent states, and again, if it is a problem, it is a problem of\n> your understanding, not a problem of the fact that the classical\n> scattering is encoded in the tree level S-matrix.\n\nI\'d say it\'s a somewhat hostile way to answer a naive question,\nif you\'ll forgive me. Can you (or someone) explain the exact\nway classical scattering encodes the S-matrix? My problem is\nthat the input/output of classical wave scattering and that of\nthe S-matrix are somewhat different and I haven\'t yet figured\nout how to covert between them.\n\n> > > I am not sure what you exactly mean by the phrase "contains observables".\n> > > It may contain some observables - like the metric at point with\n> > > coordinates (x,y,z,t)\n> >\n> > This is not an observable since it is not diffeomorphism\n> > invariant.\n>\n> It depends whether you define observables only as those that are\n> gauge-invariant, or all of them.\n\nWell, the word "observable" implies something that can be\nobserved. You can only observe gauge invariant objects.\nSometimes people speak of "non-gauge invariant observables"\nbut this is a slight abuse of language.\n\n> > I.e. you\'re saying that the reason QFT contains\n> > observables beyond the S-matrix whereas string\n> > theory does not is that we can insert external\n> > currents into the QFT action and make the\n> > S-matrix depend on them?\n>\n> Yes, it\'s one of the things and interpretations I am saying. In quantum\n> gravity (string theory), you cannot put (couple) external point-like\n> sources consistently which is another reason why we should not expect\n> off-shell local correlators.\n>\n\n(skipping a discussion about the meaning of the words\n"first", "second" and "both" that has little to do\nwith physics)\n\n> By the way, even in this new posting, you have not quite explained what\n> you mean by the "second theory" - probably (nonrenormalizable) SUGRA?\n\nI feel there is a certain confusion with terminology here.\nYou use the word "SUGRA" both for the classical theory\nand the nonrenormalizable QFT and I don\'t always follow\nwhere is which. I believe this usage is indeed typical to\nthe community, however, in this specific discussion it is\nespecially confusing (since both meanings are relevant).\n\n> Well, yes: perturbative calculations only show perturbative statements.\n> But it does not mean that we can\'t demonstrate various similar claims\n> non-perturbatively. For example, graviton scattering can be calculated in\n> M(atrix) theory or AdS/CFT, and it agrees with SUGRA in the classical\n> limit.\n\nIt is interesting how you plug non-trivial backgrounds here.\nWhat does it mean computing scattering in Matrix theory on\na non-trivial (trivial = Minkowski x compact) background.\n\n(skipping a discussion regarding who is more obvious)\n\n> > So is the S-matrix the only observable or not?\n>\n> It\'s the only "nice" observable - the statement that it is the only\n> gauge-invariant observable usually means that the off-shell correlators\n> (Green\'s functions) of local operators - the correlators we like to use in\n> QFT - can\'t be defined in quantum gravity - because there are no local\n> gauge-invariant operators in quantum gravity.\n\nThis doesn\'t get along with your later statements.\nLets clear something out: if you manage to completely\nfix gauge freedom and define observables in the gauge\nfixed picture, these observables are gauge invariant.\n\nA simple example:\n\nConsider a U(1) gauge field coupled to a complex scalar Phi with\ncharge 1. Pick the unitary gauge (Phi real and positive). In\nreality there would be problems, of course, since ambiguities\nmay result, especially when is equal to zero. In fact, for the\nlater reason the unitary gauge is only a good idea if Phi has\nsomething like a Mexican hat potential in addition, but let\'s\nignore these problems for a moment. In the gauge fixed picture,\none has the observable "Phi". In fact, this observable is\ncompletely gauge invariant since it corresponds to |Phi| in the\noriginal picture! So: if, as you claim, there are ways to gauge\nfix diffeomorphism invaraince, you can certain define lots of\ngauge invariant observables. Or maybe your gauge fixing is not\ncomplete?\n\n> > The reason I\'m saying this is as follows: my perceptions,\n> > as a person, at a given moment of my life have to be an exactly (not\n> > approximately) defined observable of any meaningful model of reality.\n>\n> That\'s a good summary of the naive classical thinking. Perceptions at a\n> given moment of a life are totally vague, approximate notions, not only\n> because the observables related to perceptions are notoriously inaccurate\n> and hard to define even in QFT, but especially because even a "given\n> moment" cannot be defined in quantum gravity.\n\n"Given moment" refers to some inner clock of the\nconsciousness, of course, which roughly agrees\nwith proper time.\n\n> > I hope you agree with me on that?!\n>\n> No, I think it is centuries (or millenia) away from the truth in quantum\n> gravity. Already quantum mechanics has showed us that many things that we\n> believed had exact and canonical and universal meaning (position and\n> velocities of an electron) were not well-defined, and quantum gravity goes\n> even further in this process. You seem to be directed in the opposite\n> direction - going from the 17th century to the 15th century.\n\nThe positon and velocities of the electron has nothing to do.\nI think we have reached a nexus here from which stems all of\nthe disagreement in this discussion. Let me get this\nstraight: you believe that the experiences, thoughts, and in\nfact the mere existence of me, you and everyone else cannot\nbe defined in an exact manner in string theory even in\nprinciple (no matter how humongously convoluted the\ndefinition is)? And that nevertheless string theory is\ncorrect?\n\n> > > On the string, light-cone gauge just means that X^+ is\n> > > set to tau (the worldsheet coordinate) and X^- (another\n> > > light-like direction) is calculated in terms of the\n> > > remaining D-2 "purely transverse" bosons X^i(sigma,tau).\n> > > In spacetime it means that only the transverse components\n> > > of various fields - such as the metric tensor fluctuation\n> > > h_{ij} - appear, and the light-like Hamiltonian is\n> > > written as a functional of these components of the fields.\n\n> > All of these things you said can be done in usual QFT\n> > and have nothing to do with gauge fixing diffeomorphism\n> > invariance.\n>\n> The most important goal of what I am explaining you is that *less* things\n> can be defined and computed in quantum gravity, not more! The only way to\n> get as much data in quantum gravity as in local QFT is to gauge-fix (or\n> break) the diffeomorphism invariance. For example, in the light-cone\n> gauge, superstring theory resembles non-gravitational local QF theories.\n\nYour quote: "gauge fixing means imposing additional\nconstraints on the fields (or other degrees of freedom) of your theory\nsuch that each class of configurations that are related by gauge\ntransformations contains at least one representative that satisfies\nthe\nadditional constraints"\n\nNow substitute "gauge transformations" by diffeomorphism. How does\nyour description refelect the content of the definition (for spacetime\ndiffeomorphisms, _not_ worldsheet ones)?\n\n> No, be sure that X^+ is not Lorentz-covariant. If you define a scalar to\n> be Lorentz invariant, then your sentence is a vacuous tautology, but\n> again, the subtlety is that there exist no Lorentz invariant local\n> gauge-invariant observables in a theory of quantum gravity.\n\nSo, you claim there is no Lorentz invariant function of the collection\nof observable you can define in the light-cone gauge? If this is\nindeed\nso, the number of observables has to be no more than the number of\nparameters in the Lorentz group, since the action of the Lorentz group\nshould be able to produce all possible combinations of their values\nfrom any given one. This doesn\'t appear to be true since there are\nlocal observables associated with various values of the transverse\ncoordinates, say, and there is an infinity of those.\n\n> Any asymptotically Minkowski state in the physical Hilbert space can be -\n> at least in the perturbative expansion - obtained via the action of the\n> light cone gauge creation operators in a unique way - there exists no\n> diffeomorphism freedom (one that would preserve the spacetime at\n> infinity), and therefore the assignment of the spacetime coordinate\n> becomes unique. But it is a very Lorentz-non-invariant procedure.\n\nThe question is what happens with this procedure in the classical\napproximation. It appears to me that the only chance for it to\nwork is that it reduces in the classical approximation to\nsomething like what I suggested but with null geodesics instead.\nIt would makes sense in the light of the description on page 4\nof "the world as a hologram" (hep-th/9409089) by Lenny Susskind.\n\n> Because many sets of coordinates can be obtained according to some\n> "gauge-invariant, physical procedures", but it is unfair to pick one such\n> procedure and call it "canonical" or "invariant". The Schwarzschild\n> geometry can be written in the Schwarzschild coordinates where "r" is\n> defined ("covariantly") from the area of the spheres (orbits of the SO(3)\n> symmetry) - but nevertheless we would not say that the precise form of\n> the metric in the Schwarzschild coordinates is something diffeomorphism\n> invariant.\n\nThis prescription only works for one specific solution (the\nSchwarzschild\nsolution), whereas to "gauge fix" you have to do something that always\nworks.\n\n> > > Nevertheless, the common sense usage of the words "gauge-invariant" leads\n> > > to the conclusion that a special choice of coordinates just *cannot* be\n> > > gauge-invariant (under coordinate redefinitions).\n> >\n> > It can if it is "priviliged" as you say.\n>\n> Some coordinates may be "priviliged" or "useful" for a particular purpose,\n> but there are no "generally priviliged" coordinates.