Helling & Policastro on GNS quantization of string [Archive]

Urs Schreiber

Sep20-04, 10:55 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>This summer in Paris at Strings_04 Robert Helling told me about having a\npaper in preparation dealing with aspects of the issue\n\nhttp://golem.ph.utexas.edu/string/archives/000369.html\n\nthat was brought up by Thomas Thiemann\'s non-standard quantization of the\nrelativistic string\n\nhttp://golem.ph.utexas.edu/string/archives/000299.html .\n\nNow this paper has appeared:\n\nRobert Helling & Guiseppe Policastro:\nString quantization: Fock vs. LQG Representations\nhep-th/0409182\n\nThe question addressed in this paper is remarkable due to the fact that it\nwas taken for granted and obvious - until Thiemann came along, that is.\n\nEverybody knows that first quantization is a mystery. Some people therefore\nlike to invoke heavy mathematical machinery to quantize something. In his\n"LQG-string" paper Thiemann has used the GNS construction to come up with a\n"quantization" of the ordinary Nambu-Goto action which was completely unlike\nthe standard way to quantize the string.\n\nWe have already talked about that at great length, but now Helling and\nPolicastro provide us with a detailed technical analysis in mathematical\nlanguage of where Thiemann\'s quantization parts company with the standard\nlore.\n\nIn a sense, the result is not too surprising, I would say: The GNS\nconstruction associates a quantization of a C* algebra of observables with a\n_choice_ of "state" (linear functional on the algebra). Helling and\nPolicastro point out that there is a continuous such state, and that it is\nsend by GNS to the standard quantum theory of the string, including the\ncorrect central charge. The state that Thomas Thiemann based his\nquantization on instead is not continuous and yields something different.\n\nWhile this may sound not too exciting, I think the paper by Helling and\nPolicastro is a very interesting contribution to the general discussion\nwhich deals with the general quantization of possibly higher dimensional\ngravity.\n\nOne crucial message it transports is: Just using methods like deformation\nquantization, GNS construction etc. does (of course) not imply that we need\nto end up with a discontinuous quantization. While these come from states\nthat have an _apparently_ desireable property, this property is not at all\nnecessary or even sufficient to obtain a meaningful quantum theory.\n\nIn a nutshell: It is not right that a sensible quantum theory with\ndiffeomorphism constraints must have a diffeomorphism invariant "ground\nstate" and in fact known sensible such quantizations don\'t.\n\nBut in this context a side remark in the above paper I found most\ninteresting: Apparently in\n\nF. Acerbi & G. Morchio & F. Strocchi:\nInfrared singular fields and nonregular representations of CCR algebras\nJ. Math. Phys. 34 (1993) 899-914\n\nit is discussed that the singular GNS state can be interpreted as a thermal\nstate of infinite temperature!\n\nI didn\'t know this before and like that insight, because it points at a way\nto understand a larger framework in which various "different" quantizations\n(of the string for instance) appear as different aspects of the same thing.\nMaybe Thiemann\'s choice of GNS state still can teach us something about\nstrings at high temperature (maybe not though, due to there is no worldsheet\nphysics beyond the Hagedorn temperature).\n\nThen I have the following comments and/or questions:\n\nRobert Helling and Giuseppe Policastro mention that by introducing the\nappropriate ghost sector the central charge vanishes, which seems to make\nthe two different quantizations appear more similar than otherwise. But\ndon\'t we need to exercise some care here? Even with the ghost sector added\nphysical states (in the standard theory) are not necessarily annihilated by\nall the symmetry generators. Instead, all that is required is that they are\nBRST-closed.\n\nThen I would like to remark that a subtle but maybe crucial issue should not\nbe overlooked: The "diffeomorphisms" discussed by Thiemann and by\nHelling&Policastro are not the "spatial diffeomorphisms" which\nreparameterize the spatial slices of the string. As such they cannot be\ncompletely compared to the spatial diffeo constraints as they appear in the\nADM constraints of (2 and higher dimensional) gravity.\n\nIn fact, one thing that was kind of odd about Thomas Thiemann\'s "LQG-string"\napproach was that he in fact did *not* follow the standard LQG presciption,\nwhich amounts to quantizing the spatial constraints non-continuously while\nquantizing the Hamiltonian constraint in the ordinary fashion. I think there\nis a simple reason for that, because in 1+1 dimensions it is easy to see\nthat when the spatial diffeos are solved no Hamiltonian constraint can be\nimposed anymore at all.\n\nFinally I have a technical question to Robert and Giuseppe Policastro\nconcerning the discussion on pp10-11 of their paper. Maybe I am mixed up,\nbut it seems to me that there at some point commuting and anti-commuting\nproperties need to be exchanged.\n\nSo let A be skew such that\n\nsigma(Af,g) = -sigma(f,Ag)\n\nand let furthermore A commute with J. Then\n\n< e^A f | e^A f>\n= sigma(e^A f, J e^A f)\n= sigma(e^A f, e^A J f)\n= sigma(e^-A e^A f, J f)\n=<f | f>\n\nRight? I am just wondering because in the text it says for this to be true A\nneeds to anti-commute with J.\n\nSimilarly, from the definition of A_1,2 on the bottom of p. 10 I find that\nA_1 commutes with J while A_2 anti-commutes (and this must be true if A_1 is\nto preserve the eigenspaces of J), but on p. 11 it is stated the other way\naround. Is that a typo or am I confused?\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 summer in Paris at Strings_04 Robert Helling told me about having a
paper in preparation dealing with aspects of the issue

http://golem.ph.utexas.edu/string/archives/000369.html

that was brought up by Thomas Thiemann's non-standard quantization of the
relativistic string

http://golem.ph.utexas.edu/string/archives/000299.html .

Now this paper has appeared:

Robert Helling & Guiseppe Policastro:
String quantization: Fock vs. LQG Representations
http://www.arxiv.org/abs/hep-th/0409182

The question addressed in this paper is remarkable due to the fact that it
was taken for granted and obvious - until Thiemann came along, that is.

Everybody knows that first quantization is a mystery. Some people therefore
like to invoke heavy mathematical machinery to quantize something. In his
"LQG-string" paper Thiemann has used the GNS construction to come up with a
"quantization" of the ordinary Nambu-Goto action which was completely unlike
the standard way to quantize the string.

We have already talked about that at great length, but now Helling and
Policastro provide us with a detailed technical analysis in mathematical
language of where Thiemann's quantization parts company with the standard
lore.

In a sense, the result is not too surprising, I would say: The GNS
construction associates a quantization of a C* algebra of observables with a
_choice_ of "state" (linear functional on the algebra). Helling and
Policastro point out that there is a continuous such state, and that it is
send by GNS to the standard quantum theory of the string, including the
correct central charge. The state that Thomas Thiemann based his
quantization on instead is not continuous and yields something different.

While this may sound not too exciting, I think the paper by Helling and
Policastro is a very interesting contribution to the general discussion
which deals with the general quantization of possibly higher dimensional
gravity.

One crucial message it transports is: Just using methods like deformation
quantization, GNS construction etc. does (of course) not imply that we need
to end up with a discontinuous quantization. While these come from states
that have an _apparently_ desireable property, this property is not at all
necessary or even sufficient to obtain a meaningful quantum theory.

In a nutshell: It is not right that a sensible quantum theory with
diffeomorphism constraints must have a diffeomorphism invariant "ground
state" and in fact known sensible such quantizations don't.

But in this context a side remark in the above paper I found most
interesting: Apparently in

F. Acerbi & G. Morchio & F. Strocchi:
Infrared singular fields and nonregular representations of CCR algebras
J. Math. Phys. 34 (1993) 899-914

it is discussed that the singular GNS state can be interpreted as a thermal
state of infinite temperature!

I didn't know this before and like that insight, because it points at a way
to understand a larger framework in which various "different" quantizations
(of the string for instance) appear as different aspects of the same thing.
Maybe Thiemann's choice of GNS state still can teach us something about
strings at high temperature (maybe not though, due to there is no worldsheet
physics beyond the Hagedorn temperature).

Then I have the following comments and/or questions:

Robert Helling and Giuseppe Policastro mention that by introducing the
appropriate ghost sector the central charge vanishes, which seems to make
the two different quantizations appear more similar than otherwise. But
don't we need to exercise some care here? Even with the ghost sector added
physical states (in the standard theory) are not necessarily annihilated by
all the symmetry generators. Instead, all that is required is that they are
BRST-closed.

Then I would like to remark that a subtle but maybe crucial issue should not
be overlooked: The "diffeomorphisms" discussed by Thiemann and by
Helling&Policastro are not the "spatial diffeomorphisms" which
reparameterize the spatial slices of the string. As such they cannot be
completely compared to the spatial diffeo constraints as they appear in the
ADM constraints of (2 and higher dimensional) gravity.

In fact, one thing that was kind of odd about Thomas Thiemann's "LQG-string"
approach was that he in fact did *not* follow the standard LQG presciption,
which amounts to quantizing the spatial constraints non-continuously while
quantizing the Hamiltonian constraint in the ordinary fashion. I think there
is a simple reason for that, because in 1+1 dimensions it is easy to see
that when the spatial diffeos are solved no Hamiltonian constraint can be
imposed anymore at all.

Finally I have a technical question to Robert and Giuseppe Policastro
concerning the discussion on pp10-11 of their paper. Maybe I am mixed up,
but it seems to me that there at some point commuting and anti-commuting
properties need to be exchanged.

So let A be skew such that

\sigma(Af,g) = -\sigma(f,Ag)

and let furthermore A commute with J. Then

< e^A f | e^A f>= \sigma(e^A f, J e^A f)= \sigma(e^A f, e^A J f)= \sigma(e^-A e^A f, J f)=<f | f>

Right? I am just wondering because in the text it says for this to be true A
needs to anti-commute with J.

Similarly, from the definition of A_1,2 on the bottom of p. 10 I find that
A_1 commutes with J while A_2 anti-commutes (and this must be true if A_1 is
to preserve the eigenspaces of J), but on p. 11 it is stated the other way
around. Is that a typo or am I confused?

