Amazing bid by Thiemann to absorb string theory into LQG

[eigenguy]

selfadjoint, do you agree that the reason that thiemann gets no anomaly is that he doesn't use the original formulation of refined algebraic quantization?

 

[selfAdjoint]

I have been digging into his quantization - BTW, all his claims of no anomaly come from the sections of his paper BEFORE quantization. That is, they are about his classical theory. But all the criticisms are that quantization brings in the central charge. It is true that Polchinski brings in the central charge at the classical level, but apparently that isn't required, and a theory that was classically clean but had the c.c. as a quantum anomaly would pass muster with the string physicists.

He uses the methods from the paper Quantization of diffeomeorphism invariant theories of connections with local degrees of freedom, gr-qc/9504018, by Ashtekar, Lewandowski, Marolf, Mourao, and Thiemann. Notice that Marolf is coauthor of the 1999 paper on group averaging (Giulini & Marolf, A uniqueness theorem for Constraint Quantization, gr-qc/9902045).

The 1995 paper also employs group averaging, but I am concerned that they explicitly DON'T do the Hamiltonian constraint (in the LQG context). They do diffeomeorphism constraints. I have been trying to work out what influence that might have on the Virasoro constraints in the string problem. It seems to be a spacelike/timelike thing.

 

[eigenguy]

Distler just posted this,

Baby & Bathwater

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.

 

[selfAdjoint]

Thiemann's quantization procedure from the 1995 paper I cited is specifically stated to be local. The problem with it is that it may not be available in case the constraints don't close. Giulini and Marolf explicitly assume that, but what their paper provides, I now see, is just the fact that the "rigging map" is unique. So there is a possible freedom there, that the quantization is not unique. But I'm still digging.

Aside from the nonlocal suggestion, I don't see what this post by Distler accomplishes. Just some more of his sarcasm, and the continuation of his phoney issue with Yang-Mills theory.

 

[eigenguy]

Originally posted by selfAdjoint
phoney issue with Yang-Mills theory

I guess he believes that all interactions should be quantized using the same procedure so if it doesn't work for YM, it's wrong.

 

[Urs]

Hi everybody -

Distler's point is that anomalies are real and important and that defining them away in testable theories like the standard model leads to conflict with experiment.

I have received an answer by Hermann Nicolai, who says he is going to have a look at this issue.

So far he only confirmed my ideas about the DDF invariants, saying that D. Bahns once gave a talk at his institute about the Pohlmeyer stuff. Afterwards he had asked her if these invariants should not be expressible in terms of DDF invariants, because that would seem to be the only possibility. Apparently she said no, but Nicolai tells me that he is sceptical about this answer.

BTW, I think it is interesting that the Pohlmeyer invariants are Wilson loops along the string for large matrix valued constant connections. This is precisely the same construction as used in the IIB/IKKT Matrix Model, e.g. see equation (2.7) of hep-th/9908038.


Regarding the string/field theory question:
The analogues of Feynman diagrams in string theory are computed using field theory on the worldsheet. This gives rise to effective theories on spacetime and on branes, which are also field theories. That's no surprise, even the theory of a single particle can be regarded as a field theory, one in 1+0 dimensions. Field theory is a pretty versatile thing.


Even string field theory reduces, when all the parts in the action that involve computations in the string's Hilbert space are integrated out, to a field theory, albeit one with infinitely many fields (one for each excitation of the string).

Usually, if somebody says "Yang-Mills theory" people will think of the respective field theory. The interesting thing in modern string theory is that apparently all supersymmetric YM theories are dual to string theory, i.e. they describe the effective theory on some brane which is embedded in a bulk in which closed string propagate.

The most prominent example is N=4 SYM which is believed to be equivalent to strings on AdS5 x S5.

 

[eigenguy]

Hi urs,

Originally posted by Urs
Regarding the string/field theory question:
The analogues of Feynman diagrams in string theory are computed using field theory on the worldsheet. This gives rise to effective theories on spacetime and on branes, which are also field theories. That's no surprise, even the theory of a single particle can be regarded as a field theory, one in 1+0 dimensions. Field theory is a pretty versatile thing.


Even string field theory reduces, when all the parts in the action that involve computations in the string's Hilbert space are integrated out, to a field theory, albeit one with infinitely many fields (one for each excitation of the string).

So was lethe correct in saying that string theory is really just 2D QFT? Or maybe it's correct to say that perturbative string theory is just 2D QFT?