\n\nIf you can completely gauge fix diffeomorphism invariance,\nthen for all practical purposes there are, the practical\npurposes being defining local gauge invariant observables.\n\n> > However, superstring theory is supposed to be the ultimate\n> > theory of everything. As such, it should contain not only\n> > the results of particle accelerator experiments (to some\n> > reasonably good approximation) but everything that can\n> > ever be said about the universe!\n>\n> Nope. By the words "theory of everything" we only mean "a theory of\n> everything that really exists in the Universe" - and therefore string\n> theory should *not* contain nonsense.\n\nSo, "everything that really exists in the Universe" is\nparticle accelerator experiments (which also exist only\nto an excellent approximation)?\n\n> > Yes, that\'s why I said "if the world is described by a QFT".\n> > Of course in reality it isn\'t (so I should really have said\n> > "if the world were described by a QFT).\n>\n> So why are you talking about it if you know that it is not true? The\n> discussion is very different and simpler in QFT than in QG.\n\nI was trying to demonstate that previous theories, like QFT,\nhave an advantage over superstring theory in this respect,\ni.e., they had no conceptual missing parts of the sort I\'m\ntalking about which would prevent them from become a TOE,\nthey were just _not_ the TOE experimentally (for instance\nby not describing gravity).\n\n> Because in QFT we compute the spectrum of states and operators, their\n> energies, dimensions, correlators, scattering amplitudes, and things that\n> follow from this list. Your "observables" are not in this list; they are\n> easy to be said in English, but it would be extremely difficult and\n> ambiguous if one tried to convert them into exact science in a quantum\n> theory. In reality, all questions and statements of the sort that you\n> describe are answered by using various classical approximations, and most\n> of the objects in such sentences make no sense outside the region of\n> validity of the classical approximation.\n\nNot at all. The objects make perfect sense, but it\nis unimaginable difficult to make them precise. In\nfact, the whole point of "approximation" is\nconstructing a simplified model which is realized\ninside the more complicated model to some accuracy,\nin some conditions. It is _not_ saying that some\nparticular sector of the complicated model (e.g.\nscattering proccesses) reduces to a particular\nsector of the simplified mode (e.g. classical\nscattering) and this somehow miraculously explains\nthe successful application of all other sectors in\nthe simplified model to reality.\n\n> > So, your claim that my complaints can be addressed to QFT\n> > in the same fashion they can be adrressed to superstring\n> > theory is plainly wrong.\n>\n> My statement was just the opposite.\n\nQuote: "The desire to calculate exact quantities of this sort\nhas more or less exact counterparts in QED"\n\n> All "loopholes" described\n> above, such as the light-cone gauge, are (and must be) fixing the\n> symmetries in a non-local fashion if they want to define gauge-invariant\n> quantities, and such quantities never transform naturally under the\n> Lorentz transformations.\n\nI never wanted observables that don\'t use non-local\ngauge fixing, in fact the observables I have suggested\n_do_ use non-local gauge fixing. The "naturality" of\nthe Lorentz action, OTOH, is an issue of esthetics,\nnot a conceptual one.\n\n> I was just pointing out that you were using incorrect classical intuition\n> for the "proper length", assuming that the "geodesics" and other stuff\n> make sense even if quantum effects are taken into account.\n\nI don\'t see any way the fluctuation of the metric tensor implies\nthere is no observable in the full quantum theory which reduces\nto the geodesic-based stuff in the classical limit.\n\n> These objects are not just "instead" of the metric. The extra fields and\n> other degrees of freedom implied by string/M-theory *complete* the metric\n> tensor into a complete picture where *everything* matters. You cannot\n> remove pieces because the structure would become inconsistent.\n>\n> If you want me to say some non-metric fields predicted by string theory,\n> for example the massive tensor\n>\n> \\alpha_{-2}^\\mu \\tilde\\alpha_{-2}^\\nu |0\\rangle.\n\nAll of this is obvious, however, I refer you again to my\nprevious statement.\n\n> > > AdS/CFT however gives you again a "preferred" choice of coordinates\n> >\n> > How does it do that?!\n>\n> It is much like in the light cone gauge or elsewhere. The asymptotic AdS\n> region at infinity - where the relative fluctuations go to zero - is\n> matched to its standard form, the rest of geometry is extrapolated to the\n> interior\n\nHow do you "extrapolate" coordinates to the interior?\nYou are talking about preferred coordinates, right?\n\n> It\'s not true. If you tried to study a textbook on string theory, you\n> would see that it allows you to calculate the S-matrix between the\n> scattering states - which themselves are adiabatic continuations of the\n> well-known free states at g=0, much like in QFT - without considering any\n> inconsistent backgrounds which would violate the equations of motion.\n>\n> I think that you must know that what you are saying most of the time is\n> simply wrong - because everyone (even some readers of the newspapers)\n> know(s) that string theory allows us to calculate a meaningful S-matrix\n> that follows the standard QFT interpretation in a more complex context,\n> and it is more, not less, consistent than the QFT S-matrices.\n\n(et cetera)\n\nApparently you think I am on some lunatic campaign to destroy\nstring theory, nevertheless, try to believe I was just asking\na simple question. How do you define the adiabatic\ncontinuation of states between different values of g without\nmaking it vary over spacetime in a manner than violates the\nequations of motion?\n\n> If you define the volume as the integral of the volume form (think about\n> \\sqrt{g}), then this integral over any finite region in the coordinate\n> space blows up if you remove the UV cutoff. Quantum mechanically, volume\n> "V" is one point (infinitesimally small neighborhood in the coordinate\n> space), and if you average over that, you won\'t cure any divergences of\n> the other operators you wanted to regulate.\n\nOK, point taken.\n\n> These infinities are sort of stopped when \\Lambda approaches the Planck\n> scale - but according to string theory, the whole concept of geometry\n> breaks down at the Planck scale, too.\n\nIt appears to me there is still hope to define something\nthat would reduce the the objects I described in the\nclassical limit. In the Planckian regime this "something"\nwould look totally different, of course.\n\nBest regards,\nSquark.\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> wrote in message news:<Pine.LNX.4.31.0406181010160.5331-100000@feynman.harvard.edu>...
> On Fri, 18 Jun 2004, Squark wrote:
> > So, you're saying that is I want to compute the scattering
> > of classical waves, I have the imagine them as coherent
> > states in Fock space and act with the tree-level S-matrix?
> > Is the result guaranteed to be a coherent state?
> ...
> The participants who can read can easily check that I did not use the
> words "coherent state" in my posting at all, and therefore I was obviously
> not saying what you are claiming now that I was saying - but if you ask
> whether the classical waves are coherent states, and whether their
> classical scattering is encoded in the tree level S-matrix, the answer to
> both questions is yes.
I was merely trying to understand the detailed mathematical
relation between the tree-level S-matrix and the classical
theory.
> No, a general interaction process does not produce exact coherent states
> out of coherent states, and again, if it is a problem, it is a problem of
> your understanding, not a problem of the fact that the classical
> scattering is encoded in the tree level S-matrix.
I'd say it's a somewhat hostile way to answer a naive question,
if you'll forgive me. Can you (or someone) explain the exact
way classical scattering encodes the S-matrix? My problem is
that the input/output of classical wave scattering and that of
the S-matrix are somewhat different and I haven't yet figured
out how to covert between them.
> > > I am not sure what you exactly mean by the phrase "contains observables".
> > > It may contain some observables - like the metric at point with
> > > coordinates (x,y,z,t)
> >
> > This is not an observable since it is not diffeomorphism
> > invariant.
>
> It depends whether you define observables only as those that are
> gauge-invariant, or all of them.
Well, the word "observable" implies something that can be
observed. You can only observe gauge invariant objects.
Sometimes people speak of "non-gauge invariant observables"
but this is a slight abuse of language.
> > I.e. you're saying that the reason QFT contains
> > observables beyond the S-matrix whereas string
> > theory does not is that we can insert external
> > currents into the QFT action and make the
> > S-matrix depend on them?
>
> Yes, it's one of the things and interpretations I am saying. In quantum
> gravity (string theory), you cannot put (couple) external point-like
> sources consistently which is another reason why we should not expect
> off-shell local correlators.
>
(skipping a discussion about the meaning of the words
"first", "second" and "both" that has little to do
with physics)
> By the way, even in this new posting, you have not quite explained what
> you mean by the "second theory" - probably (nonrenormalizable) SUGRA?
I feel there is a certain confusion with terminology here.
You use the word "SUGRA" both for the classical theory
and the nonrenormalizable QFT and I don't always follow
where is which. I believe this usage is indeed typical to
the community, however, in this specific discussion it is
especially confusing (since both meanings are relevant).
> Well, yes: perturbative calculations only show perturbative statements.
