Remaining problems w LQG (and cosmology application)

1. Jul 28, 2013

marcus

Tom gave an honest assessment just now in another thread of the real situation with LQG which in effect includes the several roads from there to cosmology (eg. Agullo, Alesci, Barrau, Engle, Vidotto, Wilson-Ewing and others).
He made a very good point which is that the problems are not different for cosmology, which after all is not formulated the same way it was back in 2005! It is increasingly formulated as a reduced version of the main theory, or embedded in the the main theory, and for that matter in formulations that are not even isotropic.
So, as Tom pointed out, is is good to look at the problems not separately but together.

So I want to comment on what he said in that post:

I think the main issue here, not mentioned by name, is the boundary formalism which comes up when one wants a general covariant quantum field theory underlying GR that is amenable to statistical mechanics/thermodynamics.

That is, if you want the theory to be general covariant then it does not make sense to ask for "transition amplitudes" between "initial" and "final" states. Instead one wants a theory that gives boundary amplitudes.

In the next few posts I'll try to explain (because it isn't immediately clear) how this addresses the question Tom raised. I think it is a good way to look at the focus where the most crucial development of LQG could be expected to occur in the next couple of years. As usual, I could be wrong

2. Jul 28, 2013

tom.stoer

I'm very curious to see your ideas ;-)

3. Jul 28, 2013

marcus

Thanks for your response! Here's what I said about the boundary formulation in another thread, just while reviewing some of the recent talks:
I suppose you could call it the Oeckl program for general covariant quantization of field theories, instead of the Dirac program. ;-)
The boundary observables represent all we can determine about the process of geometry and matter which occurs in the region within the boundary. We do not know the processes occurring within, only that whatever else they must be compatible with what we observe on the outside.
Morally it is just a simple generalization of the initial/final state idea, modified just enough to make general covariance easy. One does not have to mess with foliations.

So in Oeckl quantization one does not have distinguished initial and final states.
One has a Hilbert space of boundary states, or alternatively a C*algebra that one can think of as operators on such a Hilbert space. And one has a state of the boundary.

(This comprises all what might be preparations of the system, measurements, predicted outcomes etc.) And one has the amplitude of that boundary state. It is, of course, not time dependent. We think of the boundary as surrounding a spacetime region.

Oeckl's talk is the first 20 minutes of the recording I linked to. It might be helpful, unless one is already familiar with boundary formalism, to watch a few minutes.

If you do this, namely check out http://pirsa.org/13070084/ , you find that a different talk is listed as coming first, but in fact Oeckl's is the first 20 minutes of the recording. He calls what he is doing the GBF (general boundary formulation) and points out that the seminal work was done by Atiyah, Segal, Witten around 1988.

Last edited: Jul 28, 2013
4. Jul 28, 2013

friend

Is this the same as asking what internal configurations will give the same observables on the boundary, and the answer might be that there is a spectrum of geometries that do this? Does this necessitate a superposition of those geometries?

5. Jul 28, 2013

marcus

I think that's basically the right idea. Instead of superposition you might say "sum" as in "sum over histories".
I have to go help with supper, back later.

Yeah, I have a minute while the spaghetti finishes cooking. The word "superposition" is normally used in connection with combining states (vectors in a Hilbert space). Here we don't have a Hilbert space for what happens in the bulk. We make measurements and observations at the boundary. What goes on in the bulk is the bulk's business.

It just has to be compatible with what we see at the boundary (which involves past during and future observations). It is the boundary we want the dynamics to provide an amplitude for. So the Hilbert space is a space of stories about the boundary (not the bulk).

This is only a matter of what words to use, I think. You have the right idea.

Last edited: Jul 28, 2013
6. Jul 28, 2013

friend

By the word "It", I take it that you mean the bulk inside the boundary. I'm assuming that the whole point is to relate what is going on in the bulk by considering the effect it has on the boundary.

Even though we are considering dynamics of the boundary, isn't this still meant to be constraints for what's happening inside? I think you mean that the dynamics on the boundary may be developed independently of considerations of the bulk, right? But I still think that this must still be somehow related to the bulk inside, or it would be useless to local situations.

7. Jul 28, 2013

marcus

I think so too.
This is just a simple generalization of the path integral where we have time, and the boundary is made of two pieces, initial and final.
Think of the doubleslit experiment where we know where the particle was at the beginning and at the end, but it is meaningless to ask where it went in between.
In that case the bulk is what is in between the initial and final.

The boundary is the interface across which the system communicates with us, or with another system.

We can picture the boundary (in the familiar situation where we have time) as a cylindrical drum or can, like a can of beans. The bottom is the initial, the top is the final, and the sides are whatever we determine or constrain or know about during the process, like the walls of the laboratory or cat-box.

But it's probably better if you just watch a few minutes at the beginning of Oeckl's talk in
http://pirsa.org/13070084/
He has pictures and is the main developer of the GBF,whereas my verbal description may in some cases be more confusing than helpful.