Originally posted by Urs
Usually, if somebody says "Yang-Mills theory" people will think of the respective field theory. The interesting thing in modern string theory is that apparently all supersymmetric YM theories are dual to string theory, i.e. they describe the effective theory on some brane which is embedded in a bulk in which closed string propagate.

The most prominent example is N=4 SYM which is believed to be equivalent to strings on AdS5 x S5.

Was lethe also correct in saying that it is incorrect to think of yang-mills theory as meaning anything other than a QFT and that the solution to the mass gap problem is necessarily a QFT solution?

Was lethe correct in attributing spacetime momentum to the non-zero modes of a string?

Was my basic point - that there may be no rigorous way to formulate QFT, and if it is just an approximation to a more fundamental theory that we shouldn't be upset or surprised by this - correct?

Thanks a bunch.

 

[marcus]

Originally posted by Urs
... D. Bahns once gave a talk at his institute about the Pohlmeyer stuff. Afterwards he had asked her...

the Dorothea Bahns that TT mentions in his acknowledgments section, I guess. Do you know where she is? at Potsdam or?

 

[lethe]

Originally posted by eigenguy
Was lethe correct in attributing spacetime momentum to the non-zero modes of a string?

see, for example, Polchinski Vol I, eqs 4.3.20-4.3.22

 

[marcus]

Originally posted by lethe
see, for example, Polchinski Vol I, eqs 4.3.20-4.3.22

Hi Lethe, would it be possible for you to start a separate
thread about these questions (D-brane, Yang-Mills) and let us
have this thread for discussion of Thomas Thiemann's paper
and the response to it by those whom Urs has contacted?

 

[marcus]

Urs too!
it would be fine if you would care to start a thread
to discuss purely string topics
and let this one be more focused on the TT paper topic

BTW thanks for sharing Nicolai's initial response with
us, hope you hear from him again soon, and AA as well

 

[Urs]

Hi eigenguy -

I don't think it helps to say that string theory is "just 2d QFT". Calculating any given order of the string perturbation series indeed involves just 2d QFT. But the fact that you need to sum up the contributions from just-2d-QFT calculations for QFT on surfaces of different genus, i.e. for different QFTs really, is something that itself is not captured by any QFT.

It is captured by string field theory, though, which can be rewritten as an ordinary field theory with infinitely many fields and interactions.

But the crucial insight is that maybe all of string theory can be rewritten in terms of some QFT with finitely many fields, anyway. Still, string theory is not a QFT, but it may be "equivalent" to one in a certain sense, this is the Maldacena conjecture.

If you read the introduction to the Clay Millenium Prize Questiion on YM and the mass gap you'll note that Witten roughly argues as follows:

Nature is described by quantum YM theory. So YM QFT is important and we need to understand it.

But YM QFT is hard. So let's maybe first understand the problem in an easier setup. Let's search the space of all possible QFTs for nice ones that are not quite as hard as YM. It turns out that supersymmetric QFTs have many properties that make them much easier to study than the non-susy ones. They sort of sit at exceptional points in the space of all QFTs.

Therefore we should be interested in SUSY QFTs, even if nature is not susy. Ok. So which susy QFT? It turns out that of these nice theories one of the nicest ist N=4 supersymmetric Yang-Mills. So let's try to understand that one first, not forgetting that we are really interested in ordeinary non-sus YM.

But now a miracle happens: SYM and N=4 SYM for the U(N) gauge group with large N in particular seems to be closely related to string theory! Maldacenas AdS/CFT conjecture says that it is indeed dual to string theory. This would confirm the old intuition by t'Hooft, who long ago argued that large N gauge theories are dominated by planar Feynman diagrams that look like string worldsheets interacting. Even apart from that, Matrix Theory tells us that maybe SYM dimensionally reduced to a line (BFSS) or even to a single point (IKKT/IIB) for N taken to infinity is the nonperturbative description of M-Theory!

So even if nature is neither susy nor stringy, there is a relation to string theory. The old hope that strings would give us the elementary particle masses uniquely seems to have vanished. The more fascinating aspects of modern string theory are its intimate relations to all kinds of gauge theories. Nobody today can be interested as a theoretician in gauge theories without coming across some string theory. String theorists are the leading figures in field theory, too. See Seiberg-Witten Theory or indeed most of what Witten has done. Witten's latest paper is about how N=4 SYM can be described by a topological string even wiothout taking N to infinity. Witten's whole work is really related to understanding ordinary YM theory, in a sense.