Lubos Motl

Sep20-04, 06:42 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 Mon, 20 Sep 2004, Urs Schreiber wrote:\n\n> Robert Helling & Guiseppe Policastro:\n> String quantization: Fock vs. LQG Representations\n> hep-th/0409182\n\nTheir description of the harmonic oscillator looks particularly strange\nbecause if we did not agree what is the physics of the quantum harmonic\noscillator, we could probably agree about nothing in the world.\n\nIf someone suggested that this "alternative" approach (leading to the\nnonsensical non-separable Hilbert space) is how many of us are thinking\nabout the harmonic oscillator, it would not sound encouraging, to say the\nleast. Well, nevertheless it looks like a fair toy model of what our\nfriends are doing with other theories.\n\nI am not understanding what is exactly the difference between this\napproach to the harmonic oscillator, worldsheet dynamics, or gravity for\nthat matter on one side, and some completely different elementary\nconceptual error - or an "original treatment" by students who get bad\ngrades from their exams - in approaching any of these theories. Most of us\nand our classmates have made many different errors like that in our\nlifetime - the main difference is that we realized that they were errors.\n\nI am sure that many of us tried to deal with divergent sums and divergent\nintegrals, and various other singular objects in various ways that can be\nproved "wrong" on physical grounds. Zeno was the first famous scholar who\nobtained wrong conclusions ("paradoxes") about the infinitesimal and\n"infinite" objects. However, later people learned how to do the calculus\nand how to sum the series properly. We learned how to deal with the\ndistributions and how to regularize and renormalize the integrals, even\nthough most of us - as well as most new students who start with physics -\nare often doing incorrect manipulations with these objects.\n\nThomas Thiemann et al. are describing these "different" manipulations not\nas rudimentary errors in thinking, but as a great piece of alternative\nscience, and we are systematically being told that this is what we should\npay attention to - or even follow - and perhaps it is even a deep thought.\nWhy don\'t we treat equally all students that submit wrong calculations of\nsome type during their exam? Well...\n\nThe very "function" f(x) that equals zero for non-zero x, and equals one\nfor x=0 (or even its functional generalization), is something that does\nnot exist in physics. The measure theory guarantees that this function is\n"physically" indistinguishable from f(x)=0. The closest nontrivial object\nto this function is Dirac\'s distribution delta(x). Indeed, by taking the\nusual integrated superposition of the delta-functions, we can obtain the\nusual Hilbert space of the usual functions.\n\nIf I were completely free to decide what sort of a physics paper XY is,\nand XY was based on this "delta function divided by infinity" that is\nnevertheless different from zero, no doubt, I would conclude that the\nauthor is a crackpot.\n\nAll Hilbert spaces obtained from these wrong assumptions are\nnon-separable, unphysical, and the only way how the non-separability can\nbe cured is if the resulting theory is completely topological and all\nthese values of "x" are eventually unphysical, perhaps except for their\nordering. The states in this model represent unphysical mixtures of a\nhugely infinite number of superselection sectors - it is another\ndescription of Helling et al. comments about "discontinuity" of\nThiemann\'s representation, I think. Each of these sectors is made of a\nsingle state. In physics, it is legitimate to study each single\nsuperselection sector separately - and if these superselection sectors are\nmade of a single state, the theory is physically vacuous.\n\nThe infinitely modified normalization of the delta-function implies that\nthese states cannot be integrated over; the expectation values and\nprobabilities cannot really be calculated as integrals; they cannot be\ncombined into meaningful superpositions; the analogous configurations\ncannot contribute finitely to the path integrals. Well, it seems that the\nproponents do not worry because their goal is not to calculate expectation\nvalues, probabilities, path integrals, or any other physical quantity in a\nphysics-like theory for that matter. Their task is to satisfy some totally\nuninteresting, unphysical, loose, and completely abstract axioms about the\nHilbert spaces - while they\'re clearly assuming that nothing else is\nneeded for a physical theory. Well, they are very, very wrong.\n\n> Everybody knows that first quantization is a mystery.\n\n"Why the world is quantum?" may be a mystery and the most counterintuitive\ninsight about the world, but the mathematical operation behind the first\nquantization does not seem mysterious in any way, and it also does not\nseem ambiguous. Moreover, I don\'t know why you chose the "first\nquantization" because even it is a mystery, it is a smaller mystery than\nthe "second quantization". ;-)\n\n> In a sense, the result is not too surprising, I would say: The GNS\n> construction associates a quantization of a C* algebra of observables with a\n> _choice_ of "state" (linear functional on the algebra). Helling and\n> Policastro point out that there is a continuous such state, and that it is\n> send by GNS to the standard quantum theory of the string, including the\n> correct central charge. The state that Thomas Thiemann based his\n> quantization on instead is not continuous and yields something different.\n\nI did not quite like their comments describing the algebras with different\ncentral charges as different "representations" (with an exclamation mark).\nThe only way how can one understand this sentence is that they claim that\nthe Virasoro algebras with different central charges are isomorphic to\neach other. Well, I think that this is simply not true - and if someone\nthinks that it is true, she should find the operator to conjugate the\ngenerators with so that it changes the central charge. The Virasoro\nalgebras with different values of "c" may have been inspired by the same\n"naive" classical symmetry, but at the mathematical level and a generic\nvalue of Planck\'s constant, they are just different algebras.\n\nThere is a common theme in Thiemann\'s papers which, I\'m afraid, may\nunfortunately be shared by Helling and Policastro - which is that they\noften look at the "classical limit" of an algebra, and treat all the\nmodifications implied by quantum mechanics as unimportant - and perhaps\nannoying? - details whose relevance for any of their conclusions goes to\nzero. It is the classical theory that already contains all the information\nabout the quantum theory, Thiemann et al. seems to believe. They are\ntrying to do a "quantization without quantization". Well, until one adopts\nthe exact quantum algebra, the exact quantum generators, Hilbert spaces,\nand so on - and as long as he thinks in terms of the classical objects -\nhe has not made the quantum leap yet. The only reason why we are doing\nquantization is that it leads to *different* conclusions than what you\ncould have expected from classical physics.\n\nVirasoro algebras with different values of "c" are simply not isomorphic\nto each other, and the central charge is a purely quantum effect that must\nbe treated seriously if we\'re doing quantum theory. The theories with\ndifferent c\'s describe mathematically different symmetries and different\nsystems although they may have the same classical limit (or classical\nstarting point to obtain them). There are various other relations between\nthem - for example, the central charge can run as the function of the\nscale - but once again, they are not isomorphic, and if we take quantum\nmechanics completely seriously, they are as non-isomorphic as any other\npair of algebras.\n\nBy the way, this purely quantum viewpoint will be even more important if\nwe want to get more insight into the (2,0) theory or even M-theory at the\ngeneric point of the moduli space - because these theories (at least in\nsome backgrounds) clearly indicate that they cannot be fully obtained from\na classical theory by quantization - and they almost certainly cannot be\nobtained from a *unique* classical theory. The very assumption that a\nquantum theory is always obtained as a quantization of a specific\nclassical theory - this assumption is not correct. And if this assumption\nis even combined with a completely wrong procedure of (non)quantization,\nit\'s pretty bad.\n\nThere are other examples in which Thiemann et al. try to make this sort of\n"quantization without quantization". They want the commutators to be\nalways equal to the Poisson brackets; they want the short distance\nsingularities in OPEs to be absent much like they are absent in classical\nfield theory; they want the anomalies to be absent in any theory by\ndefinition; and so forth. But these requirements are simply incorrect in a\nquantum (field) theory. The commutator only reduces to the Poisson bracket\nin the classical limit, in which \\hbar^2 can be completely neglected, so\nto say. The singular OPEs can simply be derived from quantum field\ntheory, and the existence of anomalies can always be deduced in many ways\n- both from the IR spectrum as well as from the regularization of the UV\ndivergences.\n\nOne may be trying to obtain a completely different framework of\n"quantization" - but there are several but\'s. First of all, this procedure\nis not really quantization because it tries to preserve those properties\nof the classical theory that *cannot* hold in what is normally called a\n"quantum theory" - such as the exact equality between the commutators and\nthe Poisson brackets. Second of all, it is not physics because no one has\ncertainly seen a Thiemannian harmonic oscillator - and no one ever will,\nsimply because non-separable Hilbert spaces cannot be "seen". There is\nabsolutely no observational reason to study this kind of incorrect\n"quantization", there is also nothing attractive about the theoretical\noutcomes of such a treatment, and therefore it does not belong to physics\nbut rather the field that a classmate of mine used to call "mathematical\nmasturbation".\n\nIf this "alternative approach to quantization" were proposed in 1926, one\ncould understand it - physicists don\'t have to get the right answers\nimmediately. But I am just not getting the point of these things in 2004.\nThese attempts obviously do not want to solve "just" some open current\nproblems (well, they seem to have nothing to say about the interesting\ncutting-edge physics questions); they want to erase most of quantum\nmechanics. Why today? What is exactly the reason why we should start to\nhave doubts about the quantization of the harmonic oscillator now in 2004?\n\n> One crucial message it transports is: Just using methods like deformation\n> quantization, GNS construction etc. does (of course) not imply that we need\n> to end up with a discontinuous quantization. While these come from states\n> that have an _apparently_ desireable property, this property is not at all\n> necessary or even sufficient to obtain a meaningful quantum theory.\n\nI find it mildly entertaining that the normal procedures of quantization -\nincluding quantization of the harmonic oscillator - are themselves\npictured as an alternative approach. Well, of course that we do not need\nnonsensical non-separable spaces to describe the harmonic oscillator. Not\nonly that: non-separable spaces do *not* describe the harmonic oscillator\nand they never did. Moreover, the standard procedure has been known since\nthe mid 1920s, and it is the only one that can give physical predictions\nthat reduce to the classical oscillator in the appropriate limit. It\'s\ngreat to rediscover this cool method of quantization in 2004, but it\nshould not be viewed as something new.\n\n> In a nutshell: It is not right that a sensible quantum theory with\n> diffeomorphism constraints must have a diffeomorphism invariant "ground\n> state" and in fact known sensible such quantizations don\'t.\n\nThat\'s nice to hear because Robert Helling was the person who patiently\nrequired (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that\n\n> RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant\n> as otherwise diffeomorphisms would be spontaneously broken.\n\n....\n\n> it is discussed that the singular GNS state can be interpreted as a thermal\n> state of infinite temperature!\n\nThere are various squeezed states (e.g. pointlike states of a string) etc.\nthat have support on various "easy" string configurations, but I think\nthat no field theorist or string theorist would ever normalize them as a\nfunction "equal to one for x=0 and equal to zero otherwise" simply because\nthese "functions" behave as zero under any measure. This approach to the\nvery idea of a "function" is something that is not usable *anywhere* in\nphysics.\n\nBy the way, the idea of the string state localized on the "point-like"\nconfiguration is certainly nothing new. You can see Green\'s papers such as\n\nhttp://arxiv.org/abs/hep-th/9403040\n\nand you will quickly realize that these states are nothing else than what\nwe use as the boundary states for the D-instanton (or a D0-brane).\n\n> I didn\'t know this before and like that insight, because it points at a way\n> to understand a larger framework in which various "different" quantizations\n> (of the string for instance) appear as different aspects of the same thing.\n\nI am not getting the purpose of these attempts. Is the goal to be nice and\nto prove that no one can ever be completely silly? You won\'t be able to\nprove this thing because it is not true. I wonder how much a hypothetical\nperson XY had to suffer when he tried to comprehend the dirty\nregularization techniques used for path integrals and Feynman diagrams if\nXY has even problems with accepting the usual definition of the\nwavefunctions and their integrals in the case of a harmonic oscillator.\n\n> Maybe Thiemann\'s choice of GNS state still can teach us something about\n> strings at high temperature (maybe not though, due to there is no worldsheet\n> physics beyond the Hagedorn temperature).\n\nHow can it ever teach you anything about the physical string theory if it\ndisagrees with every detail of it? The large energy/temperature behavior\nof a string is dictated by the Hagedorn density of states, and the density\ndepends on the central charge. Recall that Thiemann even wants to put all\nthe central charges equal to zero. His machinery just cannot know anything\nabout the real behavior of strings - if you think that there might be\nsomething in it, could you be more specific which insight may be relevant?\nNo, it is not an alternative theory of anything - it is just a\nmisunderstanding of the procedure of quantization and of the role of\nintegrals and measures in quantum physics.\n\n> Robert Helling and Giuseppe Policastro mention that by introducing the\n> appropriate ghost sector the central charge vanishes, which seems to make\n> the two different quantizations appear more similar than otherwise. But\n> don\'t we need to exercise some care here? Even with the ghost sector added\n> physical states (in the standard theory) are not necessarily annihilated by\n> all the symmetry generators. Instead, all that is required is that they are\n> BRST-closed.\n\nI agree with you, Urs, as long as you use a generic value of the ghost\nnumber (e.g. the physical one). With the bc ghosts included, we usually\nstudy the physical Hilbert space as the BRST cohomology. However, one\nvirtue of the system *with* the FP ghosts is that you *may* also require\n*all* the Virasoro generators (not just the positive frequency ones) to\nannihilate your physical states. Unfortunately it probably only works for\nthe states with an unphysical values of the ghost number.\n\nFirst, note that there is no Lie algebraic obstruction. Because with the\nbc ghosts, the naive c=0 Virasoro algebra is restored, the commutator of\ntwo Virasoro generators is again a Virasoro generator, so you won\'t derive\nany contradiction if both of them annihilate your state (for c>0 you could\nderive that "c" - a c-number - annihilates your physical state).\n\nSecond, well, let\'s admit that these are not the most natural\nrepresentatives associated with the old covariant quanrization, for\nexample. But let\'s look: the generator L_n can be written as the\nanticommutator {Q,b_n} of the BRST charge with the Fadeev-Popov antighost.\nWell, if you can find, from the family of BRST-closed-and-equivalent\nstates, a representative annihilated by all b_n\'s, then it\'s also\nannihilated by L_n. Being annihilated by all b_n\'s is something unnatural\n- it is like Dirac\'s sea that is not filled - but as far as I know, you\ncan always find such a state with the desired properties, although\nunfortunately the condition "being annihilated by all b_n\'s" implies a\n(bad) value of the ghost number. And it is possible that the BRST\ncohomology at this ghost number is trivial, in which case my comment is\ncompletely useless. ;-)\n\nOn the other hand, maybe, this construction of a representative\nannihilated by all L_n\'s could be generalizable to any values of the ghost\nnumber, as long as you can excite the states from the previous paragraph\nby Virasoro-invariant functionals of c(\\sigma). If someone knows the full\nanswer, it would be nice to hear about it.\n_____________________________________________ _________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 20 Sep 2004, Urs Schreiber wrote:

> Robert Helling & Guiseppe Policastro:
> String quantization: Fock vs. LQG Representations
> http://www.arxiv.org/abs/hep-th/0409182

Their description of the harmonic oscillator looks particularly strange
because if we did not agree what is the physics of the quantum harmonic
oscillator, we could probably agree about nothing in the world.

