Current status of canonical LQG and SF models

[tom.stoer]

From time to time I try to get an overview regarding the current status of LQG. Unfortunately there seem to be a lot of interesting ideas and new proposals, but I am missing a summary paper which addresses and summarizes both main achievements and main open issues.

What about status of discretization, SF vertex and measure, consistent implementations of constraints including (second class) simplicity constraints, Holst vs. Nie-Yan action, Immirzi parameter, different gauges and different spin networks, Hamiltonian constraint, relation between canonical and spin foam models, finiteness and renormalization, quantization based on "dust reference frames", semiclassical limit, ...

Is there a recent review paper addressing these topics?

 

[marcus]

I would guess that there is not a current review paper that covers all these topics. I will list a few recent review papers, that cover PARTS of the picture.

As I think you know, Springer Press is bringing out a Handbook of Spacetime edited by Abhay Ashtekar. This will have a number of chapters on QG including Loop and Spin Foam QG, but not restricted to that!

http://arxiv.org/abs/1303.4636
Spin foams
Jonathan Engle
(Submitted on 19 Mar 2013)
The spin foam framework provides a way to define the dynamics of canonical loop quantum gravity in a spacetime covariant way, by using a path integral over histories of quantum states which can be interpreted as `quantum space-times'. This chapter provides a basic introduction to spin foams aimed principally at beginning graduate students and, where possible, at broader audiences.
Comments: 32 pages, 14 figures, 2 tables, to appear as a chapter of "The Springer Handbook of Spacetime," edited by A. Ashtekar and V. Petkov (Springer-Verlag, at Press)

http://arxiv.org/abs/1302.3833
Loop Quantum Cosmology
Ivan Agullo, Alejandro Corichi
(Submitted on 15 Feb 2013)
This Chapter provides an up to date, pedagogical review of some of the most relevant advances in loop quantum cosmology. We review the quantization of homogeneous cosmological models, their singularity resolution and the formulation of effective equations that incorporate the main quantum corrections to the dynamics. We also summarize the theory of quantized metric perturbations propagating in those quantum backgrounds. Finally, we describe how this framework can be applied to obtain a self-consistent extension of the inflationary scenario to incorporate quantum aspects of gravity, and to explore possible phenomenological consequences.
Comments: To appear as a Chapter of "The Springer Handbook of Spacetime," edited by A. Ashtekar and V. Petkov. (Springer-Verlag, at Press). 52 pages, 5 figures

 

[atyy]

Engle's take on the "extraneous" terms used to be different from Rovelli's. I don't know whether Rovelli has come round to Engle's view.

Here's Engle's original paper in which he described how his view differed from others in the literature at that time. http://arxiv.org/abs/1107.0709

I'd be interested to know whether his view is now representative in the field.

 

[marcus]

From time to time I try to get an overview regarding the current status of LQG...

Given that (as I think) we do not have an up-to-date summary review paper at the moment, there is one other way to get an idea of the situation, and that is to listen to Wolfgang Wieland's talk tomorrow.

Tuesday, Apr. 9th
Wolfgang Wieland, CPT Marseille
Title: Hamiltonian spinfoam gravity

The reason is that Wolfgang comes from Vienna and he is seriously interested in addressing the basic troubles and inconsistencies with the full LQG/SF theory. Also he happens to be getting his PhD at Marseille this year. Also he has talent and sense of humor, in my opinion. Which are important qualities. It should be a pretty good talk and a way of getting a feel for where things stand. I will be listening for who asks questions at the seminar tomorrow. Maybe Matteo Smerlak will speak up, someone I am always curious to hear what he has to say.

You know my attitude about the program---which is different from yours. I think it makes just as good sense to look at the quantum cosmology "application" even if the main theory is not in final shape. This I think is just as reasonable, in my view, as doing what Wolfgang is doing which amounts to getting under the car and rebuilding the transmission instead of taking a drive in the countryside and visiting the start of cosmic distance growth.

 

[tom.stoer]

thanks marcus, I was thinking about Wolfgang's recent papers as well.

of course it makes sense to consider physical applications; the only problem is that in quantum gravity we cannot assess the validity of theories on the basis of experimental tests, but we must place greater value on consistency; think about particle physics where the W- and Z-boson mass are out of reach for colliders: the only way to rule out the Fermi-theory for weak interactions would be mathematical inconsistency

thanks anyway; do you have other hints regarding summary papers? what about the Erlangen group and their program regarding "dust fields"?

 

[marcus]

...I was thinking about Wolfgang's recent papers as well.
...

a propos Wolfgang's recent work, the PDF of the talk he gives today is now online:
http://relativity.phys.lsu.edu/ilqgs/wieland040913.pdf

It is not the status "summary" that we would like to have (I don't know of a summary of where the dust approach is at present). But today's talk is about a central issue so it does provide perspective on the current state of theory.

==quote Wieland's slides==
Conceptual unity of loop quantum gravity?
What is the relation between canonical loop quantum gravity and spinfoam gravity?

