Amazing bid by Thiemann to absorb string theory into LQG

[eforgy]

Originally posted by Urs


For more details on the quantization of the KG particle and its relations to non-relativistic QM you might want to have a look at http://www-stud.uni-essen.de/~sb0264/TimeInQM.html

Nice! :)

I just read this and highly recommend it.

Eric

PS: Urs, this constraint business is something I haven't thought much about. It gives me a new interpretation of the subspace of paths for which \partial^2 = 0. This is like the "physical" space.

 

[ranyart]

Originally posted by lethe
i infer from this expression that you have an operator on your Hilbert space \hat x^0 that acts on states like multiplication by t. this is the straightforward application of the nonrelativistic quantization to the relativistic particle.

but something that i have learned from reading s.p.r is that there is no such operator. since learning that fact, it has been a big question mark in my mind as to whether there actually exists a theory that could really be called relativistic quantum mechanics of a particle.

i have wanted to understand what is going on with that for a while. since we were doing quantization of the relativistic particle, i thought i would toss in a question about that for you, but i can certainly appreciate that it is a bit off topic for the current discussion.

When one moves from a Three Dimensional Relativistic network, down into the Quantum Mechanical 'Hidden-Variable-Network', the equations themselves have to be able to transform one set of data (with observations) back-and-forth. This is where the problem lay, and you clearly have touched upon a deeper meaning in your post quoted above?

What 'single' type of formula's can trancend both QM and GR?..will there be a single formula that can both describe an event in GR and QM without changing the Mathematical formula?..the answer is no, both theories are by their very nature uncompatable, you cannot Unify events in 3+1 dimensions with events in 1+1 dimensional reduced fields.

The transformation has to occur with the Mathematical interpretations, for instance a 3+1 Network has collisions in space, dimensionally 'whole'?.. Particles move around and collide in the 3+1 network that allows them this freedom. In reduced QM Dimensional Networks, this cannot happen as there are no 3 Dimensional 'Whole-Particles' that can exist similtainiously in 3-D and 2-D!

If one uses a formula that traces a Particle wherever it goes (which is what einstein formulated in GR!), then there comes a point where not only does the formula cease to exist, the Particle itself has been removed from the 3+1 network, as the point in Spacetime(3+1)is reduced to a point in just Space..no TIME = no Observation = Hidden Vaiables = no Collisions = just space/fields = dimensional backgrounds = different (Special) formula's!

Now the bigger picture can be viewed by many 'theorists' into whatever formula's takes their fancy, for instance a simplyfied String Theorist would create 'extra' formulas to exist in 'extra' dimensions, all of which are technically sound, as they have no need of verifacation, and cannot-be verified by observations, the removal of 'SpaceTime' is a natural consequence of the removal of observations. The extra dimensions that some Mathematicians 'create' just simply do not exist.

From a dimensional perspective within GR and SR, one can go from 3+1 (4-D), to 2+1 (3-D) to 1+1 (2-D) TO A SINGULARITY NETWORK that is 0+1...1+0 . Of course the energies that are replacing Particles in 'Identity' terms as one reduces the Particles into Fields, also end up as Creationary Energies when one reaches the simplistic 1+1 AREA NETWORK! around a Blackhole, which happens to reside at the Core of every single Galaxy. Some would offer an einstein that Science needs a Dimensional perspective alteration to the Existence of our place within a Spacetime Galaxy (3-D+t), surrounded by Fields of QM Networks that is Electro-Magnetic-Vacuum Space (2+1) with no further need of Mathematical Extensions to Reality.

 

[lethe]

Originally posted by ranyart
When one moves from a Three Dimensional Relativistic network, down into the Quantum Mechanical 'Hidden-Variable-Network',

hidden variable network? wtf?

 

[Urs]

hidden variable network? wtf?

Maybe, maybe, maybe he is thinking of Smolin's latest attempt at merging Nelson's stochastic QM with quantum gravity

http://xxx.uni-augsburg.de/abs/gr-qc/0311059

where spin networks are indeed used as 'hidden variables' to produce QM dynamics from classical statistics.

Last time Lee Smolin tried the same with BFSS Matrix Theory

http://xxx.uni-augsburg.de/abs/hep-th/0201031 .

