# Thomas Larsson's post on LQG-String

1. Feb 22, 2004

### marcus

As part of the on-going discussion of Thiemann's "Loop-String" paper, the following was posted by Thomas Larsson, on 21 February at SPR (sci.physics.research) and also in an earlier version at Jacques Distler's board, the String Coffee Table.

Today I checked both places---SPR and Distler's board---and did not find any response. Maybe it is too early. Or perhaps Larsson's post was overlooked.

At String Coffee it is about halfway down a rather long page
http://golem.ph.utexas.edu/string/archives/000300.html
and possible to miss (I found it only on the second pass, scrolling
down that page).

I'm hoping for some comment.

This is an expanded version of a post to the string coffee table,
http://golem.ph.utexas.edu/string/archives/000300.html . It is in
response to a post by K-H Rehren, who pointed out the crucial
algebraic difference between LQG representations and lowest-energy
representations. This explains the absense of anomalies in Thiemann's
approach and IMO settles the status of LQG as a quantum theory.

K-H Rehren:
>Dorothea Bahns has shown in her diploma thesis, that if one quantizes
>classical invariant observables (Pohlmeyer charges) by embedding them
>into the oscillator algebra via normal odering (N.O.), then the N.O.
>invariants fail to commute with N.O. Virasoro constraints, and
>commutators of N.O. invariants among themselves yield other N.O.
>invariants plus "quantum corrections" which are #not# quantized
>classical invariants. Thus, the quantum algebra has not only
>relations differing from the classical ones by hbar corrections
>(which everybody expects), but it would have #more generators# than
>the classical algebra. This feature ("breakdown of the principle of
>correspondence") is worse than a central extension, because the
>latter is a multiple of one, and as such #is# a quantized classical
>observable, suppressed by hbar. This feature is a property of the
>quantization, i.e., the very choice of the quantum algebra by
>replacing classical invariants by N.O. ones. One may or may not
>appreciate the oscillator quantization with features like this.

The correspondence principle is not necessarily violated. To
construct extensions of the diffeomorphism algebra in more than 1D,
one must first expand all fields in a Taylor series around some point
q. There are no conceptual problems to express classical physics in
terms of Taylor data (q and the Taylor coefficients) rather than in
terms of fields, although there may be problems with convergence.

The reason why such a trivial reformulation is necessary is that the
higher-dimensional generalizations of the Virasoro cocycle (there are
two of them) depend on the expansion point q. The relevant extensions
are thus non-linear functions of data already present classically,
which seems consistent with the correspondence principle.

>A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0)
>representation of Diff without central charge must be trivial
>(one-dimensional). Put otherwise: c=0 is only possible if one
>abandons the positive-definite Hilbert space metric (ghosts), or
>positive energy, or unitarity.

Hence we can not maintain non-triviality, anomaly freedom, positive
energy, unitarity and ghost freedom at the same time. It seems to me
that giving up anomaly freedom makes least damage, especially since
we know that anomalous conformal symmetry is important in 2D
statistical models, such as the Ising and tricritical Ising models.
It is important to realize that such models have been realized
experimentally (e.g. in a monolayer of argon atoms on a graphite
substrate) and that the non-zero conformal anomaly has been measured
(perhaps only in computer experiments). Hence anomalous conformal
symmetry is not intrinsically inconsistent.

A clarification is in order at this point. The multi-dimensional
Virasoro algebra is a kind of gravitational anomaly, but no such
anomalies exist in 4D within a field theory framework. However, it
turns out that the phrase "within a field theory framework" is a
critical assumption. As I explained above, the relevant cocycles
depend on the expansion point q, and thus they can not be expressed
in terms of the fields which are independent of q.

>Thiemann seeks for the quantum algebra within an LQG type auxiliary
>algebra which has a unitary representation of Diff. The latter is
>#not# subject to the positive-energy condition.

The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.

Let me elaborate on the difference between lowest-energy (LE) reps
and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
by Gaul and Rovelli. The canonical variables are A(x) and E(x) and

[E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.

In the paper, the operators are smeared over suitable submanifolds
and have some index decorations, but that is irrelevant for the
argument. The canonical variables act of functionals Psi(A), with
E(x) = d/dA(x), and the constant functional 1 can be identified with
a vacuum |vac> by

E(x) |vac> = 0 for all x.

Bilinears of the form

A(x)E(y)

generate a gl(infinity), and we can embed diffeomorphism and current
algebras into this gl(infinity). The key point to observe is that
these bilinears are already normal ordered w.r.t. the vacuum |vac>.
Normal ordering means by definition that things that annihilate the
vacuum are moved to the right, and E(y) does annihilate the vacuum.
Since no further normal ordering is necessary, no anomalies arise.