Both approaches share their kinematical structure, the Hilbertspace with operators representing area, angles and parallel transport.
Does this relation extend beyond kinematics? Are these two approaches just different views of the same underlying quantum theory?

Within the reduced setting of a fixed discretisation of space-time, this talk will answer this question in the affirmative. Canonical quantisation techniques can be used to recover the spinfoam transition amplitudes.
==endquote==

The talk starts in five minutes from now, I believe. When audio is eventually posted online, and we can listen, it will be interesting to hear what questions are asked by the audience.

 

[marcus]

The talk will now have been given but on past experience it could take a day, or several days, before the audio is online at http://relativity.phys.lsu.edu/ilqgs/
The talk has 3 parts:
==quote Wieland's slides==
Three points:
1 Spinors as covariant variables for loop quantum gravity
2 Partial continuum limit and Hamiltonian formulation
3 Quantisation
We will use the spinorial framework of the theory. This is useful for us because it embeds the non-linear loop gravity phase space into a large phase space with canonical Darboux coordinates.
==endquote==

I have to say that in terms of clear organized communication, and careful logic, this is one of the highest quality sets of ILQGS slides I have seen. It is very comprehensible just looking at the slides. Not being an expert myself, I appreciate the pedagogical value.

btw at the end, right before "conclusions" he raises a question: something he is not sure about.
Tom, you may want to check that out.

Part 1 is kinematic, and reviews some of the same material that was covered in an earlier ILQGS talk by Simone Speziale.
Part 2 develops a Hamiltonian approach to the dynamics.

Whoa! The audio is already online!
http://relativity.phys.lsu.edu/ilqgs/wieland040913.wav

 

[torsten]

Thanks marcus for the link of the slides. The talk must be very inetresting.
But I have one conceptional problem: quantum gravity was seen as a theory which requires topological transitions of the space. Even the first approach ADM and Wheelers geometrodynamics implied it (in particular the spacetime foam). But fixed time slices like $\Sigma\times\mathbb{R}$ are contradict this picture. Where does a topological transition come from? Or does the opinion changed?

 

[tom.stoer]

I do not see why there should be a topological transition

But I agree that it's strange to fix the topology just to let the math work ...

 

[torsten]

The functinal integral approach includes all field configurations and integrate over it. But GR does not depend on topology. Therefore one has to include also all field configurations over all possible topologies. For instance: a homogenous metric on the 3-sphere (cross time) and a homogenous metric on the Poincare sphere (cross time) are indistinguishable, both spaces have positive scalar curvature (or spherical geometry in Thurstons sense). Therefore one can consider a transition from a 3-sphere to a Poincare sphere and quantum gravity should be able to calculate the transition probability.

 

[tom.stoer]

Thanks for the example.

My problem with the PI is that I have either seen ill-defined PIs or PIs derived from the canonical approach. But as said, I agree that this may be the Achilles' heel of the canonical approach.

 

[marcus]

Don't let me interrupt this line of discussion. I just want to report that the presentation went very smoothly until minute 39,well into part 2 of the outline. Then Wolfgang got a slew of questions especially from Abhay and Lee, Bianca Dittrich and Daniele Oriti were also commenting and others whose voices I could not recognize. The plenitude of questions slowed the completion of part 2, so that there was only time for an abbreviated treatment of the third part (Quantization).

 

[DimReg]

The functinal integral approach includes all field configurations and integrate over it. But GR does not depend on topology. Therefore one has to include also all field configurations over all possible topologies. For instance: a homogenous metric on the 3-sphere (cross time) and a homogenous metric on the Poincare sphere (cross time) are indistinguishable, both spaces have positive scalar curvature (or spherical geometry in Thurstons sense). Therefore one can consider a transition from a 3-sphere to a Poincare sphere and quantum gravity should be able to calculate the transition probability.

What does a transition probability between two physically indistinguishable states mean? Especially in the case of LQG where the spacetime is supposed to be emergent, I'm inclined to think that such a transition would be entirely meaningless, since the topology wouldn't be fundamental. (though in non emergent theories, I can see how it might be important)

Does the choice of topology (other than global hyperbolicity) change the result of the quantization? Can you tell from the quantized theory what kind of topology was used to quantize it's classical counterpart? (I honestly don't know the answers to these questions, if someone does know the answers it would definitely help me understand the problem of topology change in LQG)

From my studies of LQG, I haven't seen anything that makes it obvious that you can't compute a transition amplitude between two spin networks that correspond to different topologies in their semi-classical limit, mostly because I haven't seen any complete work on taking semi-classical limits. At least in the spin foam approach, which the above discussed paper derives an agreement with, it seems rather easy to write down transition amplitudes between different topologies (though maybe not different topologies with the same metric).

