Prospects of the canonical formalism in loop quantum gravity

[marcus]

That last one does not interest me so much, at least at the moment. It is very much at the toy model stage.

I hope you will glance at the reference to a Wen problem here, and at the next slide where Bianchi proposes a dual formulation of LQG which does not use spin networks and SF.
I think you are already familiar with this, but let's refresh.
http://pirsa.org/11090125

I am talking about slides 23/24 and 24/24. The penult and last slides of PIRSA 11090125.
In the PDF, so you can go directly to them without watching the video, they are on pages
46/48 and 48/48 of the PDF.

==quote Bianchi's last slide [slightly elucidated :)]==
Summary: Loop Gravity [as the Dynamics of] Topological Defects

* Dual formulation of Loop Gravity:
not in terms of Spin Networks and Spin Foams
[but instead as] local Quantum Field Theory with topological defects

* Derivation of the Loop Gravity functional measure via QFT methods

* New light on the main technical assumptions of Loop gravity
the microscopic d.o.f. of classical and quantum Loop Gravity are
gravitational connections A with distributional magnetic field on defects
==endquote==

See also the earlier slide 6/24 or PDF page 12/48
where he says "Canonical Quantization as above + require also:
[a flatness constraint on the connection in the bulk of the 3-manifold]"

In other words he says that the Canonical Q of HIS version of LQG can be just like the Canonical Q of the OLD version of LQG, if you please, except that his 3-manifold is shot thru with a web of hairline fractures and the connection is required to be trivial except (distributionally) on the defects.

Laurent Schwartz distributions. Takes me back to 1960s grad school days. Happy. Bianchi is a talented mathematician as well as a smart creative physicist. I guess he is postdoc at Perimeter now and might team up with Freidel on some work.

 

[atyy]

==quote Bianchi's last slide==
Summary: Loop Gravity [as the Dynamics of] Topological Defects

* Dual formulation of Loop Gravity:
not in terms of Spin Networks and Spin Foams
[but instead as] local Quantum Field Theory with topological defects

* Derivation of the Loop Gravity functional measure via QFT methods

* New light on the main technical assumptions of Loop gravity
the microscopic d.o.f. of classical and quantum Loop Gravity are
gravitational connections A with distributional magnetic field on defects
==endquote==

What does dual mean here? Does it mean unitarily equivalent? Or is it one of the potentially many inequivalent theories that tom.stoer (and others) bring up?

 

[marcus]

What does dual mean here? Does it mean unitarily equivalent? Or is it one of the potentially many inequivalent theories that tom.stoer (and others) bring up?

I don't know how Eugenio would paraphrase. To me it just means another approach.

Hopefully as different as possible!

Particle theory has become highly ritualized, or so it seems to me.

 

[atyy]

I don't know how Eugenio would paraphrase. To me it just means another approach.

Hopefully as different as possible!

Particle theory has become highly ritualized, or so it seems to me.

Seems to be dual in the sense of unitarily equivalent - at least from Rovellli's commentary on it in the Zakopane lectures http://arxiv.org/abs/1102.3660, p7?

I remember being quite excited about Rovelli's attempt to frame the new spin foams as TQFTs, but remember later thinking it unnatural. Of course I may be wrong. Witten would pick up any successful push in this direction quickly - it's his childhood dream

Edit 1: Link corrected - thanks, marcus.

Edit 2: Oh yes, an indication I'm wrong is that FGZ actually took the time to write a paper about it AND the link between spin foams and canonical LQG. I have no understanding how FGZ fits into the big picture at the moment. I didn't even know the problem the were trying to solve existed!

 

[marcus]

Seems to be dual in the sense of unitarily equivalent - at least from Rovellli's commentary on it in the Zakopane lectures http://arxiv.org/abs/1102.3660, p7?

I remember being quite excited about Rovelli's attempt to frame the new spin foams as TQFTs, but remember later thinking it unnatural. Of course I may be wrong. Witten would pick up any successful push in this direction quickly - it's his childhood dream

I remember some discussion of the "defects" formulation of LQG in the Zakopane lectures
I will go review page 7 of 1102.3660.