However, this is not what I would call a LE rep. Rather, I would call
it a "lowest-A-number" rep; the A-number operator \int dx A(x)E(x) is
always positive. This space is essentially classical in nature, so it
is not so surprising that there is no anomaly.

Let us contrast this type of rep with LE reps. For simplicity, let us
assume that x and y are points in 1D; the higher-dimensional case
requires a passage to jet space which complicates things, although
not in an essential way. We can now expand A(x) and E(x) in a Fourier
series, and the Fourier components A_m and E_m satisfy the CCR

[E_m, A_n] = delta_m+n,0 , [E_m, E_n] = [A_m, A_n] = 0.

The LQG vacuum satisfies

E_m |vac> = 0 for all m.

The LE vacuum |0>, OTOH, is defined by

E_-m |0> = A_-m |0> = 0 for all -m < 0.

In other words, it is the modes of negative frequency, i.e. those
that travel backwards in time, that annihilate this vacuum.
The bilinears that generate gl(infinity),

A_m E_n ,

are normal ordered w.r.t. the LQG vacuum |vac> but not w.r.t. the
LE vacuum |0>. To normal order w.r.t. the latter, we need to move
negative-frequency modes to the right:

:A_m E_n: = A_m E_n m >= n

E_n A_m m < n

This is the main algebraic difference between the LQG "lowest-A-number" reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
--------------end of post-----------------

Last edited: Feb 23, 2004
2. Feb 22, 2004

### jeff

I'll simply tell you that Larsson is in agreement with urs, distler, thiemann and ashtekar that LQG quantization is physically inequivalent to standard quantization (in what is most probably a rather unfortunate way as far as mother nature is concerned) and he is merely expanding on this. He also points out that his remarks about the correspondence principle don't apply to thiemann's or bahn's papers and don't change anything wrt the above basic point for full LQG.

For reasons that I'm quite sure you'll understand, I'm afraid you'll to have to ask me specifically for the details.

3. Feb 23, 2004

### marcus

Today Urs replied to Larsson's post on SPR by quoting a portion and
saying "Sorry, but this is not true" and then setting the record straight according to his own view, with many links to Distler's coffee table:

===portion of Larsson's post quoted by Urs===
The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.

Let me elaborate on the difference between lowest-energy (LE) reps
and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
by Gaul and Rovelli. The canonical variables are A(x) and E(x) and
[E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.
In the paper, the operators are smeared over suitable submanifolds
and have some index decorations, but that is irrelevant for the
argument. The canonical variables act of functionals Psi(A), with
E(x) = d/dA(x), and the constant functional 1 can be identified with
a vacuum |vac> by
E(x) |vac> = 0 for all x.
Bilinears of the form
A(x)E(y)
generate a gl(infinity), and we can embed diffeomorphism and current
algebras into this gl(infinity). The key point to observe is that
these bilinears are already normal ordered w.r.t. the vacuum |vac>.
Normal ordering means by definition that things that annihilate the
vacuum are moved to the right, and E(y) does annihilate the vacuum.
Since no further normal ordering is necessary, no anomalies arise.
==========end of quote==============
Urs takes over here:

Last edited: Feb 23, 2004
4. Feb 23, 2004

### Urs

Marcus wrote, regarding finding messages at the String Coffee Table:
Use an RSS Newsreader to always have a complete updated list of read/unread messages at the entire Coffee Table site. Many such readers for all demands are available. I have compiled a bunch of helpful lnks for how to read and participate in the Coffee Table discussion here.

It costs just about two clicks to install any RSS reader and it makes reading the Coffee Table even more comfortable than reading a USENET newsgroup or this forum here, I'd say.

Also note that last week a 'mathplayer' plugin appeared, which allows to read the formulas at the Coffee Table from within MS-IE. You don't need to install Mozilla!

5. Feb 23, 2004

### marcus

Thanks Urs,
as you may have guessed I am a creature of habit and
have grown accustomed to this (PF) place.

I am glad to know that as a user of MS-Internet Explorer I can now read the formulas at the Coffee Table site. But this does not diminish my hope that some of the discussion of Thiemann's paper and related matters will come here where it is familiar and comfortable.

Any further explication and comment would be warmly appreciated.
What you point to here, not only in Thiemann's paper but also in
one by Ashetekar et al, is what IIRC selfAdjoint called "the awful non-standardness of LQG". I am not clear as to whether this non-standardness is a feature of the development in Rovelli's book "Quantum Gravity". I noticed that you cited a paper by Rovelli and Gaul (LQG and the meaning of diffeomorphism invariance) suggesting that it might. Would it be possible for you to say simply and briefly where this non-standardness enters in LQG and what it is about?

6. Feb 23, 2004

### Urs

The non-standardness

So what is the 'non-standardness'?

