Thiemann's quantization of the Nambu-Goto action

January 27, 2004

Thiemann’s quantization of the Nambu-Goto action

Posted by urs

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Last year there was a symposium called Strings meet Loops at the AEI in Potsdam at which researchers in the fields of String Theory and Loop Quantum Gravity were supposed to learn about each other’s approaches. In his introductory remarks H. Nicolai (being a string theorist) urged the LQG theorists to try to better understand how their quantization approach compares to known results.

Since the worldsheet theory of the (super)string is nothing but (super)gravity in 1+1 dimensions coupled to other fields it would be an ideal laboratory to compare the results of LQG in this setting to the usual lore, which in particular features the central extension of the Virasoro algebra as well as consistency conditions on the number of target space dimensions.

How does this model fit into the framework of canonical and loop quantum gravity?

Nicolai asked.

A search on the arXive showed that so far only one paper had appeared which did address aspects of this simple and yet somewhat decisive question:

Artem Starodubtsev, String theory in a vertex operator representation: a simple model for testing loop quantum gravity.

Starodubtsev concluded:

The suggested [LGQ-like] version of the Hamiltonian constraint leaves us with a theory which is considerably different from ordinary string theory. There are several indications that string theory in its usual form can probably not be recovered from the model obtained. […] the first version of Hamiltonian constraint is anomaly-free and the same is true of the diffeomorphism constraint.

When, after the symposium, I mentioned this reference to A. Ashtekar, a leading figure in LQG, he told me that he meanwhile was aware of this result and planning to analyze the problem in more detail.

Apparently this has borne fruit by now, since yesterday a paper by Th. Thiemann appeared on the arXive

Th. Thiemann, The LQG-String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space

which gives a detailed analysis of an LQG inspired canonical quantization of the 1+1 dimensional Nambu-Goto action for flat target space. The approach is a little different from that by Starodubtsev, but the results are similar in their unorthodoxy: Thiemann finds

- no sign of a critical dimension

- no ghost states

- no anomaly, no central charge

- no tachyon (and, indeed, not the rest of the usual string spectrum).

The claim is that all this is possible due to a quantization ambiguity that has not been noticed or not been investigated before: Instead of using the usual Fock/CFT representation and imposing the constraints as operator equations, Thiemann uses families of abstract representations of the operator algebra obtained by the GNS construction and solves the quantum constraints by a method called group averaging, or its more sophisticated cousin, the so-called Direct Integral Method.

Since these are the same methods used in LQG for quantizing the gravitational field in 3+1 dimensions it is somewhat interesting to see how vastly different the results obtained this way are from the standard lore. One might hence take this as a sign that the LQG approach to quantization is odd. But in some circles this is interpreted in just the opposite way, dreaming of the possibility that the new quantization method might improve on the standard approach to quantization in string theory. Indeed Thiemann himself speculates in his conclusions that his quantization prescription might

- solve the cosmological constant problem

- clarify tachyon condensation [?]

- solve the vaccum degeneracy puzzle

- help finding a working phenomenological model

- help proving perturbative finiteness beyond two loops .

To my mind these are surprisingly bold speculations.

I would much rather like to understand conceptually the nature of the apparent quantization ambiguity (if it really is one) that is the basis for all this. Do we really have this much freedom in quantizing the NG action? Why then do several different quantization schemes (BRST, path integral, lightcone quantization) all yield the standard result which strongly disagrees with the one obtained by Thiemann? What is the crucial assumption in Thiemann’s quantization that makes it different from the ordinary one?

I believe that these questions are what originally motivated H. Nicolai to initiate this investigation and their answer should teach us something.

In the remainder of this entry I shall try to look at some of the technical details of Thiemann’s paper, trying to understand what exactly it is that is going on.

We all know from Edward Nelson that

First quantization is a mystery.

But it should be possible to understand how precisely it is mysterious and how it is not.

[Note added later on:]

After an intensive discussion and some false attempts to explain what is going on inThomas Thiemann’s paper, he finally chimed in himself and we could clarify the issue at the technical level. The crucial point is the following:

Thomas Thiemann does not perform a canonical quantization of the Virasoro constraints if we want to understand under canonical quantization that a theory with classical first-class constraints $C_{I}$ is quantized by demanding

(1)

〈 phys ∣ {\hat{C}}_{I} ∣ phys 〉 = 0 .

What Thomas Thiemann instead does (by his own account) is the following:

1) Find a representation ${\hat{U}}_{φ}$ of the classical symmetry group elements $φ$ on some Hilbert space. (Here the ${\hat{U}}_{φ}$ need not have anything to do with the quantized ${\hat{C}}_{I}$ , and in the case of the ‘LQG-string they don’t have anything to do with them.)

2) Demand that physical states are invariant under the action of the ${\hat{U}}_{φ}$ .

It is clear that this method explicitly translates the classical symmetry group to the ‘quantum’ theory and hence cannot, by its very construction, ever find any anomalies and related quantum effects.

An interesting aspect of this is that exactly the same method is used with respect to the spatial diffeomorphism constraints in Loop Quantum Gravity (while the Hamiltonian constraint is quantized more in the usual way). It must therefore be emphasized that LQG is not canonical quantization in the sense that the classical first-class constraints are not promoted to hold as expectation value equations in the quantum theory.

For me, this is the crucial insight of this discussion, and it shows that Hermann Nicolai’s question did address precisely the right problem. In the toy example laboratory of the Nambu-Goto string it is much easier for non-experts (like me) to follow the details and implications of what is being done, than in full fledged LQG. And it turns out, to my surprise, that what is being done is a speculative proposal for an alternative to standard quantum theory. This is not only my interpretation, but Thomas Thiemann himself says that the procedure, sketched above, for dealing with the constraints, should be compared to experiment to see if nature favors it over standard Dirac/Gupta-Bleuler quantization.

I am open-minded and can accept this in principle, but this has not been obvious to me at all, before. It means that, in the strinct sense of the word ‘canonical’, LQG is not canonical at all but rather similar in spirit to other proposed modifications of quantum theory, like for instance those proposed to explain away the black hole information loss problem by modifying Schroedinger’s equation.

I have tried to discuss some of these insights here.

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So let me try to recapitulate the key idea in Thiemann’s quantization of the Nambu-Goto action, as far as I understand it.

Let $π_{μ}$ be the canonical momentum to the embedding variable $X^{μ}$ . The usual left and right-moving bosonic fields are (pointwise)

(1)

Y_{\pm}^{μ} : = η^{μ ν} π_{μ} \pm X^{' μ} .

Smearing them over an interval $I$ of the circle and contracting with some reak $k_{μ}$ yields

(2)

Y_{\pm}^{k} (I) : = \int_{I} d σ k_{μ} Y_{\pm}^{μ} .