> But it does not mean that we can't demonstrate various similar claims
> non-perturbatively. For example, graviton scattering can be calculated in
> M(atrix) theory or AdS/CFT, and it agrees with SUGRA in the classical
> limit.
It is interesting how you plug non-trivial backgrounds here.
What does it mean computing scattering in Matrix theory on
a non-trivial (trivial = Minkowski x compact) background.
(skipping a discussion regarding who is more obvious)
> > So is the S-matrix the only observable or not?
>
> It's the only "nice" observable - the statement that it is the only
> gauge-invariant observable usually means that the off-shell correlators
> (Green's functions) of local operators - the correlators we like to use in
> QFT - can't be defined in quantum gravity - because there are no local
> gauge-invariant operators in quantum gravity.
This doesn't get along with your later statements.
Lets clear something out: if you manage to completely
fix gauge freedom and define observables in the gauge
fixed picture, these observables are gauge invariant.
A simple example:
Consider a U(1) gauge field coupled to a complex scalar \Phi with
charge 1. Pick the unitary gauge (\Phi real and positive). In
reality there would be problems, of course, since ambiguities
may result, especially when is equal to zero. In fact, for the
later reason the unitary gauge is only a good idea if \Phi has
something like a Mexican hat potential in addition, but let's
ignore these problems for a moment. In the gauge fixed picture,
one has the observable "\Phi". In fact, this observable is
completely gauge invariant since it corresponds to |\Phi| in the
original picture! So: if, as you claim, there are ways to gauge
fix diffeomorphism invaraince, you can certain define lots of
gauge invariant observables. Or maybe your gauge fixing is not
complete?
> > The reason I'm saying this is as follows: my perceptions,
> > as a person, at a given moment of my life have to be an exactly (not
> > approximately) defined observable of any meaningful model of reality.
>
> That's a good summary of the naive classical thinking. Perceptions at a
> given moment of a life are totally vague, approximate notions, not only
> because the observables related to perceptions are notoriously inaccurate
> and hard to define even in QFT, but especially because even a "given
> moment" cannot be defined in quantum gravity.
"Given moment" refers to some inner clock of the
consciousness, of course, which roughly agrees
with proper time.
> > I hope you agree with me on that?!
>
> No, I think it is centuries (or millenia) away from the truth in quantum
> gravity. Already quantum mechanics has showed us that many things that we
> believed had exact and canonical and universal meaning (position and
> velocities of an electron) were not well-defined, and quantum gravity goes
> even further in this process. You seem to be directed in the opposite
> direction - going from the 17th century to the 15th century.
The positon and velocities of the electron has nothing to do.
I think we have reached a nexus here from which stems all of
the disagreement in this discussion. Let me get this
straight: you believe that the experiences, thoughts, and in
fact the mere existence of me, you and everyone else cannot
be defined in an exact manner in string theory even in
principle (no matter how humongously convoluted the
definition is)? And that nevertheless string theory is
correct?
> > > On the string, light-cone gauge just means that X^+ is
> > > set to \tau (the worldsheet coordinate) and X^- (another
> > > light-like direction) is calculated in terms of the
> > > remaining D-2 "purely transverse" bosons X^i(\sigma,\tau).
> > > In spacetime it means that only the transverse components
> > > of various fields - such as the metric tensor fluctuation
> > > h_{ij} - appear, and the light-like Hamiltonian is
> > > written as a functional of these components of the fields.
> > All of these things you said can be done in usual QFT
> > and have nothing to do with gauge fixing diffeomorphism
> > invariance.
>
> The most important goal of what I am explaining you is that *less* things
> can be defined and computed in quantum gravity, not more! The only way to
> get as much data in quantum gravity as in local QFT is to gauge-fix (or
> break) the diffeomorphism invariance. For example, in the light-cone
> gauge, superstring theory resembles non-gravitational local QF theories.
Your quote: "gauge fixing means imposing additional
constraints on the fields (or other degrees of freedom) of your theory
such that each class of configurations that are related by gauge
transformations contains at least one representative that satisfies
the
additional constraints"
Now substitute "gauge transformations" by diffeomorphism. How does
your description refelect the content of the definition (for spacetime
diffeomorphisms, _not_ worldsheet ones)?
> No, be sure that X^+ is not Lorentz-covariant. If you define a scalar to
> be Lorentz invariant, then your sentence is a vacuous tautology, but
> again, the subtlety is that there exist no Lorentz invariant local
> gauge-invariant observables in a theory of quantum gravity.
So, you claim there is no Lorentz invariant function of the collection
of observable you can define in the light-cone gauge? If this is
indeed
so, the number of observables has to be no more than the number of
parameters in the Lorentz group, since the action of the Lorentz group
should be able to produce all possible combinations of their values
from any given one. This doesn't appear to be true since there are
local observables associated with various values of the transverse
coordinates, say, and there is an infinity of those.
> Any asymptotically Minkowski state in the physical Hilbert space can be -
> at least in the perturbative expansion - obtained via the action of the
> light cone gauge creation operators in a unique way - there exists no
> diffeomorphism freedom (one that would preserve the spacetime at
> infinity), and therefore the assignment of the spacetime coordinate
> becomes unique. But it is a very Lorentz-non-invariant procedure.
The question is what happens with this procedure in the classical
approximation. It appears to me that the only chance for it to
work is that it reduces in the classical approximation to
something like what I suggested but with null geodesics instead.
It would makes sense in the light of the description on page 4
of "the world as a hologram" (http://www.arxiv.org/abs/hep-th/9409089) by Lenny Susskind.
> Because many sets of coordinates can be obtained according to some
> "gauge-invariant, physical procedures", but it is unfair to pick one such
> procedure and call it "canonical" or "invariant". The Schwarzschild
> geometry can be written in the Schwarzschild coordinates where "r" is
> defined ("covariantly") from the area of the spheres (orbits of the SO(3)
> symmetry) - but nevertheless we would not say that the precise form of
> the metric in the Schwarzschild coordinates is something diffeomorphism
> invariant.
This prescription only works for one specific solution (the
Schwarzschild
solution), whereas to "gauge fix" you have to do something that always
works.
> > > Nevertheless, the common sense usage of the words "gauge-invariant" leads
> > > to the conclusion that a special choice of coordinates just *cannot* be
> > > gauge-invariant (under coordinate redefinitions).
> >
> > It can if it is "priviliged" as you say.
>
> Some coordinates may be "priviliged" or "useful" for a particular purpose,
> but there are no "generally priviliged" coordinates.
If you can completely gauge fix diffeomorphism invariance,
then for all practical purposes there are, the practical
purposes being defining local gauge invariant observables.
> > However, superstring theory is supposed to be the ultimate
> > theory of everything. As such, it should contain not only
> > the results of particle accelerator experiments (to some
> > reasonably good approximation) but everything that can
> > ever be said about the universe!
>
> Nope. By the words "theory of everything" we only mean "a theory of
> everything that really exists in the Universe" - and therefore string
> theory should *not* contain nonsense.
So, "everything that really exists in the Universe" is
particle accelerator experiments (which also exist only
to an excellent approximation)?
> > Yes, that's why I said "if the world is described by a QFT".
> > Of course in reality it isn't (so I should really have said
> > "if the world were described by a QFT).
>
> So why are you talking about it if you know that it is not true? The
> discussion is very different and simpler in QFT than in QG.
I was trying to demonstate that previous theories, like QFT,
have an advantage over superstring theory in this respect,
i.e., they had no conceptual missing parts of the sort I'm
talking about which would prevent them from become a TOE,
they were just _not_ the TOE experimentally (for instance
by not describing gravity).
> Because in QFT we compute the spectrum of states and operators, their
> energies, dimensions, correlators, scattering amplitudes, and things that
> follow from this list. Your "observables" are not in this list; they are
> easy to be said in English, but it would be extremely difficult and
> ambiguous if one tried to convert them into exact science in a quantum
> theory. In reality, all questions and statements of the sort that you
> describe are answered by using various classical approximations, and most
> of the objects in such sentences make no sense outside the region of
> validity of the classical approximation.
Not at all. The objects make perfect sense, but it
is unimaginable difficult to make them precise. In
fact, the whole point of "approximation" is
constructing a simplified model which is realized
inside the more complicated model to some accuracy,
in some conditions. It is _not_ saying that some
particular sector of the complicated model (e.g.
scattering proccesses) reduces to a particular
sector of the simplified mode (e.g. classical
scattering) and this somehow miraculously explains
the successful application of all other sectors in
the simplified model to reality.
> > So, your claim that my complaints can be addressed to QFT
> > in the same fashion they can be adrressed to superstring
> > theory is plainly wrong.
>
> My statement was just the opposite.
Quote: "The desire to calculate exact quantities of this sort
has more or less exact counterparts in QED"
> All "loopholes" described
> above, such as the light-cone gauge, are (and must be) fixing the
> symmetries in a non-local fashion if they want to define gauge-invariant
> quantities, and such quantities never transform naturally under the
> Lorentz transformations.