If someone suggested that this "alternative" approach (leading to the
nonsensical non-separable Hilbert space) is how many of us are thinking
about the harmonic oscillator, it would not sound encouraging, to say the
least. Well, nevertheless it looks like a fair toy model of what our
friends are doing with other theories.

I am not understanding what is exactly the difference between this
approach to the harmonic oscillator, worldsheet dynamics, or gravity for
that matter on one side, and some completely different elementary
conceptual error - or an "original treatment" by students who get bad
grades from their exams - in approaching any of these theories. Most of us
and our classmates have made many different errors like that in our
lifetime - the main difference is that we realized that they were errors.

I am sure that many of us tried to deal with divergent sums and divergent
integrals, and various other singular objects in various ways that can be
proved "wrong" on physical grounds. Zeno was the first famous scholar who
obtained wrong conclusions ("paradoxes") about the infinitesimal and
"infinite" objects. However, later people learned how to do the calculus
and how to sum the series properly. We learned how to deal with the
distributions and how to regularize and renormalize the integrals, even
though most of us - as well as most new students who start with physics -
are often doing incorrect manipulations with these objects.

Thomas Thiemann et al. are describing these "different" manipulations not
as rudimentary errors in thinking, but as a great piece of alternative
science, and we are systematically being told that this is what we should
pay attention to - or even follow - and perhaps it is even a deep thought.
Why don't we treat equally all students that submit wrong calculations of
some type during their exam? Well...

The very "function" f(x) that equals zero for non-zero x, and equals one
for x=0 (or even its functional generalization), is something that does
not exist in physics. The measure theory guarantees that this function is
"physically" indistinguishable from f(x)=0. The closest nontrivial object
to this function is Dirac's distribution \delta(x). Indeed, by taking the
usual integrated superposition of the \delta-functions, we can obtain the
usual Hilbert space of the usual functions.

If I were completely free to decide what sort of a physics paper XY is,
and XY was based on this "\delta function divided by infinity" that is
nevertheless different from zero, no doubt, I would conclude that the
author is a crackpot.

All Hilbert spaces obtained from these wrong assumptions are
non-separable, unphysical, and the only way how the non-separability can
be cured is if the resulting theory is completely topological and all
these values of "x" are eventually unphysical, perhaps except for their
ordering. The states in this model represent unphysical mixtures of a
hugely infinite number of superselection sectors - it is another
description of Helling et al. comments about "discontinuity" of
Thiemann's representation, I think. Each of these sectors is made of a
single state. In physics, it is legitimate to study each single
superselection sector separately - and if these superselection sectors are
made of a single state, the theory is physically vacuous.

The infinitely modified normalization of the \delta-function implies that
these states cannot be integrated over; the expectation values and
probabilities cannot really be calculated as integrals; they cannot be
combined into meaningful superpositions; the analogous configurations
cannot contribute finitely to the path integrals. Well, it seems that the
proponents do not worry because their goal is not to calculate expectation
values, probabilities, path integrals, or any other physical quantity in a
physics-like theory for that matter. Their task is to satisfy some totally
uninteresting, unphysical, loose, and completely abstract axioms about the
Hilbert spaces - while they're clearly assuming that nothing else is
needed for a physical theory. Well, they are very, very wrong.

> Everybody knows that first quantization is a mystery.

"Why the world is quantum?" may be a mystery and the most counterintuitive
insight about the world, but the mathematical operation behind the first
quantization does not seem mysterious in any way, and it also does not
seem ambiguous. Moreover, I don't know why you chose the "first
quantization" because even it is a mystery, it is a smaller mystery than
the "second quantization". ;-)

> In a sense, the result is not too surprising, I would say: The GNS
> construction associates a quantization of a C* algebra of observables with a
> _choice_ of "state" (linear functional on the algebra). Helling and
> Policastro point out that there is a continuous such state, and that it is
> send by GNS to the standard quantum theory of the string, including the
> correct central charge. The state that Thomas Thiemann based his
> quantization on instead is not continuous and yields something different.

I did not quite like their comments describing the algebras with different
central charges as different "representations" (with an exclamation mark).
The only way how can one understand this sentence is that they claim that
the Virasoro algebras with different central charges are isomorphic to
each other. Well, I think that this is simply not true - and if someone
thinks that it is true, she should find the operator to conjugate the
generators with so that it changes the central charge. The Virasoro
algebras with different values of "c" may have been inspired by the same
"naive" classical symmetry, but at the mathematical level and a generic
value of Planck's constant, they are just different algebras.

There is a common theme in Thiemann's papers which, I'm afraid, may
unfortunately be shared by Helling and Policastro - which is that they
often look at the "classical limit" of an algebra, and treat all the
modifications implied by quantum mechanics as unimportant - and perhaps
annoying? - details whose relevance for any of their conclusions goes to
zero. It is the classical theory that already contains all the information
about the quantum theory, Thiemann et al. seems to believe. They are
trying to do a "quantization without quantization". Well, until one adopts
the exact quantum algebra, the exact quantum generators, Hilbert spaces,
and so on - and as long as he thinks in terms of the classical objects -
he has not made the quantum leap yet. The only reason why we are doing
quantization is that it leads to *different* conclusions than what you
could have expected from classical physics.

Virasoro algebras with different values of "c" are simply not isomorphic
to each other, and the central charge is a purely quantum effect that must
be treated seriously if we're doing quantum theory. The theories with
different c's describe mathematically different symmetries and different
systems although they may have the same classical limit (or classical
starting point to obtain them). There are various other relations between
them - for example, the central charge can run as the function of the
scale - but once again, they are not isomorphic, and if we take quantum
mechanics completely seriously, they are as non-isomorphic as any other
pair of algebras.

By the way, this purely quantum viewpoint will be even more important if
we want to get more insight into the (2,0) theory or even M-theory at the
generic point of the moduli space - because these theories (at least in
some backgrounds) clearly indicate that they cannot be fully obtained from
a classical theory by quantization - and they almost certainly cannot be
obtained from a *unique* classical theory. The very assumption that a
quantum theory is always obtained as a quantization of a specific
classical theory - this assumption is not correct. And if this assumption
is even combined with a completely wrong procedure of (non)quantization,
it's pretty bad.

There are other examples in which Thiemann et al. try to make this sort of
"quantization without quantization". They want the commutators to be
always equal to the Poisson brackets; they want the short distance
singularities in OPEs to be absent much like they are absent in classical
field theory; they want the anomalies to be absent in any theory by
definition; and so forth. But these requirements are simply incorrect in a
quantum (field) theory. The commutator only reduces to the Poisson bracket
in the classical limit, in which \hbar^2 can be completely neglected, so
to say. The singular OPEs can simply be derived from quantum field
theory, and the existence of anomalies can always be deduced in many ways
- both from the IR spectrum as well as from the regularization of the UV
divergences.

One may be trying to obtain a completely different framework of
"quantization" - but there are several but's. First of all, this procedure
is not really quantization because it tries to preserve those properties
of the classical theory that *cannot* hold in what is normally called a
"quantum theory" - such as the exact equality between the commutators and
the Poisson brackets. Second of all, it is not physics because no one has
certainly seen a Thiemannian harmonic oscillator - and no one ever will,
simply because non-separable Hilbert spaces cannot be "seen". There is
absolutely no observational reason to study this kind of incorrect
"quantization", there is also nothing attractive about the theoretical
outcomes of such a treatment, and therefore it does not belong to physics
but rather the field that a classmate of mine used to call "mathematical
masturbation".

If this "alternative approach to quantization" were proposed in 1926, one
could understand it - physicists don't have to get the right answers
immediately. But I am just not getting the point of these things in 2004.
These attempts obviously do not want to solve "just" some open current
problems (well, they seem to have nothing to say about the interesting
cutting-edge physics questions); they want to erase most of quantum
mechanics. Why today? What is exactly the reason why we should start to
have doubts about the quantization of the harmonic oscillator now in 2004?

> One crucial message it transports is: Just using methods like deformation
> quantization, GNS construction etc. does (of course) not imply that we need
> to end up with a discontinuous quantization. While these come from states
> that have an _apparently_ desireable property, this property is not at all
> necessary or even sufficient to obtain a meaningful quantum theory.

I find it mildly entertaining that the normal procedures of quantization -
including quantization of the harmonic oscillator - are themselves
pictured as an alternative approach. Well, of course that we do not need
nonsensical non-separable spaces to describe the harmonic oscillator. Not
only that: non-separable spaces do *not* describe the harmonic oscillator
and they never did. Moreover, the standard procedure has been known since
the mid 1920s, and it is the only one that can give physical predictions
that reduce to the classical oscillator in the appropriate limit. It's
great to rediscover this cool method of quantization in 2004, but it
should not be viewed as something new.

> In a nutshell: It is not right that a sensible quantum theory with
> diffeomorphism constraints must have a diffeomorphism invariant "ground
> state" and in fact known sensible such quantizations don't.

That's nice to hear because Robert Helling was the person who patiently
required (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that

> RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant
> as otherwise diffeomorphisms would be spontaneously broken.

....

> it is discussed that the singular GNS state can be interpreted as a thermal
> state of infinite temperature!

There are various squeezed states (e.g. pointlike states of a string) etc.
that have support on various "easy" string configurations, but I think
that no field theorist or string theorist would ever normalize them as a
function "equal to one for x=0 and equal to zero otherwise" simply because
these "functions" behave as zero under any measure. This approach to the
very idea of a "function" is something that is not usable *anywhere* in
physics.

By the way, the idea of the string state localized on the "point-like"
configuration is certainly nothing new. You can see Green's papers such as

http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9403040

and you will quickly realize that these states are nothing else than what
we use as the boundary states for the D-instanton (or a D0-brane).

> I didn't know this before and like that insight, because it points at a way
> to understand a larger framework in which various "different" quantizations
> (of the string for instance) appear as different aspects of the same thing.

I am not getting the purpose of these attempts. Is the goal to be nice and
to prove that no one can ever be completely silly? You won't be able to
prove this thing because it is not true. I wonder how much a hypothetical
person XY had to suffer when he tried to comprehend the dirty
regularization techniques used for path integrals and Feynman diagrams if
XY has even problems with accepting the usual definition of the
wavefunctions and their integrals in the case of a harmonic oscillator.

> Maybe Thiemann's choice of GNS state still can teach us something about
> strings at high temperature (maybe not though, due to there is no worldsheet
> physics beyond the Hagedorn temperature).