This are the fields that we want to represent as operators on some Hilbert space with commutation relation given by

(3)

[{\hat{Y}}_{\pm}^{μ} (σ), {\hat{Y}}_{\pm}^{ν} (σ^{'})] = 2 i δ^{'} (σ, σ^{'}) .

From these one obtains bounded operators by exponentiation

(4)

{\hat{W}}_{\pm}^{k} (I) : = \exp (i {\hat{Y}}_{\pm}^{k} (I)) .

The point is that for these bounded operators the GNS construction applies which tells us how to represent any unital *-algebra by bounded operators on some Hilbert space $ℋ$ , which will be called the kinematical Hilbert space (up to some details).

Now, the crucial difference to the usual Dirac quantization ,where the constraints $C_{I}$ are imposed as

(5)

{\hat{C}}_{I} ∣ ψ 〉 = 0

seems to be that instead the technique of group averaging imposes the exponentiation of this, namely

(6)

\exp ({\hat{C}}_{I}) ∣ ψ 〉 = ∣ ψ 〉

(in the weak sense discussed between eqs. (5.4) and (5.5) of Thiemann’s paper). Naively this might appear to be the same thing, but it is not at all!

As an example, consider the commutator of one of the Virasoro constraints ${\hat{V}}_{\pm} (ξ)$ with ${\hat{W}}_{\pm}^{k} (I)$ . There is an operator ordering issue and dealing with that yields the usual result that the conformal dimension of these ${\hat{W}}_{\pm}^{k}$ depends on $k$ . But now instead look at the exponentiated expression

(7)

e^{{\hat{V}}_{\pm}} {\hat{W}}_{\pm}^{k} (I) e^{- {\hat{V}}_{\pm}} = e^{{\hat{V}}_{\pm}} \exp (i {\hat{Y}}_{\pm}^{k} (I)) e^{- {\hat{V}}_{\pm}} = \exp (i e^{{\hat{V}}_{\pm}} {\hat{Y}}_{\pm}^{k} (I) e^{- {\hat{V}}_{\pm}}) = \exp (i {\hat{Y}}_{\pm}^{k} (ϕ (I))) = {\hat{W}}_{\pm}^{k} (ϕ_{\pm} (I)),

where $ϕ_{\pm}$ here denotes the group element of $Diff (S^{1})$ associated with $V_{\pm} = V_{\pm} (ξ)$ ( $ξ$ is some smearing function).

The exponentiation in a sense removes all operator ordering ambiguities, since the conjugation operation (the similarity transform) $e^{{\hat{V}}_{\pm}} \cdot e^{- {\hat{V}}_{\pm}}$ acts on every ${\hat{Y}}_{\pm}^{k}$ seperately and there is no operator ordering issue in the commutator $[{\hat{V}}_{\pm}, {\hat{Y}}_{\pm}^{k} (I)]$ .

Without this operator ordering issue there is no anomaly, hence no critical dimension, no tachyon, etc.

I therefore believe that the quantum ambiguity between the two sides of

(8)

{\hat{C}}_{I} ∣ ψ 〉 = 0 \leftrightarrow \exp ({\hat{C}}_{I}) ∣ ψ 〉 = ∣ ψ 〉

is what is at the heart of the difference between Thiemann’s quantization and the usual OCQ/BRST quantization.

Am I wrong?

Even if this is about right, there is something related which I don’t quite understand yet. Somehow the center-of-mass degree of freedom of the string is missing from Thiemann’s original Hilbert space. In section 6.4 he re-incorporates it by using a D-parameter familiy of his original Hilbert space, which hence clearly was just that of string oscillations. What I am puzzled about is that the 0-mode of the momentum operator does not seem to be the same thing as $π_{μ} (p_{ν})$ above equation (6.36). It seems to me that the two should be identified, somehow, and that then the question whether there is a tachyon or not should be addressed by actually constructing group-averaged and hence physical states.

Posted at January 27, 2004 3:34 PM UTC

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Re: Thiemann’s quantization of the Nambu-Goto action

Here is a copy of Luboš’ answer to a related post of mine on sci.physics.research:

On 27 Jan 2004, Urs Schreiber wrote:

> I was trying to figure out what exactly it is in Th. Thiemanns
> quantization hep-th/0401172 of what he calls the ‘LQG-string’ that
> makes it so different from the usual quantization. I now believe that
> the crucial issue is how to impose the constraints.

Exactly. If physics is done properly, the (Virasoro) constraints are not
arbitrary constraints that are added by hand. They are really Einstein’s
equations, derived as the equations of motion from the action if it is
varied with respect to the metric - in this case the worldsheet metric.
The term R_{ab}-R.g_{ab}/2 vanishes identically in two dimensions, and
T_{ab}=0 is the only term in the equation that imposes the constraint. The
constraints are really Einstein’s equations, once again.

Moreover, because the (correct) theory is conformal, the trace
T_{ab}g^{ab} vanishes indentically, too, and therefore the three
components of the symmetric tensor T_{ab} actually reduce to two
components, and those two components impose the so-called Virasoro
constraints (which are easiest to be parameterized in the conformal gauge
where the metric is the standard flat metric rescaled by a
spacetime-dependent factor). For closed strings, there are independent
holomorphic and independent antiholomorphic generators - and they become
left-moving and right-moving observables on the Minkowski worldsheet
after we Wick-rotate.

Thomas Thiemann does not appreciate the logic behind all these things, and
he wants to work directly with the (obsolete) Nambu-Goto action to avoid
conformal field theory that he finds too difficult. Of course, the
Nambu-Goto action has no worldsheet metric, and therefore one is not
allowed to impose any further constraints. They simply don’t follow and
can’t follow from anything such as the equations of motion.

Thiemann does not give up, and imposes “the two” constraints by hand. It
is obvious from his paper that he thinks that one can add any constraints
he likes. Of course, there are no “the two” constraints. If he has no
worldsheet metric, the stress energy tensor has three components, and
there is no way to reduce them to two. Regardless of the effort one makes,
two tensor constraints in a general covariant nonconformal theory can
never transform properly as a tensor - because a symmetric tensor simply
has three components - and therefore his constraints won’t close upon an
algebra. His equations are manifestly general non-covariant, in contrast
with his claims.

Equivalently, because he obtained these constraints by artificially
imposing them, they won’t behave as conserved currents. (In a general
covariant theory without the worldsheet metric, we can’t even say what
does it mean for a current to be conserved, because the conservation law
nabla_a T^{ab} requires a metric to define the covariant derivative.) If
they don’t behave as conserved currents, they don’t commute with the
Hamiltonian, and imposing these constraints at t=0 will violate them at
nonzero “t” anyway (the constraint is not conserved).

