<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:<Pine.LNX.4.31.0406191012250.6405-100000@feynman.harvard.edu>...\n\n> If you compute the S-matrix via path integrals, the tree-level diagrams\n> simply correspond to approximating the path integral by its stationary\n> points which are the classical solutions of the corresponding classical\n> equations of motion. This is how the perturbative calculation in powers of\n> hbar works in any quantum theory.\n\nOf course, but it\'s not entirely clear to me how\nto realize this formal fact about hbar dependance\nas an explicit relation between the quantum and\nthe classical theory.\n\n> For example, the path integral for the evolution between the initial\n> configuration <I| and the final configuration |F> equals, in the classical\n> limit, to the contribution \\exp(i S_{classical}) of the classical solution\n> satisfying these initial conditions\n\nWhat initial conditions precisely? |I> and |F> are quantum states.\n\n> > You use the word "SUGRA" both for the classical theory\n> > and the nonrenormalizable QFT ...\n>\n> Yes, it\'s because both of them are SUGRA, and the relation between the\n> classical theory and the tree-level approximation of the quantum theory\n> is trivial.\n\nFirstly, the quantum theory doesn\'t really exist,\nit is merely a formal perturbative expansion which becomes\ndivergent quick enough.\nSecondly, it appears to me there might be a relation\nbetween the description of scattering proccesses, but\nclassical SUGRA contains lots of other stuff which I don\'t\nknow how to define in the (nonexistent) quantum theory.\nAt the least classical SUGRA has lots of solutions which\nare not asymptotically Minkowski.\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>\n> Could you be more specific about the statement where there was any\n> confusion?\n\nI wrote:\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\nYou wrote:\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. This fact identifies the local classical limit of both\n> theories.\n\nThis wasn\'t clear to me since one of the "both theories"\nis classical, so in what sense are talking about its\nclassical limit? I guessed you meant the "perturvative\nQFT" quantization of SUGRA and tried to verify my guess.\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>\n> We don\'t know how to define Matrix theory on general backgrounds, which is\n> the main handicap that slowed down progress in Matrix theory. Matrix\n> theory however describes all physics on asymptotically Minkowski (times an\n> acceptable manifold) backgrounds.\n\nThis is clear to me. However, I was referring to scattering\non a non-trivial asymptotically Minkowski background (a 2D\nCFT of the right kind), since you were bringing such\nbackgrounds as an example of SUGRA being the classical limit.\n\n> It contains very complicated geometries\n> that are nevertheless asymptotically Minkowski; by locality, you may say\n> that physics of any 11D geometry is included because you can extend it to\n> an asymptotically Minkowski spacetime.\n\nThis is a very far fetched statement. The precise\nmeaning of locality in string theory is highly\nnon-trivial.\n\n> If you dress your operators to make them gauge-invariant, you will have\n> gauge-invariant operators, but they will be non-locally related to the\n> original ones.\n\nThis is obvious. As I said, the same is true\nof the observables I suggested.\n\n> Consciousness has no exact "inner clock" ;-)\n\nIt does, as far as I can tell.\n\n> > The positon and velocities of the electron has nothing to do.\n>\n> Be sure that it does! It\'s one half of the phrase "quantum gravity".\n> Quantum gravity is a theory where all these principles, such as the\n> uncertainty principle, are also applied to spacetime geometry.\n\nThis is entirely obvious and entirely unrelated to my point.\n\n> Thinking about the metric tensor that has a well-defined value at some\n> point as well as a well-defined time-derivative - which is what you are\n> doing essentially in every single sentence of your postings\n\nI\'m doing no such thing.\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>\n> Definitely, this is how the whole physics works. Most notions, statements,\n> objects that can be defined in human every-day language have no exact\n> counterparts in a physical theory.\n> ...\n> People thought that they could always say whether two events are\n> simultaneous - no, because of relativity; they also thought that it was\n> always possible to say where an electron is and how quickly it moves - no,\n> because of the uncertainty principle; or they thought that that could have\n> said how many positrons and electrons were in a given region - no, because\n> of QED fluctuations below the Compton wavelength.