So will the solution to the mass-gap problem be a QFT solution? I don't know! Maybe the crucial clues will come from string theory, that's quite plausible. But of course, due to the duality, it will then also be understandable in a pure QFT kind of way.

It is not clear yet that string theory is related to the experimentally accessible universe. But it is already clear that string theory is discovering new continents in our universe of theories.

 

[Haelfix]

Urs,

thats the most concise explanation I've ever read on why String theory should be important. to theoreticians, even if you are an abject skeptic.

In my opinion, Wittens work on the subject of the space of connections modulo gauge transformations is his primary accomplishments as a physicist. Subsequent development of topological field theory, its applications to Morse theory and Gromow-Witten invariants gave him the fields medal, (which he richly deserved).

Back to the topic though.

Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories. I suppose you could argue that the central charge present has a different topological character, and hence inapplicable to say YM.

But I don't see a good reason a priori to restrict this scheme to only live in quantum gravity scenarios.

 

[marcus]

Originally posted by Haelfix
...Back to the topic though.

Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories...

Haelfix, thanks for redirecting the discussion back on topic!

To anyone with other string questions (not directly connected with TT's paper) it would be great if you would start a thread for them---we could use a good string thread.

 

[eigenguy]

Originally posted by Urs
I don't think it helps to say that string theory is "just 2d QFT"...

Fantastic. Thanks!

 

[selfAdjoint]

Originally posted by Haelfix
Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories. I suppose you could argue that the central charge present has a different topological character, and hence inapplicable to say YM.

But I don't see a good reason a priori to restrict this scheme to only live in quantum gravity scenarios.

I didn't say it was bogus to apply the quantiszation to Y-M or other theories, I said Distler's harping on quantizing Y-M was bogus because he refused to look directly at TT's theory. Perhaps I was over excited.

I just found a good paper on the background of the TT quantization on the arxiv. It's hep-th/0402097, Lectures on Integrable Hierarchies and Vertex Operators, by A. A. Vladimirov. It is intended for undergraduates, and after scanning it over I am truly impressed by those Russian undergraduates!

 

[eigenguy]

What's the simplest system one could play with in which the same basic issues being discussed here arise?

 

[ranyart]

Originally posted by eigenguy
What's the simplest system one could play with in which the same basic issues being discussed here arise?

Probably a single Kodama State embedded into a Zero Dimensional Phase(Zero-Worldsheet).

 

[eigenguy]

Originally posted by ranyart
a Kodama State embedded in a Zero Dimensional Phase...

A rabbi and priest in a rowboat...

 

[Urs]

eigenguy wrote:

What's the simplest system one could play with in which the same basic issues being discussed here arise?

Ok, here is a simple exercise that everybody who has followed our discussion should be able to solve:

1) Consider the Nambu-Goto action in 1+0 dimensions, which describes the free relativistic particle in Minkowski space (alternatively, for those who enjoy a bigger challenge: the charged particle in curved space with an electromagnetic field turned on)

2) Compute the single constraint of the theory.

3) Do a Dirac quantization by promoting this single constraint to an operator equation. Discuss the resulting quantum equation.

4) Now subject this system instead to the method used in Thomas Thiemann's paper. Discuss the ambiguity that one encounters and the differences and/or similarities to 3).

 

[ranyart]

Originally posted by eigenguy
A rabbi and priest in a rowboat...

Indeed two 'Ed's' are better than one?

 

[Urs]

There is another simple example:

Look at the LQG-like quantization of the 1d non-relativistic point is
http://xxx.uni-augsburg.de/abs/gr-qc/0207106 . Then note equation (IV.5) and the one above it. In these equations care is taken that the usual quantum algebra is carried over to the LQG-like quantization scheme. If one were to remove the term $$\exp(-\alpha^2)/2$$ one would get instead the classical algebra, which corresponds to the way Thomas Thiemann 'quantizes' the LQG-string.

 

[selfAdjoint]

Urs, I was interested in Rehren's email that you posted on the Coffee Table, with its presentation of the various things that can happen, and the origin of the different levels of anomalies, central charges, and broken invariances. It was very educational. Do you have any comments on it? Does it have any bearing on these examples you give us from other quantizations?

 

[Urs]

selfAdjoint -

maybe we should carry this discussion to the Coffee Table where Rehren can see it.