===============================

Yes, what you point to on page 7 is such a concise account of the Bianchi "topological defects" presentation of LQG that I'm inclined just to quote the whole thing:

==Zako 1102.3660 page 7==
There is another very interesting way of interpreting the Hilbert space H_Γ, pointed out by Eugenio Bianchi [40]. Consider a Regge geometry in three (euclidean) dimensions. That is, consider a triangulation (or, more in general, a cellular decomposition) of a 3d manifold M, where every cell is flat and curvature, determined by the deficit angles, is concentrated on the bones. Let ∆₁ be one-skeleton of the cellular decomposition, namely the union of all the bones.
Notice that the spin connection of the Regge metric is flat everywhere except on ∆₁. Consider the space M^∗ = M − ∆₁ obtained removing all the bones from M. Let A be the moduli space of the flat connections on M^∗ modulo gauge trasformations.
A moment of reflection will convince the reader that this is precisely the configuration space [SU(2)^L/SU(2)^N] considered above, determined by the graph Γ which is dual to the cellular decomposition. This is the graph obtained by representing each cell by a node and connecting any two nodes by a link if the corresponding cells are adjacent. It is the graph capturing the fundamental group of M^∗.
Therefore the Hilbert space H_Γ is naturally a quantization of a 3d Regge geometry. Since Regge geometries can approximate Riemanian geometries arbitrarily well, this can be seen as a way to capture quantum states of 3d geometries.
The precise relation between these variables and geometry becomes more clear in light of the Ashtekar formulation of GR. Ashtekar has shown that GR can be formulated using the kinematics of an SU(2) YM theory. The canonical variable is an SU(2) connection and the corresponding conjugate momentum is the triad field. Accordingly, we might expect that the quantum derivative operators on the wave functions on H_Γ represent the triad, namely metric information. We’ll see below that this in indeed the case.
A word of caveat: in the Ashtekar formalism, the SU(2) connection is not the spin connection Γ of the triad: it is a linear combination of Γ and the extrinsic curvature. Therefore the momentum conjugate the connection will code information about the metric, while the information about the conjugate variable, namely the extrinsic curvature, is included in the connection itself, or, in the discretization, in the group elements h_l.
==endquote==

 

[marcus]

Edit 2: Oh yes, an indication I'm wrong is that FGZ actually took the time to write a paper about it AND the link between spin foams and canonical LQG. I have no understanding how FGZ fits into the big picture at the moment. I didn't even know the problem the were trying to solve existed!

I am just as in the dark as you or, I suspect, more so. I am very happy to be in a state of suspenseful incomprehension in this case. Something to look forward to: a future ahah.

I also suspect that the FGZ paper has in (perhaps small) part an unspoken cultural or tribal purpose. In my eyes it legitimizes the Aharonov-Bohm version of LQG. It makes it official that this has entered into the worthy brotherhood of versions of LQG.

An advisor can now, if he or she so desires, suggest thesis problems to grad students that they may investigate various aspects of the Aharo-Bo LQG, it is on the "interesting problems" board. If I did not realize it before, I am now awake to the respectability of this version of Loop.

I do not know that it is "unitarily equivalent" or even that it should be. From a genepool evolutionary standpoint it might be better for everybody if theories were slightly different, to increase the chances of success.

 

[atyy]

I am just as in the dark as you or, I suspect, more so. I am very happy to be in a state of suspenseful incomprehension in this case. Something to look forward to: a future ahah.

I also suspect that the FGZ paper has in (perhaps small) part an unspoken cultural or tribal purpose. In my eyes it legitimizes the Aharonov-Bohm version of LQG. It makes it official that this has entered into the worthy brotherhood of versions of LQG.

An advisor can now, if he or she so desires, suggest thesis problems to grad students that they may investigate various aspects of the Aharo-Bo LQG, it is on the "interesting problems" board. If I did not realize it before, I am now awake to the respectability of this version of Loop.

I do not know that it is "unitarily equivalent" or even that it should be. From a genepool evolutionary standpoint it might be better for everybody if theories were slightly different, to increase the chances of success.