\n\nYou seem to be missing a fundumental difference between the notions\nI\'m talking about and the examples you brought. The later are all\nparts of _models_ of reality, models that were eventually proven\nwrong. Whereas the consciousness is the very definition of reality\nitself. All other manifestions of reality can only be perceived by\nus though our own consciousness. A solipcist would conclude the\nconsciousness is the only thing that exist. A reasonable person\nwould say the solipcist is doing vacuous philosophy and instead\nstart constructing models of external reality. These models would\ninclude the consciousness itself and thus be applicable to\nempirical tests (since consciousness is the only thing that can be\nobserved directly). Of course no physical model ever really\ncontained a complete mathemtical description of the human mind, or\nanything close. Instead, other concepts (like the indicications of\nmeasuremet apparatuses) were linked to the mathematical model\n(usually in an approximate fashion). These concepts are themselves\nparts of a simple intermediate "model" which links our perceptions\nwith the immediate observed reality (that includes measurement\nappearetuses). However, it was always implicit that such a\ndescription could be performed in principle once a fine enough\nimprovement of our model of physics is constructed.\nYou, on the other hand, dismiss this principle and are apparently\ncontent with accepting any abstract mathematical construction as\na physical model, without an accompanying explanation of the\nrelation of this mathematical construction with the immediately\nobserved.\n\n> ...\n> Once again, there exist no local Lorentz- and gauge-invariant observables\n> in a theory of quantum gravity which is a trivial consequence of the\n> diffeomorphism invariance.\n\nSo, once again, in what sense is the S-matrix the only\nobservable? It is not the only local Lorentz covariant\nand gauge invariant observable since it\'s hardly local!\n\n> > The question is what happens with this procedure in the classical\n> > approximation.\n>\n> The procedure (of generating the spectrum in the light-cone gauge) *was*\n> classical in spacetime because it considers free string theory at g=0.\n\nSo the question is what happens with this procedure at finite g.\nYou might object that finite g means non-classical physical. To\nwhich I will object that the realistic theory (with finite g)\nmust still incorporate classical physics in some approximate\nmanner. In particular, semiclassical solutions must have\ncorresponding quantum states in the full superstring theory (as\nlong as they have the right asymptotics, don\'t carry too much\ncurvature without a covering horizon etc.).\n\n> Not sure which relation between Lenny\'s paper and light-cone gauge string\n> oscillators you exactly want to investigate.\n\nI\'m suggesting that light-cone localized operators are really\nlocalized on the light-like geodesic Susskind uses to descibe\nhologrophy.\n\n> No gauge fixing can work for the *whole* theory (string theory) because\n> the identity of the useful gauge symmetry itself is changing from point to\n> point, and therefore one needs different things to be gauge-fixed. Gauge\n> fixing can only work universally around some region in the\n> configuration/moduli space.\n\nSo the light-cone gauge only works for some region? For which?\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>\n> If you can do a random thing, it does not mean that it is priviliged -\n> on the contrary, most likely it is not.\n\nReread my explanation of what property of notion "priviliged"\ndo I have in mind. Local in this case being local w.r.t. these\ncoordinates.\n\n> > > > it (string theory) 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> > Not at all. The objects make perfect sense, but it\n> > is unimaginable difficult to make them precise.\n>\n> There is no unique and canonical way to make them precise - simply because\n> the domain of their validity is limited, much like the domain of validity\n> of virtually any concept in science and reality. One can perhaps define\n> some convoluted exact notions that are valid under (almost) any\n> circumstances and that reduce to the old notions in the appropriate limit,\n> but such a generalization cannot be unique.\n\nIt will be unique to the relevant accuracy.\n\n> In most cases, one can show\n> that a generalization that would satisfy some extra conditions simply\n> can\'t exist at all.\n\nIf you use conditions there\'s no reason for it to satisfy,\nthen, yes, it might turn out the generalization can\'t exist.\n\n> What you say is the opposite of the truth, at least in modern physics. Our\n> more general theories that extend the previous approximations nearly\n> always use a different language, and the old concepts are simply *not*\n> generalized and they can only be defined in the limit in which the old\n> approximation is acceptable.