Rehren argues that what Thiemann does is formally a form of 'quantization'. But let's not argue about words. The paper by Ashtekar, Fairhurst and Willis shows that with such a general notion of quantization one does not obtain the correct results even for the nonrelativistic 1d particle. The same holds true when applying this 'generalized' notion of quantization to the Maxwell field or the free relativistic particle. In each of these cases the results differ drastically from known physics (even if the large ambiguity in how to impose the exponentiated operator equations is used in a way that closely follows the usual quantization instead of following the classical theory).

This is not disputed by Ashtekar, Fairhurst and Willis. Their argument is the same as that by Thiemann: Maybe this drastically different notion of quantization is the correct one for gravity. Right, maybe it is. But maybe not. What I would like to see is some sort of motivation for why a radical departure from usual quantization is the right thing to do in quantum gravity. It is certainly not the physically right thing to do in the other systems that have been studied by this method.

 

[lethe]

Originally posted by Urs

Ok, here is a simple exercise that everybody who has followed our discussion should be able to solve:

lemme see..

1) Consider the Nambu-Goto action in 1+0 dimensions, which describes the free relativistic particle in Minkowski space (alternatively, for those who enjoy a bigger challenge: the charged particle in curved space with an electromagnetic field turned on)

start with Minkowski space, no background field:

$$S=m\int ds=m\int\sqrt{\eta_{\mu\nu}\dot{x}^\mu\dot{x}^\nu}d\tau$$

canonical momentum:
$$p_\mu=\frac{\delta L}{\delta\dot{x}^\mu}=\frac{m\dot{x}_\mu}{\sqrt{\dot{x}^\nu\dot{x}_\nu}}$$

2) Compute the single constraint of the theory.

from the expression for the canonical momentum, we see that

$$p^2=m^2$$

this is a first class constraint, since the equations of motion were not invoked.

3) Do a Dirac quantization by promoting this single constraint to an operator equation. Discuss the resulting quantum equation.

given a state $|\psi>$, look at its Fourier transform in Minkowski space

$$|\psi>=\int d^dk e^{ikx}|k>$$

since this is a generic Fourier transform over d-dimensional Minkowski space, the components of k are independent.

i guess i want to say something about enforcing only the expectation value of the constraint here, instead of the operator version of the constraint.

but i don't see why i have to do that here. let me see...

 

[Urs]

Hi lethe -

you can make a Fourier transformation, of course, but probably that's not necessary to make the point.

You have derived the classical constraint. Quantize it. Then impose Dirac quantization of constraints. Alternatively, impose LQG quantization of constraints. What do you get?

 

[Urs]

1st class constraints

By the way:

First-class constraints are those whose Poisson bracket closes on the set of constraints, i.e. is a linear combination of any of the constraints of the theory. Second class constraints are those whose Poisson brackte does not give another constraint.

In other words, the Poisson-bracket of 1st class constraints vanishes "weakly" or "on shell". The Poisson-bracket of second-class constraints does not.

Obviously only 1st class constraints have a chance to give a consistent set of operator constraint equations when quantized. Second class constraints must be eliminated by introducing "Dirac-brackets". This is a deformation of the usual Poisson bracket engineered in just such a way that all constraints become first class with respect to this new bracket. Dirac quantization really consists of replacing Dirac-brackets by commutators and sending the classical constraints to operator equations.

In any case, since for the relativistic particle there is only a single constraint it is trivially 1st class and we can quantize it immediately without worrying about 2nd class subtleties.

See for instance http://www.math.ias.edu/QFT/fall/faddeev4.ps

 

[eforgy]

Originally posted by Urs

You have derived the classical constraint. Quantize it. Then impose Dirac quantization of constraints. Alternatively, impose LQG quantization of constraints. What do you get?

Thanks lethe for starting to solve this exercise. It's been so long since I've looked at QM that I wouldn't have had a chance to get as far as you did.

But picking up where you left off, I think Urs simply wants you to look at the equation

$$p^2|\psi\rangle = m^2|\psi\rangle$$

and note that this is the Klein-Gordon equation.