I never wanted observables that don't use non-local
gauge fixing, in fact the observables I have suggested
_do_ use non-local gauge fixing. The "naturality" of
the Lorentz action, OTOH, is an issue of esthetics,
not a conceptual one.
> I was just pointing out that you were using incorrect classical intuition
> for the "proper length", assuming that the "geodesics" and other stuff
> make sense even if quantum effects are taken into account.
I don't see any way the fluctuation of the metric tensor implies
there is no observable in the full quantum theory which reduces
to the geodesic-based stuff in the classical limit.
> These objects are not just "instead" of the metric. The extra fields and
> other degrees of freedom implied by string/M-theory *complete* the metric
> tensor into a complete picture where *everything* matters. You cannot
> remove pieces because the structure would become inconsistent.
>
> If you want me to say some non-metric fields predicted by string theory,
> for example the massive tensor
>
> \alpha_{-2}^\mu \tilde\alpha_{-2}^\nu |0\rangle.
All of this is obvious, however, I refer you again to my
previous statement.
> > > AdS/CFT however gives you again a "preferred" choice of coordinates
> >
> > How does it do that?!
>
> It is much like in the light cone gauge or elsewhere. The asymptotic AdS
> region at infinity - where the relative fluctuations go to zero - is
> matched to its standard form, the rest of geometry is extrapolated to the
> interior
How do you "extrapolate" coordinates to the interior?
You are talking about preferred coordinates, right?
> It's not true. If you tried to study a textbook on string theory, you
> would see that it allows you to calculate the S-matrix between the
> scattering states - which themselves are adiabatic continuations of the
> well-known free states at g=0, much like in QFT - without considering any
> inconsistent backgrounds which would violate the equations of motion.
>
> I think that you must know that what you are saying most of the time is
> simply wrong - because everyone (even some readers of the newspapers)
> know(s) that string theory allows us to calculate a meaningful S-matrix
> that follows the standard QFT interpretation in a more complex context,
> and it is more, not less, consistent than the QFT S-matrices.
(et cetera)
Apparently you think I am on some lunatic campaign to destroy
string theory, nevertheless, try to believe I was just asking
a simple question. How do you define the adiabatic
continuation of states between different values of g without
making it vary over spacetime in a manner than violates the
equations of motion?
> If you define the volume as the integral of the volume form (think about
> \sqrt{g}), then this integral over any finite region in the coordinate
> space blows up if you remove the UV cutoff. Quantum mechanically, volume
> "V" is one point (infinitesimally small neighborhood in the coordinate
> space), and if you average over that, you won't cure any divergences of
> the other operators you wanted to regulate.
OK, point taken.
> These infinities are sort of stopped when \Lambda approaches the Planck
> scale - but according to string theory, the whole concept of geometry
> breaks down at the Planck scale, too.
It appears to me there is still hope to define something
that would reduce the the objects I described in the
classical limit. In the Planckian regime this "something"
would look totally different, of course.
Best regards,
Squark.
Lubos Motl
Jun19-04, 12:08 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 Sat, 19 Jun 2004, Squark wrote:\n\n> I was merely trying to understand the detailed mathematical\n> relation between the tree-level S-matrix and the classical\n> theory.\n\nWell, my feeling has been the opposite one - namely that you were trying\nto claim at least in five long postings that no such a relation exists, so\nthat you can avoid all attempts of those who are trying to explain you\nthis important and elementary relation. ;-)\n\nIf you compute the S-matrix via path integrals, the tree-level diagrams\nsimply correspond to approximating the path integral by its stationary\npoints which are the classical solutions of the corresponding classical\nequations of motion. This is how the perturbative calculation in powers of\nhbar works in any quantum theory.\n\nThe loop diagrams (contributions to the amplitudes) are proportional to\nadditional positive powers of hbar.\n\nFor example, the path integral for the evolution between the initial\nconfiguration <I| and the final configuration |F> equals, in the classical\nlimit, to the contribution \\exp(i S_{classical}) of the classical solution\nsatisfying these initial conditions, and all amplitudes in a quantum\ntheory are just various perturbative functional expansions of this\nclassical amplitude - the derivatives of the classically calculated action\nwith respect to the final conditions determine the final velocities, and\nso on. The tree-level S-matrix therefore contains all the information\nabout classical (field-theoretical) physics that depends on specifying the\nconditions at |t|=infinity, and vice versa.\n\nIt\'s just obvious that the tree-level and classical field theory limits of\na QFT contain the same information, because both of them contain all\ninformation about the hbar=0 limit of all QFT observables.\n\n> I feel there is a certain confusion with terminology here.\n\nI don\'t think so.\n\n> You use the word "SUGRA" both for the classical theory\n> and the nonrenormalizable QFT ...\n\nYes, it\'s because both of them are SUGRA, and the relation between the\nclassical theory and the tree-level approximation of the quantum theory\nis trivial. There is no reason to use different words for the same thing.\n\n> and I don\'t always follow where is which. I believe this usage is\n> indeed typical to the community, however, in this specific discussion\n> it is especially confusing (since both meanings are relevant).\n\nCould you be more specific about the statement where there was any\nconfusion? Well, if someone does not want to get rid of the idea that\nthere is a huge difference between the data about a classical theory and a\nquantum theory in the tree-level approximation, he may have problems with\nunderstanding the usage of the word SUGRA. But it is not a mere problem of\nterminology - it is mainly a problem of his understanding of the relations\nbetween a classical and quantum theory.\n\nInstead of repeatedly suggesting that there is a problem with the whole\ncommunity and with the whole string theory or perhaps the whole\ntheoretical physics and all these things that you have written at least 6\ntimes, is not it a better idea to first consider the possibility that\nthere may be a problem with *your* assumptions and *your* understanding of\nphysics? If you think that you are smarter than all physicists, then I\napologize for having insulted you, especially because I am very far from\nsharing this opinion of yours.\n\n> It is interesting how you plug non-trivial backgrounds here.\n> What does it mean computing scattering in Matrix theory on\n> a non-trivial (trivial = Minkowski x compact) background.\n\nWe don\'t know how to define Matrix theory on general backgrounds, which is\nthe main handicap that slowed down progress in Matrix theory. Matrix\ntheory however describes all physics on asymptotically Minkowski (times an\nacceptable manifold) backgrounds. It contains very complicated geometries\nthat are nevertheless asymptotically Minkowski; by locality, you may say\nthat physics of any 11D geometry is included because you can extend it to\nan asymptotically Minkowski spacetime.\n\nOne can in principle derive Matrix theory for some mildly curved\nbackgrounds with light-like Killing vectors as well.\n\n> This doesn\'t get along with your later statements.\n\nIt gets along very well.\n\n> Lets clear something out: if you manage to completely\n> fix gauge freedom and define observables in the gauge\n> fixed picture, these observables are gauge invariant.\n\nIf you dress your operators to make them gauge-invariant, you will have\ngauge-invariant operators, but they will be non-locally related to the\noriginal ones.\n\n> A simple example:\n> Consider a U(1) gauge field coupled to a complex scalar Phi with\n> charge 1. Pick the unitary gauge (Phi real and positive). In ...\n\nI agree with this. The light-cone gauge-fixing *is* complete, by the way.\n\n> "Given moment" refers to some inner clock of the\n> consciousness, of course, which roughly agrees\n> with proper time.\n\nConsciousness has no exact "inner clock" ;-), and the proper time only\nmakes sense if you define a classical worldline where it\'s measured. These\nnotions are very useful, but they are only well-defined in the classical\napproximation in which the quantum fluctuations of the metric and the\nworldlines are neglected.\n\n> The positon and velocities of the electron has nothing to do.\n\nBe sure that it does! It\'s one half of the phrase "quantum gravity".\nQuantum gravity is a theory where all these principles, such as the\nuncertainty principle, are also applied to spacetime geometry. Even the\nloop quantum gravity people would agree that you are wrong, and you bet\nthat such a statement can hardly be viewed as a compliment. ;-)\n\nMuch like X and P do not commute in mechanics,\ng_{23} and \\partial_0 g_{23} do not commute in quantum gravity. It is\nvirtually an exact analogy.\n\nThinking about the metric tensor that has a well-defined value at some\npoint as well as a well-defined time-derivative - which is what you are\ndoing essentially in every single sentence of your postings - is\nincompatible with the basic principle of quantum theory called the\nuncertainty principle.\n\n> I think we have reached a nexus here from which stems all of\n> the disagreement in this discussion.\n\nI am afraid that we have reached it already at the beginning when I\ndecided to answer any question asked by someone who obviously *never*\nwishes to accept even the most basic principles of quantum mechanics.\n\n> Let me get this straight: you believe that the experiences, thoughts,\n> and in fact the mere existence of me, you and everyone else cannot be\n> defined in an exact manner in string theory even in principle (no\n> matter how humongously convoluted the definition is)? And that\n> nevertheless string theory is correct?