How can it ever teach you anything about the physical string theory if it
disagrees with every detail of it? The large energy/temperature behavior
of a string is dictated by the Hagedorn density of states, and the density
depends on the central charge. Recall that Thiemann even wants to put all
the central charges equal to zero. His machinery just cannot know anything
about the real behavior of strings - if you think that there might be
something in it, could you be more specific which insight may be relevant?
No, it is not an alternative theory of anything - it is just a
misunderstanding of the procedure of quantization and of the role of
integrals and measures in quantum physics.

> Robert Helling and Giuseppe Policastro mention that by introducing the
> appropriate ghost sector the central charge vanishes, which seems to make
> the two different quantizations appear more similar than otherwise. But
> don't we need to exercise some care here? Even with the ghost sector added
> physical states (in the standard theory) are not necessarily annihilated by
> all the symmetry generators. Instead, all that is required is that they are
> BRST-closed.

I agree with you, Urs, as long as you use a generic value of the ghost
number (e.g. the physical one). With the bc ghosts included, we usually
study the physical Hilbert space as the BRST cohomology. However, one
virtue of the system *with* the FP ghosts is that you *may* also require
*all* the Virasoro generators (not just the positive frequency ones) to
annihilate your physical states. Unfortunately it probably only works for
the states with an unphysical values of the ghost number.

First, note that there is no Lie algebraic obstruction. Because with the
bc ghosts, the naive c=0 Virasoro algebra is restored, the commutator of
two Virasoro generators is again a Virasoro generator, so you won't derive
any contradiction if both of them annihilate your state (for c>0 you could
derive that "c" - a c-number - annihilates your physical state).

Second, well, let's admit that these are not the most natural
representatives associated with the old covariant quanrization, for
example. But let's look: the generator L_n can be written as the
anticommutator {Q,b_n} of the BRST charge with the Fadeev-Popov antighost.
Well, if you can find, from the family of BRST-closed-and-equivalent
states, a representative annihilated by all b_n's, then it's also
annihilated by L_n. Being annihilated by all b_n's is something unnatural
- it is like Dirac's sea that is not filled - but as far as I know, you
can always find such a state with the desired properties, although
unfortunately the condition "being annihilated by all b_n's" implies a
(bad) value of the ghost number. And it is possible that the BRST
cohomology at this ghost number is trivial, in which case my comment is
completely useless. ;-)

On the other hand, maybe, this construction of a representative
annihilated by all L_n's could be generalizable to any values of the ghost
number, as long as you can excite the states from the previous paragraph
by Virasoro-invariant functionals of c(\sigma). If someone knows the full
answer, it would be nice to hear about it.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Robert C. Helling

