The role of time in Loop Quantum gravity

[Micha]

As far as I understand, loop quantum gravity treats space and time on very different footings. What I get is that in LQG time acts like a CPU clock in a computer representing a counter for the subsequent changes in the descrete structure of space.
What worries me about this picture is, when time and space are treated so different in the foundation of this theory, why is it, that the role of time and space are almost completely symmetrical on the level of the Lorentz metric (only a difference in sign appearing)?
As I understand, a goal of LQG must be to derive Lorentz invariance as an approximate feature of certain spin foam models. Not knowing, what the status of this is, does anybody know, how the symmetry between space and time comes about in this theory then?

 

[MeJennifer]

why is it, that the role of time and space are almost completely symmetrical on the level of the Lorentz metric (only a difference in sign appearing)?

They are not.
In the theory of relativity the metric represents time, which is usually called proper time.

In Galilean relativity and in the Newton/Cartan formulation does the t dimension represent time but not in the theory of relativity.

As an illustation, if you look at the hypersurfaces of constant proper time in flat space-time you will notice that they are hyperbolic, only when c is infinite do they overlap the flat hypersurfaces of constant t.

 

[Micha]

Thank you for your answer. I think, what you write, is correct. But I don't think, it answers my question.
Time t (and not only proper time) is a meaningful concept in special relativity and it appears completely symmetrical in the expression for the metric (which is an expression for the proper time, yes).
And in general relativity, you can always rediscover Minkowski space at least locally by tranforming to a freely falling observer. So time t is a meaningful concept here as well and the symmetry is also present.
So my question still is, how does this symmetry come about in LQG?
Or, if you want to put the question into the language of general relativity:
Why is it, according to LPG, that in the concept of general covariance space and time can be completely transformed into each other?

 

[marcus]

Hi Micha, if you have the patience let me give a little history (which might help other people who read too)

the puzzle I think you point to is shared by several theories going back to 1967 or earlier and may actually be OK (if the theory does not fail for other reasons). it is connected to CANONICAL FORMULATION of Gen Rel

==exerpt Wiki Canonical general relativity===

In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt[1] in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann[2] using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac[3]. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle-Hawking state, Regge calculus, the Wheeler-DeWitt equation and loop quantum gravity

The quantization is based on decomposing the metric tensor...[now it talks about Lapse and Shift]...Hamiltonian constraint...
...
...DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque.

Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom...
...
References

1. ↑ B. S. DeWitt (1967). "Quantum theory of gravity. I. The canonical theory". Phys. Rev. 160: 1113–48.
2. ↑ see, e.g. P. G. Bergmann, Helv. Phys. Acta Suppl. 4, 79 (1956) and references.
3. ↑ P. A. M. Dirac (1950). "Generalized Hamiltonian dynamics". Can. J. Math. 2: 129–48. P. A. M. Dirac (1964). Lectures on quantum mechanics. New York: Yeshiva University.
===

As this little bit of history suggests, any attempt to use Dirac-style canonical quantization on Gen Rel gives up obvious covariance. It may be really covariant, but this is opaque, or non-obvious.

You have probably heard of the famous Wheeler-DeWitt equation of Bryce DeWitt and John Archibald Wheeler. That goes back to DeWitt 1967 work or earlier. When LQG came along in 1986-1987, they were just doing the programme already well-established by DeWitt and others---Dirac-style canonical quantization.

That kind of approach inherently leads to a "frozen time formalism".
In that approach, you can really have covariance, and yet it may not be manifestly obvious. The formalism obscures it because things are defined on a spatial hypersurface and the Hamiltonian constraint controls them to make sure that they COULD evolve correctly in time. But you do not explictly show the evolution.

this is not to say that the standard 1990s version of LQG does not have problems with the Hamiltonian constraint! It sure does.
And this even forced a lot of LQG-community to go over to spinfoam, which is more manifestly covariant because it is a 4D path integral approach.

But we should be clear about what the problem is. It is not fatally wrong to use Dirac canonical quantization with a Hamiltonian constraint! That is lovely, even though it may obscure the covariance. It is not wrong to follow in DeWitt and Wheeler footsteps and do canonical Gen Rel and canonical quantization of Gen Rel. However in LQG case, in the late 1990s the particular Hamiltonian constraints that they tried did not work.