 

[torsten]

The quantization of non-simply connected manifolds showed the dependence of the quantization procedure (the fundamental group). Examples are also discussed by Witten in case of 2+1 dimensional gravity. In particular I studied it in several papers to show that the exotic smoothness strutures are connected to quantum gravity.
For your question, let's consider Wittens approach to 2+1 dim gravity. The quantization procedure has to include the mapping class group, i.e. the quantization procedure depends on this group.
I had always the problem to understand the emergence of the spacetime in LQG. Ok, there is a spin network, the graph of holonomies representing the SU(2) conection. Then one constructs the dual graph but a dual needs a fixed space where one direction is the spin network and the dual graph perpendicular to it. In particular, everyone argues that the spin network implies that the space has to be discrete. But then one interprets the dual graph as part of a triangulation of the space. But a triangulation is a discrete representation (the piecewise-linear structure) of a continuous space (the line between two points does exists, every point of the surface formed by three lines also exists). So, where is the emergence of space in LQG?

 

[tom.stoer]

Isn't there a related question? The dual of a triangulation is always a graph, but the the dual of an arbitrary (!) graph is not always a triangulation. Where does the restriction of graphs come from?

 

[QGravity]

Does the same critical issues that critics raised against String Theory is applicable about LQG? for example the critics says that string theory give us nothing new beyond standard model and GR except some highly hypothetical phenomenological ideas such as extra dimensions, does the same critic can be raised against LQG that it give us nothing beyond the above two theories despite 27 years of activity? do other critical issues apply?

Also I found the following two critical view on LQG interesting in this regard although I coundn't understand them fully because I have an elemenatry knowledge of LQG at the level of Rovelli's book:

Critical Overview of Loops and Foams [arXiv:1009.4475]
Abstract : This is a review of the present status of loop and spin foam approaches to quantization of four-dimensional general relativity. It aims at raising various issues which seem to challenge some of the methods and the results often taken as granted in these domains. A particular emphasis is given to the issue of diffeomorphism and local Lorentz symmetries at the quantum level and to the discussion of new spin foam models. We also describe modifications of these two approaches which may overcome their problems and speculate on other promising research directions

Spin Foams and Canonical Quantization [arXiv:1112.1961]
This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization \`a la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possesses in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.

it would be great if someone can explain the issues raised in these articles!

thanks!

 

[tom.stoer]

Does the same critical issues that critics raised against String Theory is applicable about LQG? for example the critics says that string theory give us nothing new beyond standard model and GR except some highly hypothetical phenomenological ideas such as extra dimensions, does the same critic can be raised against LQG that it give us nothing beyond the above two theories despite 27 years of activity? do other critical issues apply?

Not the same critical issues, but similar ones:
- no well-defined, consistent quantization scheme (see below)
- no experimental check (which applies to all QG theories, I suppose)

The difference to string theory is that LQG
- does not aim for a fundamental, theory but "only" for QG
- does not add a huge number of "physical" entities like extra dimensions, SUSY, new particles, ...

There is however a kind of landscape problem, not in the solution space (like for strings) but in the construction of the models.

Critical Overview of Loops and Foams [arXiv:1009.4475]

http://arxiv.org/abs/1009.4475
Is outdated afaik, b/c i does not address the new EPRL and FK models

Spin Foams and Canonical Quantization [arXiv:1112.1961]

http://arxiv.org/abs/1112.1961
Alexandrov is partially right (we had a discussion of these articles, if I remember correctly it was me who started the thread ;-) It is true that the quantization, especially the implementation of the constraints, the second-class issues like modification of the algebra / the measure, the fact that the discretization affects the algebra, the different mechanisms to implement the constraints, ... is not fully understood, neither in the canonical nor in the SF models. It is well-known that - after 25 years - there are still conceptiual issues

http://arxiv.org/abs/1302.7142v1
Holonomy Operator and Quantization Ambiguities on Spinor Space

http://arxiv.org/abs/1301.5859
Hamiltonian spinfoam gravity

 

[DimReg]

So, where is the emergence of space in LQG?

This is why I brought up the fact that the procedure of taking a semi-classical limit is not well understood. It's not clear to me at all that the quantization results in a Hilbert space with states only corresponding to one topology, because it's not clear to me how a state produces a topology. As Tom mentions, the quantization produces states that aren't even triangulations, do these states even have a (semi-classsical) topology? If they don't, doesn't that imply that a quantization using a fixed topology doesn't produce states with only one topology?

The homotopy group example you gave me is a good one, I forgot about that. I think it is clear that using a fixed choice in the quantization can miss important feature of the resulting quantum theory, such as a non trivial fundamental group.

As far as I understand it, this is the philosophy of "emergent spacetime". You can define your quantum theory without using the topology or metric used to obtain the quantized theory (for example, you can't do this in QED, both the classical and quantum theory need a background metric). Since you don't have to define the theory using a topology or metric, you simply don't, and hope that you can recover the topology and metric as some kind of derived quantity. Since we don't know how to recover the topology, I just don't think we are in a position to understand topology change, or lack there of, in LQG.

Anyways, I'm just thinking out loud, because the only comments I've heard about topology change in LQG is just critics claiming it doesn't exist, with no argument to say why.