Sep21-04, 05:18 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0409201737310.2887-100000@feynman.harvard.edu>...\n\n> On Mon, 20 Sep 2004, Urs Schreiber wrote:\n>\n> > Robert Helling & Guiseppe Policastro:\n> > String quantization: Fock vs. LQG Representations\n> > hep-th/0409182\n\nThanks for noting our paper. Unfortunately, I am about to leave\nCambridge (my next postdoc is at IU Bremen, back in Germany) and all\nmy papers and notes are stored away in boxes and unaccesible to me at\nthe moment. So I cannot answer Urs\' qustions about signs and\n(anti)-commutativity. He might well be right and we screwed those up\nbut those would be just typos and wouldn\'t change anything substantial\nin the conclusions. Furthermore, I don\'t have my laptop\'s network\nconnection currently running, thus I have to use google groups rather\nthan my regular news reader.\n\n> Their description of the harmonic oscillator looks particularly strange\n> because if we did not agree what is the physics of the quantum harmonic\n> oscillator, we could probably agree about nothing in the world.\n\nMybe you missed that point but our philosophy was to say "this is what\nyou get when you apply LQG methods to the harmonic oscillator". Be\ncareful with them im general because the _physical_ consequences (esp.\nthe spectrum) are not what you meassure in this easy example. It was\nimportant to us not just to say that functions that jump have no place\nin physics because you could never observe them. That would be too\neasy (and in fact plain wrong: If you describe a D-brane (whose\nphysical existence I understand Lubos does not doubt) by a skyscraper\nsheef then this is done exactly by a function (of the transverse\ncoordinates) that is zero everywhere except at one point where it has\na finite value).\n\n[Moderator\'s note: Unfortunately I can\'t agree at all. The principle to\ndeal with the wavefunctions in the "right" way and not the "discontinuous" way\nis completely universal, and it surely applies to D-branes as well.\nClassically you may imagine that the position of a D-brane is a\nwell-defined classical number, but it is not correct to extrapolate\nthis picture to the full theory - in which you include the loop diagrams\n- and it is also inappropriate to imagine that the wavefunctions like\nsqrt(delta(x)) or delta(x)/delta(0) are usable for the particular\nstate of the D-brane.\n\nOn the contrary, dynamics dictates you that there\nare new collective degrees of freedom - the coordinates of the D-brane\n(think about a D0-brane for concreteness), and these degrees of freedom\nsimply must be quantized if you want to compute things quantum\nmechanically and not just in a classical approximation. Consequently,\nthe position of the D0-brane is described, at low energies, by the same\nwavefunctions analogous to the "standard" treatment of quantum mechanics,\nfor example to your harmonic oscillator.\n\nWell, I am sure that you know this stuff because you have worked on\nMatrix theory. The wavefunctions of *any* objects in quantum physics\n- strings, D0-branes, any other objects you may think of - follow the\nrules of quantum mechanics and never Thiemann\'s "quantum" mechanics, and\nif one only works in the classical approximation, she should never think\nabout the wavefunctions for the classical degrees of freedom because it\nis simply a wrong mixture of ideas. If the sheaves were a way to\nintroduce these "alternative" incorrectly normalized delta-functions to\nphysics, as you suggest, then the sheaves would be physically\nincorrect, too. Well, I guess that some controversy may exist concerning\nthe last sentence. ;-) LM]\n\nSo the point of our discussion of the harmonic oscillator is that there\nare measurable consequences of doingit \'the wrong way\'.\n\n[Moderator\'s note: Of course, the most important consequence is that\nphysics as we know it can\'t exist in this wrong treatment. ;-) LM]\n\n> I am sure that many of us tried to deal with divergent sums and divergent\n> integrals, and various other singular objects in various ways that can be\n> proved "wrong" on physical grounds.\n\nActually, this is a major reason for why this formalism is more\ninvolved than the usual one: In the algebraic language (and this is\njust mathemtically more careful language, no physical difference to\nthe usual approach) great care is taken to avoid divergent (and\nsimilar) sums etc so no ambiguities (or ways to do it wrong) appear\nfrom that. This is why careful people deal with the Weyl operators\ne^ix and e^ip instead of the usual x and p: The Weyl operators are\nbounded and thus problems with domains of definition etc do not arise.\nFor example, in the position representation p is the deriviative. But\nnot all functions in L^2(R) are differentiable. Only a dense subset\nis. But all are translateble. Thus one saves some complications (if\nones intend is to be careful) if one uses the better defined Weyl\noperators instead. I am not saying that it cannot be done with x and\np, it\'s just you either close your eyes to mathematical subtleties\n(which is what we physicists do most of the time and it works fine\nmost of the time) or you have to deal with limits and that stuff.\n\n[Moderator\'s note: This is just quantum mechanics or 0+1-dimensional\nquantum field theory, and whoever wants to look for rigorous subtleties can\ndo it. Nevertheless these rigorous tricks to solve these pseudoproblems\ndo not apply to quantum field theory in 2 dimensions or more. The QFT\ndivergences are real, and an approach that wants to eliminate them\ncompletely - for example, to set the singular OPEs to zero - is not "more\ncareful"; it is rather physically incorrect. LM]\n\n> All Hilbert spaces obtained from these wrong assumptions are\n> non-separable, unphysical, and the only way how the non-separability can\n> be cured is if the resulting theory is completely topological and all\n> these values of "x" are eventually unphysical, perhaps except for their\n> ordering.\n\nThe separability is not the issue.\n\n[Moderator\'s note: Everyone can have her own opinion, but I would say that\nit is the issue, and it is the main reason that makes these alternative\nquantizations unphysical. LM]\n\nIn fact, there are (accepted) physical systems with a non-separable\nHilbert space: One (as we remark in a footnote) are Bloch electrons.\nThat is electrons in a periodic potential.\n\n[Moderator\'s note: Again, I completely disagree with these statements\nabout non-separability. Non-separable Hilbert space simply do not have\nany room in physics, and Bloch electrons certainly *are* physics.\n\nYou either describe the Bloch electrons using the full wavefunctions,\nthen you deal with the usual separable Hilbert space of quantum\nmechanics, or you describe them in the approximation in which they are\nonly allowed to sit in the minima - in which case the basis of the\nHilbert space is given by {|n>} where n is an integer label of the atom.\nIn both cases, you obtain ordinary separable Hilbert spaces - which\nmeans Hilbert spaces with a countable orthonormal basis - well, I\nhave constructed it for you.\n\nIt is very important that the continuum of the Bloch electrons is the\nstandard smooth quantum mechanics, and it is completely separable.\n\nI see only one reason that can lead to you see anything nonseparable\nabout the Bloch electrons\' Hilbert space - the reason is that you *are*\nthinking in terms of the unphysical Thiemann-like wavefunctions. But even\nif you imagined that these wavefunctions are legitimate descriptions\nof the electrons sitting in the minima - I would recommend you\nsqrt(delta(x)) instead of delta(x)/delta(0) - you will *not* get\nany nonseparable Hilbert spaces simply because the position of the atoms\nmust be discrete in this case. LM]\n\nThen you know that the wave function is periodic as well. Ahem no, not\nquite, only the physics is periodic. So the wave function is periodic up\nto a phase. And by doing the intergral over the whole infinite crystal,\nyou find that two wave functions with different phases are orthogonal. So\nfor each point in the interval [0,2pi) of phases there is an orthogonal\nsector in the Hilbert space. Thus the total Hilbert space is kind of the\nL^2 of the unit cell to times the number of points in that interval,\nclearly a non-separable space.\n\n[Moderator\'s note: I suppose that you are only joking because it is not\neasy to believe that you would write this seriously. The Hilbert space\nof electrons on a one-dimensional lattice is definitely separable. You\ncan either describe it as the L^2 space of complex-valued functions\nof a single variable "p" on the interval [0,2\\pi], or you can describe\nit as a collection of amplitudes a_n for the electron to be stuck\nnear the atom number #n. These two spaces are isomorphic and they\ncertainly should not be multiplied with one another, and I can find\nan introduction to Fourier expansion if you need one.\n\nIf the electron is associated with a single atom (position), then\nit cannot have a well-defined momentum - sorry but this is called the\nuncertainty principle. Your products of the position Hilbert space and the\nmomentum space are exactly the things that do *not* exist in quantum\nphysics, because of its very basic defining principle. The only limit\nhow can you avoid the uncertainty principle is the classical limit,\nbut the classical limit does *not* define a *quantum* theory, and it is\njust not correct to think about its Hilbert space. LM]\n\nYou could say that this happens only in the infinite crystal size\nlimit.\n\n[Moderator\'s note: What you say *never* happens with electrons, not\neven for infinite crystals, and non-separable Hilbert spaces never\noccur in real physics. LM]\n\nBut this idealization people usually are happy to make. Otherwise (with an\nIR cut-off) there would for example be no phase transitions. But that is a\ndifferent matter.\n\n> The states in this model represent unphysical mixtures of a\n> hugely infinite number of superselection sectors - it is another\n> description of Helling et al. comments about "discontinuity" of\n> Thiemann\'s representation, I think. Each of these sectors is made of a\n> single state. In physics, it is legitimate to study each single\n> superselection sector separately - and if these superselection sectors are\n> made of a single state, the theory is physically vacuous.\n\nAt least in the mathematical sense (and that is supposed to coincide\nwith the physical sense), a super-selection sector is a representation\nof the quantum algebra. To states are in different sectors if they\nare in inequivalent representations.\n\n> > Everybody knows that first quantization is a mystery.\n>\n> "Why the world is quantum?" may be a mystery and the most counterintuitive\n> insight about the world, but the mathematical operation behind the first\n> quantization does not seem mysterious in any way, and it also does not\n> seem ambiguous. Moreover, I don\'t know why you chose the "first\n> quantization" because even it is a mystery, it is a smaller mystery than\n> the "second quantization". ;-)\n\nWas this just a joke? If not, here is why people say this (and usualy\nthis continues with "second quantization is a functor"): Of course if\nyour classical system has R^n as configuration space and its cotangent\nbundle as the symplectic space then every child knows how to quntize:\nTake L^2(R^n) as your Hilbert space and replace all x\'s by\nmultiplication operators and all p\'s by derivatives. Oh, and when\nthere\'s an ordering ambiguity, follow one or the other prescription\n(but do that consistently).\n\nHowever, what do you do, if I just give you some symplectic space and\ndon\'t tell you which are the simple prefered position and momentum\ncoordinates (and that\'s what x and p are, just coordinates). And as\nthe real world does not come with coordinates written on everything\none should have some recepy how to deal with this more general\nsituation. And then check that this reduces to the usual story (or an\nequivalent one) in the simple situation. And this is what we have done\nin the paper.\n\nIt is known, that there is no unique way to do this map from a\nsymplectic space to a Hilbert space with operators in general. There\nare further choices involved.\n\n[Moderator\'s note: OK, so it should have been said that "geometric\nquantization" is a subtle subject or something like that. You can\ninvent complicated classical phase spaces that will create problems\nwith finding its quantum counterpart, but these can either be solved,\nin which case you obtain a straightforward quantization, or they cannot\nbe canonically or easily solved (or they cannot be solved at all), in\nwhich case you have simply chosen a wrong problem based on incorrect\nassumptions. I am not interested in such a problem because if there is\nno indication that it may be interesting.\n\nWell, this is another thing that seems to be a common theme\nin the loop quantum gravity community. They take for granted that if\nthere exists a classical theory, there must be a corresponding quantum\ntheory, and therefore it always makes sense to ask "what is the\nquantization". But this is just a misconception. We know many independent\nreasons why a meaningful classical theory often has no meaningful quantum\ncounterpart - gauge anomalies are probably the most definitive example.\n\nI would still say that the first quantization, whenever is meaningful and\nunique, is a mathematically transparent procedure.\n\nLoop quantum gravity, and the "algebraic quantization" in general, starts\nwith the assumption that any classical theory can be "quantized". Well,\nthen they find a (usually unphysical) structure of equations that is good\nenough to call it a "representation of an algebra". Finally they\ncelebrate that they showed that a theory can be easily quantized. Come\non. This is exactly the description of the GIGO-scheme (garbage in,\ngarbage out). LM]\n\n> I did not quite like their comments describing the algebras with different\n> central charges as different "representations" (with an exclamation mark).\n\nWhen we say algebra, we mean the C*-algebra of the observables. And\nthose are indeed the same (and only the represenations differ).\n\n[Moderator\'s note: Conventional approaches to quantum physics immediately\nimply a very specific "representation" or a Hilbert space once you define\nyour observables; the knowledge of the "representation" is a part of\nthe definition of a quantum theory. LM]\n\nThese algebras are the algebras of the X\'s (or the W(f) after some\nmassaging). Then this Weyl algebra has representations. And on those\nrepresentations there is a symmetry (Lie-)algebra acting by unitary\noperators. And this symmetry algebra is some Virasoro alegbra in both\ncases. But these symmetry algebras have different central charges in\nthe two representations of the Weyl algebra. So: There are two kinds\nof algebras, don\'t confuse them.\n\nFurthermore, even if we didn\'t talk about it, in the Virasoro algebra\nthe central charge is just an abstract element usally called c. It\ncommutes with everything, so in an irreducible representation it is\nrepresented by a number. And again, this number depends on the\nrepresentation. This number, together with h, the eigenvalue of L_0 in\nthat representation, label a highest weight representation of the\nalgebra.\n\n[Moderator\'s note: I agree; if you replace "c" by a new generator\nof an extended algebra, then there is a single algebra only - and it\ncan have different manifestations with different eigenvalues of c.\nBut then we should not talk about any isomorphism because there is only\none extended algebra. At any rate, your statement seems either vacuous,\nor incorrect. LM]\n\n> The only way how can one understand this sentence is that they claim that\n> the Virasoro algebras with different central charges are isomorphic to\n> each other.\n\nNobody claimed that. As I just said: In the algbra, c is an abstract\nelement, it becomes a number only in a representation. And nobody\nclaims that representations with different c are equivalent.\n\n[Moderator\'s note: Good. LM]\n\n> There is a common theme in Thiemann\'s papers which, I\'m afraid, may\n> unfortunately be shared by Helling and Policastro - which is that they\n> often look at the "classical limit" of an algebra, and treat all the\n> modifications implied by quantum mechanics as unimportant - and perhaps\n> annoying? - details whose relevance for any of their conclusions goes to\n> zero.\n\nCould you be more specific with this claim?\n\n[Moderator\'s note: It is easy to be specific - you repeated, even in this\nposting, that you always think about "quantization" to be a well-defined,\nalthough potentially difficult, task that must have a solution. I claim\nthat it is not true, and that it is more or less equivalent to the\nstatement that quantum physics does not bring anything really new to the\noriginal classical theory. LM]\n\n> By the way, this purely quantum viewpoint will be even more important if\n> we want to get more insight into the (2,0) theory or even M-theory at the\n> generic point of the moduli space - because these theories (at least in\n> some backgrounds) clearly indicate that they cannot be fully obtained from\n> a classical theory by quantization - and they almost certainly cannot be\n> obtained from a *unique* classical theory.\n\nQuantization is a game that always involves a classical system.\nHowever nobody claimed that every qunatum system arises from the\nquantization of a classical theory.\n\n[Moderator\'s note: The word "nobody" is certainly an exaggeration. In the\nold times, for example, it used to be believed by everyone. The main\nincorrect prejudice that I am trying to point out is however the opposite\none: it is believed by many that "quantization" is something that must\nmake sense for any classical system. No, it is not true. LM]\n\n> There are other examples in which Thiemann et al. try to make this sort of\n> "quantization without quantization". They want the commutators to be\n> always equal to the Poisson brackets;\n\nI hope you don\'t include us in "et al". We impose the Poisson goes to\ncommutator rule only for linear combinations of what would be x and p,\nnot for higher powers. And I doubt Thomas would do commit that crime\neither.\n\n[Moderator\'s note: I am convinced that he does. See e.g. 0401172 around equation\n(5.1). Well, this is also about his setting the central charge to zero.\nI don\'t know whether he would do such things in all cases, but he\ncertainly does it in many cases in which the "physical" treatment of\nthese theories leads to a different commutator. By the way, I am not the\nonly one who thinks that Thiemann sets the commutators equal to Poisson\nbrackets. Let me cite a mail from Urs: LM]\n\n>> UrS: Let me say that I do think that his construction of a Hilbert space\n>> and of the quantum operators is well defined and that indeed the\n>> commutators that he considers reproduce just the classical Poisson\n>> brackets. I have tried to indicate why this can be true and why there are\n>> no higher order Wick contractions in Thiemann\'s framework in this comment.\n\n> One may be trying to obtain a completely different framework of\n> "quantization" - but there are several but\'s.\n\nWe tried hard to spell out the general framework of the quantization\nprocedure used in the two approaches. We say, that you can include the\npolymer quantization if you do not impose the at first rather\ntechnical condition of weak continuity. But then this has huge\nobservable consequences. So don\'t confuse framework and consequences.\n\n[Moderator\'s note: In consistent physical theories there should be no\ndifference and dependence on the choice of "continuities": the dynamics\nchooses its own notion of continuity. I also don\'t understand\nhow you want to separate the framework from its consequences. LM]\n\nCould you spell out the rules for your framework that clearly rules\nout the LQG one? It should be a machine that turns a symplectic space\nwith its observables into a Hilbertspace with its operators.\n\n[Moderator\'s note: I think that the very idea that the search for\nquantum theory should be a "machine that turns a symplectic space\ninto a Hilbert space" is a very narrow-minded idea, and it is in fact one\nof the ideas that lead to LQG. It only makes sense to look for a Hilbert\nspace arising from a classical theory if the theory also makes sense at the\nquantum level. And if a meaningful predictive quantum theory cannot\nbe constructed, then the task to construct it from an arbitrary\nclassical starting point is simply a crappy question to ask. An\ninteresting quantum theory to study is a theory with a finite number of\nparameters such that any other modification of the theory breaks a\nfundamental principle (e.g. a symmetry), and a theory that must be\nable to predict an infinite amount of phenomena. This constraint has\nnothing to do with the question whether it is obtained as a geometric\nquantization of a classical theory - the latter requirement is not too\ninteresting from a physical point of view.\n\nClassical pure GR may have an infinite-dimensional symplectic phase\nspace, but it does not lead to a meaningful (renormalizable) quantum\ntheory, and therefore this theory simply should *not* be considered\nat the quantum level. The infinite amount of the counterterms can\nbe proved and we should not try to avoid this conclusions simply because\nit is true and solid. Quantizing pure GR is not a deep task, but it is\na wrong task.\n\nI am not arguing against the geometric quantization of Calabi-Yaus\nand other things relevant for string theory with a B-field or anything\nlike that, but I certainly argue against "quantization" of a generic\nclassical theory. LM]\n\n> First of all, this procedure\n> is not really quantization because it tries to preserve those properties\n> of the classical theory that *cannot* hold in what is normally called a\n> "quantum theory" - such as the exact equality between the commutators and\n> the Poisson brackets.\n\nNobody imposes that. We only ask for a unitary representation of the\ndiffeomorphism symmetry. And those might obey the group law of the\ndiffeo group or not (because of an anomaly).\n\n[Moderator\'s note: Unitary representation of the diffeomorphism symmetry\nis not a physically important question because the physical states must be\ninvariant under the (normalizable part) of the diffeomorphism group - so\nthat the physically interesting representation is always the trivial one,\nwhich is of course unitary. (Many copies of the trivial representation;\nthose copies encode the other, physical degrees of freedom.)\nMoreover, the Hilbert space is not a representation of the "large"\ndiffeomorphisms that change the asymptotics at all. LM]\n\n> Second of all, it is not physics because no one has\n> certainly seen a Thiemannian harmonic oscillator\n\nRight. That we meant by "(un)physical" in the abstract.\n\n- and no one ever will,\n> simply because non-separable Hilbert spaces cannot be "seen".\n\nSee above.\n\n> I find it mildly entertaining that the normal procedures of quantization -\n> including quantization of the harmonic oscillator - are themselves\n> pictured as an alternative approach.\n\nWhere?\n\n[Moderator\'s note: The last sentence of your article, for example, states\nthat "But at least it is demonstrated that there is a viable alternative\nto the singular representations based on polymer states." If I understand\nwell, the "viable alternative" is the standard quantum mechanics. The\nonly thing I claimed is that you call standard quantum mechanics\nan "alternative", and I think that you\'ve been proved that it is the\ncase. OK? LM]\n\nWe describe both quantizations in a single framework. There is\none choice to be made. And that has physical consequences. In the\nmechanics example, those consequnces are unphysical, so the choice was\nwrong. Everybody is free to deduce something about the choice in the\nstring case.\n\n> Well, of course that we do not need\n> nonsensical non-separable spaces to describe the harmonic oscillator. Not\n> only that: non-separable spaces do *not* describe the harmonic oscillator\n> and they never did. Moreover, the standard procedure has been known since\n> the mid 1920s, and it is the only one that can give physical predictions\n> that reduce to the classical oscillator in the appropriate limit. It\'s\n> great to rediscover this cool method of quantization in 2004, but it\n> should not be viewed as something new.\n\nWe didn\'t say there is anything wrong with the standard harmonic\noscillator. Rather we used it as a test bed for the quantization\nprocedure. This was to counter arguments along the lines of \'nobody\nhas yet seen a string in nature".\n\n> That\'s nice to hear because Robert Helling was the person who patiently\n> required (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that\n>\n> > RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant\n> > as otherwise diffeomorphisms would be spontaneously broken.\n\nLet\'s face it: In that thread you didn\'t get it that I was playing the\ndevil\'s advocate.\n\n[Moderator\'s note: Well, sometimes I am confused who is your devil and who\nis your God, or whoever is your devil\'s "alternative". ;-)\n\n> > it is discussed that the singular GNS state can be interpreted as a thermal\n> > state of infinite temperature!\n\nOne should be carefule as this is world sheet temperature and not\ntarget space.\n\n> > I didn\'t know this before and like that insight, because it points at a way\n> > to understand a larger framework in which various "different" quantizations\n> > (of the string for instance) appear as different aspects of the same thing.\n>\n> I am not getting the purpose of these attempts. Is the goal to be nice and\n> to prove that no one can ever be completely silly?\n\nIf you like to express it that way...\n\nSorry, right now, I do not have more time to reply to the more polemic\nparts of your post.\n\nRobert\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.4.31.0409201737310.2887-100000@feynman.harvard.edu>...