I used to consider this interesting,

http://groups.google.de/groups?selm=ahe52s$1a2q$1@rs04.hrz.uni-essen.de

though I am not so sure anymore.

Anyway, this is what ranyart's avant-garde poetry reminded me of. As with every piece of modern art, you have to search the answer within yourself. ;-)

 

[lethe]

Originally posted by Urs

As far as I remember the argument is that there cannot, because two operators satisfying a CCR as will act like multiplication/differentiation with respect to each other's eigenvalues and hence be unbounded from below and from above. But the spectrum of a decent Hamiltonian is supposed to be bounded from below (have a ground state), so it cannot satisfy any CCR.

right, this is what i had in mind.

But this argument doe not apply to systems which do not have an ordinary Hamiltonian. For instance the KG particle that we were discussing is governed by a constraint, not a Hamiltonian evolution. Here time is on par with the spatial dimenions.

ok, this is interesting

If you wish, you can regard the constraint of the KG particle as the Hamiltonian with respect to parameter evolution, where the parameter is an auxiliary variable along the worldline of the particle. This plays formally the role of time in non-relativistic QM and the above argument would show that there is not operator associated with the worldline parameter which has the CCR with the constraint.

i think i can see that now. thank you, that was very helpful for me.

Regarding your summary of the Stone-vonNeumann theorem I do not quite agree. I think the message is that there are many reps of the Weyl algebra and that if and only if these reps are weakly continuous does the Heisenberg algebra exist and then the Weyl rep is the exponentiation of the Heisenberg algebra.

LQG like approaches play with the possibility that even if the Heisenberg algebra does not have a rep still a rep of the Weyl algebra exists.

before i think about your point here, I am confused as to what you are referring to when you say "Weyl algebra". i am thinking it should be the set of operators you get after exponentiation, but do these things form an algebra? i expect them to form a group, but i wouldn't expect the sum of two of these guys to be another one of these guys.

in short, the operators in the Weyl relation are the Lie group corresponding to the Lie algebra spanned by the operators in the canonical commutation relations (the Heisenberg algebra)

 

[Urs]

Hi lethe -

in short, the operators in the Weyl relation are the Lie group corresponding to the Lie algebra spanned by the operators in the canonical commutation relations (the Heisenberg algebra)

Wait, we have to get out nomenclature in sync.

What I am calling a Weyl algebra are operators U(a),V(a) which satisfy
U(a)V(b) = \exp(i 2\pi ab/\hbar)V(b)U(a). These
need not come from exponentiating elements of a Heisenberg algebra. But the Stone-vonNeumann theorem tells us that iff U and V are weakly-continuous, then they do come from an exponentiated Heisenberg algebra. Otherwise they don't. If they are weakly continuous, then you are right that U and V give the Lie group of the Heisenberg algebra, namley the Heisenberg group.

LQG is based on throwing away the Heisenberg algebra and concentrating on reps of the Weyl algebra U and V which are not weakly continuous.

 

[ranyart]

Originally posted by Urs
Maybe, maybe, maybe he is thinking of Smolin's latest attempt at merging Nelson's stochastic QM with quantum gravity

http://xxx.uni-augsburg.de/abs/gr-qc/0311059

where spin networks are indeed used as 'hidden variables' to produce QM dynamics from classical statistics.

Last time Lee Smolin tried the same with BFSS Matrix Theory

http://xxx.uni-augsburg.de/abs/hep-th/0201031 .

I used to consider this interesting,

http://groups.google.de/groups?selm=ahe52s$1a2q$1@rs04.hrz.uni-essen.de

though I am not so sure anymore.

Anyway, this is what ranyart's avant-garde poetry reminded me of. As with every piece of modern art, you have to search the answer within yourself. ;-)

Ah!..with a little detective work I see what you mean (which is not a literal reference to me seeing into your mind!)

 

[arivero]

Hmm yes, "QM from QG".

 

[Urs]

QM from QG

Right, that's what the Smolin paper is called. Unfortunately the statement implied by the title is either a tautology or circular.

 

[arivero]

No, if the idea is that QG is a new theory above QM, so that any classical field or particle defined inside the QG theory will magically be a quantum field or particle. Still, the paper does not go as fas as his title, because Nelson stochasticity is imposed, not deduced.