I think I have said that many times already, but maybe I am not expressing myself clearly.

The easiest way to say it is: LQG-like quantization is not canonical quantization.

In LQG-like quantization the canonical data, i.e. coordinates and momenta, are not (both) represented as operators on a Hilbert space. (Open any book on elementary QM to see why this is non-standard.)

If there are constraints, they are not (all) represented as operators in LQG-like quantization. Instead one tries to find an operator representation of the group that these constraints generate.

P.S.

Concerning the RSS readers and habits: I don't want to deprive anybody of his or her habits. But since you were complaining that it is hard to find messages at the Coffee Table I just pointed out that using an RSS reader makes that easy. An easy way to keep up-to-date with stuff at the Coffee Table should be closer to your habits than a tedious way. ;-)

7. Feb 23, 2004

### marcus

Urs, I appreciate your taking the trouble of making
a brief clear summary like this!
As I understand this idea of non-standardness, it applies
also to the development in Rovelli's forthcoming book.

There too, for instance, part of the constraints (spatial diffeo inv.) are realized algebraically by quotienting a hilbert space of
quantum states, and not imposed via an operator.

Since Rovelli's book is likely to become a standard reference
that many people have access to, it might be interesting if you or
someone could correctly refer each of your objections (regarding the quantization procedure of LQG) to sections and pages of that book.

These seem to be potentially important disagreements and doing this page-referencing would make them more widely accessible, or so I think.

8. Feb 23, 2004

### jeff

Re: Re: Thomas Larsson's post on LQG-String

My point was that larsson was not disagreeing on the basic point about LQG quantization being quite different than canonical quantization (I'll explain why I interpreted his remarks this way even though his arguments were faulty). I chose to say only this because I believed - and still do - that this was really all you were interested in. Like I said, if you wanted the details you needed to ask me specifically, again for obvious reasons.

Notice that at the bottom of larsson's post he points out that because were dealing with an infinite dimensional algebra, the reordering of modes he described required to define normal ordering wrt what he called the "LE" vacuum produces an inequivalent theory. Thus despite the errors in his argument, I think he meant he didn't believe that this was still just ordinary quantization.

9. Feb 24, 2004

### Urs

Marcus -

didn't I already give you some page numbers in that other thread? Please look them up again.

The point is that whener the diffeomorphism constraints in LQG are 'solved', the procedure is non-standard, because it does not follow Gupta-Bleuler quantization. The constraints themselves are not even represented on the LQG Hilbert space. With this in mind you can easily find all the page numbers that you want by just looking at the table of contents.

Do you think you understand a bit of what we have talked about in the LQG-string thread? It's best if you try to understand it yourself, then you won't have to rely on others giving you page numbers. The basic ideas are not too difficult, I think.

The basic idea is that in standard quantization there are constraints $$C_I$$ and their quantization looks like
$$\langle \psi \hat C_I |\phi\rangle = 0 \,.$$

The most important point is to understand that this equation is not even defined in the LQG approach. That's why it is non-standard. Everything else are technical details.

10. Feb 24, 2004

### marcus

I originally suggested looking at page 173 of Rovelli for the realization of of the sp. diff. constraint as a quotient.

Jan 29 in the Thiemann thread. But this was my pointer to a page in the book, not yours. I've been looking for your page refs to Rovelli but havent found them yet. Its a long thread

What I am hoping to get from you is specific references to Rovelli's book illustrating a non-standard approach to quantization which you feel characterizes LQG in general (not TT's paper as a separate case).

I dont recall your providing so far any page refs to Rovelli besides what I already mentioned----if you did please remind me!

If you don't have any pointers to spots in the book besides that business around page 170, then that is OK. It should be possible to decide if taking a quotient Hilbert space (reducing the states to equivalence classes) is actually "non-standard" or problematical in any way. Or whether the mountain is actually a molehill.

But I would really like it if you could point me to other places in Rovelli's book where you think he deviates from the right path! Perhaps other cases will occur to you as you think about it.

Last edited: Feb 24, 2004
11. Feb 24, 2004

### marcus

Urs, it occurs to me that maybe all you are talking about is how on page 173 Rovelli defines the kinematic state space as

H/Diff

essentially by identifying spin network states that are equivalent under diffeomorphism (that is, calling two states equivalent if one can be smoothly deformed into the other)

an equivalence class of networks is an abstract knot
so the Hilbert space is essentially one of (labeled) knots.
So the kinematic Hilbert space turns out to have a countable basis consisting of abstract (labeled) knots.

I dont want to misunderstand you. Is this your general criticism of Loop Quantum Gravity? I really want to know if there is more to it, or whether this is the "non-standardness" you have in mind.