> On Mon, 20 Sep 2004, Urs Schreiber wrote:
>
> > Robert Helling & Guiseppe Policastro:
> > String quantization: Fock vs. LQG Representations
> > http://www.arxiv.org/abs/hep-th/0409182

Thanks for noting our paper. Unfortunately, I am about to leave
Cambridge (my next postdoc is at IU Bremen, back in Germany) and all
my papers and notes are stored away in boxes and unaccesible to me at
the moment. So I cannot answer Urs' qustions about signs and
(anti)-commutativity. He might well be right and we screwed those up
but those would be just typos and wouldn't change anything substantial
in the conclusions. Furthermore, I don't have my laptop's network
connection currently running, thus I have to use google groups rather
than my regular news reader.

> Their description of the harmonic oscillator looks particularly strange
> because if we did not agree what is the physics of the quantum harmonic
> oscillator, we could probably agree about nothing in the world.

Mybe you missed that point but our philosophy was to say "this is what
you get when you apply LQG methods to the harmonic oscillator". Be
careful with them im general because the _physical_ consequences (esp.
the spectrum) are not what you meassure in this easy example. It was
important to us not just to say that functions that jump have no place
in physics because you could never observe them. That would be too
easy (and in fact plain wrong: If you describe a D-brane (whose
physical existence I understand Lubos does not doubt) by a skyscraper
sheef then this is done exactly by a function (of the transverse
coordinates) that is zero everywhere except at one point where it has
a finite value).

[Moderator's note: Unfortunately I can't agree at all. The principle to
deal with the wavefunctions in the "right" way and not the "discontinuous" way
is completely universal, and it surely applies to D-branes as well.
Classically you may imagine that the position of a D-brane is a
well-defined classical number, but it is not correct to extrapolate
this picture to the full theory - in which you include the loop diagrams
- and it is also inappropriate to imagine that the wavefunctions like
\sqrt(\delta(x)) or \delta(x)/\delta(0) are usable for the particular
state of the D-brane.

On the contrary, dynamics dictates you that there
are new collective degrees of freedom - the coordinates of the D-brane
(think about a D0-brane for concreteness), and these degrees of freedom
simply must be quantized if you want to compute things quantum
mechanically and not just in a classical approximation. Consequently,
the position of the D0-brane is described, at low energies, by the same
wavefunctions analogous to the "standard" treatment of quantum mechanics,
for example to your harmonic oscillator.

Well, I am sure that you know this stuff because you have worked on
Matrix theory. The wavefunctions of *any* objects in quantum physics
- strings, D0-branes, any other objects you may think of - follow the
rules of quantum mechanics and never Thiemann's "quantum" mechanics, and
if one only works in the classical approximation, she should never think
about the wavefunctions for the classical degrees of freedom because it
is simply a wrong mixture of ideas. If the sheaves were a way to
introduce these "alternative" incorrectly normalized \delta-functions to
physics, as you suggest, then the sheaves would be physically
incorrect, too. Well, I guess that some controversy may exist concerning
the last sentence. ;-) LM]

So the point of our discussion of the harmonic oscillator is that there
are measurable consequences of doingit 'the wrong way'.

[Moderator's note: Of course, the most important consequence is that
physics as we know it can't exist in this wrong treatment. ;-) LM]

> I am sure that many of us tried to deal with divergent sums and divergent
> integrals, and various other singular objects in various ways that can be
> proved "wrong" on physical grounds.

Actually, this is a major reason for why this formalism is more
involved than the usual one: In the algebraic language (and this is
just mathemtically more careful language, no physical difference to
the usual approach) great care is taken to avoid divergent (and
similar) sums etc so no ambiguities (or ways to do it wrong) appear
from that. This is why careful people deal with the Weyl operators
e^{ix} and e^{ip} instead of the usual x and p: The Weyl operators are
bounded and thus problems with domains of definition etc do not arise.
For example, in the position representation p is the deriviative. But
not all functions in L^2(R) are differentiable. Only a dense subset
is. But all are translateble. Thus one saves some complications (if
ones intend is to be careful) if one uses the better defined Weyl
operators instead. I am not saying that it cannot be done with x and
p, it's just you either close your eyes to mathematical subtleties
(which is what we physicists do most of the time and it works fine
most of the time) or you have to deal with limits and that stuff.

[Moderator's note: This is just quantum mechanics or 0+1-dimensional
quantum field theory, and whoever wants to look for rigorous subtleties can
do it. Nevertheless these rigorous tricks to solve these pseudoproblems
do not apply to quantum field theory in 2 dimensions or more. The QFT
divergences are real, and an approach that wants to eliminate them
completely - for example, to set the singular OPEs to zero - is not "more
careful"; it is rather physically incorrect. LM]

> All Hilbert spaces obtained from these wrong assumptions are
> non-separable, unphysical, and the only way how the non-separability can
> be cured is if the resulting theory is completely topological and all
> these values of "x" are eventually unphysical, perhaps except for their
> ordering.

The separability is not the issue.

[Moderator's note: Everyone can have her own opinion, but I would say that
it is the issue, and it is the main reason that makes these alternative
quantizations unphysical. LM]

In fact, there are (accepted) physical systems with a non-separable
Hilbert space: One (as we remark in a footnote) are Bloch electrons.
That is electrons in a periodic potential.

[Moderator's note: Again, I completely disagree with these statements
about non-separability. Non-separable Hilbert space simply do not have
any room in physics, and Bloch electrons certainly *are* physics.

You either describe the Bloch electrons using the full wavefunctions,
then you deal with the usual separable Hilbert space of quantum
mechanics, or you describe them in the approximation in which they are
only allowed to sit in the minima - in which case the basis of the
Hilbert space is given by {|n>} where n is an integer label of the atom.
In both cases, you obtain ordinary separable Hilbert spaces - which
means Hilbert spaces with a countable orthonormal basis - well, I
have constructed it for you.

It is very important that the continuum of the Bloch electrons is the
standard smooth quantum mechanics, and it is completely separable.

I see only one reason that can lead to you see anything nonseparable
about the Bloch electrons' Hilbert space - the reason is that you *are*
thinking in terms of the unphysical Thiemann-like wavefunctions. But even
if you imagined that these wavefunctions are legitimate descriptions
of the electrons sitting in the minima - I would recommend you
\sqrt(\delta(x)) instead of \delta(x)/\delta(0) - you will *not* get
any nonseparable Hilbert spaces simply because the position of the atoms
must be discrete in this case. LM]

Then you know that the wave function is periodic as well. Ahem no, not
quite, only the physics is periodic. So the wave function is periodic up
to a phase. And by doing the intergral over the whole infinite crystal,
you find that two wave functions with different phases are orthogonal. So
for each point in the interval [0,2pi) of phases there is an orthogonal
sector in the Hilbert space. Thus the total Hilbert space is kind of the
L^2 of the unit cell to times the number of points in that interval,
clearly a non-separable space.

[Moderator's note: I suppose that you are only joking because it is not
easy to believe that you would write this seriously. The Hilbert space
of electrons on a one-dimensional lattice is definitely separable. You
can either describe it as the L^2 space of complex-valued functions
of a single variable "p" on the interval [0,2\pi], or you can describe
it as a collection of amplitudes a_n for the electron to be stuck
near the atom number #n. These two spaces are isomorphic and they
certainly should not be multiplied with one another, and I can find
an introduction to Fourier expansion if you need one.

If the electron is associated with a single atom (position), then
it cannot have a well-defined momentum - sorry but this is called the
uncertainty principle. Your products of the position Hilbert space and the
momentum space are exactly the things that do *not* exist in quantum
physics, because of its very basic defining principle. The only limit
how can you avoid the uncertainty principle is the classical limit,
but the classical limit does *not* define a *quantum* theory, and it is
just not correct to think about its Hilbert space. LM]

You could say that this happens only in the infinite crystal size
limit.

[Moderator's note: What you say *never* happens with electrons, not
even for infinite crystals, and non-separable Hilbert spaces never
occur in real physics. LM]

But this idealization people usually are happy to make. Otherwise (with an
IR cut-off) there would for example be no phase transitions. But that is a
different matter.

> The states in this model represent unphysical mixtures of a
> hugely infinite number of superselection sectors - it is another
> description of Helling et al. comments about "discontinuity" of
> Thiemann's representation, I think. Each of these sectors is made of a
> single state. In physics, it is legitimate to study each single
> superselection sector separately - and if these superselection sectors are
> made of a single state, the theory is physically vacuous.

At least in the mathematical sense (and that is supposed to coincide
with the physical sense), a super-selection sector is a representation
of the quantum algebra. To states are in different sectors if they
are in inequivalent representations.

> > Everybody knows that first quantization is a mystery.
>
> "Why the world is quantum?" may be a mystery and the most counterintuitive
> insight about the world, but the mathematical operation behind the first
> quantization does not seem mysterious in any way, and it also does not
> seem ambiguous. Moreover, I don't know why you chose the "first
> quantization" because even it is a mystery, it is a smaller mystery than
> the "second quantization". ;-)

Was this just a joke? If not, here is why people say this (and usualy
this continues with "second quantization is a functor"): Of course if
your classical system has R^n as configuration space and its cotangent
bundle as the symplectic space then every child knows how to quntize:
Take L^2(R^n) as your Hilbert space and replace all x's by
multiplication operators and all p's by derivatives. Oh, and when
there's an ordering ambiguity, follow one or the other prescription
(but do that consistently).

However, what do you do, if I just give you some symplectic space and
don't tell you which are the simple prefered position and momentum
coordinates (and that's what x and p are, just coordinates). And as
the real world does not come with coordinates written on everything
one should have some recepy how to deal with this more general
situation. And then check that this reduces to the usual story (or an
equivalent one) in the simple situation. And this is what we have done
in the paper.

It is known, that there is no unique way to do this map from a
symplectic space to a Hilbert space with operators in general. There
are further choices involved.

[Moderator's note: OK, so it should have been said that "geometric
quantization" is a subtle subject or something like that. You can
invent complicated classical phase spaces that will create problems
with finding its quantum counterpart, but these can either be solved,
in which case you obtain a straightforward quantization, or they cannot
be canonically or easily solved (or they cannot be solved at all), in
which case you have simply chosen a wrong problem based on incorrect
assumptions. I am not interested in such a problem because if there is
no indication that it may be interesting.