 

[ranyart]

This paper has some relevence to this post.

http://arxiv.org/PS_cache/hep-th/pdf/0403/0403108.pdf

 

[arivero]

Originally posted by ranyart
This paper has some relevence to this post.

http://arxiv.org/PS_cache/hep-th/pdf/0403/0403108.pdf

But quantisation of the strings is a very different matter, isn't it? To begin with, the string is already by itself a many-particle entity, so first quantisation should be enough.

 

[pelastration]

Originally posted by arivero
... the string is already by itself a many-particle entity

many-particle ... can you explain more?

 

[selfAdjoint]

Thomas Larson's SPR post

Thomas Larson has today posted a possible way forward for LQG of sci.physics research, here . Recall that Urs had said LQG required a factor in the commutator that he called V to obtain the Virasoro algebra, and noted that LQG theorists set V = 1.

Now Larson points us to a paper on the math-ph arxiv which discusses a great many (all?) the possible candidates for V.

 

[marcus]

More about Thiemann's LQG-string
(the project to realize string theory within the context of LQG)

Robert C. Helling, Giuseppe Policastro
String quantization: Fock vs. LQG Representations
19 pages
http://arxiv.org/abs/hep-th/0409182

---abstract---
We set up a unified framework to compare the quantization of the bosonic string in two approaches: One proposed by Thiemann, based on methods of loop quantum gravity, and the other using the usual Fock space quantization. Both yield a diffeomorphism invariant quantum theory. We discuss why there is no central charge in Thiemann's approach but a discontinuity characteristic for the loop approach to diffeomorphism invariant theories. Then we show the (un)physical consequences of this discontinuity in the example of the harmonic oscillators such as an unbounded energy spectrum. On the other hand, in the continuous Fock representation, the unitary operators for the diffeomorphisms have to be constructed using the method of Gupta and Bleuler representing the diffeomorphism group up to a phase given by the usual central charge.
---end quote---

 

[marcus]

Robert Helling's papers go back to 1998, he has co-authored with
Hermann Nicolai, has been much of the time at Albert Einstein Institute,
Potsdam MPI, has specialized in M-theory (from the looks of it)

It looks like Hermann Nicolai, a director at AEI Potsdam, who organized last year's String meet Loop conference (with Abhay Ashtekar), has perhaps encouraged Thiemann to try this merger of theories in the first place

this was what the StringMeetLoop conference last October was supposedly to lay the groundwork for. Nicolai does String/M and particle theory and he co-organized it with Ashtekar who does Loop and is a relativist.

At the conference they wanted to get the HEP people---the particle physicists---talking to the relativists---the General Relativity people. Both being concerned with quantizing gravity in their respective fashions.

But after, when Thiemann took the first step the reception was not so hopeful or encouraging, as I thought. More like the bluejays in the front yard when the neighbor cat comes to visit. All kinds of reasons offered why it could not possibly be right.

This Robert Helling article has a different tone of voice

here is the original Thiemann article, in case you have not already seen it:

The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
Thomas Thiemann
46 pages
http://arxiv.org/hep-th/0401172

 

[selfAdjoint]

"...at least in the case of the quantum oscillator the polymer state is unphysical." That seems to be the death-knell fot the Thiemann approach. I wasn't aware of the fierce properties of the polymer state (no momenta), which as the authors say, is much used in LQG derivations. Back to the drawing board.

 

[marcus]

Helling's post without the LM comment

---quote from Robert Helling SPS post---
Thanks for noting our paper. Unfortunately, I am about to leave
Cambridge (my next postdoc is at IU Bremen, back in Germany) and all
my papers and notes are stored away in boxes and unaccesible to me at
the moment. So I cannot answer Urs' qustions about signs and
(anti)-commutativity. He might well be right and we screwed those up
but those would be just typos and wouldn't change anything substantial
in the conclusions. Furthermore, I don't have my laptop's network
connection currently running, thus I have to use google groups rather
than my regular news reader.