\n\nDefinitely, this is how the whole physics works. Most notions, statements,\nobjects that can be defined in human every-day language have no exact\ncounterparts in a physical theory. The more we know, the more we\nunderstand that our previous concepts do not exist in the exact theory.\nPeople thought that they could always say whether two events are\nsimultaneous - no, because of relativity; they also thought that it was\nalways possible to say where an electron is and how quickly it moves - no,\nbecause of the uncertainty principle; or they thought that that could have\nsaid how many positrons and electrons were in a given region - no, because\nof QED fluctuations below the Compton wavelength.\n\nAll these things are simply caused by incorrect prejudices, by wrong\nmodels, by unjustifiable extrapolations of some approximate insights that\nare nearly true in the every-day life, but they become invalid further.\nWhat I say is not specific to string theory; it holds for any theory in\nmodern physics. The models that you are using are *extremely* incorrect\nbecause you even want to contradict the uncertainty principle for the\nmetric.\n\nThe more correct and complete models we deal with, the more they can show\nthat the previous concepts and descriptions were unusable, inaccurate, or\nwrong in general.\n\n(By the way, again, the light-cone gauge is an extreme case where we can\nmake the commutator of the fields and their light-cone-time derivatives\nvanish - exactly because the "time" is null.)\n\nPlease don\'t ask me again this question whether some formerly meaningful\nconcepts must be meaningful forever - because you have already done it 6\ntimes, and I answered 6 times. Yes, I think it is probably very difficult\nto understand modern physics without the understanding of the fact that\nsome concepts from the old theories can become meaningless in a more\ncomplete theory.\n\n> Your quote: "gauge fixing means imposing additional constraints on the\n> fields (or other degrees of freedom) of your theory such that each\n> class of configurations that are related by gauge transformations\n> contains at least one representative that satisfies the additional\n> constraints"\n>\n> Now substitute "gauge transformations" by diffeomorphism.\n\nDiffeomorphism fixing means imposing additional constraints on the fields\n(or other degrees of freedom) of your theory such that each class of\nconfigurations that are related by gauge (coordinate) transformations\ncontains at least one representative that satisfies the additional\nconstraints.\n\nSatisfied?\n\n> How does your description refelect the content of the definition (for\n> spacetime diffeomorphisms, _not_ worldsheet ones)?\n\nVery well, thank you for asking.\n\n> > No, be sure that X^+ is not Lorentz-covariant. If you define a scalar to\n> > be Lorentz invariant, then your sentence is a vacuous tautology, but\n> > again, the subtlety is that there exist no Lorentz invariant local\n> > gauge-invariant observables in a theory of quantum gravity.\n>\n> So, you claim there is no Lorentz invariant function of the collection\n> of observable you can define in the light-cone gauge?\n\nYou forgot the word "local".\n\n> If this is indeed so, the number of observables has to be no more than\n> the number of parameters in the Lorentz group, since the action of the\n> Lorentz group should be able to produce all possible combinations of\n> their values from any given one.\n\nYou forgot the word "local".\n\n> This doesn\'t appear to be true since there are local observables\n> associated with various values of the transverse coordinates, say, and\n> there is an infinity of those.\n\nOnce again, there exist no local Lorentz- and gauge-invariant observables\nin a theory of quantum gravity which is a trivial consequence of the\ndiffeomorphism invariance.\n\n> > Any asymptotically Minkowski state in the physical Hilbert space can be -\n> > at least in the perturbative expansion - obtained via the action of the\n> > light cone gauge creation operators in a unique way - there exists no\n> > diffeomorphism freedom (one that would preserve the spacetime at\n> > infinity), and therefore the assignment of the spacetime coordinate\n> > becomes unique. But it is a very Lorentz-non-invariant procedure.\n>\n> The question is what happens with this procedure in the classical\n> approximation.\n\nThe procedure (of generating the spectrum in the light-cone gauge) *was*\nclassical in spacetime because it considers free string theory at g=0. If\nyou want to talk about a classical theory on the worldsheet, then you will\nget nothing like a local field theory in spacetime and nothing like\nspacetime diffeomorphism invariance (or other local symmetries).\n\nTo get this local field-theory-like physics in spacetime, the strings must\nbe quantized on the worldsheet. Classical strings are just classical\npieces of rubber band that propagate on a fixed and unchangeable\nbackground geometry. All the things like "predicting gravity,\nelectromagnetism, particles" and all local symmetries only arise once the\nworldsheet is quantized.\n\n> It appears to me that the only chance for it to\n> work is that it reduces in the classical approximation to\n> something like what I suggested but with null geodesics instead.\n> It would makes sense in the light of the description on page 4\n> of "the world as a hologram" (hep-th/9409089) by Lenny Susskind.\n\nNot sure which relation between Lenny\'s paper and light-cone gauge string\noscillators you exactly want to investigate.\n\n> This prescription only works for one specific solution (the\n> Schwarzschild solution), whereas to "gauge fix" you have to do\n> something that always works.\n\nNo gauge fixing can work for the *whole* theory (string theory) because\nthe identity of the useful gauge symmetry itself is changing from point to\npoint, and therefore one needs different things to be gauge-fixed. Gauge\nfixing can only work universally around some region in the\nconfiguration/moduli space. By the way, the gauge-fixing described for the\nSchwarzschild can be generalized - although it is not easy to define the\nexact rules - to a big class of topologically trivial asymptotically\nMinkowski metrics. Even though it can be done, it does not change anything\nabout the fact that it is just one possible gauge-fixing and it cannot be\n"universally priviliged".\n\n> > Some coordinates may be "priviliged" or "useful" for a particular purpose,\n> > but there are no "generally priviliged" coordinates.\n>\n> If you can completely gauge fix diffeomorphism invariance,\n> then for all practical purposes there are, the practical\n> purposes being defining local gauge invariant observables.\n\nIf you can do a random thing, it does not mean that it is priviliged -\non the contrary, most likely it is not.\n\n> > Nope. By the words "theory of everything" we only mean "a theory of\n> > everything that really exists in the Universe" - and therefore string\n> > theory should *not* contain nonsense.\n>\n> So, "everything that really exists in the Universe" is\n> particle accelerator experiments (which also exist only\n> to an excellent approximation)?\n\nString theory should (and, at least qualitatively, it does) reproduce the\nevents seen at the particle accelerators, but it in no way reproduces your\nnaive, incorrect models how the accelerators or other things in the\nUniverse work - models that contradict the uncertainty principle, among\nmany other things.\n\n> > So why are you talking about it if you know that it is not true? The\n> > discussion is very different and simpler in QFT than in QG.\n>\n> I was trying to demonstate that previous theories, like QFT,\n> have an advantage over superstring theory in this respect,\n\nIt\'s not a real advantage, it is just simplicity. Simplicity implied by\nthe absence of general covariance. It\'s the same simplicity that implies\nthat QFT are not unique, while the bigger constraints in quantum gravity\nimply that a theory of quantum gravity probably *is* unique\n(string/M-theory).\n\nYou could also say that the classical theories have an "advantage" over\nthe quantum theories because many more things can be said\n(positions+velocities), and so on. In reality, it is over-compensated by\nthe disadvantage that the classical theories simply disagree with reality\n(and they would not admit the existence of life as we know it).\n\nSimilarly, the real world contains gravity, and you may scream that\nnon-gravitational QFTs have an advantage, but all this screaming is just\ntotally irrelevant because we know that these theories don\'t describe the\nwhole Universe.\n\n> i.e., they had no conceptual missing parts of the sort I\'m\n> talking about which would prevent them from become a TOE,\n> they were just _not_ the TOE experimentally (for instance\n> by not describing gravity).\n\nA theory of everything does not mean a theory of nonsensical concepts\nthat are incompatible with the basic principles of physics. A theory, if\nit should have a chance to be a theory of everything, is not allowed to\ncontain anything of the type that you say simply because what you say is\ninconsistent rant based on 17th century models of reality.\n\n> Not at all. The objects make perfect sense, but it\n> is unimaginable difficult to make them precise.\n\nThere is no unique and canonical way to make them precise - simply because\nthe domain of their validity is limited, much like the domain of validity\nof virtually any concept in science and reality. One can perhaps define\nsome convoluted exact notions that are valid under (almost) any\ncircumstances and that reduce to the old notions in the appropriate limit,\nbut such a generalization cannot be unique. In most cases, one can show\nthat a generalization that would satisfy some extra conditions simply\ncan\'t exist at all.\n\n> In fact, the whole point of "approximation" is constructing a\n> simplified model which is realized inside the more complicated model\n> to some accuracy, in some conditions. It is _not_ saying that some\n> particular sector of the complicated model (e.g. scattering\n> proccesses) reduces to a particular sector of the simplified mode\n> (e.g. classical scattering) and this somehow miraculously explains the\n> successful application of all other sectors in the simplified model to\n> reality.