Well, this is another thing that seems to be a common theme
in the loop quantum gravity community. They take for granted that if
there exists a classical theory, there must be a corresponding quantum
theory, and therefore it always makes sense to ask "what is the
quantization". But this is just a misconception. We know many independent
reasons why a meaningful classical theory often has no meaningful quantum
counterpart - gauge anomalies are probably the most definitive example.

I would still say that the first quantization, whenever is meaningful and
unique, is a mathematically transparent procedure.

Loop quantum gravity, and the "algebraic quantization" in general, starts
with the assumption that any classical theory can be "quantized". Well,
then they find a (usually unphysical) structure of equations that is good
enough to call it a "representation of an algebra". Finally they
celebrate that they showed that a theory can be easily quantized. Come
on. This is exactly the description of the GIGO-scheme (garbage in,
garbage out). LM]

> I did not quite like their comments describing the algebras with different
> central charges as different "representations" (with an exclamation mark).

When we say algebra, we mean the C*-algebra of the observables. And
those are indeed the same (and only the represenations differ).

[Moderator's note: Conventional approaches to quantum physics immediately
imply a very specific "representation" or a Hilbert space once you define
your observables; the knowledge of the "representation" is a part of
the definition of a quantum theory. LM]

These algebras are the algebras of the X's (or the W(f) after some
massaging). Then this Weyl algebra has representations. And on those
representations there is a symmetry (Lie-)algebra acting by unitary
operators. And this symmetry algebra is some Virasoro alegbra in both
cases. But these symmetry algebras have different central charges in
the two representations of the Weyl algebra. So: There are two kinds
of algebras, don't confuse them.

Furthermore, even if we didn't talk about it, in the Virasoro algebra
the central charge is just an abstract element usally called c. It
commutes with everything, so in an irreducible representation it is
represented by a number. And again, this number depends on the
representation. This number, together with h, the eigenvalue of L_0 in
that representation, label a highest weight representation of the
algebra.

[Moderator's note: I agree; if you replace "c" by a new generator
of an extended algebra, then there is a single algebra only - and it
can have different manifestations with different eigenvalues of c.
But then we should not talk about any isomorphism because there is only
one extended algebra. At any rate, your statement seems either vacuous,
or incorrect. LM]

> The only way how can one understand this sentence is that they claim that
> the Virasoro algebras with different central charges are isomorphic to
> each other.

Nobody claimed that. As I just said: In the algbra, c is an abstract
element, it becomes a number only in a representation. And nobody
claims that representations with different c are equivalent.

[Moderator's note: Good. LM]

> There is a common theme in Thiemann's papers which, I'm afraid, may
> unfortunately be shared by Helling and Policastro - which is that they
> often look at the "classical limit" of an algebra, and treat all the
> modifications implied by quantum mechanics as unimportant - and perhaps
> annoying? - details whose relevance for any of their conclusions goes to
> zero.

Could you be more specific with this claim?

[Moderator's note: It is easy to be specific - you repeated, even in this
posting, that you always think about "quantization" to be a well-defined,
although potentially difficult, task that must have a solution. I claim
that it is not true, and that it is more or less equivalent to the
statement that quantum physics does not bring anything really new to the
original classical theory. LM]

> By the way, this purely quantum viewpoint will be even more important if
> we want to get more insight into the (2,0) theory or even M-theory at the
> generic point of the moduli space - because these theories (at least in
> some backgrounds) clearly indicate that they cannot be fully obtained from
> a classical theory by quantization - and they almost certainly cannot be
> obtained from a *unique* classical theory.

Quantization is a game that always involves a classical system.
However nobody claimed that every qunatum system arises from the
quantization of a classical theory.

[Moderator's note: The word "nobody" is certainly an exaggeration. In the
old times, for example, it used to be believed by everyone. The main
incorrect prejudice that I am trying to point out is however the opposite
one: it is believed by many that "quantization" is something that must
make sense for any classical system. No, it is not true. LM]

> There are other examples in which Thiemann et al. try to make this sort of
> "quantization without quantization". They want the commutators to be
> always equal to the Poisson brackets;

I hope you don't include us in "et al". We impose the Poisson goes to
commutator rule only for linear combinations of what would be x and p,
not for higher powers. And I doubt Thomas would do commit that crime
either.

[Moderator's note: I am convinced that he does. See e.g. 0401172 around equation
(5.1). Well, this is also about his setting the central charge to zero.
I don't know whether he would do such things in all cases, but he
certainly does it in many cases in which the "physical" treatment of
these theories leads to a different commutator. By the way, I am not the
only one who thinks that Thiemann sets the commutators equal to Poisson
brackets. Let me cite a mail from Urs: LM]

>> UrS: Let me say that I do think that his construction of a Hilbert space
>> and of the quantum operators is well defined and that indeed the
>> commutators that he considers reproduce just the classical Poisson
>> brackets. I have tried to indicate why this can be true and why there are
>> no higher order Wick contractions in Thiemann's framework in this comment.

> One may be trying to obtain a completely different framework of
> "quantization" - but there are several but's.

We tried hard to spell out the general framework of the quantization
procedure used in the two approaches. We say, that you can include the
polymer quantization if you do not impose the at first rather
technical condition of weak continuity. But then this has huge
observable consequences. So don't confuse framework and consequences.

[Moderator's note: In consistent physical theories there should be no
difference and dependence on the choice of "continuities": the dynamics
chooses its own notion of continuity. I also don't understand
how you want to separate the framework from its consequences. LM]

Could you spell out the rules for your framework that clearly rules
out the LQG one? It should be a machine that turns a symplectic space
with its observables into a Hilbertspace with its operators.

[Moderator's note: I think that the very idea that the search for
quantum theory should be a "machine that turns a symplectic space
into a Hilbert space" is a very narrow-minded idea, and it is in fact one
of the ideas that lead to LQG. It only makes sense to look for a Hilbert
space arising from a classical theory if the theory also makes sense at the
quantum level. And if a meaningful predictive quantum theory cannot
be constructed, then the task to construct it from an arbitrary
classical starting point is simply a crappy question to ask. An
interesting quantum theory to study is a theory with a finite number of
parameters such that any other modification of the theory breaks a
fundamental principle (e.g. a symmetry), and a theory that must be
able to predict an infinite amount of phenomena. This constraint has
nothing to do with the question whether it is obtained as a geometric
quantization of a classical theory - the latter requirement is not too
interesting from a physical point of view.

Classical pure GR may have an infinite-dimensional symplectic phase
space, but it does not lead to a meaningful (renormalizable) quantum
theory, and therefore this theory simply should *not* be considered
at the quantum level. The infinite amount of the counterterms can
be proved and we should not try to avoid this conclusions simply because
it is true and solid. Quantizing pure GR is not a deep task, but it is
a wrong task.

I am not arguing against the geometric quantization of Calabi-Yaus
and other things relevant for string theory with a B-field or anything
like that, but I certainly argue against "quantization" of a generic
classical theory. LM]

> First of all, this procedure
> is not really quantization because it tries to preserve those properties
> of the classical theory that *cannot* hold in what is normally called a
> "quantum theory" - such as the exact equality between the commutators and
> the Poisson brackets.

Nobody imposes that. We only ask for a unitary representation of the
diffeomorphism symmetry. And those might obey the group law of the
diffeo group or not (because of an anomaly).

[Moderator's note: Unitary representation of the diffeomorphism symmetry
is not a physically important question because the physical states must be
invariant under the (normalizable part) of the diffeomorphism group - so
that the physically interesting representation is always the trivial one,
which is of course unitary. (Many copies of the trivial representation;
those copies encode the other, physical degrees of freedom.)
Moreover, the Hilbert space is not a representation of the "large"
diffeomorphisms that change the asymptotics at all. LM]

> Second of all, it is not physics because no one has
> certainly seen a Thiemannian harmonic oscillator

Right. That we meant by "(un)physical" in the abstract.

- and no one ever will,
> simply because non-separable Hilbert spaces cannot be "seen".

See above.

> I find it mildly entertaining that the normal procedures of quantization -
> including quantization of the harmonic oscillator - are themselves
> pictured as an alternative approach.

Where?

[Moderator's note: The last sentence of your article, for example, states
that "But at least it is demonstrated that there is a viable alternative
to the singular representations based on polymer states." If I understand
well, the "viable alternative" is the standard quantum mechanics. The
only thing I claimed is that you call standard quantum mechanics
an "alternative", and I think that you've been proved that it is the
case. OK? LM]

We describe both quantizations in a single framework. There is
one choice to be made. And that has physical consequences. In the
mechanics example, those consequnces are unphysical, so the choice was
wrong. Everybody is free to deduce something about the choice in the
string case.

> Well, of course that we do not need
> nonsensical non-separable spaces to describe the harmonic oscillator. Not
> only that: non-separable spaces do *not* describe the harmonic oscillator
> and they never did. Moreover, the standard procedure has been known since
> the mid 1920s, and it is the only one that can give physical predictions
> that reduce to the classical oscillator in the appropriate limit. It's
> great to rediscover this cool method of quantization in 2004, but it
> should not be viewed as something new.

We didn't say there is anything wrong with the standard harmonic
oscillator. Rather we used it as a test bed for the quantization
procedure. This was to counter arguments along the lines of 'nobody
has yet seen a string in nature".

> That's nice to hear because Robert Helling was the person who patiently
> required (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that
>
> > RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant
> > as otherwise diffeomorphisms would be spontaneously broken.

Let's face it: In that thread you didn't get it that I was playing the
devil's advocate.

[Moderator's note: Well, sometimes I am confused who is your devil and who
is your God, or whoever is your devil's "alternative". ;-)

> > it is discussed that the singular GNS state can be interpreted as a thermal
> > state of infinite temperature!

One should be carefule as this is world sheet temperature and not
target space.

> > I didn't know this before and like that insight, because it points at a way
> > to understand a larger framework in which various "different" quantizations
> > (of the string for instance) appear as different aspects of the same thing.
>
> I am not getting the purpose of these attempts. Is the goal to be nice and
> to prove that no one can ever be completely silly?

If you like to express it that way...

Sorry, right now, I do not have more time to reply to the more polemic
parts of your post.

Robert

Aaron Bergman

Sep22-04, 02:44 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <408ef233.0409211224.263dfa1b-100000@posting.google.com>,\n"Robert C. Helling" <helling@atdotde.de> wrote:\n\n> Lubos Motl <motl@feynman.harvard.edu> wrote in message\n> news:<Pine.LNX.4.31.0409201737310.2887-100000@feynman.harvard.edu>...\n>\n> > On Mon, 20 Sep 2004, Urs Schreiber wrote:\n> >\n> > > Robert Helling & Guiseppe Policastro:\n> > > String quantization: Fock vs. LQG Representations\n> > > hep-th/0409182\n>\n> Thanks for noting our paper\n\nI\'m a little confused about your paper, actually. What, exactly, are you\nquantizing? I think you claim that your theory has central charge one.\nIf so, you\'re quantizing the CFT, not the gravity theory on the\nworldsheet, I think. If you\'re quantizing the theory including the\nmetric in the path integral, one does not \'add\' in the bc theory at any\npoint. Instead, it\'s forced upon us from the FP determinant when we fix\nthe diffxWeyl invariance on the world sheet (leaving the conformal\ninvariance left over).\n\nThis is what forces bosonic string theory to live in 26 dimensions,\nafter all. Otherwise, we could just throw in some other CFT to cancel\nthe central charge of the Xs (as sort of happens in the superstring\nwhere ghost system has central charge -15).\n\nIf this procedure is to really be a quantiztation of gravity, I think\nyou should be able to start with the 2D quantum gravity containing 26 X\nfields and quantize it all the way through without ever encountering an\nanomaly.\n\nAaron\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <408ef233.0409211224.263dfa1b-100000@posting.google.com>,
"Robert C. Helling" <helling@atdotde.de> wrote:

> Lubos Motl <motl@feynman.harvard.edu> wrote in message
> news:<Pine.LNX.4.31.0409201737310.2887-100000@feynman.harvard.edu>...
>
> > On Mon, 20 Sep 2004, Urs Schreiber wrote:
> >
> > > Robert Helling & Guiseppe Policastro:
> > > String quantization: Fock vs. LQG Representations
> > > http://www.arxiv.org/abs/hep-th/0409182
>
> Thanks for noting our paper

I'm a little confused about your paper, actually. What, exactly, are you
quantizing? I think you claim that your theory has central charge one.
If so, you're quantizing the CFT, not the gravity theory on the
worldsheet, I think. If you're quantizing the theory including the
metric in the path integral, one does not 'add' in the bc theory at any
point. Instead, it's forced upon us from the FP determinant when we fix
the diffxWeyl invariance on the world sheet (leaving the conformal
invariance left over).