If one summarizes the situation, these constraints simply contradict the
equations of motion. It is not surprising. We are only allowed to derive
*one* equation of motion for each degree of freedom i.e. each component of
X, and this equation was derived from the action. Any further constraint
is inconsistent with such equations unless we add new degrees of freedom.

I hope that this point is absolutely clear. The equations of motion don’t
allow any new arbitrarily added constraints unless it is possible to
derive them from extra terms in the action (that can contain Lagrange
multipliers). The Lagrange multipliers for the Virasoro constraints *are*
the components of worldsheet metric, and omitting one component of g_{ab}
makes his theory explicitly non-covariant (even if Thiemann tries to
obscure the situation by using the letters C,D for the two components of
the metric in eqn. (3.1)).

The conformal symmetry is absolutely paramount in the process of solving
the theory and identifying the Virasoro algebra - isolating the two
generators T_{zz} and T_{zBAR zBAR} per point from the general symmetric
tensor. Conformal/Virasoro transformations are those that fix the
conformal gauge - i.e. the requirement that the metric is given by the
unit matrix up to an overall rescaling. Conformal theories give us T_{z
zBAR} (the trace) equal to zero, and this is necessary to decouple T_{zz}
and T_{z zBAR}. In two dimensions, the conformal transformations -
equivalently the maps preserving the angles - are the holomorphic maps
(with possible poles), and the holomorphic automorphisms of a closed
string’s worldsheet are generated by two sets of the Virasoro generators.

This material - why it is necessary to go from the Nambu-Goto action to
the Polyakov action and to conformal field theory in order to solve the
relativistic string and quantize it - is a basic material of chapter 1 or
chapter 2 of all elementary books about string theory and conformal field
theory. I think that a careful student should first try to understand this
basic stuff, before he or she decides to write “bombastic” papers boldly
claiming the discovery of new string theories and invalidity of all the
constraints (such as the critical dimension) that we have ever found.

In fact, I think that a careful student should first try to go through the
whole textbook first, before he publishes a paper on a related topic.
Thomas Thiemann is extremely far from being able to understand the chapter
3 about the BRST quantization, for example.

Thiemann’s theory has very little to do with string theory, and very
little to do with real physics, and unlike string theory, it is
inconsistent and misled. String theory is a very robust and unique theory
and there is no way to “deform it” from its stringiness, certainly not in
these naive ways.

> This may seem like essentially the same thing, but the crucial issue is
> apparently that the latter form allows to deal quite differently with
> operator ordering, which completely changes the quantization. In particular,
> it seems to allow Thiemann, in this case, to have no operator re-ordering at
> all, which is the basis for him not finding an anomaly, hence no tachyon and
> no critical dimension.

A problem is that you don’t know what you’re averaging over because his
“group” is not a real symmetry of the dynamics.

By the way, if you want to define physical spectrum by a
Gupta-Bleuler-like method, you must have a rule for a state itself that
decides whether the state is physical or not. In Gupta-Bleuler old
quantization of the string, “L_0 - a” and “L_m” for m>0 are required
to annihilate the physical states. This implies that the matrix element of
any L_n is zero (or “a” for n=0) because the negative ones annihilate the
bra-vector.

It is important that we could have defined the physical spectrum using a
condition that involves the single state only. If you decided to define
the physical spectrum by saying that all matrix elements of an operator
(or many operators) between the physical states must vanish, you might
obtain many solutions of this self-contained condition. For example, you
could switch the roles of L_7 and L_{-7}. However all consistent solutions
would give you an equivalent Hilbert space to the standard one.

The modern BRST quantization allows us to impose the conditions in a
stronger way. All these subtle things - such as the b,c system carrying
the central charge c=-26 - are extremely important for a correct
treatment of the strings, and they can be derived unambiguously.

> If this is true and Group averaging on the one hand and Gupta-Bleuler
> quantization on the other hand are two inequivalent consistent quantizations
> for the same constrained classical system I would like to understand if they
> are related in any sense.

No, they are not. What is called here the “group averaging” is a naive
classical operation that does not allow one any sort of quantization. You
can simply look that at his statements - such as one below eqn. (5.2) -
that in his treatment, the “anomaly” (central charge) in the commutation
relations (of the Virasoro algebra, for example) vanishes, are never
justified by anything. They are only justified by their simple intuition
that things should be simple. This incorrect result is then spread
everywhere, much like many other incorrect results. It is equally wrong as
simply saying that we have constructed a different representation of
quantum mechanics where the operators “x” and “p” commute with one
another.

The central charge - the c-number that appears on the right hand side of
the Virasoro algebra - is absolutely real and unique determined by the
type of field theory that we study (and the theory must be conformal,
otherwise it is not possible to talk about the Virasoro algebra). It can
be calculated in many ways and any treatment that claims that the Virasoro
generators constructed out of X don’t carry any central charge is simply
wrong.

There is absolutely no ambiguity in quantization of the perturbative
string. Knowing the background is equivalent to knowing the full theory,
its spectrum, and its interactions. There is no doubt that Thiemann’s
paper - one with the big claims about the “ambiguities” of the
quantization of the string - is plain wrong, and exhibits not one, but a
plenty of elementary misunderstanding by the author about the role of
constraints, symmetries, anomalies, and commutators in physics.

Let me summarize a small part of his fundamental errors again. He believes
many very incorrect ideas, for example that

* artificially chosen constraints can be freely imposed on your Hilbert
space, without ruining the theory and contradicting the equations of motion
* two constraints in 2 dimensions can transform as a general symmetric
tensor, and having a tensor with a wrong number of components does not
spoil the general covariance
* he also thinks that the Virasoro generators have nothing to do with the
conformal symmetry and they have the same form in any 2D theory
* in other words, he believes that you can isolate the Virasoro generators
without going to a conformal gauge
* classical Poisson brackets and classical reasoning is enough to
determine the commutators in the corresponding quantum theory
* anomalies in symmetries, carried by various degrees of freedom,
can be ignored or hand-waved away
* there is an ambiguity in defining a representation of the algebra of
creation and annihilation operators
* the calculation of the conformal anomaly does not have to be treated
seriously
* the tools of the so-called axiomatic quantum field theory are useful
in treating two-dimensional field theories related to
perturbative string theory
* if a set of formulae looks well enough to him, it must be OK and the
consistent stringy interactions and everything else must follow

Once again, all these things are wrong, much like nearly all of his
conclusions (and completely all “new” conclusions).

Thiemann himself admits that this is the same type of “methods” that they
have also applied to four-dimensional gravity. Well, probably. My research
of the papers on loop quantum gravity confirms it with a high degree of
reliability. Every time one can calculate something that gives them an
interesting but inconvenient result, they claim that in fact we don’t need
to calculate it, and it might be ambiguous, and so on. No, this is not
what we can call science. In science, including string theory, we have
pretty well-defined rules how to calculate some class of observables, and
all things calculated according to these rules must be treated seriously.
If a single thing disagrees, the theory must be rejected.

