Thomas Larsson's post on LQG-String

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In summary, Thiemann's "Loop-String" paper seeks to find a quantum algebra within an LQG type auxiliary algebra which has a unitary representation of Diff. Rehren's post points out the crucial algebraic difference between LQG representations and lowest-energy representations, which explains the absense of anomalies in Thiemann's approach. The correspondence principle is not necessarily violated.
  • #1
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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.

---------Larsson's post---------

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
the CCR read

[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-----------------
 
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  • #2
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
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
the CCR read
[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:
Sorry, but this is not true.
I made the same mistake here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000527

was corrected by Jacques Distler here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000529

and later posted the correct details here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000572 .

The point is that there is _no_ quantization whatsoever of the Virasoro algebra which avoids the m^3/12 term of the anomaly, no matter which ordering you choose.

Furthermore note that this is not how Thomas Thiemann claims to get rid of the anomaly, according to himself:
http://golem.ph.utexas.edu/string/archives/000299.html#c000577 .

Also note that in LQG (e.g. gr-qc/9910079) we do _not_ have canonical
operators A(x) and E(x). That's because the classical A(x) is represented as an operator _only_ in its exponentiated form of a holonomy U[A,\gamma]. (e.g. equation (36)).

The analog holds true in Thomas Thiemann's 'LQG-string': Neither canonical coordinates nor canonical momenta of the string are represented on his Hilbert space:
http://golem.ph.utexas.edu/string/archives/000299.html#c000549 .

It follows that hence the Virasoro algebra is _not_ reprented (and not
claimed to be represented) on the 'LQG-string' Hilbert space, with or
without anomaly. The quantization method in LQG is _not_ canonical in the usual strict sense, i.e. first class constraints of the classical theory are_not_ represented as operators on some Hilbert space. This is made very clear by Thomas Thiemann here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000554
http://golem.ph.utexas.edu/string/archives/000299.html#c000588 .


The fact that not both of canonical coordinates and momenta are represented as operators on a Hilbert space in LQG-like quantizations is also emphasized in
A. Ashtekar, S. Fairhurst & J. Willis, Quantum gravity, shadow states and quantum mechanics, gr-qc/0207106.

I have summarized some of the aspects of this paper with an eye on the
'LQG-string' here:
http://golem.ph.utexas.edu/string/archives/000299.html#c000647 .

The bottom line seems to be the following, roughly:
LQG-like quantization searches operator representations for the group, not the algebra.

As shown by the various references mentioned, the result is in general
radically different from what I would call canonical quantization and
furthermore ambiguous, even more ambiguous than ordinary first quantization, that is.

To make this point quite clear consider the toy version of the 'LQG-string', the 'LQG Klein-Gordon particle'
http://golem.ph.utexas.edu/~distler/blog/archives/000307.html#c000637 .
The 1+0 dimensional Nambu-Goto action of the KG particle has a single
classical constraint which hence generates the group R, or U(1) if you wish. So according to the prescription "represent the group, not the generators", every operator rep of U(1) would be an 'LQG-like' quantization of the KG particle.

While this sounds extreme, I don't think that it misses the point. When one look at the equation right above (IV.5) in Ashtekar,Fairhurst&Willis gr-qc/0207106 one will see that the crucial factor exp(-\alpha^2/2) which is encluded to ensure at least some similarity with the ordinary quantization of the 1d nonrelativistic particle, is chosen completely arbitraryly. One could pick any other value and in particular use a vanishing exponent.

The latter would copy the _classical_ relations to the operator algebra and is in fact precisely what Thomas Thiemann is doing in the 'LQG-string'. There an operator representation of the _classical_ conformal group is constructed by fiat. The same is true for the diffeomorphism constraints of 3+1 dimensional gravity:

http://golem.ph.utexas.edu/string/archives/000299.html#c000559

Maybe it makes sense to technically call such an approach an 'alternative quantization' as for instance K.-H. Rehren argues in

http://golem.ph.utexas.edu/string/archives/000300.html#c000649
http://golem.ph.utexas.edu/string/archives/000300.html#c000674 .

Thomas Thiemann says that only experiment can show if maybe this
'alternative quantization' is the one followed by nature:

http://golem.ph.utexas.edu/string/archives/000299.html#c000588 .