> Their description of the harmonic oscillator looks particularly strange
> because if we did not agree what is the physics of the quantum harmonic
> oscillator, we could probably agree about nothing in the world.[/color]

Mybe you missed that point but our philosophy was to say "this is what
you get when you apply LQG methods to the harmonic oscillator". Be
careful with them I am general because the _physical_ consequences (esp.
the spectrum) are not what you meassure in this easy example. It was
important to us not just to say that functions that jump have no place
in physics because you could never observe them. That would be too
easy (and in fact plain wrong: If you describe a D-brane (whose
physical existence I understand Lubos does not doubt) by a skyscraper
sheef then this is done exactly by a function (of the transverse
coordinates) that is zero everywhere except at one point where it has
a finite value).

[Moderator's note:...LM]

So the point of our discussion of the harmonic oscillator is that there
are measurable consequences of doingit 'the wrong way'.

[Moderator's note:... LM]

> I am sure that many of us tried to deal with divergent sums and divergent
> integrals, and various other singular objects in various ways that can be
> proved "wrong" on physical grounds.[/color]

Actually, this is a major reason for why this formalism is more
involved than the usual one: In the algebraic language (and this is
just mathemtically more careful language, no physical difference to
the usual approach) great care is taken to avoid divergent (and
similar) sums etc so no ambiguities (or ways to do it wrong) appear
from that. This is why careful people deal with the Weyl operators
e^{ix} and e^{ip} instead of the usual x and p: The Weyl operators are
bounded and thus problems with domains of definition etc do not arise.
For example, in the position representation p is the deriviative. But
not all functions in L^2(R) are differentiable. Only a dense subset is. But all are translateble. Thus one saves some complications (if
ones intend is to be careful) if one uses the better defined Weyl
operators instead. I am not saying that it cannot be done with x and
p, it's just you either close your eyes to mathematical subtleties
(which is what we physicists do most of the time and it works fine
most of the time) or you have to deal with limits and that stuff.

[Moderator's note: ... LM]

> All Hilbert spaces obtained from these wrong assumptions are
> non-separable, unphysical, and the only way how the non-separability can
> be cured is if the resulting theory is completely topological and all
> these values of "x" are eventually unphysical, perhaps except for their
> ordering.[/color]

The separability is not the issue.

[Moderator's note:... LM]

In fact, there are (accepted) physical systems with a non-separable
Hilbert space: One (as we remark in a footnote) are Bloch electrons.
That is electrons in a periodic potential.

[Moderator's note:...LM]

Then you know that the wave function is periodic as well. Ahem no, not
quite, only the physics is periodic. So the wave function is periodic up
to a phase. And by doing the intergral over the whole infinite crystal,
you find that two wave functions with different phases are orthogonal. So
for each point in the interval [0,2pi) of phases there is an orthogonal
sector in the Hilbert space. Thus the total Hilbert space is kind of the
L^2 of the unit cell to times the number of points in that interval, clearly a non-separable space.

[Moderator's note:... LM]

You could say that this happens only in the infinite crystal size
limit.

[Moderator's note:... LM]

But this idealization people usually are happy to make. Otherwise (with an
IR cut-off) there would for example be no phase transitions. But that is a
different matter.

> The states in this model represent unphysical mixtures of a
> hugely infinite number of superselection sectors - it is another
> description of Helling et al. comments about "discontinuity" of
> Thiemann's representation, I think. Each of these sectors is made of a
> single state. In physics, it is legitimate to study each single
> superselection sector separately - and if these superselection sectors are
> made of a single state, the theory is physically vacuous.[/color]

At least in the mathematical sense (and that is supposed to coincide
with the physical sense), a super-selection sector is a representation
of the quantum algebra. To states are in different sectors if they
are in inequivalent representations.

> > Everybody knows that first quantization is a mystery.[/color]
>
> "Why the world is quantum?" may be a mystery and the most counterintuitive
> insight about the world, but the mathematical operation behind the first
> quantization does not seem mysterious in any way, and it also does not
> seem ambiguous. Moreover, I don't know why you chose the "first
> quantization" because even it is a mystery, it is a smaller mystery than
> the "second quantization". ;-)[/color]

Was this just a joke? If not, here is why people say this (and usualy
this continues with "second quantization is a functor"): Of course if
your classical system has R^n as configuration space and its cotangent
bundle as the symplectic space then every child knows how to quntize:
Take L^2(R^n) as your Hilbert space and replace all x's by
multiplication operators and all p's by derivatives. Oh, and when
there's an ordering ambiguity, follow one or the other prescription
(but do that consistently).