The inevitability of conformal symmetry for a controlled quantization of
the relativistic string - and for isolation (in fact, the definition) of
the Virasoro generators - is real. The theorems of CFT about its being
uniquely determined by certain data are also real. The conformal anomalies
of certain fields are also real. The two-loop divergent diagrams in
ordinary GR are also real. We know how to compute and prove all these
things, and propagating fog and mist can only obscure these
well-established facts from those who don’t want to see the truth.

I guess that this paper will demonstrate to most theoretical physicists -
even those who have not been interested in these “alternative” fields -
how bad the situation in the loop quantum gravity community has become.
There are hundreds of people who understand the quantization of a free
string very well, and they can judge whether Thiemann’s paper is
reasonable or not and whether funding of this “new kind of science”
should continue.

All the best
Lubos

Posted by: Urs Schreiber on January 28, 2004 3:01 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

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Hi Luboš,

thanks for your answer!

I see your general point, but would like to look at some of the issues you raised in more detail.

You say that the Nambu-Goto action is ‘obsolete’. But of course the NG action is classically equivalent to the Polyakov action and I think that in the critical number of dimensions the equivalence extends to the quantum theory. Furthermore, the Nambu-Goto action for the string is essentially the Dirac-Born-Infeld action (up to the worldsheet gauge field) of the D-string.

As far as I can see the constraints that Thiemann arrives at in equation (2.4) of his paper follow from standard canonical reasoning. One finds that the canonical momenta $π_{μ}$ of the Nambu-Goto action as well as of the DBI action classically satisfy two identities which can be identified as constraints. At the classical level these constraints are precisely the (classical) Virasoro constraints that one also obtains by varying the worldsheet metric in the Polyakov action. Since the two actions are classically equivalent this is no surprise.

My point is that there should be a priori nothing wrong with looking at the Nambu-Goto action when studying the string. Indeed this is frequently done for instance when F-strings and D-strings are considered at the same time, as for instance in

Y. Igarashi, K. Itoh, K. Kamimura, R. Kuriki, Canonical equivalence between super D-string and type IIB superstring.

In equations (2.3) and (2.4) of this paper the authors in particular give the same two bosonic constraints of the Nambu-Goto action that Thiemann arrives at. Their action also involves superfields and the worldsheet gauge field, but this does not affect the general result that the Virasoro constraints follow from a canonical analysis of the Nambu-Goto action. I have spelled out the derivation (for the bosonic DBI action) in a recent entry. (By setting the worldsheet gauge field and the $C$ fields to zero this derivation directly restricts to that for the ordinary Nambu-Goto action).

My point is that it is maybe not fair to say that Thiemann artificially or freely chooses the constraints - at least not at the classical level. The constraints that he uses are, classically, the Virasoro constraints of the closed bosonic string.

My suspicion is rather that Thiemann devitates from standard reasoning when he defines what he wants to understand under quantizing the Virasoro constraints. Would you agree with this?

Let’s ignore the way on which we arrived at the classical Virasoro constraints (by starting from one of various classically equivalent actions) and concentrate on the question what it means to quantize them.

The standard procedure is to make Gupta-Bleuler quantization and use either creation/annihilation operator normal ordering or CFT techniques to make sense of the quantum representation of the classical Virasoro generators. This leads in the usual way to the anomaly, the shift a in (L_0 - a) and so on.

Thiemann claims (based on a large literature on quantization of constrained systems that is also the basis for loop quantum gravity) that there is an at least superficially different technique that can also be addressed as quantization of the Virasoro constraints. In the simple case at hand this is imposing the constraint the way mentioned right below equation (5.4), which essentially says that

(1)

〈 ψ ∣ \exp (constraints) ∣ ψ^{'} 〉 = 〈 ψ ∣ ψ^{'} 〉,

where the Hilbert space and the representation of the operators is not necessarily the usual Fock representation.

This is not equivalent to and not even implied by saying that

(2)

〈 ψ ∣ constraints ∣ ψ^{'} 〉 = 0 .

Of course when I write this I am ignoring issues of what we really mean by writing $\exp (some operator)$ , i.e. whether this is supposed to be normal ordered or regulated or what. I am trusting that this is taken care of by Thiemann’s rigorous construction of Hilbert spaces and operators on them, but I guess that Luboš disagrees with this. :-)

Posted by: Urs Schreiber on January 28, 2004 4:14 PM | Permalink | Reply to this

Huh?

Thomas Thiemann does not appreciate the logic behind all these things, and he wants to work directly with the (obsolete) Nambu-Goto action to avoid conformal field theory that he finds too difficult. Of course, the Nambu-Goto action has no worldsheet metric, and therefore one is not allowed to impose any further constraints. They simply don’t follow and can’t follow from anything such as the equations of motion.

As I teach my students in the first days of my String Theory class, the Virasoro constraints follow straightforwardly from a canonical treatment of the Nambu-Goto string.

That is hardly the issue.

Posted by: Jacques Distler on January 29, 2004 2:58 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

Here is another reply by Luboš:

Dear Urs,

Concerning your comments that you can get rid of all ordering constants by
exponentiating something, I hope that you don’t really believe it because
this would be a complete misunderstanding of the singularities in quantum
field theory. The exponentials of something always store the same
information as “something”, and if one of them has some ordering constant
contribution, you see it in the other as well.

For example, X(z) X(0) have logarithmic OPEs. This implies that
exp(i.K.X(z)) has a power law OPE with exp(-i.K.X(z)). It’s totally
nonsensical at quantum level to imagine that exp(-i.K.X(z)) is an inverse
operator to exp(i.K.X(z)). Do you understand why? This is a very
important point.

While for the Virasoro group without the central charge you would be able
to write the explicit “exponentiated” elements of the reparameterization
group and - because they have a clear geometric interpretatino, you could
invert them without anomalies, it is simply not true for the Virasoro
operators generating the reparameterization of X’s. Because of the term
c/z^4 in the OPE of two stress energy tensors, you must know very well
that exp(-V) can’t be treated as the inverse of exp(+V). You can only
imagine that exp(V) is an honest element of a group if the OPEs of V with
itself - and all other “V“‘s that you want to use - only have the 1/z
term, corresponding to the commutator. Recall that

O1(z) O2(0) ~ [O1,O2] (z) / z

the coefficient of 1/z is schematically the commutator of the two
operators. If you integrate a stress energy tensor etc., it is also OK to
have the 1/z^2 term in the OPEs of the stress energy tensor because it
reflects the worldsheet dimension of the stress energy tensor and tells
you how should you integrate it to get scalars etc.