Eric

PS: By the way, why do you call this a "constraint"? Remember, it's been eons :)

PPS: I'd have no idea how to loop quantize this and I HAVE been following this thread :)

 

[eforgy]

Originally posted by eforgy

PPS: I'd have no idea how to loop quantize this and I HAVE been following this thread :)

I can do a little better than this. Looking back at the earlier posts here, I'd write down

$$e^{p^2-m^2}|\psi\rangle = |\psi\rangle.$$

Since $$p^2$$ commutes with $$m^2$$, I guess we could write this as

$$e^{p^2} |\psi\rangle = e^{m^2} |\psi\rangle,$$

but I don't know if this buys us anything. Would it make sense to look at

$$e^{\epsilon(p^2 - m^2)}|\psi\rangle = |\psi\rangle$$

and combine terms of the same order in $$\epsilon$$? If we did this, I think we'd end up with

$$(p^2 - m^2)^n |\psi\rangle = 0$$

for all $$n> 0$$. Otherwise, we'd end up with a mess.

Eric

 

[Urs]

Hi Eric -

thanks for chiming in.

Ok, let me give it away:

As you said, the operator version of the single constraint of the free relativistic particle is obtained by the usual correspondence rule
$$p^\mu \to \hat p^\mu = -i \hbar \frac{\partial}{\partial x^\mu}$$ and yields nothing but the Klein-Gordon equation
$$\partial^\mu \partial_\mu \phi = -m^2 \phi$$
(up to factors of $$c,\hbar$$). I am calling this a constraint because, as lethe has derived, it is the operator version of the classical constraint $$\varphi = p^2 + m^2 = 0$$ of the action of the free relativistic particle. The classical free relativistic particle cannot move in all of its phase space, but has to stick to the subspace given by this equation, which is incidentally called the mass shell.

Ordinary quantization rules, found by Dirac and others, tell us that the quantum theory is given by demanding that this holds as an operator equation, which is nothing but the Klein-Gordon equation. This equation, together with its fermionic cousin, the Dirac equation, is very well tested experimentally, since it is the very basis on which all of QFT that can be measured in any accelerator is built.

Now let's see how LQG tells us to quantize the free relativistic particle:

There we are told not to consider the constraint $$\varphi$$ itself but the group which is generated by it by means of Poisson brackets. I.e. we are supposed to look at the group elements
$$U(\tau) = \exp\left([\phi,\cdot]_\mathrm{PB}\right)$$
where [\cdot,\cdot]_\mathrm{PB}$ is the Poisson bracket and this guy is supposed to act on classical observables, i.e. functions on phase space.

But because there is just a single constraint this group is nothing but the group of real numbers under addition:
$$U(\tau_1)\circ U(\tau_2) = U(\tau_1 + \tau_2) \,.$$

So LQG-like quantization consists of finding an operator representation $\hat U(\tau)$ of the group of real numbers on some Hilbert space so that the operator product satisfies
$$\hat U(\tau_1) \hat U(\tau_2) = \hatU(\tau_1 + \tau_2) \,.$$

Of course, due to the simpliciy of this example, we could just choose the usual Hilbert space of the free relativistic particle and set
$$\hat U(\tau) := \exp(i \tau (\hat p^\mu \hat p_\mu + m^2)) \,.$$
But we could also choose something very different. This is the great ambiguity that I was referring to. For instance, if we followed the tretament by Ashtekar, Fairhurst and Willis of the LQG-like quantization of the 1d nonrelativistic particle, than we'd want to use a nonseparable Hilbert space on which the momentum operator $\hat p$ is not representable. In this case, which is the precise analog of what Thomas Thiemann does in the 'LQG-string' the above choice for $$\hat U$$ is not an option.

But of course, due to the great simplicity of this toy example, if you pick any (hermitian) operator $$\hat O$$ on any Hilbert space, you can set
$$\hat U(\tau) := \exp(\tau i \hat O)$$
and get a representation of the group of real numbers. You can furthermore choose operators that don't come from exponentiating other operators, of course.

LQG tells us that physical states are those invariant under the action of this group. Clearly, by choosing $$\hat U$$ appropriately you can find an enormous number of states that satisfy the 'equation of motion'
$$\hat U(\tau)|\psi\rangle = |\psi\rangle \,.$$
Almost anything goes.

For instance I could choose $$\hat O$$ to be any projector. Than every state that is projected out by $$\hat O$$ is physical, according to the LQG-like quantization prescription.

Or I could use strange Hilbert spaces, like, say, the 2-dimensional Hilbert space C^2.

The weirdest things are possible. The reason is, that by ignoring the form of the constraint $$\phi$$ and just looking at the abstract classical group that it generates, we are loosing a lot of crucial physical information.

Thomas Thiemann similarly ignores the form of the Virasoro constraints and just cares about the classical group they generate. The result is no less strange than the above toy example. In fact, the above toy example is a subset of the full string quantization, namely that corresponding to strings that have no wiggly excitations.