This is what forces bosonic string theory to live in 26 dimensions,
after all. Otherwise, we could just throw in some other CFT to cancel
the central charge of the Xs (as sort of happens in the superstring
where ghost system has central charge -15).

If this procedure is to really be a quantiztation of gravity, I think
you should be able to start with the 2D quantum gravity containing 26 X
fields and quantize it all the way through without ever encountering an
anomaly.

Aaron

Urs Schreiber

Sep22-04, 03:05 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 22 Sep 2004, Aaron Bergman wrote:\n\n> In article <408ef233.0409211224.263dfa1b-100000@posting.google.com>,\n> "Robert C. Helling" <helling@atdotde.de> wrote:\n>\n> > Lubos Motl <motl@feynman.harvard.edu> wrote in message\n> > news:<Pine.LNX.4.31.0409201737310.2887-100000@feynman.harvard.edu>...\n> >\n> > > On Mon, 20 Sep 2004, Urs Schreiber wrote:\n> > >\n> > > > Robert Helling & Guiseppe Policastro:\n> > > > String quantization: Fock vs. LQG Representations\n> > > > hep-th/0409182\n> >\n> > Thanks for noting our paper\n>\n> I\'m a little confused about your paper, actually. What, exactly, are you\n> quantizing? I think you claim that your theory has central charge one.\n> If so, you\'re quantizing the CFT, not the gravity theory on the\n> worldsheet, I think. If you\'re quantizing the theory including the\n> metric in the path integral, one does not \'add\' in the bc theory at any\n> point. Instead, it\'s forced upon us from the FP determinant when we fix\n> the diffxWeyl invariance on the world sheet (leaving the conformal\n> invariance left over).\n>\n> This is what forces bosonic string theory to live in 26 dimensions,\n> after all. Otherwise, we could just throw in some other CFT to cancel\n> the central charge of the Xs (as sort of happens in the superstring\n> where ghost system has central charge -15).\n>\n> If this procedure is to really be a quantiztation of gravity, I think\n> you should be able to start with the 2D quantum gravity containing 26 X\n> fields and quantize it all the way through without ever encountering an\n> anomaly.\n\nI don\'t want to answer for Robert, but would like to comment anyway:\n\nI think as Robert points out in that paper the ghosts are easily included\nin his framework and don\'t change much about the conclusions. Even with\nghosts the GNS state associated with the ordinary quantization is not\nrep-invariant but continuous, I think.\n\nNext here is a comment which is at the same time really a question: When\nyou take the Polyakov action without the worldsheet metric fixed in any\nway (i.e. no conformal gauge chosen) and compute its ADM constraints, you\nfind that these still generate, classically, the Virasoro algebra\n(without central charge at this classical point). Quantizing this algebra\nshould hence be precisely the analog of the canonical quantization attempted\nfor higher dimensional gravity.\n\nIn fact, precisely the same form of the constraints is also obtained when\none starts with the NG action, which is what Thomas Thiemann\nconsidered in his "LQG-string" paper. The NG worldsheet action is not\ngravity but still a rep-invariant system. Since it does not have any\n"auxiliary" metric that could be gauge fixed, there is no reason for any\nghosts to appear in this context at all - or is there?\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, 22 Sep 2004, Aaron Bergman wrote:

> In article <408ef233.0409211224.263dfa1b-100000@posting.google.com>,
> "Robert C. Helling" <helling@atdotde.de> wrote:
>
> > Lubos Motl <motl@feynman.harvard.edu> wrote in message
> > news:<Pine.LNX.4.31.0409201737310.2887-100000@feynman.harvard.edu>...
> >
> > > On Mon, 20 Sep 2004, Urs Schreiber wrote:
> > >
> > > > Robert Helling & Guiseppe Policastro:
> > > > String quantization: Fock vs. LQG Representations
> > > > http://www.arxiv.org/abs/hep-th/0409182
> >
> > Thanks for noting our paper
>
> I'm a little confused about your paper, actually. What, exactly, are you
> quantizing? I think you claim that your theory has central charge one.
> If so, you're quantizing the CFT, not the gravity theory on the
> worldsheet, I think. If you're quantizing the theory including the
> metric in the path integral, one does not 'add' in the bc theory at any
> point. Instead, it's forced upon us from the FP determinant when we fix
> the diffxWeyl invariance on the world sheet (leaving the conformal
> invariance left over).
>
> This is what forces bosonic string theory to live in 26 dimensions,
> after all. Otherwise, we could just throw in some other CFT to cancel
> the central charge of the Xs (as sort of happens in the superstring
> where ghost system has central charge -15).
>
> If this procedure is to really be a quantiztation of gravity, I think
> you should be able to start with the 2D quantum gravity containing 26 X
> fields and quantize it all the way through without ever encountering an
> anomaly.

I don't want to answer for Robert, but would like to comment anyway:

I think as Robert points out in that paper the ghosts are easily included
in his framework and don't change much about the conclusions. Even with
ghosts the GNS state associated with the ordinary quantization is not
rep-invariant but continuous, I think.

Next here is a comment which is at the same time really a question: When
you take the Polyakov action without the worldsheet metric fixed in any
way (i.e. no conformal gauge chosen) and compute its ADM constraints, you
find that these still generate, classically, the Virasoro algebra
(without central charge at this classical point). Quantizing this algebra
should hence be precisely the analog of the canonical quantization attempted
for higher dimensional gravity.

In fact, precisely the same form of the constraints is also obtained when
one starts with the NG action, which is what Thomas Thiemann
considered in his "LQG-string" paper. The NG worldsheet action is not
gravity but still a rep-invariant system. Since it does not have any
"auxiliary" metric that could be gauge fixed, there is no reason for any
ghosts to appear in this context at all - or is there?

Robert C. Helling

Oct1-04, 06:21 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, 20 Sep 2004 11:55:48 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> Then I would like to remark that a subtle but maybe crucial issue should not\n> be overlooked: The "diffeomorphisms" discussed by Thiemann and by\n> Helling&Policastro are not the "spatial diffeomorphisms" which\n> reparameterize the spatial slices of the string. As such they cannot be\n> completely compared to the spatial diffeo constraints as they appear in the\n> ADM constraints of (2 and higher dimensional) gravity.\n\nCorrect. Although we start with a canonical 1+1 split, we end up\ndiscussing the left and right chiral halfs j^\\pm separately. So the\nVirasoro that we discuss is the chiral symmetry algebra (the total\nconformal symmetry being Vir_left + Vir_right).\n\n> Finally I have a technical question to Robert and Giuseppe Policastro\n> concerning the discussion on pp10-11 of their paper. Maybe I am mixed up,\n> but it seems to me that there at some point commuting and anti-commuting\n> properties need to be exchanged.\n\nRight again. In the text, exchange commuting and anti-commuting once\nand everything should be fine. All formulas as far as I can see are correct.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 20 Sep 2004 11:55:48 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

> Then I would like to remark that a subtle but maybe crucial issue should not
> be overlooked: The "diffeomorphisms" discussed by Thiemann and by
> Helling&Policastro are not the "spatial diffeomorphisms" which
> reparameterize the spatial slices of the string. As such they cannot be
> completely compared to the spatial diffeo constraints as they appear in the
> ADM constraints of (2 and higher dimensional) gravity.

Correct. Although we start with a canonical 1+1 split, we end up
discussing the left and right chiral halfs j^\pm separately. So the
Virasoro that we discuss is the chiral symmetry algebra (the total
conformal symmetry being Vir_left + Vir_right).

> Finally I have a technical question to Robert and Giuseppe Policastro
> concerning the discussion on pp10-11 of their paper. Maybe I am mixed up,
> but it seems to me that there at some point commuting and anti-commuting
> properties need to be exchanged.

Right again. In the text, exchange commuting and anti-commuting once
and everything should be fine. All formulas as far as I can see are correct.

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

Urs Schreiber

Oct1-04, 08:11 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2s4q0iF1g8iftU1-100000@uni-berlin.de...\n> On Mon, 20 Sep 2004 11:55:48 -0400, Urs Schreiber\n<Urs.Schreiber@uni-essen.de> wrote:\n>\n> > Then I would like to remark that a subtle but maybe crucial issue should\nnot\n> > be overlooked: The "diffeomorphisms" discussed by Thiemann and by\n> > Helling&Policastro are not the "spatial diffeomorphisms" which\n> > reparameterize the spatial slices of the string. As such they cannot be\n> > completely compared to the spatial diffeo constraints as they appear in\nthe\n> > ADM constraints of (2 and higher dimensional) gravity.\n>\n> Correct. Although we start with a canonical 1+1 split, we end up\n> discussing the left and right chiral halfs j^\\pm separately. So the\n> Virasoro that we discuss is the chiral symmetry algebra (the total\n> conformal symmetry being Vir_left + Vir_right).\n\n\nYes. Nothing wrong with that. I just pointed it out because the whole\ndiscussion is latently about how to quantize gravity and I believe it is\nimportant to note when we are talking about spatial diffeomorphisms and when\nabout something that just looks like these.\n\n\n> > Finally I have a technical question to Robert and Giuseppe Policastro\n> > concerning the discussion on pp10-11 of their paper. Maybe I am mixed\nup,\n> > but it seems to me that there at some point commuting and anti-commuting\n> > properties need to be exchanged.\n>\n> Right again. In the text, exchange commuting and anti-commuting once\n> and everything should be fine. All formulas as far as I can see are\ncorrect.\n\n\nYes, sorry, I did not mean to imply that anything is wrong with your\ncomputations. I was just checking if I correctly followed your derivations.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2s4q0iF1g8iftU1-100000@uni-berlin.de...
> On Mon, 20 Sep 2004 11:55:48 -0400, Urs Schreiber
<Urs.Schreiber@uni-essen.de> wrote:
>
> > Then I would like to remark that a subtle but maybe crucial issue should
not
> > be overlooked: The "diffeomorphisms" discussed by Thiemann and by
> > Helling&Policastro are not the "spatial diffeomorphisms" which
> > reparameterize the spatial slices of the string. As such they cannot be
> > completely compared to the spatial diffeo constraints as they appear in
the
> > ADM constraints of (2 and higher dimensional) gravity.
>
> Correct. Although we start with a canonical 1+1 split, we end up
> discussing the left and right chiral halfs j^\pm separately. So the
> Virasoro that we discuss is the chiral symmetry algebra (the total
> conformal symmetry being Vir_left + Vir_right).

Yes. Nothing wrong with that. I just pointed it out because the whole
discussion is latently about how to quantize gravity and I believe it is
important to note when we are talking about spatial diffeomorphisms and when
about something that just looks like these.

> > Finally I have a technical question to Robert and Giuseppe Policastro
> > concerning the discussion on pp10-11 of their paper. Maybe I am mixed
up,
> > but it seems to me that there at some point commuting and anti-commuting
> > properties need to be exchanged.
>
> Right again. In the text, exchange commuting and anti-commuting once
> and everything should be fine. All formulas as far as I can see are
correct.

Yes, sorry, I did not mean to imply that anything is wrong with your
computations. I was just checking if I correctly followed your derivations.