But the OPE of the stress energy tensor (of the X^mu CFT) with itself
contains an extra 1/z^4 term. This is just a fact that you can calculate
in many ways, and this simply means that exp(V) where V is a Virasoro
generator, or some integrated combination of the stress energy tensor,
does not behave as an honest element of some group, and exp(-V) is not in
any naive sense inverse to exp(V) because these two *operators* have
singularities.

Note that his naive operation, involving the (wrong) application of the
formula

exp(C.D.C^{-1}) = C exp(D) C^{-1}

which is OK for matrices, is incorrect in our “usual” representation of
CFT, because of singularities between C and C itself. You can’t imagine
that C^{-1} is inverse to C - there are just no meaningful operators on
the Hilbert space that would look like C=exp(V) and were inverse to one
another. Because C^{-1}.C is not really one, you can’t derive the formula
you derived either, unless c=0. Note that it even requires you, for
C=exp(V), to consider exp(exp(V)…). These are heavily singular
operators, and all these confusions simply come from his/their wrong
intuition that you can work with the operators in CFT as with ordinary
classical numbers. They don’t understand where the normal ordering terms
come from, they don’t understand singularities of operators in quantum
field theories, they don’t understand the difference between classical and
quantum field theory.

It’s just totally pathetic, and every student in theoretical physics
should be able to identify all these errors.

All the best
Lubos

Posted by: Urs Schreiber on January 28, 2004 4:17 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

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Hi again, Luboš!

Yes, I understand everything that you say here. I know that $: \exp (- V) :$ is not the inverse to $: \exp (V) :$ in CFT and I do understand where the $1 / z^{4}$ terms come from. When you go back to my original entry you’ll see that I address precisely this phenomenon by mentioning that things like $: \exp (k \cdot X) :$ have conformal dimension depending on $k$ in CFT, which is another aspect of this phenomenon.

But, yes, I was taking for granted that Thiemann is using a rep of his operators that allows him to ignore all normal ordering issues and work with them as with matrices and hence not as in CFT. He is referring to lot’s of mathematical theorems, using the GNS construction etc. (that I obviously haven’t checked myself and I am trusting that he applies them correctly) and even though he does not say so explicitly I deduced from his paper, in particular from the the third paragraph on p. 20, that he does use

(1)

\exp (C \cdot D \cdot C^{- 1}) = C \exp (D) C^{- 1} .

I do understand that this does not make sense in CFT (or even any other quantum field theory in the usual sense) but I also believe that a large number of mathematically versed people in the LQG camp do think that this can be given good meaning by using all these mathematical constructions that Thiemann alludes to. Unfortunately I am not an expert on this stuff.

I think the key ingredient is the GNS construction, which tells you that a unital *-algebra can be represented faithfully i.e. without normal ordering issues just like matrices on some Hilbert space. That’s the content of the relation in the 9th line from below on p.15:

(2)

[a] [b] = [ab] .

On the right hand side is the classical multiplication of the algebra, on the left hand side we have operator multiplication. Whenever this is true we do have

(3)

{(\exp (π_{ω} (a)))}^{- 1} = \exp (- π_{ω} (a)) .

There is some fine print to this construction which I am maybe not fully aware of. In particular things need to be bounded for this to make sense. That’s why Thiemann uses the operators $\hat{W} = \exp (i \hat{Y})$ instead of the $\hat{Y}$ themselves, because these would be unbounded.

Posted by: Urs Schreiber on January 28, 2004 4:41 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

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For the general discussion of Thiemann’s paper I think it is important to realize that much of the usual lore of QFT is not supposed to apply. In particular, there is, as far as I understand, nothing like a double Wick contraction in the commutator of two Virasoro generators.

Let me spell this out in detail:

Assume that we have operators $Y (σ)$ , $σ \in S^{1}$ which have the commutator

(1)

[Y (σ), Y (σ^{'})] = - δ^{'} (σ, σ^{'}),

as in equation (6.4) of Thiemanns paper. Next assume that one can make sense of products of these operators $Y (σ) Y (σ)$ at equal points, without introducing any notion of normal ordering. This can be either thought of as pertaining to the classical Poisson algebra or, according to Thiemann et. al (if I understand correctly), by using a special representation on a special Hilbert space $ℋ_{ω}$ obtained by the GNS construction. Anyway, assume that the following expression makes sense:

(2)

L_{ξ} : = \frac{1}{2} \int_{S^{1}} d σ ξ (σ) Y (σ) Y (σ) .

The point is not to worry, for the moment, how this object is supposed to act on some state, but merely to regard its algebraic relations.

These all follow from

(3)

[L_{ξ}, Y (σ)] = {(ξ (σ) Y (σ))}^{'} .

This is nothing but what one also gets by using classical Poisson brackets, too.

For convenience, let me introduce some notation: For a general field $A (σ)$ let $w (A)$ be the classical conformal weight of $A (σ)$ iff

(4)

[L_{ξ}, A (σ)] = ξ (σ) A^{'} (σ) + w (A) ξ^{'} (σ) A (σ) .

It is easy to check that

(5)

w (A (σ) B (σ)) = w (A (σ)) + w (B (σ))

so that

(6)

w (Y (σ)) = 1

and

(7)

w (Y (σ) Y (σ)) = 2

and so on.

Now, denote for any field $A (σ)$ and any complex-valued function $ξ$ on $S^{1}$ the $ξ$ -mode of $A$ by $A_{ξ}$ , i.e.

(8)

A_{ξ} : = \int d σ ξ (σ) A (σ) .

Using again the naive quantum mechanical commutation relations (or Poisson brackets) one finds the following transformation of such modes

(9)

[L_{ξ_{1}}, A_{ξ_{2}}] = A_{(w - 1) ξ_{1}^{'} ξ_{2} - ξ_{1} ξ_{2}^{'}} .

This implies in particular that

(10)

[L_{ξ_{1}}, L_{ξ_{2}}] = L_{ξ_{1}^{'} ξ_{2} - ξ_{1} ξ_{2}^{'}} .

This is of course nothing but the usual relation known from classical Poisson brackets of the classical Virasoro constraints, as reviewed for instance by Thiemann in his equation (3.3). There is no anomaly because one assumed to have no need to consider normal ordering as in $: L_{ξ} :$ or the like and all operator products are assumed to behave like classical products. But the important point seems to be that Thiemann claims that secion 6.2 of his paper gives us a way to make sense of the above algebraic expressions as relations between operators that are well defined on some Hilbert space $ℋ_{ω}$ . This is how he gets a representation of the conformal group on his Hilbert space without having a conformal anomaly.

Posted by: Urs Schreiber on January 28, 2004 6:48 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

I do not understand Thiemann’s paper at all so I’m going to ask totally naive questions. It seems clear that if Thiemann’s construction is consistent, this new theory is nothing like a 1+1 dimensional field theory, so it might be that my questions will not even make sense in this framework.