Of course nobody really knows what might happen at the Planck scale. But one should be aware of the following points:

- LQG-like quantization in the above sense is not canonical in the usual sense

- applied to systems which we can test experimentally, like the 1d
nonrelativistic particle or the free electromagnetic field, it produces results drastically different from the standard ones.

- there seem to be no arguments why this modification of the quantum
principle should be the one used for quantum gravity excetp that 'only
experiment will show'

At least that's what I have learned from the discussion concerning the
'LQG-string'. Admittedly, I am a little surprised to find myself left with such a rather drastic conclusion, but it seems to be confirmed by many sources.
 
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  • #4
Marcus wrote, regarding finding messages at the String Coffee Table:
and possible to miss (I found it only on the second pass, scrolling

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
Originally posted by Urs

...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...to read the formulas at the Coffee Table from within MS-IE. You don't need to install Mozilla!

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.

Many thanks, also, for your valuable comments on Larsson's post.
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
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
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


Originally posted by marcus
Today Urs replied to Larsson's post on SPR by quoting a portion and
saying "Sorry, but this is not true"

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
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 [tex]C_I[/tex] and their quantization looks like
[tex]
\langle \psi \hat C_I |\phi\rangle = 0
\,.
[/tex]

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
Originally posted by Urs


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

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

https://www.physicsforums.com/showthread.php?s=&postid=137778#post137778

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 haven't found them yet. Its a long thread :wink:

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 don't 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.
 
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  • #11
Originally posted by Urs
...
The point is that whenever the diffeomorphism constraints in LQG are 'solved', the procedure is non-standard, because it does not follow Gupta-Bleuler quantization...

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 don't 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
Marcus -

see page 19 of the 'Amazing bid' thread:

https://www.physicsforums.com/showthread.phps=&threadid=13263&perpage=12&pagenumber=19 [Broken]

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! :-)
 
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  • #13
Originally posted by Urs
Marcus -

see page 19 of the 'Amazing bid' thread:

https://www.physicsforums.com/showthread.phps=&threadid=13263&perpage=12&pagenumber=19 [Broken]

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! :-)

Great!
this is something solid to chew on!
unfortunately I have to go out. but will be back later this morning.
thanks
 
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  • #14
Hello Urs,
So far I don't 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
to your objection.

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
Originally posted by marcus

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.

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.


Eric
 
  • #16
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
Originally posted by eforgy
Hi Marcus,

... The constraints are not even representable as operators on the Hilbert space of LQG.
...

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
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
Originally posted by 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.

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 [Broken]

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
 
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  • #20
Originally posted by Urs
Marcus -

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.

Originally posted by marcus
Hello Urs,

...I don't see how to reply...I cannot see how things
could be constructed in a different order...I'm not convinced that the way Rovelli does things now is faulty

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.

Originally posted by marcus
...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...the implementation of spatial diff invariance is in flux in LQG

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.
 
  • #21
Originally posted by eforgy
...

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.

...

Eric, I think Urs is misinformed in general about the field of LQG as a whole, since (in some developments) spatial diff invariance IS imposed by defining an operator on the hilbert space.

To take a classic paper as an example there is the 1995 paper of
Ashtekar, Lewandowski, Marolf, Mourao, Thiemann

"Quantization of diffeomorphism invariant theories of connections with local degrees of freedom"

http://arxiv.org./gr-qc/9504018 [Broken]

--------exerpt from page one----------
1. We will construct the quantum configuration space A/G and select the measure [mu] on it for which L2(A/G; d[mu]) can serve as the auxiliary Hilbert space Haux, i.e., can be used to incorporate the kinematical reality conditions of the classical phase space.

2. Introduce the diffeomorphism constraints as well-defined operators on Haux, and demonstrate that there are no anomalies in the quantum theory.

3. Construct a dense subspace &Phi; of Haux, with the required properties and obtain a complete set of solutions of the diffeomorphism constraints in its topological dual &Phi;'. We will also characterize the solutions in terms of generalized knots (i.e., diffeomorphism invariance classes of certain graphs) and obtain
the Hilbert spaces of physical states by introducing the inner products which ensure that real physical observables are represented by self-adjoint operators.
--------end of quote------

This is only my private opinion. However it seems that Urs and some of the others make very general statements about in LQG the diffeo constraint not being implemented by an "honest" operator but instead by some (less honest?) algebraic means.