However, what do you do, if I just give you some symplectic space and
don't tell you which are the simple preferred position and momentum
coordinates (and that's what x and p are, just coordinates). And as
the real world does not come with coordinates written on everything
one should have some recepy how to deal with this more general
situation. And then check that this reduces to the usual story (or an
equivalent one) in the simple situation. And this is what we have done
in the paper.

It is known, that there is no unique way to do this map from a
symplectic space to a Hilbert space with operators in general. There
are further choices involved.

[Moderator's note:... LM]

> I did not quite like their comments describing the algebras with different
> central charges as different "representations" (with an exclamation mark).[/color]

When we say algebra, we mean the C*-algebra of the observables. And
those are indeed the same (and only the represenations differ).

[Moderator's note:... LM]

These algebras are the algebras of the X's (or the W(f) after some
massaging). Then this Weyl algebra has representations. And on those
representations there is a symmetry (Lie-)algebra acting by unitary
operators. And this symmetry algebra is some Virasoro alegbra in both
cases. But these symmetry algebras have different central charges in
the two representations of the Weyl algebra. So: There are two kinds
of algebras, don't confuse them.

Furthermore, even if we didn't talk about it, in the Virasoro algebra
the central charge is just an abstract element usally called c. It
commutes with everything, so in an irreducible representation it is
represented by a number. And again, this number depends on the
representation. This number, together with h, the eigenvalue of L_0 in that representation, label a highest weight representation of the
algebra.

[Moderator's note:... LM]

> The only way how can one understand this sentence is that they claim that
> the Virasoro algebras with different central charges are isomorphic to
> each other.[/color]

Nobody claimed that. As I just said: In the algbra, c is an abstract
element, it becomes a number only in a representation. And nobody
claims that representations with different c are equivalent.

[Moderator's note:... LM]

> There is a common theme in Thiemann's papers which, I'm afraid, may
> unfortunately be shared by Helling and Policastro - which is that they
> often look at the "classical limit" of an algebra, and treat all the
> modifications implied by quantum mechanics as unimportant - and perhaps
> annoying? - details whose relevance for any of their conclusions goes to
> zero.[/color]

Could you be more specific with this claim?

[Moderator's note:... LM]

> By the way, this purely quantum viewpoint will be even more important if
> we want to get more insight into the (2,0) theory or even M-theory at the
> generic point of the moduli space - because these theories (at least in
> some backgrounds) clearly indicate that they cannot be fully obtained from
> a classical theory by quantization - and they almost certainly cannot be
> obtained from a *unique* classical theory.[/color]

Quantization is a game that always involves a classical system.
However nobody claimed that every qunatum system arises from the
quantization of a classical theory.

[Moderator's note:... LM]
-----end quote---

 

[marcus]

Part 2 of Helling's post

there was so much mod comment inserted into Helling's text that it was hard to see the overall intent of his post, I have elided the mod comment,
as in the preceding, to get a sense of the original.

---quote from Helling's SPS post---
> There are other examples in which Thiemann et al. try to make this sort of
> "quantization without quantization". They want the commutators to be
> always equal to the Poisson brackets;[/color]

I hope you don't include us in "et al". We impose the Poisson goes to
commutator rule only for linear combinations of what would be x and p,
not for higher powers. And I doubt Thomas would do commit that crime
either.

[Moderator's note:... LM]


> One may be trying to obtain a completely different framework of
> "quantization" - but there are several but's.[/color]

We tried hard to spell out the general framework of the quantization
procedure used in the two approaches. We say, that you can include the
polymer quantization if you do not impose the at first rather
technical condition of weak continuity. But then this has huge
observable consequences. So don't confuse framework and consequences.

[Moderator's note:... LM]

Could you spell out the rules for your framework that clearly rules
out the LQG one? It should be a machine that turns a symplectic space
with its observables into a Hilbertspace with its operators.