I have pointed out the place at which this huge ambiguity appears in the Ashtekar, Fairhurst and Willis papers. There, too, one could choose almost anything else and get weird results. AF&W do choose to use a representation of the group operators that is very close to the usual one, but instead they choose a very strange Hilbert space. Other choices are possible. Nowehere in their paper is it explained why one choice should be preferred over the other.

 

[Urs]

Hi Eric -

yes, in your second post you demonstrate that exponentiating the KG constraint and demanding invariance yields the same thing as usual. The point is that LQG-like quantization allows you to exponentiate anything else, on any other Hilbert space and call it a quantization of the relativistic particle.

 

[eforgy]

Originally posted by Urs
Hi Eric -

yes, in your second post you demonstrate that exponentiating the KG constraint and demanding invariance yields the same thing as usual. The point is that LQG-like quantization allows you to exponentiate anything else, on any other Hilbert space and call it a quantization of the relativistic particle.

Hi Urs,

Let me state in my words how I am beginning to understand this. I am sure I am just repeating what you've been saying all along (assuming what I say is correct that is).

In the "usual" approach, you take your constraints and "quantize" them directly as operators on some hilbert space

$$C_I|\psi\rangle = 0.$$

Now if you take this and exponentiate it, you get something like

$$e^{C_I} |\psi\rangle = |\psi\rangle,$$

but this is still just an operator on the same Hilbert space we started with. These exponentiated operators form a group I suppose? An algebra?

Then, are you saying that the loop quantization procedure then looks for any group or algebra of operators that satisfy the same rules on any old Hilbert space and call this a quantization?

I'm sure I don't have that right.

Eric

 

[Urs]

Hi Eric -

yes, that's the idea - almost! :-)

There is a crucial subtlety:

Let me write $$C_I$$ for the classical constraints, i.e. these are functions on phase space and the system is restricted to be at points in phase space on which $$C_I = 0$$.

The $$C_I$$ are, already classically, generators of the gauge transformations of the system and physical observables must be gauge invariant and hence Poisson-commute with the constraints. We can write either
$$[C_I,A]_\mathrm{PB} = 0$$
or
$$\exp([r^I C_I,\cdot]_\mathrm{PB})A = A \,.$$
Classically this is both well defined and equivalent. The latter form makes us notice that the exponentiated constraints form a group (since they are 1st class)
$$\exp([r^I C_I,\cdot]_\mathrm{PB}) \exp([s^I C_I,\cdot]_\mathrm{PB}) = \exp([(r \times s)^I C_I,\cdot]_\mathrm{PB}) \,.$$
This group is the gauge group of the system with elements
$$U(r^I) := \exp([r^I C_I,\cdot]_\mathrm{PB}) \,.$$

In standard Dirac quantization the classical constraints $$C_I$$ are promoted to operators $$\hat C_I$$ and the equations of motion become
$$\langle \psi | \hat C_I |\psi \rangle = 0 \,.$$
In simple cases this is equivalent to
$$\hat C_I |\psi \rangle = 0 \,.$$

In LQG-like quantization people like to choose a Hilbert space on which the $$C_I$$ are not representable as operators. Then they claim that they can still 'quantize' the equation
$$\exp([r^I C_I,\cdot]_\mathrm{PB})A = A$$
by doing the following:

- Find operators
$$\hat U(r^I)$$
such that
$$\hat U(r^I) \hat U(s^I) = \hat U((r \times s)^I) \,.$$

- Demand that physical states satisfy
$$\hat U(r^I)|\psi \rangle = 0 \,.$$

In previous posts I have mentioned that this presription introduces a large arbitrariness in the choice of the $$\hat U(r^I)$$.
The example of the KG particle show that the physical information contained only in the abstract structure of the group generated by the classical constraints is way too little to reconstruct the physical system that one started with

 

[selfAdjoint]

And do the LQG people ever exhibit such a pretend quantisation of the Klein Gordon particle? I've never seen any but I'm a long way from completely famliar with their literature. It seems to me that you have to ask in each case, as Reheren does in his email, what are the degrees of freedom of the various things you are working with. Proving that the particle is too, umm, sparse to be quantized their way doesn't prove that their way is false in other cases.

And has anyone ever seen a KG particle, quantized or otherwise?

 

[lethe]

Originally posted by selfAdjoint

And has anyone ever seen a KG particle, quantized or otherwise?

yeah, i think they saw the pion already in the 30s