12. Feb 24, 2004

### Urs

Marcus -

see page 19 of the 'Amazing bid' thread:

Yes, there is this page 170 in Rovelli's book. I also gave you page and formula number in another review.

And, yes, the problem is in how the H/Diff construction. You keep emphasizing that there is a modding out by an equivalence relation. Sure there is. But the problem is the choice of equivalence relation. The choice they are using does not follow from standard quantum theory but only from classical reasoning.

In LQG the spin-network states are constructed and then smeared by classical diffeomorphisms. But the example of the LQG-string shows that already in 1+1 dimensions the quantum constraints do not generate classical diffeomorphisms. So why should this be true in 1+3 dimensions? If Rovelli can answer that I'll stop talking of a problem´- promised! :-)

13. Feb 24, 2004

### marcus

Great!
this is something solid to chew on!
unfortunately I have to go out. but will be back later this morning.
thanks

14. Feb 25, 2004

### marcus

Hello Urs,
So far I dont see how to reply.
Part of the trouble is I cannot see how things
could be constructed in a different order so as to respond

I've been re-reading pages 170-173 and trying to
imagine how the construction could be done in a way
that might satisfy you.

I'm not convinced that the way Rovelli does things now is faulty,
but I would like to understand better how you would wish
the approach to be different.

maybe I will be able to formulate this as a question to you

15. Feb 25, 2004

### eforgy

Hi Marcus,

I'm no expert either, but I can tell you what I understand of Urs' stance on this issue. He can elaborate more if I miss the point.

I don't think anyone thinks Rovelli's (or Thiemann's) stuff is mathematically faulty. The only thing Urs is claiming is that LQG represents a DRASTIC modification to what one normally thinks of a quantum theory. Furthermore, he doesn't think the term "canonical quantization" is appropriate to describe what they are doing because a canonical quantization would involve promoting the constraints to operators on some Hilbert space. This is NOT what is done in LQG so it is NOT canonical. The constraints are not even representable as operators on the Hilbert space of LQG.

I think all parties agree at this moment that the only test of who is right is going to have to be experiment. On the other hand, the trouble we saw with the simple KG equations suggests that things are even worse than this.

Once again, the mathematics is not under question. Rather, the physics is under question here.

Best regards,
Eric

16. Feb 25, 2004

### Urs

Hi Eric -

many thanks, yes, that's the point. I feel that I have tried to say this so many times now that I don't know how to further reformulate it! :-)

17. Feb 25, 2004

### marcus

thanks Eric, Urs,

and would it be correct to narrow it down still further in the case of Rovelli's development and say that it is
only the spatial diffeomorphism constaint which is not represented as an operator?

you see after the kinematic Hilbert space is constructed (as in pages 170-173) then operators are defined on it
and several constraints are implemented (by operator equations)

so I would like to say that in the normal LQG development a la Rovelli this strategy which you regard as nonstandard is confined to implementing the spatial diffeomorphisms

if I am mistaken and Rovelli applies it more generally please let me know!

18. Feb 25, 2004

### Urs

Yes, that's what Thomas Thiemann said: Only the spatial constraints are dealt with in the non-standard way. The Hamiltonian constraint is an honest operator.

19. Feb 25, 2004

### marcus

what I think is a brief and up-to-date discussion of
the issue of spatial diff invariance is contained in a summary of LQG
in an article posted this week by Velhinho

"On the structure of the space of generalized connections"

(page 19 and a bit on page 18)

http://arxiv.org/math-ph/0402060

he indicates several directions that are being explored, for
realizing spatial diff invariance, and he indicates some possible
problems

Velhinho's description is the most mathematically elegant (or conceptually efficient) of LQG I have seen so far. I just became aware of him. Perhaps (since he has co-authored with Thiemann in the past) you know him?

the implementation of spatial diff invariance is in flux in LQG and
it is an interesting topic-----which your constructive critique of TT's Loop-String paper has brought into focus

20. Feb 25, 2004

### jeff

Marcus,

Generally, ideas can't be critically assessed by simply identifying their logical flow. Seeing that in some sense C follows from B which follows from A etc isn't enough: logical consistency doesn't imply validity. One must be able to identify the assumptions underlying an argument and appreciate their implications. But the requisite insight must originate outside the arguments being analyzed, and it's difficult to gain that kind of perspective by bypassing the basics and going straight to the cutting edge.

You really need to step back from this. To improve your understanding I recommend solving exercises found in textbooks. Start with undergraduate level problems in classical mechanics, electrodynamics, and quantum mechanics etc. If you get stuck, just post a question. You certainly seem to have the time for it. I mean no offence by any of this.

Not according to this paper, which like most LQG papers is just another review and doesn't bear on the the basic point urs has tried to help you appreciate. Also, I don't think it's fair to other members to be constantly posting reviews of papers you don't actually understand.