I hope the future ahah is as good as the past one :)

My understanding of unitarily inequivalent is clearest in the case of Asymptotic Safety. In that approach, we assume "classical gravity" is ok as a quantum theory, just that one needs to look for a non-trivial fixed point. But which classical variables does one use? If the fixed point exists, it seems that the metric variables and the Holst variables give different fixed points http://arxiv.org/abs/1012.4280, so the quantum versions of the theories are not the same, even though their classical limits are.

 

[tom.stoer]

The simplest model of non-equivalent quantizations is the free particle on a circle. The basis is

$$\psi_{n,\beta}(x) = e^{i(n+\beta)x}$$

The boundary condition reads

$$\psi_{n,\beta} (x+2\pi)= e^{2 \pi i \beta} \psi_{n,\beta}(x)$$

which shows that beta is some kind of winding (for irrational beta this interpretation becomes difficult). You can move beta from the wave functions to the momentum operator; beta affects the spectrum of certain operators; beta cannot be removed by any unitary transformation.

The operator 'x' does not exist on this space as it violates the boundary condition already for the simplest wave function

$$\psi_{0,0} = 1$$

$$x \psi_{0,0} = x$$

$$x \psi_{0,0}|_{x=2\pi} = 2\pi \neq 0$$

Therefore one has to use

$$U = e^{ix}$$

and has to quantize this U, using the Poisson brackets for {U,p}.

So I think there are some simple examples where these inequivalent quantizations show up; there is no preferred choice for beta . There is not even a reasonable explanation why beta should 'scale'.

I think we have to live with the fact that quantization classical theories, e.g. GR may result in inequivalent quantum theories. Then there are two possibilities:
a) different quantum theories reproduce GR in the IR
b) different quantum theories lead to diferent classical theories in the IR
In order to understand that and in order to rule out b) we have to take care about the quantization.

$$\text{GR} \quad \stackrel{\text{quantization}}{\Longrightarrow} \quad \text{QG 1, 2, ...} \quad \quad \stackrel{\text{classical limit}}{\Longrightarrow} \quad \text{GR, ...}$$

As far as I understand Rovelli he picks one 'QG x' - but I am sure that in the current state of (L)QG we still have to pay more attention to the first arrow instead of starting with one specific model. I do do say that his model 'QG x' is not reasonable, that it cannot be motivated, that it is not correct physically. But limiting the focus to exactly this model would be the wrong way to proceed.

 

[marcus]

I suppose that the 3 of us who recently posted would have different views of the situation. Maybe it would help if I just indicate my perspective.

I think that we do not know if NOW is the right time to invent the canonical formalism for LQG. It could be! It could also be the right moment to invent the quantization problem that LQG solves, the continuous phase space from which one quantizes.

We have not agreed on a standard classical config or phase space for LQG, so the discussion of equiv or inequivalent quantizations is a bit abstract and academic.

What the FGZ paper is doing is primarily to make precise what is the right picture of the classical phase BEFORE one quantizes.

What I like very much about Freidel's paper is that it disentangles two operations which are truly different and should not be confused! It separates "discretize" from "quantize".
It discretizes classical GR first, before anything quantic happens. This gives a proposed LQG phase space. I think it is right. Better than Regge. Regge limitations show up.

And yet the Freidel et al title is "Continuous formulation of the LQG phase space". That is how close it is to continuous GR. It is quasi continuous blending into discrete. Right on the "cusp". And they keep the map, so that they can go back from classic discrete back to classic continuous.

I am speaking impressionistically and carelessly about very careful math work. So be it.

 

[marcus]

Anyway, the basic situation in Loop now is that one has a definite quantum theory: Zako LQG. It may not be so forever but it is prevailing for the time being.

And one of the things we would like to do is "reverse-quantize" it. Like "reverse-engineering" one might do with a new consumer product out of the box. Zako LQG did not appear as the result of quantizing anything, it just appeared.

Now one would like to go back and figure out what it could be the quantization of.

That will undoubtably be very instructive and will lead to new discoveries!

