### Thiemann’s quantization of the Nambu-Goto action

#### Posted by urs

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

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

1) Find a representation ${\hat{U}}_{\phi}$ of the classical symmetry group elements $\phi $ on some Hilbert space. (Here the ${\hat{U}}_{\phi}$ 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 ${\hat{U}}_{\phi}$.

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.

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 ${\pi}_{\mu}$ be the canonical momentum to the embedding variable ${X}^{\mu}$. The usual left and right-moving bosonic fields are (pointwise)

Smearing them over an interval $I$ of the circle and contracting with some reak ${k}_{\mu}$ yields

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

From these one obtains
*bounded* operators by exponentiation

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 $\mathscr{H}$, 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

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

(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 ${\hat{V}}_{\pm}(\xi )$ 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

where ${\varphi}_{\pm}$ here denotes the group element of $\mathrm{Diff}({\mathrm{S}}^{1})$ associated with ${V}_{\pm}={V}_{\pm}(\xi )$ ($\xi $ is some smearing function).

The exponentiation in a sense removes all operator ordering ambiguities, since the conjugation operation (the similarity transform) ${e}^{{\hat{V}}_{\pm}}\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{e}^{-{\hat{V}}_{\pm}}$ 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

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 ${\pi}_{\mu}({p}_{\nu})$ 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.

## 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