The String Coffee Table

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

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

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 $Y(\sigma)$, $\sigma \in S^1$ which have the commutator

as in equation (6.4) of Thiemanns paper. Next assume that one can make sense of products of these operators $Y(\sigma)Y(\sigma)$ 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 $\mathcal{H}_\omega$ obtained by the GNS construction. Anyway, assume that the following expression makes sense:

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

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 $A(\sigma)$ let $w(A)$ be the classical conformal weight of $A(\sigma)$ iff

It is easy to check that

so that

and

and so on.

Now, denote for any field $A(\sigma)$ and any complex-valued function $\xi$ on $S^1$ the $\xi$-mode of $A$ by $A_\xi$, i.e.

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

This implies in particular that

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 $:L_\xi:$ 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 $\mathcal{H}_\omega$. This is how he gets a representation of the conformal group on his Hilbert space without having a conformal anomaly.

Hopefully Thomas will join this discussion himself. But for now let me say that he tells me that

1) he is aware of the fact (which took me a while to appreciate) that one cannot get a quantization of the $L_m$ without anomaly, no matter which ordering is chosen

2) this does not affect his approach because he defines the action of the operators representing the exponentiated constraints by

Here $U_\pm(\phi)$ (in Thomas’ paper this is a \varphi) is the operator which represents the exponentiated Virasoro element $\exp(\sum_n c^n L_n^\pm)$ which again implements the diffeomorphism $\phi$ on one half ($+$ or $-$) of the algebra.

$\pi_\omega(W_\pm(s))$ is the operator version of $W_\pm(s)$, which is essentially an exponentiated oscillator.

$\Omega_\omega$ is sort of a vaccuum in the GNS-Hilbert space (all states are obtained by acting on $\Omega_\omega$ with the $\pi_\omega(W_\pm(s))$.

Thomas says that the oscillators themselves are not represented on his Hilbert space, only the exponentiated oscillators are. Also the Virasoro generators $L_m$ are not represented on the Hilbert space, only the exponentiated $U_\pm(\phi)$ are, he says.

My question would be:

Is it ok to just define the action of the $U_\pm(\phi)$ as above, without writing them out in terms of the canonical coordinates-momenta/oscillators?

If this were ok, could’t I just do the same in the usual Fock quantization of the string by simply declaring that

(where now $\alpha_{-m}$ is a worldsheet oscillator and hats $\hat \cdot$ distinguish Poisson algebra elements from operators and $\{,\}$ is the Poisson bracket.

Thomas Thiemann has asked me to forward the following email message to the Coffee Table discussion.

In reply to an email by myself he answers (the quoted text is from my original mail):

[begin forwarded text]

let me try to rephrase the objections that have been raised in terms of the following question:

By the logic of your paper, what keeps you from constructing analogues of the operators $U_\pm(\varphi)$ on the standard Fock space of string states?

nothing, however, they I am worried that they won’t act unitarily because they mix the standard annihilation and creation operators. in other words, the operators $U_\pm$ are defined in the standard Fock rep. of the string but I am not sure whether they define a unitary rep. of the reparameterization group. That’s a good question, see below.

There certainly exist these operators (defined by their very action on the states) which represent the conformal group without anomaly on this Fock space. But they are not expressible in terms of the $L_n$. So in which sense could one claim that these $U_\pm(\varphi)$ are obtained from Dirac’s quantization scheme, if they are not expressible in terms of the quantized first class constraints?

Good question. Here is the simple answer: Suppose you have a classical phase space and a constraint function $C$ on it which generates infinitesimal gauge transformations as

where $F$ is any function on phase space and $\{.,.\}$ is the Poisson bracket. One can exponentiate this infinitesimal action to the Hamiltonian flow of $C$ given by the automorphisms

The Dirac observables of the system are those functions which are gauge invariant, that is,

In a quantization you want to find a representation $\pi$ of the functions $F$ as operators $\pi(F)$ on a Hilbert space $H$ with a cyclic “vacuum” $|0\rangle$. We now define a representation of the one parameter group of automorphisms by

It follows that $\Omega$ is invariant since $\pi(1)=1$, it is a physical state. More generally, physical states are defined by the condition

There is a beautiful interplay between physical states and Dirac observables because they obviously map $\Omega$ to physical states.

Your question now boils down to asking whether one can define a self-adjoint operator by

First of all this works at best only if $U(t)$ are unitary operators as otherwise $\pi(C)$ cannot be self-adjoint. If $\pi(C)$ is not self-adjoint you have violated a basic quantization principle, namely that real valued functions $C$ should be represented as self-adjoint operators. The necessary and sufficient criterion for $U(t)$ to be unitary is to check whether the functional

is $\alpha_t$ – invariant, that is

[Editor’s note: The original message here has a \circ instead of a \cdot $\cdot$.]

If that is the case, Stone’s theorem of functional analysis says that $\pi(C)$ exists if and only if the one parameter unitary group $t\mapsto U(t)$ is weakly continuous, that is

In my representation this condition is violated. However, this is unimportant because obviously $U(t)$ is a bona fide quantization of $\alpha_t$ and secondly one can use the $U(t)$ in order to define physical states, the $\pi(C)$ are not needed for that.

Notice that all of this is standard knowledge in constraint quantization, it is not my invention. The beauty of the construction is that you get everything for free once you have a positive linear functional $\omega$. The functional of standard string theory is positive only in $D=26$ and $a=1$, so you need the tachyon there. It is a good exercise to check whether that functional is invariant under the Virasoro group or only under the algebra. I have not done that calculation yet.