It will almost surely lead to a Hamiltonian because everybody wants one very badly .
So far the Alesci-Rovelli paper is a first step because it incorporates 4-valent and the 1-4 Pachner. It is a recent hopeful sign. But I would guess that the researchers will first listen to FGZ and other papers like that decide what is the phase space, what is the thing to be quantized, and then they will work out a Hamiltonian from that which involves the Alesci-Rovelli idea. And the eventual canonical theory will be compatible with Zako LQG or whatever path integral formalism it has changed into.

It is a fantasy that the researchers have agreed on something to quantize. So talking about "inequivalent quantizations" is irrelevant.

Zako LQG is very tight. What other alternative has been constructed that is sufficiently like it so one could make a meaningful comparison.

When you tell me by what quantization procedure it could have arisen, from what phase space, then we can see what ambiguities and inequivalent variants we can find. That will be fun! and instructive! But one is not yet at that point. First one must reverse-quantize. As I said, that is where the paper of Freidel Geiller Ziprick comes in.

 

[tom.stoer]

I suppose that the 3 of us who recently posted would have different views of the situation.

At least you (marcus) and me :-)

I think that we do not know if NOW is the right time to invent the canonical formalism for LQG.

As early as possible b/c it shows most clearly if something goes wrong; it does not allow for
any sleight of hand.

We have not agreed on a standard classical config or phase space for LQG, ...

Yes, there are two ambiguities; first the different classical phase spaces (as Alexandrov shows there is a two-parameter family of connections); second the quantization ambiguites itself which again fall into different categories, namely ordering ambiguities and genuine inequivalent quantizations as my S^toy model demonstrates.

This gives a proposed LQG phase space.

One of many, unfortunately ... and each individual choice can lead to different inequivalent quatizations.

 

[marcus]

I've written some stuff responding to some of these issues right before your post, one the previous page. Also Finbar made a related observation in the "What's happening with Loop?" thread, and I just amplified on what he said.

One way to say the essential point is that before we talk about different quantizations we need to establish what is Loop Classical Gravity.

That is a classical form of General Relativity with finite degrees of freedom. (finite d.o.f. so it can be quantized.)

I'm not sure you have a well-defined LCG in mind, as a starting point for quantization. Would you like to describe what LCG is, as you see it? And since it is a matter of consensus what we call LCG, what assurance have we that the Loop community (Marseille, Perimeter, Penn State...) would accept it? They might or might not. I don't know what classical d.o.f. you have in mind.

In case anyone else is interested I will get the links to my previous posts, and Finbar's.

Here's what I said (#80) about "reverse-quantizing" by analogy with "reverse engineering"
https://www.physicsforums.com/showthread.php?p=3588751#post3588751

Here's Finbar's remark, which makes an important point about disentangling the two logically separate processes of discretizing and quantizing
https://www.physicsforums.com/showthread.php?p=3588133#post3588133

Another post of mine (#79) a fragment of which you quoted:
https://www.physicsforums.com/showthread.php?p=3588716#post3588716

 

[tom.stoer]

One way to say the essential point is that before we talk about different quantizations we need to establish what is Loop Classical Gravity.

Of course you are right, but look at this

 

[marcus]

Of course you are right,

Thanks. We both see the need for the Loop community to agree on what is going to be its Loop Classical Gravity

but look at this

However I notice your picture leaves professional consensus out. It is an important element. After all, the picture is not about Nature or about some God-given mathematical absolute. GR is at bottom a human artifact and serves here as heuristic. We do not know that it is right, or how it will be changed as it metamorphoses into a quantum theory.

There are no formal rules to discovering a theory of nature. It is a community function---the self-selecting professional community guides the process by argument and consensus.

So although GR is extremely important, ultimately the community which we call LQG will decide what is the agreed-on classical GR formulation with finite d.o.f. the LCG.

Freidel has made his bid to define it. As I read and reread the FGZ paper, I become persuaded that this LCG will play in Penn State, Perimeter, and Marseille. I confess to being very excited by this and it may be affecting my judgment. I think that a tipping point has been reached.