And this blanket statement about the whole of LQG is based on their own idea of what they have heard from Thomas Thiemann!

And on top of that they say the quantization should not be called canonical. But this is not a normal use of language since everybody writes about LQG as a canonical quantization of General Relativity!
This is because it generally follows Dirac's plan of quantizing with constraints-----implemented (with some exceptions) by well-defined operators. But this seems to me to be a semantic argument about what shall we call genuine "canonical".

My private preference is to go along with what words the experts in the field use----with what seems like plenty of traditional justification---and not redefine words like "canonical", which I think is being done in some of our threads.

But I don't have any interest in arguing. So if they want to say that in LQG the diffeo constraint is never implemented with an operator in the usual way! or that LQG is not "really" canonical (which is its whole purpose to be a canonical quantiz. of GR) or that it does not "really" follow the basic program of Dirac in quantizing a cl. theory with constraints, this is OK with me :smile:

I just want to find out what the others think, and ordinarily not contradict or criticize (which rarely does any good)

also I find I like you and Urs both because you are mostly quite patient and unhostile---which I think means you want to find out things and not have fights (as I too)
 
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  • #22
Originally posted by 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! :-)

Hi Urs, I think you are right that Eric has summarized very well what you have been trying to say about LQG as a whole. You said earlier that you had written email to Ashtekar asking if LQG should perhaps not be called "canonical" because, if I remember right, of this nonstandardness. Did he reply by any chance?

I could be mistaken but I think you may have a misconception about the field as a whole. This in any case is mostly just a matter of words. There seem to be several different ways being tried for imposing diffeo contraints.

Originally posted by 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.

Thanks for the reply. You have more confidence in that damned Hamiltonian than I do:wink: I have sometimes suspected that it is a very dishonest operator.

You have gotten me looking at the different approaches to realizing diffeo invariance and it is fascinating.
Velhinho's recent paper briefly indicates the range of methods
currently being tried. If you are curious, page 6 of
http://arxiv.org/math-ph/0402060 [Broken]
 
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  • #23
Here's a brief exerpt from page 6 of Velhinho's paper to give the flavor:

----quote----
The above facts alone are sufficient to justify the absolutely central role of the H0 representation in...the canonical loop quantum gravity programme. The H0 representation is the kinematical representation used in loop quantum gravity; virtually all further developments are based upon it.

The seemingly unique status of the H0 representation was recently reinforced by a detailed analysis of the representation theory of the kinematical algebra [27, 28, 29, 30]. Although the uniqueness of the H0 representation was not established, it was shown [29, 30] that an a priori large class of representations, that also support a unitary implementation of the group of analytic diffeormophisms, contains in fact only reducible representations, and that every irreducible component is equivalent to the H0 representation.

Finally, for completeness, to avoid confusion and to give the reader an indication as to where loop quantum gravity is going, let us stress that the H0 representation is not, by far, the end of the quantization process.

Important as it is, H0 is the starting point for the hardest and most interesting part of the quantization, and this is precisely the reason why it is so important that H0, and therefore A-bar, are well defined and well understood.

Once the constraints are represented, one must, of course, solve them.
As already mentioned, the Gauss constraint is easily dealt with. It can be solved before or after solving the other constraints...
----end quote----

Velhinho is explicit that the quantization process he is describing is canonical---and follows the Dirac program of quantization with constraints---and that the constraints are imposed via operators defined on the hilbertspace.
 
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  • #24
I note that Velhinho does not describe the quantization itself, his interest lies in the algebraic structure of [tex]\bar{\mathcal{A}}[/tex].

In his preliminary discussion he exhibits two loose ends, the uniqueness of the kinematic space [tex]H_0[/tex] and the separability of [tex]H_{DIFF}[/tex]. In both case he reports (what we have seen from the Sahlmann and other papers) that partial and encouraging results have been achieved, but not rigor.

He does, however, characterize the quantization as canonical, in the tradition of Dirac, and I think this should weigh somewhat among all the assertions we are hearing that Thiemann's quantization of the string, and even LQG quantization, are not properly describable as canonical or Dirac quantizations. Evidently these names are used differently in different branches of research.
 