8. Jul 28, 2013

marcus

There is a generalization of Hilbertspace called KREIN space, named after the mathematician Mark Krein, born 1907 in Ukraine.
http://arxiv.org/abs/1208.5038
Free Fermi and Bose Fields in TQFT and GBF
Robert Oeckl
(Submitted on 24 Aug 2012 (v1), last revised 5 Apr 2013 (this version, v2))
We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.
46 pages, published in SIGMA 9 (2013)

My comment: Definition and some facts about Krein space are on page 7. Basically it is just like a Hilbert space except split into two pieces one where the the inner product is negative definite, that is:
<x,x> is negative instead of positive as would be with a normal inner product.

Last edited: Jul 29, 2013
9. Jul 29, 2013

tom.stoer

Boundary formalism, PI, SF, ... doesn't help per se.

Either you have to define a consistent, anomaly-free constraint algebra incl. Hamiltonian constraint,
or you have to define a consistent, anomaly-free path integral including measure, effective action, ...

I don't see that either of these approaches has succeeded in providing such a consistent definition. For all constructions it is unclear whether this is the case.

10. Jul 29, 2013

marcus

At one level we agree that the theorists still have work to do. However you seem focused on just two alternative goals: either they should achieve a Dirac-style canonical quantization or they should achieve a satisfactory path integral (which would require a measure on the space of paths, with which to integrate). I think you have a very clear idea of each of these two alternative goals.

As you doubtless understand (especially if you have watched a few minutes of Oeckl's talk at http://pirsa.org/13070084/ ) I would like to add a third one to the list of alternative destinations.
This may actually be a more difficult goal to reach.

I don't feel comfortable with either of your two alternatives because neither seems particularly thermodynamics-friendly---where does temperature fit in? where is there room for statistical mechanics? where does Jacobson 1995 fit? You may have answers and can explain this to me but by myself I don't see it.

From where I stand, I know that geometry must be a general covariant qft, AND I am constantly seeing indications that geometry has temperature and entropy.
So I think that the goal must be a general covariant QFT which is, synonymously so to say, a general covariant statistical mechanics.

So I am willing to consider a third goal which is slightly different from the two you have in mind, and which I cannot picture so clearly. I think it may be along the lines that Oeckl was discussing.

11. Jul 29, 2013

atyy

Bianchi gave a fascinating talk about thermodynamics and spin foams. http://pirsa.org/13070048/ There's a gap to the spin foam part. The more interesting thing is that he says he's filled in some gaps in the idea that black hole entropy is entanglement entropy, which is an idea people like Srednicki worked on as long ago as 1993. His recent paper about the boundary being mixed tries to tie together Oeckl's formulation with thermodynamics. I don't think he's got spin foams, thermo, and boundary all in one paper yet, but he's clearly thinking about it.

I'm not so worried about the algebra not closing, since that could be provided by string theory.

Another interesting talk with Dittrich's. http://pirsa.org/13070079/ She said, suppose we want to actually do a calculation with the boundary formalism, how would we concretely do it. She can't do it yet it with the current spin foam models, but she really tries to get the nitty gritty in a toy model.

Last edited: Jul 29, 2013
12. Jul 30, 2013

tom.stoer

I don't know what you mean.

In LQG we have an algebra of objects (A,E) defining directly the spacetime symmetry (G, D, H). I do not see how to formulate this algebra at all using string theory.

13. Jul 30, 2013

atyy

Not literally:) I think maybe LQG is formally not the right way to go, but aspects of its language like spin foams and Oeckl's boundary formalism may be interesting even in another theory of quantum gravity like string theory. I think Bianchi's and Dittrich's talks could be seen in this light.

I know it is still hoped that EPRL or FK could solve the Hamiltonian constraints after projecting, but I didn't mean that string theory would be helpful there. I'm not even sure that E, F or K still think those two theories are good.

Last edited: Jul 30, 2013
14. Jul 30, 2013

tom.stoer

I have never seen PIs doing a better in fundamental constructions than the canonical approach. The latter one either succeeds or fails - and tells you why, PIs a good in hiding problems, but if you don't succeed you don't know why. One MUST understand what exactly goes wrong in the canonical approach. Abandoning it w/o understanding its fundamental problem w/o understanding them is a no-go.

15. Jul 30, 2013

atyy

How about something like http://arxiv.org/abs/1307.5885 ?

16. Jul 30, 2013

tom.stoer

Yes. I saw that paper a couple of days ago and will certainly study it

17. Jul 30, 2013

marcus

Last edited: Jul 30, 2013
18. Jul 30, 2013

MTd2

In the paper spotted by marcus, AMELINO-CAMELIA also begins talking about Born duality. I suspect that Freidel will get curved momentum space time from his theory.

It seems also by the argument above eq. 4 of http://arxiv.org/abs/1307.7080, Freidel seems to direct his thought to try to fix LQG anomaly.

19. Jul 31, 2013

tom.stoer

I don't see the relation to LQG

20. Jul 31, 2013

tom.stoer

Is it fair to say that Hillary fixed Mallory's failure when climbing the Mount Everest??