First, how does he calculate the spectrum? He says in some place that the graviton state is gauge-dependent, which just boggles me out. What are the observables?

Secondly, can he write down the operator corresponding to X, and see what its commutation relations are?

Let me say I also share Lubos’ view about such grandiose claims. It doesn’t improve my confidence in the paper when he blithely disregards all the previous literature about quantizing the string.

Posted by: Arvind on January 28, 2004 11:17 PM | Permalink | Reply to this

Re: Thiemann’s quantization of the Nambu-Goto action

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Just a very quick and brief comment for the moment: Thiemann claims to be able to construct an operator representation $\hat{W}$ of the classical observables $W (ξ) = \exp (\int d σ ξ (σ) (i \frac{δ}{δ X (σ)} \pm X^{'} (σ))$ that essentially behaves just as the classical $W$ . This way the Pohlmeyer charges, uncontroversial classical invariants of the string, become quantum ‘charges’ for him.

I believe that if instead of Pohlmeyer charges we use classical DDF states (in the hopefully obvious sense) this construction would even give something similar to the usual string spectrum (up to the offset $a$ and the existence of null states).

We should (at least I should) try to understand if and why the claim about $\hat{W}$ can be correct. This is where the mystery lies, I believe.

Posted by: Urs Schreiber on January 29, 2004 2:01 AM | Permalink | Reply to this

There’s no place like home

Why don’t I just close my eyes, click my heels and wish away all anomalies?

What are the rules here?

It is well known that it is impossible to preserve all of the relations of the classical Poisson-bracket algebra as operator relations in the quantum theory.

What principle allows Thiemann to decide which relations will be carried over into the quantum theory?

Where does he discuss which relations fail to carry over?

Posted by: Jacques Distler on January 29, 2004 3:20 AM | Permalink | Reply to this

Re: There’s no place like home

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Maybe the issue is seperability of the Hilbert space.

Is the Hilbert space Thiemann constructs in his paper seperable? Unless I am missing something it is apparently not. This might explain why things work very different on this Hilbert space than on the ordinary seperable one.

Compare the situation in the what is called “loop quantum cosmology”. There, after the dust has settled, what is done is essentially an ordinary quantization of the Wheelder-DeWitt equation but on a nonseperable Hilbert space, where, if $a$ is the scale factor of the universe, all states of the form $\exp (ip a)$ for real $p$ are orthonormal simply by postulating a non-standard scalar product $〈 \cdot ∣ \cdot 〉$ with respect to which

(1)

〈 e^{i p_{1} a} ∣ e^{i p_{2} a} 〉 : = 1 if p_{1} = p_{2} and = 0 otherwise .

This is what makes the operator $\partial / \partial a$ technically have a discrete spectrum (where eigenstates are normalizable) even though its eigenvalues have a continuous range. (I am doing this from memory. Probably I mix up some details. Maybe the role of $\hat{a}$ and $\partial / \partial a$ in the above has to be exchanged.) This is the basis on which loop quantum cosmology obtains a discrete evolution of the scale factor, somehow (unfortunately I didn’t understand how precisely this follows when hearing a talk about this once).

So quantization on a non-seperable Hilbert space leads to radically different quantum theories. Maybe that’s what happens in Thiemann’s paper on the quantization of the string?

Posted by: Urs Schreiber on January 29, 2004 7:17 AM | Permalink | Reply to this

Re: There’s no place like home

Hi all,

I can’t follow any of this in detail at the moment, but my impression from talking to some of Ashtekar’s student is that non-separable Hilbert space is generic in their quatization. So, Urs may be on the correct reasoning here. Now , then the issue is how to recover the ordinary classical world from it. I know Josh Willis did some work (and still working on it) on the issue.

Demian Cho

Posted by: Demian on January 29, 2004 2:16 PM | Permalink | Reply to this

Re: There’s no place like home

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Hi Demian,

I have just finished posting a lengthy message to s.p.r. explaining why I think Thiemann’s Hilbert space is indeed non-seperable - when your comment comes in! :-)

Good, so my memory was correct that LQG usually deals with non-seperable Hilbert spaces. I didn’t fully realize this until I heard a talk by Bojowald at ‘Strings meet Loops’ where he mentioned that this is a crucial issue in his ‘loop quantum cosmology’.

You write:

Now , then the issue is how to recover the ordinary classical world from it. I know Josh Willis did some work (and still working on it) on the issue.

Hm. Maybe I am confused, but right now it seems that there is way too much classicality in Thiemann’s paper and that we’d rather like to understand how to recover the ordinary quantum world in his approach! :-)

For instance, he emphasizes that his choice of inner product = choice of $ω$ is just meant to be a simple example (although he also mentions that other examples may be hard to come by). Would other $ω$ maybe yield seperable Hilbert spaces?

As far as I understand this could be possible, but it does not appear to be likely. Maybe Josh Willis could comment on this point?

Posted by: Urs Schreiber on January 29, 2004 2:45 PM | Permalink | Reply to this

Re: There’s no place like home

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On p. 115 of

T. Thiemann, Introduction to Modern Canonical Quantum General Relativity

it says indeed

We remark that the spin-network basis is not countable because the set of graphs in $σ$ is not countable, whence $ℋ^{0}$ is not separable. We will see that this is even the case after moding out by spatial diffeomorphisms although one can argue that after moding out by diffeomorphisms the remaining space is an orthogonal, uncountably infinite sum of superselected, mutually isomorphic, separable Hilbert spaces.

Posted by: Urs Schreiber on January 29, 2004 3:01 PM | Permalink | Reply to this

Re: There’s no place like home

Too many forums! :)

I just posted some of my thoughts at the Physics Forum

http://www.physicsforums.com/

(I had trouble getting the direct link to work here)

Which is the preferred place to discuss this: there, here, spr? :)

The point is, Urs, I think our work may be relevent to “fixing” the problem of Hilbert spaces in LQG.

You and I haven’t talked much about topology in our approach, but of course, any topology we have will be non-Hausdorff. However, it does have a property that I sometimes think of as being “weakly Hausdorff”. I hope I am not clashing with standard terminology there. Points in our space are not separable in general, but they are weakly separable. What I mean by that is that two points are separable if they are not contained in the same D-diamond. This gives a kind of “blurriness” down at the level of individual cells, but carries the usual notion of separability as long as you back away from the cells a bit.

Very exciting stuff :)

Eric

Posted by: Eric on January 29, 2004 3:15 PM | Permalink | Reply to this

Re: There’s no place like home

Hi Eric!

Here, there, everywhere (imagine the respective Beatles tune :-)

I would vote for discussion here at the Coffee Table. You can be sure that I read this, while I will not regularly check the Physics Forum, in general.