[Moderator's note:... LM]

> First of all, this procedure
> is not really quantization because it tries to preserve those properties
> of the classical theory that *cannot* hold in what is normally called a
> "quantum theory" - such as the exact equality between the commutators and
> the Poisson brackets.[/color]

Nobody imposes that. We only ask for a unitary representation of the
diffeomorphism symmetry. And those might obey the group law of the
diffeo group or not (because of an anomaly).

[Moderator's note:...LM]

> Second of all, it is not physics because no one has
> certainly seen a Thiemannian harmonic oscillator[/color]

Right. That we meant by "(un)physical" in the abstract.

- and no one ever will,
> simply because non-separable Hilbert spaces cannot be "seen".[/color]

See above.

> I find it mildly entertaining that the normal procedures of quantization -
> including quantization of the harmonic oscillator - are themselves
> pictured as an alternative approach.[/color]

Where?

[Moderator's note:... LM]

We describe both quantizations in a single framework. There is
one choice to be made. And that has physical consequences. In the
mechanics example, those consequnces are unphysical, so the choice was
wrong. Everybody is free to deduce something about the choice in the
string case.

> Well, of course that we do not need
> nonsensical non-separable spaces to describe the harmonic oscillator. Not
> only that: non-separable spaces do *not* describe the harmonic oscillator
> and they never did. Moreover, the standard procedure has been known since
> the mid 1920s, and it is the only one that can give physical predictions
> that reduce to the classical oscillator in the appropriate limit. It's
> great to rediscover this cool method of quantization in 2004, but it
> should not be viewed as something new.[/color]

We didn't say there is anything wrong with the standard harmonic
oscillator. Rather we used it as a test bed for the quantization
procedure. This was to counter arguments along the lines of 'nobody
has yet seen a string in nature".

> That's nice to hear because Robert Helling was the person who patiently
> required (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that
>
> > RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant
> > as otherwise diffeomorphisms would be spontaneously broken.[/color][/color]

Let's face it: In that thread you didn't get it that I was playing the
devil's advocate.

[Moderator's note: Well, sometimes I am confused who is your devil and who
is your God, or whoever is your devil's "alternative". ;-)

> > it is discussed that the singular GNS state can be interpreted as a thermal
> > state of infinite temperature![/color][/color]

One should be carefule as this is world sheet temperature and not
target space.

> > I didn't know this before and like that insight, because it points at a way
> > to understand a larger framework in which various "different" quantizations
> > (of the string for instance) appear as different aspects of the same thing.[/color]
>
> I am not getting the purpose of these attempts. Is the goal to be nice and
> to prove that no one can ever be completely silly?[/color]

If you like to express it that way...

Sorry, right now, I do not have more time to reply to the more polemic
parts of your post.

Robert
---end quote---

 

[marcus]

"...at least in the case of the quantum oscillator the polymer state is unphysical." That seems to be the death-knell fot the Thiemann approach. I wasn't aware of the fierce properties of the polymer state (no momenta), which as the authors say, is much used in LQG derivations. Back to the drawing board.

Your take on this was shared by Thomas Larsson today
https://www.physicsforums.com/showthread.php?t=44495
He says the way (according to Helling/Policastro) that LQG handles
the harmonic oscillator is fatal for LQG.

In effect, he tells the Loop Gravitists to go "back to the drawing board".

the first thing I notice is I am pleased with our reflexes
I posted notice of H/P on 19 September
then it appeared (I think the next day) on SPS
and now discussion has started (24 September) on SPR

You were sadly shaking your head 5 days before Thomas Larsson.

By now I am accustomed to surprises so I am waiting to see how this turns out and cannot really give a reaction.

I recall that Rovelli (and Daniele Colosi) had a paper last year about the Harmonic Oscillator. Their paper was called:
A simple background-independent hamiltonian quantum model
"...Our main tool is the kernel of the projector on the solutions of Wheeler-de Witt equation, which we analyze in detail..."
http://arxiv.org/abs/gr-qc/0306059

this is not to say that their paper has any bearing on the fatal harmonic oscillator disease discovered by Helling/Policastro (wouldnt seem to but I suppose it might)

 

[selfAdjoint]

Well, the Colosi-Rovelli paper doesn't contain the word polymer at least!