\n\nWhat you say is the opposite of the truth, at least in modern physics. Our\nmore general theories that extend the previous approximations nearly\nalways use a different language, and the old concepts are simply *not*\ngeneralized and they can only be defined in the limit in which the old\napproximation is acceptable.\n\nIn non-relativistic physics, you can ask what is the time difference\nbetween two events. Special relativity explains that there cannot be any\nuniversal answer because the answer depends on the observer. Classical\nphysics allows us to talk about the exact positions and velocities, and\nunique predictions of their future values. These numbers just don\'t make\nany sense in quantum mechanics.\n\nThe strength of the electromagnetic force - the fine structure constant -\nwas thought to be constant and one could have asked how much it exactly\nis, but people realized that it runs as a function of the energy scale,\nand at shorter distances, a bigger fine structure constant is relevant.\nNew refined interactions and particles are being added, and the old\ntheories are just not good anymore - even their language becomes\ninappropriate because they are showed to be built on many assumptions that\nare simply not true - only approximately true.\n\nI find it highly surprising that this basic aspect of progress in physics\n- replacing the old concepts by totally new ones, reducing the role of the\nwhole old language to a mere approximation - could be still hidden from\nsomeone who spends so much time with physics newsgroups. You\'re not\nquestioning just string theory, you\'re questioning the meaning of all of\ntheoretical physics.\n\n> > > So, your claim that my complaints can be addressed to QFT\n> > > in the same fashion they can be adrressed to superstring\n> > > theory is plainly wrong.\n> >\n> > My statement was just the opposite.\n>\n> Quote: "The desire to calculate exact quantities of this sort\n> has more or less exact counterparts in QED"\n\nI used it in a very specific context - "this sort" meant a very specific\nthing, and you should not be surprised that if you replace "this sort" by\na completely different sort, you may again obtain wrong conclusions.\n\nIn "this sort" sentence I was trying to tell you that your construction\nviolated some properties of quantum gravity such that it could have been\nmade even in QFT - you simply violate the uncertainty principle. I was\nalso trying to explain you that quantum gravity is much more demanding -\nmany things that could be well-defined even in as complex framework as\nQFT become meaningless in quantum gravity.\n\nI don\'t exactly see where you see any contradiction. I am telling you that\nyour thinking is too naive even in ordinary QFT (or QM), and therefore it\nis guaranteed that it is naive in QG, too, because QG is much more\nsophisticated and constraining than QFT.\n\n> I never wanted observables that don\'t use non-local\n> gauge fixing, in fact the observables I have suggested\n> _do_ use non-local gauge fixing. The "naturality" of\n> the Lorentz action, OTOH, is an issue of esthetics,\n> not a conceptual one.\n\nThe question whether one wants to compute the gravitational S-matrix or\nsome pseudolocal correlators of some operators defined in a convoluted\ngauge-fixing scheme is also mostly an issue of esthetics, and most people\nprefer the former.\n\n> > I was just pointing out that you were using incorrect classical intuition\n> > for the "proper length", assuming that the "geodesics" and other stuff\n> > make sense even if quantum effects are taken into account.\n>\n> I don\'t see any way the fluctuation of the metric tensor implies\n> there is no observable in the full quantum theory which reduces\n> to the geodesic-based stuff in the classical limit.\n\nBecause in the QFT description, these fluctuations become infinitely big,\nif you remove all approximations and cutoffs, and all quantities you talk\nabout are either infinity or zero in this full treatment. You can use a\nbeyond-QFT description of string theory, but string theory just does not\nallow you to define various objects that you want to define, and it is\nimportant that it does not allow one to do so.\n\n> > \\alpha_{-2}^\\mu \\tilde\\alpha_{-2}^\\nu |0\\rangle.\n>\n> All of this is obvious, however, I refer you again to my\n> previous statement.\n\nOK, so I did not get the meaning of your statement "instead of the metric"\nif it is obvious to you.\n\n> How do you "extrapolate" coordinates to the interior?\n\nI am not doing anything like that by hand. AdS/CFT does similar things\nautomatically. No one knows exactly how to derive the local phenomena\ninside the AdS space from the boundary CFT description (how to define the\napproximate locality in the holographic dimension, in particular),\nnevertheless everyone is convinced that this local physics in the bulk is\ncontained in the CFT and explicit tests of everything we can calculate\nconfirm it.\n\n> You are talking about preferred coordinates, right?\n\nThe AdS/CFT map for the local phenomena in the bulk is very nontrivial,\nbut the correspondence does not change anything about the fact that the\nphysics in the bulk is equivalent to a theory that exhibits general\ncovariance at low energies - a theory that is most naturally described in\nterms of degree of freedom with no preferred coordinates - and it does not\nchange the fact that the metric in the bulk is a quantum mechanical,\nfluctuating degree of freedom.\n\nThe AdS/CFT computations are analogous to the S-matrix - the AdS boundary\nplays the role of t=-infinity and t=+infinity in the S-matrix treatment.\nIn this sense, you can imagine that "anything can happen" and "any\ncoordinates can be used" in the bulk - the only place where the\ncoordinates are required to match a "standard" is near the boundary\n(analogy of |t|=infinity).\n\n> Apparently you think I am on some lunatic campaign to destroy\n> string theory, nevertheless, try to believe I was just asking\n> a simple question. How do you define the adiabatic\n> continuation of states between different values of g without\n> making it vary over spacetime in a manner than violates the\n> equations of motion?\n\nJust like in QFT. You infinitely slowly change the coupling constant, and\nlook what\'s happening with the state. Be sure that if the evolution of the\ndilaton is slow, I can even prove the existence of the required solution\nof the dilaton equations of motion and this solution will be arbitrarily\nclose to simply "phi=epsilon.t" for a very small epsilon.\n\nThe smaller the coupling constant it, the easier and less controversial\nsuch an adiabatic "turning on" will be. For bigger "g" the perturbative\nexpansions break down anyway.\n\nEven if there were some subtle problems with any procedure described 2\nparagraphs ago, we just don\'t need to define the external states in this\nway - the identity of a graviton, for example, is totally determined by\nits momentum and polarizations. A graviton with momentum "p" and\npolarization tensor "ij" simply *exists*, and your attempts to define it,\nredefine it and mix with many other concepts are not well-motivated.\n\nEven if we could not imagine adiabatic changes of the coupling and similar\npsychological games, it would be a perfectly acceptable and complete to\nhave a theory with a known list of allowed external states, and the tools\nto calculate their cross sections.\n\n> It appears to me there is still hope to define something\n> that would reduce the the objects I described in the\n> classical limit. In the Planckian regime this "something"\n> would look totally different, of course.\n\nBut there is no reason why such generalized objects should be defined\nuniquely, and there is no reason why such notions should be useful to\ndescribe the Planckian physics. We only know that they are useful to\ntalk about low-energy, classical physics, but the physics at the Planck\nscale is a very different beast.\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: +1-617/496-8199 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 Sat, 19 Jun 2004, Squark wrote:
> I was merely trying to understand the detailed mathematical
> relation between the tree-level S-matrix and the classical
> theory.
Well, my feeling has been the opposite one - namely that you were trying
to claim at least in five long postings that no such a relation exists, so
that you can avoid all attempts of those who are trying to explain you
this important and elementary relation. ;-)
If you compute the S-matrix via path integrals, the tree-level diagrams
simply correspond to approximating the path integral by its stationary
points which are the classical solutions of the corresponding classical
equations of motion. This is how the perturbative calculation in powers of
\hbar works in any quantum theory.
The loop diagrams (contributions to the amplitudes) are proportional to
additional positive powers of \hbar.
For example, the path integral for the evolution between the initial
configuration <I| and the final configuration |F> equals, in the classical
limit, to the contribution \exp(i S_{classical}) of the classical solution
satisfying these initial conditions, and all amplitudes in a quantum
theory are just various perturbative functional expansions of this
classical amplitude - the derivatives of the classically calculated action
with respect to the final conditions determine the final velocities, and
so on. The tree-level S-matrix therefore contains all the information
about classical (field-theoretical) physics that depends on specifying the
conditions at |t|=infinity, and vice versa.
It's just obvious that the tree-level and classical field theory limits of
a QFT contain the same information, because both of them contain all
information about the \hbar=0 limit of all QFT observables.
> I feel there is a certain confusion with terminology here.
I don't think so.
> You use the word "SUGRA" both for the classical theory
> and the nonrenormalizable QFT ...
Yes, it's because both of them are SUGRA, and the relation between the
classical theory and the tree-level approximation of the quantum theory
is trivial. There is no reason to use different words for the same thing.
> and I don't always follow where is which. I believe this usage is
> indeed typical to the community, however, in this specific discussion
> it is especially confusing (since both meanings are relevant).