\n\nThis much is obvious, however, all of the old\nconcepts that were relevant before _can_ be\ndefined (in this "limit", i.e. for states that\nhave some special properties).\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> I don\'t exactly see where you see any contradiction. I am telling you that\n> your thinking is too naive even in ordinary QFT (or QM), and therefore it\n> is guaranteed that it is naive in QG, too, because QG is much more\n> sophisticated and constraining than QFT.\n\nBut didn\'t you just said your statement was the opposite of that?\n\n> The question whether one wants to compute the gravitational S-matrix or\n> some pseudolocal correlators of some operators defined in a convoluted\n> gauge-fixing scheme is also mostly an issue of esthetics, and most people\n> prefer the former.\n\nIt\'s not since not all experiments probe the S-matrix.\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>\n> Because in the QFT description, these fluctuations become infinitely big,\n> if you remove all approximations and cutoffs, and all quantities you talk\n> about are either infinity or zero in this full treatment.\n\nBut it is only in the classical limit that I use these\nquantities, where the approximations and cutoffs are valid.\n\n> No one knows exactly how to derive the local phenomena\n> inside the AdS space from the boundary CFT description (how to define the\n> approximate locality in the holographic dimension, in particular)\n\nExcellent, this is precisely the thing I was talking about\nall along! Surprisingly, you admit it, even though you kept\narguing with me before.\n\n> nevertheless everyone is convinced that this local physics in the bulk is\n> contained in the CFT and explicit tests of everything we can calculate\n> confirm it.\n\nMe too. But believing in victory is different than winning.\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>\n> Just like in QFT. You infinitely slowly change the coupling constant, and\n> look what\'s happening with the state. Be sure that if the evolution of the\n> dilaton is slow, I can even prove the existence of the required solution\n> of the dilaton equations of motion and this solution will be arbitrarily\n> close to simply "phi=epsilon.t" for a very small epsilon.\n\nAs I said, this is a nice argument, but it\'s not obvious how to\nmake it exact nonperturbatively.\n\n> The smaller the coupling constant it, the easier and less controversial\n> such an adiabatic "turning on" will be. For bigger "g" the perturbative\n> expansions break down anyway.\n\nYes but the S-matrix should make sense beyond perturbation theory.\n\n> Even if we could not imagine adiabatic changes of the coupling and similar\n> psychological games, it would be a perfectly acceptable and complete to\n> have a theory with a known list of allowed external states, and the tools\n> to calculate their cross sections.\n\nYes, but the list is infinite.\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>\n> But there is no reason why such generalized objects should be defined\n> uniquely, and there is no reason why such notions should be useful to\n> describe the Planckian physics.\n\nThey will be useful to understand the boundary region between usual\nand Planckian physics.\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:<Pine.LNX.4.31.0406191012250.6405-100000@feynman.harvard.edu>...
> 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.
Of course, but it's not entirely clear to me how
to realize this formal fact about \hbar dependance
as an explicit relation between the quantum and
the classical theory.
> 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
What initial conditions precisely? |I> and |F> are quantum states.
> > 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.
Firstly, the quantum theory doesn't really exist,
it is merely a formal perturbative expansion which becomes
divergent quick enough.
Secondly, it appears to me there might be a relation
between the description of scattering proccesses, but
classical SUGRA contains lots of other stuff which I don't
know how to define in the (nonexistent) quantum theory.
At the least classical SUGRA has lots of solutions which
are not asymptotically Minkowski.
> > 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?
I wrote:
> That is OK, however, it is still unclear to me in precisely in what sense
> do you get SUGRA as the classical limit.
You wrote:
> 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.
This wasn't clear to me since one of the "both theories"
is classical, so in what sense are talking about its
classical limit? I guessed you meant the "perturvative
QFT" quantization of SUGRA and tried to verify my guess.