Regarding your point on separability: Is the notion of separability of a Hilbert space really related to separability of points in the sense of Hausdorff/non-Hausdorff? I think with respect to Hilbert spaces separability simplye means ‘has a countable basis’. Is this related to a Hausdorff property, somehow?

Posted by: Urs Schreiber on January 29, 2004 3:31 PM | Permalink | Reply to this

Re: There’s no place like home

Regarding your point on separability: Is the notion of separability of a Hilbert space really related to separability of points in the sense of Hausdorff/non-Hausdorff? I think with respect to Hilbert spaces separability simplye means ‘has a countable basis’. Is this related to a Hausdorff property, somehow?

Hi Urs,

Don’t forget that my math sucks :) I don’t know of any theorem that relates the two ideas, but it feels right. What you wrote (somewhere between here, there, and everywhere :)) made me think that the two illnesses were related. For example, you said

the W(I) are not sensitive to ‘neighbouring’ W(J): The Hilbert space is by construction so large that W({x}) and W({x+epsilon}) can sit right next to each other without noticing each other.

I am probably totally off and should just shut up :) At least I can say I’m having fun :)

The real point is that their inner product seems to be sick, and I think that our work (or maybe Harrison’s) could be step in the direction of trying to fix it. Maybe not.

Eric

Posted by: Eric on January 29, 2004 3:45 PM | Permalink | Reply to this

Re: There’s no place like home

What’s a separable Hilbert space, and why does it matter?

Posted by: Arvind on January 29, 2004 4:45 PM | Permalink | Reply to this

Re: There’s no place like home

A Separable Hilbert Space is one with a countable basis.

I don’t think it matters a whit.

Posted by: Jacques Distler on January 29, 2004 5:14 PM | Permalink | Reply to this

Re: There’s no place like home

Hm, don’t you think that the reason that Thiemann can work with his operators essentially as if dealing with a classical Poisson algebra is due to the peculiar nature of the Hilbert space that he chooses?

Posted by: Urs Schreiber on January 29, 2004 5:19 PM | Permalink | Reply to this

Re: There’s no place like home

No, I don’t believe there’s any quantization scheme that takes the full Poisson-bracket algebra of the classical theory and carries it over — unaltered — into the operator algebra of the quantum theory.

Depending on the quantization scheme, you may be able to carry over some subalgebra (the prototypical example, being the CCRs).

Posted by: Jacques Distler on January 29, 2004 7:34 PM | Permalink | Reply to this

Re: There’s no place like home

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I don’t believe either that the full Poisson algebra carries over. (There is a theorem showing that this cannot work in general.) That’s why I added the qualifier ‘essentially’. My point is, which I have been discussing with Luboš here, that in Thiemann’s quantization many more properties of the classical algebra carry over than just the CCR. That, of course, not the entire classical algebra is reproduced is the content of section 6.6 of Thiemann’s paper, where he discusses the quantum deformations of the classical invariant algebra of the Pohlmeyer charges.

But what is crucial for Thiemann’s removal of the anomaly is that things like

(1)

{(\exp ((π^{μ} + i X^{' μ})))}^{- 1} = \exp (- (π^{μ} + i X^{' μ}))

do hold true in his quantization, as oppsosed to the analogous normal-ordered relations in CFT. You can see this explicitly in his equation (6.7) and implicitly in the absolutely crucial relation

(2)

α (W (Y_{\pm})) = W (α (Y_{\pm}))

in the third paragraph on p. 20. (Here $α$ is the action of the exponentiated Virasoro generators.)

As far as I can see Thiemann’s quantization is technically correct (no mathematical errors). So there must be some physical assumption which makes him part company with the usual lore.

I think that it is crucial that he allows himself to work on non-separable Hilbert spaces. His construction of a Hilbert space by applying the GNS theorem to the Weyl algebra of $W$ operators is what allows the above-mentioned non-standard quantum relations, but it also leads to a non-separability of the Hilbert space.

Of the kinematical Hilbert space that is. It is not too surprising that the physical Hilbert space is separable again, because it is obviously much ‘smaller’ in general. But, when comparing his quantization with the OCQ or BRST quantization (instead of the LCQ, where only the physical Hilbert space appears because the constraints are solved before quantization) we have to look at the kinematical Hilbert space, because the Hilbert space on which the CFT operators are represented in the usual approach is also kinematical (contains non-physical states). The physical Hilbert space in the usual approach is that generated by the DDF operators acting on physical massless/tachyonic states.

Of course non-separable Hilbert spaces do appear in practice from time to time, but then we are always dealing with uncountably many superselection secors, each of which is separable. Thiemann’s non-countable Hilbert space (and, by the way, I have just received email by him confirming that the kinematical Hilbert space in his paper is non-separable) does however not separate into superselction sectors each of which would carry a representation of the constraints.

I think there are two alternatives:

1) Either there is a technical, mathematical error in Thiemann’s paper and hence his conclusions are wrong. If you believe that this is the case, that his quantization in particular is flawed, then please point out where you think the mistake lies.

2) Or the math is correct (which I am pretty convinced that it is). In this case we need to talk about if the assumptions that are made before the crank of the formalism is turned are viable. I am suggesting that the assumption of a non-separable kinematical Hilbert space may be a physically non-viable assumption.

Posted by: Urs Schreiber on January 30, 2004 12:16 PM | Permalink | Reply to this

Re: There’s no place like home

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OK, so you (he) claim(s) that there is a quantization in which the commutation relations of $X' (σ)$ , $Π (σ)$ , $T_{+ +} (σ)$ and $T_{- -} (σ)$ are carried over from the classical Poisson-bracket algebra, unaltered (i.e., the commutators of the $T$ ’s do not pick up a central term)?

Certainly, that’s not true if the $T$ ’s lie in the universal enveloping algebra generated by $X' (σ)$ , $Π (σ)$ — as is conventionally the case.

Posted by: Jacques Distler on January 30, 2004 3:17 PM | Permalink | Reply to this

Re: There’s no place like home

So what is wrong with this?

Posted by: Urs Schreiber on January 30, 2004 3:26 PM | Permalink | Reply to this

Re: There’s no place like home

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You mean aside from the fact that none of the symbols are well-defined?

Look, this is elementary stuff.

We can expand everything in Fourier modes. If $T_{+ +}$ is in the universal enveloping algebra of the Fourier modes of $X'$ and $Π$ , then its Fourier modes (conventionally called $L_{n}$ ) are some expressions quadratic in those modes.

Since the Fourier modes of $X'$ and $Π$ (the “oscillators”) don’t commute, you need to specify an ordering. I don’t care what ordering you choose, but I insist that you choose one.

Now compute the commutator of two $L_{n}$ ’s. Again, you will obtain something which is at most quadratic in oscillators (there will, in general, also be a piece $0^{th}$ -order in oscillators). And it must be re-ordered to agree with your original definition of the $L_{n}$ s.