  • #25
Originally posted by selfAdjoint
I note that Velhinho does not describe the quantization itself, his interest lies in the algebraic structure of [tex]\bar{\mathcal{A}}[/tex].

In his preliminary discussion he exhibits two loose ends, the uniqueness of the kinematic space [tex]H_0[/tex] and the separability of [tex]H_{DIFF}[/tex]. In both case he reports (what we have seen from the Sahlmann and other papers) that partial and encouraging results have been achieved, but not rigor.

Hi sA,
on page 173 Rovelli says flatly that H-Diff is separable
because of the extended group he uses. Have wondered about this.
Tend to trust Rovelli but would like to see a proof. or at least a good explanation.

Interesting topic.

In fact Velhinho says more than just "suggestive". Look on page 19

"For instance it was shown in [37] that the inclusion of piecewise analytic transformations is sufficient to achieve
separabililty.
..."

That looks like he is saying not "partial and encouraging", but
"rigor".

He could be wrong of course. But we are talking about what Vel. says.
 
  • #26
Originally posted by selfAdjoint
I note that Velhinho does not describe the quantization itself...

It strikes me that for two pages---pages 4 and 5---he is doing nothing else but that.

It is condensed and he cites the prior work he is following like
"Quantization of diffeomorphism invariant theories of connections..."
by Ashtekar, Lewandowski, Marolf, Mourao, Thiemann (1995). But
he seems to touch the bases himself. True his main interest is in describing his own work and other recent developments, but he gives something of a thumbnail in the first 8 pages.

See from the top of page 4:
"Before we go into any details, let us fix the particular framework..."

to the top of page 6:
"...The H0 representation is the ...used in loop quantum gravity; virtually all further developments are based upon it."

Touches a lot of bases in those 2 pages, tho very condensed and perhaps could be said not to touch on enough to constitute a description of the quantization of the gravitational field.
 
  • #27
The conversation between Thomas Larsson and Urs continues at SPR
(do I hear a note that it is reaching conclusion?)

--------quote----

from <Urs.Schreiber@uni-essen.de>

"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb I am Newsbeitrag
news:24a23f36.0402250142.3074cde2@posting.google.com


> I think it should be possible to approximately understand LQG
> quantization of the string within the standard framework, even
> if one has to resort to formal manipulations with ill-defined
> operators. Just saying that Thiemann does something different
> is not very satisfactory.

Right. And we know exactly what it is that is different. Thomas Thiemann constructs a Hilbert space on which he defines operators U_\pm(phi) which by definition represent the diff x diff group without anomaly. The Virasoro generators are not represented on his Hilbert space, only these U operators are and they are _defined_ to produce the classical group without anomaly.

Such operators do exist, no problem, they just don't drop out of usual
quantization prescriptions.

The same is done in LQG for the spatial diffeomorphism constraints. There Operators U(phi) are defined which represent the spatial diffeo group on the space of spin network states.

Only the Hamiltonian constraint is really quantized itself and imposed as a Dirac constraints.

----end quote---

Urs, an interesting divergence is appearing between how you (and people who talk like you) use words as compared with how the main Loop Gravity people use words!
The latter group are explicit about following the Dirac program of quantization with constraints.

Since the early 1990s Loop Gravity people have defined their approach as "canonical quantization of General Relativity" and they have referred to the constraints (Gaussian, diffeomorphism, Hamiltonian)
as "Dirac constraints".

To me it seems kind of maybe ten years too late to tell them they should refine the definitions and the criteria so that one
may say their constraints are not "really" Dirac
and their quantization proceedure is not "really" canonical.

e.g. check out the 1995 paper by Ashtekar, Lewandowski, Mourao, Marolf, and Thiemann---hardly need to give link, everyone concerned must know it.

But this is, in part, what academic specialization accomplishes. It creates enclaves of language---split-off groups of specialists which can use words in special ways to mean special things, as they choose.

It is an interesting process to watch---just one of many fascinating changes going on at present!
 
  • #28
I think that specialized splitting of language is related to a comment a few posts back. It's not necessarily bad! One just has to be aware of the divisions in jargon and the potential for confusion.