A related question comes to mind: In the LQG quantization of 1+3d gravity, is the representation of the constraints there similar to those in the ‘LQG-string’?

yes and no. the spatial diffeomorphism group is represented very much in analogy to the diffeomorphism group of the circle for the lqg string. The Hamiltonian constraint is represented in the form $\pi(C)$.

Hope that helps. Best,

Thomas

PS: Maybe you can upload this to the coffee table, I somehow can’t make this work.

end forwarded text

Since I have been asked by others about it and generally for the record, I’d like to present the calculation, recommended by Jacques Distler as a worthwhile exercise and of some importance for the present discssion, of the Virasoro anomaly by means of using regulated generators. In retrospect all this is pretty obvious and I am sufficiently ashamed to have been confused about it, but that’s life.

Everybody knows the various derivations of the anomaly as done in in GSW and Polchinski. Here I am going to discuss what could be called the canonical, functional perspective, because this is a perspective that might confuse one into missing the anomaly - as I have unfortunately demonstrated.

So let there be a canonical coordinate field $X(\sigma)$ on the circle, with canonical momentum $\pi(\sigma) = -i\frac{\delta}{\delta X(\sigma)}$ such that

From these the ‘chiral’ field

is constructed, which has the commutator

The task is to make sense of the commutator algebra of squared $Y$.

Naively one might write “$[\frac{1}{2}Y(\sigma)Y(\sigma),\frac{1}{2}Y(\sigma^\prime)Y(\sigma^\prime)] = -i \delta^\prime(\sigma,\sigma^\prime) \left( Y(\sigma)Y(\sigma^\prime) + i\frac{1}{2} \delta^\prime(\sigma,\sigma^\prime) \right)$”, where the second term on the right is due to reordering, is hence classically absent and essentially the quantum anomaly - except for the fact that all this is not well defined since it involves products of distributions.

To deal with these, the fields have to be smeared appropriately or, equivalently, their sum over modes have to be truncated. There are many possible smearings and truncations. After some experimenting the most convenient one I found is obtained by using

where $f(\sigma-\sigma^\prime)$ is an approximation for $\delta(\sigma-\sigma^\prime)$:

This one turns the $Y$-$Y$ commutator into

and the regulated modes of the Virasoro generators $L_m^{(N)}$ are simply

The commutators in question are now computed formally just as for the ill-defined expression mentioned above:

The second term in the integral, which is again due to the reordering, should essentially give the sought-after anomaly. Indeed, it can easily be evaluated explicitly, which yields

where $F(N)$ is a polynomial in $N$ which is independent of $m$. The first term is the standard anomaly

The second is a shift that can be re-absorbed into the definition of $L^{(N)}_0$:

so that the non-normal-ordered $\tilde L^{(N)}_0$ has finite expectation value in the Fock vacuum.

The term $F(N)$ would diverge when the regulator is removed (when $N$ is sent to infinity). It should somehow cancel. To see that we need to look at the first term of the above $[L^{(N)}_m,L^{(N)}_{-m}]$. From the Jacobi identity it follows that the anomaly can contain only $m^3$ and $m^1$ terms. Therefore it makes sense to look at terms containing different powers of $m$:

For even powers of $m$ the coefficient in front of the $Y$s is an odd function of $\sigma-\kappa$. This means that for even powers of $m$ we may replace $Y^{(N)}(\sigma)Y^{(N)}(\kappa)$ in the integrand with $\frac{1}{2}[Y^{(N)}(\sigma),Y^{(N)}(\kappa)] = -if^\prime(\sigma-\kappa)$.

For $m^0$ this yields

which indeed precisely cancels the previously found potentially diverging term $F(N)$. From the Jacobi identity it now follows that all other even powers of $m$, which could give c-numbers, will disappear. Since we are implicitly using symmetric ordering (no (normal-)reordering in the Fourier decomposition) the odd powers of $m$ don’t give rise to further c-number terms, either.

This means that the regulator can now be removed, which finally yields

Since this topic is all-but-beaten to death now, I thought it might be fun apply Thomas’s methods to a theory people actually care about (nobody gives a crap about the bosonic string).

Classical Yang-Mills theory is dilatation-invariant.

Exercise 1: Construct the generator, $D$, of dilatations in classical Yang-Mills.

The action of dilatations (just like the action of Poincaré) doesn’t quite commute with the Hamiltonian. Rather, the Poisson-bracket of $D$ with the Hamiltonian is proportional to $H$. Equivalently, the action of a 1-parameter group of dilatations is to rescale the Hamiltonian, $H\to \lambda H$.

Exercise 2: Now, apply Thomas’s procedure to construct the action of this 1-parameter group of dilatations in the quantum theory (“just like Poincaré”, as Thomas would say).

Excellent! Since $H$ rescales under the action of the dilatation group, we have proven that the spectrum of $H$ in the quantum theory is continuous near zero.

In other words, the quantum theory “has no mass-gap.”

Bzzzt! Do not pass GO, do not collect $1 million!

So now, the party line is that Thiemann’s quantization is some clever new method of quantization, completely unrelated to canonical quantization, that no one has thought of before.

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.

It is well-known that if one is willing to abandon locality, one has great lattitude to “cancel” the anomalies which arise in local QFT. A charitable interpretation of Thiemann’s procedure is that it correponds precisely to such a nonlocal modification of local field theory.

There are reasons to reject nonlocal modification of the worldsheet theory of the bosonic string — to do with getting consistent string interaction, a problem on which Thiemann is clueless, as he has, at best, made a failed attempt to construct the free bosonic string.

However, it is quite clear why Thiemann does not wish to apply his methods to Quantum Field Theories people care about, like Yang-Mills Theory. There, we know quite clearly whose side Mother Nature has come down on.