Could you be more specific about the statement where there was any
confusion? Well, if someone does not want to get rid of the idea that
there is a huge difference between the data about a classical theory and a
quantum theory in the tree-level approximation, he may have problems with
understanding the usage of the word SUGRA. But it is not a mere problem of
terminology - it is mainly a problem of his understanding of the relations
between a classical and quantum theory.
Instead of repeatedly suggesting that there is a problem with the whole
community and with the whole string theory or perhaps the whole
theoretical physics and all these things that you have written at least 6
times, is not it a better idea to first consider the possibility that
there may be a problem with *your* assumptions and *your* understanding of
physics? If you think that you are smarter than all physicists, then I
apologize for having insulted you, especially because I am very far from
sharing this opinion of yours.
> It is interesting how you plug non-trivial backgrounds here.
> What does it mean computing scattering in Matrix theory on
> a non-trivial (trivial = Minkowski x compact) background.
We don't know how to define Matrix theory on general backgrounds, which is
the main handicap that slowed down progress in Matrix theory. Matrix
theory however describes all physics on asymptotically Minkowski (times an
acceptable manifold) backgrounds. It contains very complicated geometries
that are nevertheless asymptotically Minkowski; by locality, you may say
that physics of any 11D geometry is included because you can extend it to
an asymptotically Minkowski spacetime.
One can in principle derive Matrix theory for some mildly curved
backgrounds with light-like Killing vectors as well.
> This doesn't get along with your later statements.
It gets along very well.
> Lets clear something out: if you manage to completely
> fix gauge freedom and define observables in the gauge
> fixed picture, these observables are gauge invariant.
If you dress your operators to make them gauge-invariant, you will have
gauge-invariant operators, but they will be non-locally related to the
original ones.
> A simple example:
> Consider a U(1) gauge field coupled to a complex scalar \Phi with
> charge 1. Pick the unitary gauge (\Phi real and positive). In ...
I agree with this. The light-cone gauge-fixing *is* complete, by the way.
> "Given moment" refers to some inner clock of the
> consciousness, of course, which roughly agrees
> with proper time.
Consciousness has no exact "inner clock" ;-), and the proper time only
makes sense if you define a classical worldline where it's measured. These
notions are very useful, but they are only well-defined in the classical
approximation in which the quantum fluctuations of the metric and the
worldlines are neglected.
> The positon and velocities of the electron has nothing to do.
Be sure that it does! It's one half of the phrase "quantum gravity".
Quantum gravity is a theory where all these principles, such as the
uncertainty principle, are also applied to spacetime geometry. Even the
loop quantum gravity people would agree that you are wrong, and you bet
that such a statement can hardly be viewed as a compliment. ;-)
Much like X and P do not commute in mechanics,
g_{23} and \partial_0 g_{23} do not commute in quantum gravity. It is
virtually an exact analogy.
Thinking about the metric tensor that has a well-defined value at some
point as well as a well-defined time-derivative - which is what you are
doing essentially in every single sentence of your postings - is
incompatible with the basic principle of quantum theory called the
uncertainty principle.
> I think we have reached a nexus here from which stems all of
> the disagreement in this discussion.
I am afraid that we have reached it already at the beginning when I
decided to answer any question asked by someone who obviously *never*
wishes to accept even the most basic principles of quantum mechanics.
> Let me get this straight: you believe that the experiences, thoughts,
> and in fact the mere existence of me, you and everyone else cannot be
> defined in an exact manner in string theory even in principle (no
> matter how humongously convoluted the definition is)? And that
> nevertheless string theory is correct?
Definitely, this is how the whole physics works. Most notions, statements,
objects that can be defined in human every-day language have no exact
counterparts in a physical theory. The more we know, the more we
understand that our previous concepts do not exist in the exact theory.
People thought that they could always say whether two events are
simultaneous - no, because of relativity; they also thought that it was
always possible to say where an electron is and how quickly it moves - no,
because of the uncertainty principle; or they thought that that could have
said how many positrons and electrons were in a given region - no, because
of QED fluctuations below the Compton wavelength.
All these things are simply caused by incorrect prejudices, by wrong
models, by unjustifiable extrapolations of some approximate insights that
are nearly true in the every-day life, but they become invalid further.
What I say is not specific to string theory; it holds for any theory in
modern physics. The models that you are using are *extremely* incorrect
because you even want to contradict the uncertainty principle for the
metric.
The more correct and complete models we deal with, the more they can show
that the previous concepts and descriptions were unusable, inaccurate, or
wrong in general.
(By the way, again, the light-cone gauge is an extreme case where we can
make the commutator of the fields and their light-cone-time derivatives
vanish - exactly because the "time" is null.)
Please don't ask me again this question whether some formerly meaningful
concepts must be meaningful forever - because you have already done it 6
times, and I answered 6 times. Yes, I think it is probably very difficult
to understand modern physics without the understanding of the fact that
some concepts from the old theories can become meaningless in a more
complete theory.
> Your quote: "gauge fixing means imposing additional constraints on the
> fields (or other degrees of freedom) of your theory such that each
> class of configurations that are related by gauge transformations
> contains at least one representative that satisfies the additional
> constraints"
>
> Now substitute "gauge transformations" by diffeomorphism.
Diffeomorphism fixing means imposing additional constraints on the fields
(or other degrees of freedom) of your theory such that each class of
configurations that are related by gauge (coordinate) transformations
contains at least one representative that satisfies the additional
constraints.
Satisfied?
> How does your description refelect the content of the definition (for
> spacetime diffeomorphisms, _not_ worldsheet ones)?
Very well, thank you for asking.
> > No, be sure that X^+ is not Lorentz-covariant. If you define a scalar to
> > be Lorentz invariant, then your sentence is a vacuous tautology, but
> > again, the subtlety is that there exist no Lorentz invariant local
> > gauge-invariant observables in a theory of quantum gravity.
>
> So, you claim there is no Lorentz invariant function of the collection
> of observable you can define in the light-cone gauge?
You forgot the word "local".
> If this is indeed so, the number of observables has to be no more than
> the number of parameters in the Lorentz group, since the action of the
> Lorentz group should be able to produce all possible combinations of
> their values from any given one.
You forgot the word "local".
> This doesn't appear to be true since there are local observables
> associated with various values of the transverse coordinates, say, and
> there is an infinity of those.
Once again, there exist no local Lorentz- and gauge-invariant observables
in a theory of quantum gravity which is a trivial consequence of the
diffeomorphism invariance.
> > Any asymptotically Minkowski state in the physical Hilbert space can be -
> > at least in the perturbative expansion - obtained via the action of the
> > light cone gauge creation operators in a unique way - there exists no
> > diffeomorphism freedom (one that would preserve the spacetime at
> > infinity), and therefore the assignment of the spacetime coordinate
> > becomes unique. But it is a very Lorentz-non-invariant procedure.
>
> The question is what happens with this procedure in the classical
> approximation.
The procedure (of generating the spectrum in the light-cone gauge) *was*
classical in spacetime because it considers free string theory at g=0. If
you want to talk about a classical theory on the worldsheet, then you will
get nothing like a local field theory in spacetime and nothing like
spacetime diffeomorphism invariance (or other local symmetries).
To get this local field-theory-like physics in spacetime, the strings must
be quantized on the worldsheet. Classical strings are just classical
pieces of rubber band that propagate on a fixed and unchangeable
background geometry. All the things like "predicting gravity,
electromagnetism, particles" and all local symmetries only arise once the
worldsheet is quantized.
> It appears to me that the only chance for it to
> work is that it reduces in the classical approximation to
> something like what I suggested but with null geodesics instead.
> It would makes sense in the light of the description on page 4
> of "the world as a hologram" (http://www.arxiv.org/abs/hep-th/9409089) by Lenny Susskind.
Not sure which relation between Lenny's paper and light-cone gauge string
oscillators you exactly want to investigate.
> This prescription only works for one specific solution (the
> Schwarzschild solution), whereas to "gauge fix" you have to do
> something that always works.
No gauge fixing can work for the *whole* theory (string theory) because
the identity of the useful gauge symmetry itself is changing from point to
point, and therefore one needs different things to be gauge-fixed. Gauge
fixing can only work universally around some region in the
configuration/moduli space. By the way, the gauge-fixing described for the
Schwarzschild can be generalized - although it is not easy to define the
exact rules - to a big class of topologically trivial asymptotically
Minkowski metrics. Even though it can be done, it does not change anything
about the fact that it is just one possible gauge-fixing and it cannot be
"universally priviliged".
> > Some coordinates may be "priviliged" or "useful" for a particular purpose,
> > but there are no "generally priviliged" coordinates.
>
> If you can completely gauge fix diffeomorphism invariance,
> then for all practical purposes there are, the practical
> purposes being defining local gauge invariant observables.
If you can do a random thing, it does not mean that it is priviliged -
on the contrary, most likely it is not.