Originally posted by selfAdjoint
...
He does, however, characterize the quantization as canonical, in the tradition of Dirac, and I think this should weigh somewhat among all the assertions we are hearing that Thiemann's quantization of the string, and even LQG quantization, are not properly describable as canonical or Dirac quantizations. Evidently these names are used differently in different branches of research.
 
  • #29
To me it seems kind of maybe ten years too late

It is indeed at least ten years too late that they started applying the LQG formalism to systems other than gravity to see what happens. If they had done this earlier the current embarrasment could have been avoided.

I have been told by several string theorists that the discussion of Thiemann's paper at the Coffee Table had been an eye-opener for them, because before that they believed that the LQG approach was an honest attempt at canonical gravity, maybe a too naive one. Now they realize that it is a very odd approach indeed.

The fact that the nature of the oddness of this approach is not emphasized in bold letters in the beginning of every LQG review is very unfrotunate. Did you notice that most of the formulas and steps in the LQG papers which we discussed and found problematic did not even have formula numbers? The very nonstandard assumptions were all mentioned by the way. This way non-LQG-expert necessarily miss them. For ten years.
 
  • #30
Originally posted by Urs
...the LQG approach was an honest attempt at canonical gravity, ...

Urs, I don't think you will get very far in clarifying this issue by phrasing it in terms of "honesty"
and, in effect, accusing other people of bad faith.
what you evidently have is a semantic issue where
two groups use some technical terms differently
 
  • #31
Originally posted by marcus
Urs, I don't think you will get very far in clarifying this issue by phrasing it in terms of "honesty"
and, in effect, accusing other people of bad faith.
what you evidently have is a semantic issue where
two groups use some technical terms differently

Forget about what either side means when they say "canonical". This is not an issue of semantics. We can even put aside the issue of scientific integrity. What matters is that LQG quantization is fundamentally different from standard canonical quantization, something which LQG researchers apparently confirm when asked directly. However, I do think the question of why no one outside of the LQG camp knew this is worth asking.
 
  • #32
The main problem is not that the terms are used differently. The problem is that one of the uses of this term refers to highly speculative physics.

And let me emphasize that by 'highly speculative' I mean something drastic. Of course every theory of quantum gravity in the absence of experiments has to be speculative. In string theory there is the single and obvious speculation that strings exist. Everything else follows. If they don't exist, they don't. Fine.

But in LQG the speculation is that at the Planck scale the quantum principle itself is radically different from everything we know so far. Maybe one can argue that the modified principle should still be called 'canonical'. Words are arbitrary. But it still refers to a concept drastically different from what is usually called canonical, outside the LQG-literature. You wouldn't claim that the LQG-like quantization of the 1d nonrelativistic particle in gr-qc/0207106 is 'canonical' would you? It's not canonical - it's weird!

I can say that with full confidence because if we know one thing for sure it is how the quantum theory of the 1d nonrel particle works. And it works very differently from the supposedly 'canonical' theory that is presented in gr-qc/0207106. Now from where comes the belief that applying this weird quantization to gravity gives something more reasonable?

If we just had a single hint that the quantum principle must be modified at the Planck scale. But do we have any? If the LQG-authors have such a hint then they at least have not published it. All that Thomas Thiemann said is that "experiment will show".

Right. Maybe Bohm trajectories are found at the Planck scale, or Jadczyks 'Event Enhanced Quantum Theory' or Smolins version of Nelson stochastics or nonunitary QM or whatnot. All this has been proposed. But all this is known and acknowledged to be highly speculative. Nobody would call a nonunitary version of QM a 'canonical' quantization. Unless, of course, he wants to risk to be misunderstood for over 10 years... :-)

But let us not get deeper in this kind of discussion. If there are any further technical issues to be discussed, in the vein of my discussion with Thomas Larsson on spr, then I am willing to participate. Otherwise there is little point in restating my assessment over and over again.
 
  • #33
Jeff -

thanks, yes, that's my point.

Oh, and apparently I must clarify my use of the word 'honest'. It was surely not supposed to question the personal or scientific integrity of anyone. I was using this in the same sense as in, for instance 'Momentum eigenstates are not honest states.' or 'x is not an honest operator for a particle on the circle'.

So this is why I said the Hamiltonian constraint is an 'honest' constraint in LQG, because it is represented as an operator as usual for quantized constraints. This is not true for the other constraints, so they are not really Dirac constraints.