> > 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.
This is clear to me. However, I was referring to scattering
on a non-trivial asymptotically Minkowski background (a 2D
CFT of the right kind), since you were bringing such
backgrounds as an example of SUGRA being the classical limit.
> 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.
This is a very far fetched statement. The precise
meaning of locality in string theory is highly
non-trivial.
> 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.
This is obvious. As I said, the same is true
of the observables I suggested.
> Consciousness has no exact "inner clock" ;-)
It does, as far as I can tell.
> > 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.
This is entirely obvious and entirely unrelated to my point.
> 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
I'm doing no such thing.
> > 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.
> ...
> 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.
You seem to be missing a fundumental difference between the notions
I'm talking about and the examples you brought. The later are all
parts of _models_ of reality, models that were eventually proven
wrong. Whereas the consciousness is the very definition of reality
itself. All other manifestions of reality can only be perceived by
us though our own consciousness. A solipcist would conclude the
consciousness is the only thing that exist. A reasonable person
would say the solipcist is doing vacuous philosophy and instead
start constructing models of external reality. These models would
include the consciousness itself and thus be applicable to
empirical tests (since consciousness is the only thing that can be
observed directly). Of course no physical model ever really
contained a complete mathemtical description of the human mind, or
anything close. Instead, other concepts (like the indicications of
measuremet apparatuses) were linked to the mathematical model
(usually in an approximate fashion). These concepts are themselves
parts of a simple intermediate "model" which links our perceptions
with the immediate observed reality (that includes measurement
appearetuses). However, it was always implicit that such a
description could be performed in principle once a fine enough
improvement of our model of physics is constructed.
You, on the other hand, dismiss this principle and are apparently
content with accepting any abstract mathematical construction as
a physical model, without an accompanying explanation of the
relation of this mathematical construction with the immediately
observed.
> ...
> 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.
So, once again, in what sense is the S-matrix the only
observable? It is not the only local Lorentz covariant
and gauge invariant observable since it's hardly local!
> > 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.
So the question is what happens with this procedure at finite g.
You might object that finite g means non-classical physical. To
which I will object that the realistic theory (with finite g)
must still incorporate classical physics in some approximate
manner. In particular, semiclassical solutions must have
corresponding quantum states in the full superstring theory (as
long as they have the right asymptotics, don't carry too much
curvature without a covering horizon etc.).
> Not sure which relation between Lenny's paper and light-cone gauge string
> oscillators you exactly want to investigate.
I'm suggesting that light-cone localized operators are really
localized on the light-like geodesic Susskind uses to descibe
hologrophy.
> 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.
So the light-cone gauge only works for some region? For which?
> > 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.
Reread my explanation of what property of notion "priviliged"
do I have in mind. Local in this case being local w.r.t. these
coordinates.
> > > > it (string theory) 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!
> > 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.
It will be unique to the relevant accuracy.
> In most cases, one can show
> that a generalization that would satisfy some extra conditions simply
> can't exist at all.
If you use conditions there's no reason for it to satisfy,
then, yes, it might turn out the generalization can't exist.
> 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.
This much is obvious, however, all of the old
concepts that were relevant before _can_ be
defined (in this "limit", i.e. for states that
have some special properties).
> > > > 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.
> ...
> 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.
But didn't you just said your statement was the opposite of that?
> 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.
It's not since not all experiments probe the S-matrix.
> > 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.
But it is only in the classical limit that I use these
quantities, where the approximations and cutoffs are valid.
> 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)
Excellent, this is precisely the thing I was talking about
all along! Surprisingly, you admit it, even though you kept
arguing with me before.
> 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.
Me too. But believing in victory is different than winning.
> > 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.
As I said, this is a nice argument, but it's not obvious how to
make it exact nonperturbatively.
> 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.
Yes but the S-matrix should make sense beyond perturbation theory.
> 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.
Yes, but the list is infinite.
> > 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.
They will be useful to understand the boundary region between usual
and Planckian physics.
Best regards,
Squark.