Carrying out this computation, you obtain the central term in the Virasoro algebra, and I believe that it is a theorem that the result is independent of what ordering you chose for the $L_{n}$ s.

Note that I never mentioned what Hilbert space I hope to represent these operators on. So I don’t see where its separability (or lack thereof) enters into the considerations.

Posted by: Jacques Distler on January 30, 2004 5:02 PM | Permalink | Reply to this

Re: There’s no place like home

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You mean aside from the fact that none of the symbols are well-defined?

I don’t see why as an algebra these symbols should not be well defined. All the caveats that I included pertained only to the representation of these things as operators.

[…] you need to specify an ordering. I don’t care what ordering you choose, but I insist that you choose one.

Ok, let me choose the ordering the way it drops out from the Fourier decomposition without reordering:

(1)

L_{m} = \frac{1}{2} \sum_{k = - ∞}^{∞} α_{m - k} α_{k} .

I could open Green, Schwarz & Witten on p. 73, where they derive the classical algebra of this object and check that in going from their (2.1.83) to (2.1.84) there is no re-ordering involved. But let me write it out here in a different way:

Using

(2)

[L_{m}, α_{k}] = - k α_{k + m}

one gets

(3)

[L_{m}, L_{n}] = \frac{1}{2} \sum_{k} [L_{m}, α_{n - k} α_{k}]

(4)

= \frac{1}{2} \sum_{k} ([L_{m}, α_{n - k}] α_{k} + α_{n - k} [L_{m}, α_{k}])

(5)

= \frac{1}{2} \sum_{k} ((k - n) α_{n + m - k} α_{k} - k α_{n - k} α_{m + k})

(6)

= \frac{1}{2} \sum_{k} ((k - n) α_{n + m - k} α_{k} + (m - k) α_{n + m - k} α_{k})

(7)

= (m - n) \frac{1}{2} \sum_{k} α_{n + m - k} α_{k}

(8)

= (m - n) L_{m + n} .

There is no reordering involved in this.

I believe that it is a theorem that the result is independent of what ordering you chose for the $L_{n}$ ’s.

Do you have a reference to this theorem?

Posted by: Urs Schreiber on January 30, 2004 6:40 PM | Permalink | Reply to this

Re: There’s no place like home

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Good God! If you’re going to be that sloppy manipulating divergent quantities, we had better quit discussing this now.

Cut off those infinite sums (i.e., rather than $\sum_{k = - ∞}^{∞}$ , consider $\sum_{k = - N}^{N}$ ) and try again.

The only $L_{n}$ with an ordering ambiguity is $L_{0}$ , so it suffices define an ordering for it. To compute central term, it suffices to compute the commutator of $[L_{n}, L_{- n}]$ .

Posted by: Jacques Distler on January 30, 2004 7:15 PM | Permalink | Reply to this

Re: There’s no place like home

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Dear Jacques -

you write:

Good God! If you’re going to be that sloppy manipulating divergent quantities, we had better quit discussing this now.

I hope I am not annoying you. I very much appreciate that you take the time to discuss these things with me.

You seem to be very convinced that Thiemann (and myself, for that matter) are confused about a very elementary point. As for myself I don’t see my mistake yet, but it is of course well possible that I am subject to misapprehensions. Certainly you don’t have infinite time to waste on this - but please be at least assured that your contributions are very valuable to me and probably to others, who are interested in Thomas Thiemann’s work.

I believe that if you, or other string theorists, can point out technical mistakes in Thiemann’s paper that this will have considerable effect on the Loop Quantum Gravity people in general. Thiemann’s quantization can be regarded as a testing ground, a laboratory, for the techniques used in LQG. The LQG camp is well known for its high esteem of mathematical rigour and it would be very important to them to be made aware of a technical mistake. Physical viability of their approach is another matter, but I do expect that they care about the consistent definition of the objects that they are dealing with.

Thomas Thiemann, as you know, is one of the more prominent people working on LQG, and he has a record of papers with a rather high technical level. I have heard string theorists criticising his papers as being games of math instead of physics. But in any case the claim is that this math is well done. So I bet that he and many others in the LQG field would highly appreciate if string theorists can spot technical mathematical mistakes in their work.

Because of this I would kindly ask you not to give up on me and my attempts to answer to your charges. I may not be the most suitable person for that task and am indeed hoping that somebody more knowledgeable will chime in to help me out. I have indeed contacted Thomas Thiemann by email and he says that next week he’ll be back from a conference and willing to discuss his paper. Surely he’ll be a better advocate of his work than I am.

That said, let me try to answer your latest comments. Unfortunately, as you will see below, I will still not be able to completely understand your criticism. Please bear with me. Thanks!

You write:

Cut off those infinite sums […] and try again.

Ok. At least I can reassure you that I do understand that cutting off these sums does modify the algebraic relation $[L_{n}, L_{m}] = (n - m) L_{n + m}$ .

But, alas, I don’t see what the cutting off of these sums has to do with the question whether there is a non-commutative algebra such that it has commutators which reproduce the Poisson-brackets of the oscillators $a_{n}$ and the generators $L_{n}$ .

You are saying that I should be more careful with manipulating divergent terms. This puzzles me a little. All I wrote down are infinite sums of products of elements of an infinite-dimensional non-commutative algebra generated by elements $a_{n}$ . Until I talk about representing these as operators on some space there are no numbers which could diverge, I think.

Of course I do understand that when I acted with the $L_{0}$ generator with the ordering as given in my previous comment on a Fock vaccum state which is annihilated by the $a_{n}$ for positive $n$ , that the result would be ill defined because it would formally contain an infinite real number multiplying the Fock vacuum.

But this leads precisely to the idea that I tried to discuss before: The claim by Thiemann is essentially that the noncommutative algebra that I indicated in my previous comment, with the ordering as given there (which is equivalent to the definitions in that other comment) can be represented on a non-separable Hilbert space in such a way that the objects that I manipulated in my last comment have a perfectly well defined action on this Hilbert space, without any divergencies.

This is the crucial claim. It is about operator representations of the abstract non-commutative algebra of the $a_{n}$ and the $L_{n}$ (in the ordering indicated before) or equivalently the smeared $Y (σ)$ and $Y (σ) Y (σ)$ , I believe. The claim is essentially that there is an operator representation where the $L_{n}$ (in the ordering that I have given, or equivalently, the $\int_{S^{1}} d σ ξ (σ) Y (σ) Y (σ)$ ) are well defined and act without producing divergencies. This applies to the $L_{n}$ with the sums going from $- ∞$ to $∞$ , because this is what one gets when Fourier-decomposing the $Y (σ)$ in $L_{n}$

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January 27, 2004