This is all I meant. I apologize if this was unclear. Honestly.
 
  • #34
No problem.
 
  • #35
Originally posted by Urs
...Otherwise there is little point in restating my assessment over and over again.

I think you are probably right about there not being much point.
Perhaps I had better take a turn and try to state my assessment instead. Or some of the others.
I would like to determine what it is that people generally understand by canonical quantization and Dirac's program of quantizing a classical theory with constraints. There is probably some breadth of interpretation as to what is expected and what is meant. I reject the idea that Ashtekar, Lewandowski were being dishonest or obtuse when they said what they were doing in 1995 was a canonical quantization.

It is not obvious that semantic issues can be resolved democratically---by a simple headcount---but I rather suspect that you (and the string theorist you have talked to about this) may be in the minority. I can understand that you must be tired of reiterating your position so many times in so many forums. You should not feel as if you are obliged to continue repeating your assessment.
 
<h2>1. What is LQG-String theory?</h2><p>LQG-String theory is a theoretical framework that attempts to reconcile two major theories in physics - Loop Quantum Gravity (LQG) and String Theory. It combines the discrete spacetime structure of LQG with the concept of strings from String Theory to provide a more complete understanding of the universe at both the macroscopic and microscopic levels.</p><h2>2. How does LQG-String theory differ from other theories of quantum gravity?</h2><p>LQG-String theory differs from other theories of quantum gravity in its approach to understanding the fundamental nature of spacetime. While other theories, such as String Theory and General Relativity, view spacetime as continuous and smooth, LQG-String theory proposes a discrete and granular structure for spacetime. It also incorporates the concept of strings, which are one-dimensional objects, to explain the behavior of matter and energy at the smallest scales.</p><h2>3. What are the main challenges of LQG-String theory?</h2><p>One of the main challenges of LQG-String theory is the lack of experimental evidence to support its predictions. As a theoretical framework, it has not yet been tested or proven through experiments. Additionally, there are still many mathematical and conceptual challenges that need to be addressed in order to fully develop and understand the theory.</p><h2>4. How does LQG-String theory relate to the search for a theory of everything?</h2><p>LQG-String theory is often considered a candidate for a theory of everything, as it attempts to unify the principles of quantum mechanics and general relativity. However, it is not yet a complete theory and many scientists believe that a true theory of everything may require additional insights and developments.</p><h2>5. What are the potential implications of LQG-String theory?</h2><p>If LQG-String theory is proven to be a valid description of the universe, it could have significant implications for our understanding of the fundamental laws of nature. It could also provide insight into the behavior of matter and energy at the smallest scales, and potentially lead to new technologies and advancements in physics and engineering.</p>

1. What is LQG-String theory?

LQG-String theory is a theoretical framework that attempts to reconcile two major theories in physics - Loop Quantum Gravity (LQG) and String Theory. It combines the discrete spacetime structure of LQG with the concept of strings from String Theory to provide a more complete understanding of the universe at both the macroscopic and microscopic levels.

2. How does LQG-String theory differ from other theories of quantum gravity?

LQG-String theory differs from other theories of quantum gravity in its approach to understanding the fundamental nature of spacetime. While other theories, such as String Theory and General Relativity, view spacetime as continuous and smooth, LQG-String theory proposes a discrete and granular structure for spacetime. It also incorporates the concept of strings, which are one-dimensional objects, to explain the behavior of matter and energy at the smallest scales.

3. What are the main challenges of LQG-String theory?

One of the main challenges of LQG-String theory is the lack of experimental evidence to support its predictions. As a theoretical framework, it has not yet been tested or proven through experiments. Additionally, there are still many mathematical and conceptual challenges that need to be addressed in order to fully develop and understand the theory.

4. How does LQG-String theory relate to the search for a theory of everything?

LQG-String theory is often considered a candidate for a theory of everything, as it attempts to unify the principles of quantum mechanics and general relativity. However, it is not yet a complete theory and many scientists believe that a true theory of everything may require additional insights and developments.

5. What are the potential implications of LQG-String theory?

If LQG-String theory is proven to be a valid description of the universe, it could have significant implications for our understanding of the fundamental laws of nature. It could also provide insight into the behavior of matter and energy at the smallest scales, and potentially lead to new technologies and advancements in physics and engineering.

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