> > Nope. By the words "theory of everything" we only mean "a theory of
> > everything that really exists in the Universe" - and therefore string
> > theory should *not* contain nonsense.
>
> So, "everything that really exists in the Universe" is
> particle accelerator experiments (which also exist only
> to an excellent approximation)?
String theory should (and, at least qualitatively, it does) reproduce the
events seen at the particle accelerators, but it in no way reproduces your
naive, incorrect models how the accelerators or other things in the
Universe work - models that contradict the uncertainty principle, among
many other things.
> > So why are you talking about it if you know that it is not true? The
> > discussion is very different and simpler in QFT than in QG.
>
> I was trying to demonstate that previous theories, like QFT,
> have an advantage over superstring theory in this respect,
It's not a real advantage, it is just simplicity. Simplicity implied by
the absence of general covariance. It's the same simplicity that implies
that QFT are not unique, while the bigger constraints in quantum gravity
imply that a theory of quantum gravity probably *is* unique
(string/M-theory).
You could also say that the classical theories have an "advantage" over
the quantum theories because many more things can be said
(positions+velocities), and so on. In reality, it is over-compensated by
the disadvantage that the classical theories simply disagree with reality
(and they would not admit the existence of life as we know it).
Similarly, the real world contains gravity, and you may scream that
non-gravitational QFTs have an advantage, but all this screaming is just
totally irrelevant because we know that these theories don't describe the
whole Universe.
> i.e., they had no conceptual missing parts of the sort I'm
> talking about which would prevent them from become a TOE,
> they were just _not_ the TOE experimentally (for instance
> by not describing gravity).
A theory of everything does not mean a theory of nonsensical concepts
that are incompatible with the basic principles of physics. A theory, if
it should have a chance to be a theory of everything, is not allowed to
contain anything of the type that you say simply because what you say is
inconsistent rant based on 17th century models of reality.
> Not at all. The objects make perfect sense, but it
> is unimaginable difficult to make them precise.
There is no unique and canonical way to make them precise - simply because
the domain of their validity is limited, much like the domain of validity
of virtually any concept in science and reality. One can perhaps define
some convoluted exact notions that are valid under (almost) any
circumstances and that reduce to the old notions in the appropriate limit,
but such a generalization cannot be unique. In most cases, one can show
that a generalization that would satisfy some extra conditions simply
can't exist at all.
> In fact, the whole point of "approximation" is constructing a
> simplified model which is realized inside the more complicated model
> to some accuracy, in some conditions. It is _not_ saying that some
> particular sector of the complicated model (e.g. scattering
> proccesses) reduces to a particular sector of the simplified mode
> (e.g. classical scattering) and this somehow miraculously explains the
> successful application of all other sectors in the simplified model to
> reality.
What you say is the opposite of the truth, at least in modern physics. Our
more general theories that extend the previous approximations nearly
always use a different language, and the old concepts are simply *not*
generalized and they can only be defined in the limit in which the old
approximation is acceptable.
In non-relativistic physics, you can ask what is the time difference
between two events. Special relativity explains that there cannot be any
universal answer because the answer depends on the observer. Classical
physics allows us to talk about the exact positions and velocities, and
unique predictions of their future values. These numbers just don't make
any sense in quantum mechanics.
The strength of the electromagnetic force - the fine structure constant -
was thought to be constant and one could have asked how much it exactly
is, but people realized that it runs as a function of the energy scale,
and at shorter distances, a bigger fine structure constant is relevant.
New refined interactions and particles are being added, and the old
theories are just not good anymore - even their language becomes
inappropriate because they are showed to be built on many assumptions that
are simply not true - only approximately true.
I find it highly surprising that this basic aspect of progress in physics
- replacing the old concepts by totally new ones, reducing the role of the
whole old language to a mere approximation - could be still hidden from
someone who spends so much time with physics newsgroups. You're not
questioning just string theory, you're questioning the meaning of all of
theoretical physics.
> > > So, your claim that my complaints can be addressed to QFT
> > > in the same fashion they can be adrressed to superstring
> > > theory is plainly wrong.
> >
> > My statement was just the opposite.
>
> Quote: "The desire to calculate exact quantities of this sort
> has more or less exact counterparts in QED"
I used it in a very specific context - "this sort" meant a very specific
thing, and you should not be surprised that if you replace "this sort" by
a completely different sort, you may again obtain wrong conclusions.
In "this sort" sentence I was trying to tell you that your construction
violated some properties of quantum gravity such that it could have been
made even in QFT - you simply violate the uncertainty principle. I was
also trying to explain you that quantum gravity is much more demanding -
many things that could be well-defined even in as complex framework as
QFT become meaningless in quantum gravity.
I don't exactly see where you see any contradiction. I am telling you that
your thinking is too naive even in ordinary QFT (or QM), and therefore it
is guaranteed that it is naive in QG, too, because QG is much more
sophisticated and constraining than QFT.
> I never wanted observables that don't use non-local
> gauge fixing, in fact the observables I have suggested
> _do_ use non-local gauge fixing. The "naturality" of
> the Lorentz action, OTOH, is an issue of esthetics,
> not a conceptual one.
The question whether one wants to compute the gravitational S-matrix or
some pseudolocal correlators of some operators defined in a convoluted
gauge-fixing scheme is also mostly an issue of esthetics, and most people
prefer the former.
> > I was just pointing out that you were using incorrect classical intuition
> > for the "proper length", assuming that the "geodesics" and other stuff
> > make sense even if quantum effects are taken into account.
>
> I don't see any way the fluctuation of the metric tensor implies
> there is no observable in the full quantum theory which reduces
> to the geodesic-based stuff in the classical limit.
Because in the QFT description, these fluctuations become infinitely big,
if you remove all approximations and cutoffs, and all quantities you talk
about are either infinity or zero in this full treatment. You can use a
beyond-QFT description of string theory, but string theory just does not
allow you to define various objects that you want to define, and it is
important that it does not allow one to do so.
> > \alpha_{-2}^\mu \tilde\alpha_{-2}^\nu |0\rangle.
>
> All of this is obvious, however, I refer you again to my
> previous statement.
OK, so I did not get the meaning of your statement "instead of the metric"
if it is obvious to you.
> How do you "extrapolate" coordinates to the interior?
I am not doing anything like that by hand. AdS/CFT does similar things
automatically. No one knows exactly how to derive the local phenomena
inside the AdS space from the boundary CFT description (how to define the
approximate locality in the holographic dimension, in particular),
nevertheless everyone is convinced that this local physics in the bulk is
contained in the CFT and explicit tests of everything we can calculate
confirm it.
> You are talking about preferred coordinates, right?
The AdS/CFT map for the local phenomena in the bulk is very nontrivial,
but the correspondence does not change anything about the fact that the
physics in the bulk is equivalent to a theory that exhibits general
covariance at low energies - a theory that is most naturally described in
terms of degree of freedom with no preferred coordinates - and it does not
change the fact that the metric in the bulk is a quantum mechanical,
fluctuating degree of freedom.
The AdS/CFT computations are analogous to the S-matrix - the AdS boundary
plays the role of t=-infinity and t=+infinity in the S-matrix treatment.
In this sense, you can imagine that "anything can happen" and "any
coordinates can be used" in the bulk - the only place where the
coordinates are required to match a "standard" is near the boundary
(analogy of |t|=infinity).
> Apparently you think I am on some lunatic campaign to destroy
> string theory, nevertheless, try to believe I was just asking
> a simple question. How do you define the adiabatic
> continuation of states between different values of g without
> making it vary over spacetime in a manner than violates the
> equations of motion?
Just like in QFT. You infinitely slowly change the coupling constant, and
look what's happening with the state. Be sure that if the evolution of the
dilaton is slow, I can even prove the existence of the required solution
of the dilaton equations of motion and this solution will be arbitrarily
close to simply "\phi=\epsilon.t" for a very small \epsilon.
The smaller the coupling constant it, the easier and less controversial
such an adiabatic "turning on" will be. For bigger "g" the perturbative
expansions break down anyway.
Even if there were some subtle problems with any procedure described 2
paragraphs ago, we just don't need to define the external states in this
way - the identity of a graviton, for example, is totally determined by
its momentum and polarizations. A graviton with momentum "p" and
polarization tensor "ij" simply *exists*, and your attempts to define it,
redefine it and mix with many other concepts are not well-motivated.
Even if we could not imagine adiabatic changes of the coupling and similar
psychological games, it would be a perfectly acceptable and complete to
have a theory with a known list of allowed external states, and the tools
to calculate their cross sections.
> It appears to me there is still hope to define something
> that would reduce the the objects I described in the
> classical limit. In the Planckian regime this "something"
> would look totally different, of course.
But there is no reason why such generalized objects should be defined
uniquely, and there is no reason why such notions should be useful to
describe the Planckian physics. We only know that they are useful to
talk about low-energy, classical physics, but the physics at the Planck
scale is a very different beast.
Cheers
Lubos
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