Some people then switched over to path integral 4D approach, as I said, but SOME STUCK IT OUT AND KEPT ON TRYING TO GET A CANONICAL QUANTIZATION. These people include Thomas Thiemann, who now has a new variant of LQG which he calls AQG that he presented at KITP Santa Barbara in February this year. It looks quite interesting. there are video talks about this you can watch.

The die-hard group that stuck with canonical quantization also proved highly successful in a simplified version applied to COSMOLOGY called LQC, where the Hamiltonian DOES work, and you get a splendid classical largescale limit.
The people pursuing LQC include Martin Bojowald and Abhay Ashtekar (one of the original LQG founders).

In LQC the time evolution works smoothly and they can run their model back thru where the BigBang singularity used to be. there is a lot of excitement about LQC, which is why KITP invited Bojo, Ashtekar, Thiemann to that 3-week workshop on how to cure classical spacetime singularities.

 

[marcus]

OK Micha, that is some history. I hope it may be helpful for you or other readers. I did not answer your questions yet and perhaps I will be only partly successful in doing that.

the basic thing is to know that the canonical formulation of classical gen rel is mathematically VALID even tho you don't see the fullblown 4D. you only see them working on a 3D hypersurface "scratchpad" policed by Ham constraint.
And recovering the full 4D covariance is MESSY and takes blackboard time in a graduate course. but expressing classical Gen Rel in this way is nevertheless valid. BTW the famous ADM (Arnowitz Deser Misner) treatment of Gen Rel was also this same canonical way. "The Dynamics of General Relativity" 1962.

Because it is messy, I don't know how to convince you that it can work. Maybe you should look at the famous 1962 ADM paper which carries thru this program in a way that a lot of people like. Many formulations of classical Gen Rel are of this type. Somehow you can define things on a 3D hypersurface and make sure that even though it looks frozen it can evolve.
and that solution on the 3D hypersurface actually determines the whole 4D history! Seems magic, doesn't it?

The 1962 ADM paper was so widely read that some thoughtful person put it on arxiv a few years ago, like maybe in 2005, to make it more accessible to the community. Just shows what a great paper.

Again, the approach (in which time seems to disappear but then can come back like grass does from bare dirt) only works if you can carry it thru to get a Hamiltonian constraint that works!

This is where the first attempt at LQG failed in the late 1990s. However there is a chance that Thomas Thiemann has now carried the program thru with his "Master Constraint" and his AQG (algebraic quantum gravity). I can't say about that. there is an interesting paper by Bianca Dittrich (an associate of Thiemann) about this. If I had to evaluate, I think the first thing I would check would be Bianca's paper. Look on arxiv for Dittrich.

If you want to watch Thiemann presenting at KITP and answering questions from the largely string audience, I will get a link.

If you simply want to try to understance how time evolution can reappear from a canonical (frozentime Hamiltonian constraint) package, then I would suggest reading LQC papers. they run computer models of time-evolution thru the big bang. They set up the hamiltonian constraint and it turns into a DIFFERENCE equation that let's them solve for the NEXT STEP and they solve and then they apply the hamiltonian constraint again, and so on...

If you want to look at LQC papers to see concretely how it works, tell me and I will get some links.

THANKS FOR THE QUESTION by the way It was fun answering

 

[william donnelly]

It's worth noting that many in the LQG community believe that Lorentz symmetry will not hold exactly. This area of research is usually called Deformed Special Relativity. The reason for this belief is that LQG has something like a minimum length, which would appear to be in conflict with relativistic length contractions. DSR is an attempt to reconcile the Lorentz transformations with the minimum length.

 

[Micha]

Marcus,
this is for sure a much more elaborate and detailed answer than I could
have hoped for. Thank you very much. Probably it will be helpful to others as well.
It only shows me, that I need some years to catch up. :-)
But why not start trying.

@ William
Yes, I read about Deformed Special Relativity and the upcoming
experiments with cosmic rays in Smolin's book and this brought up
my interest in the LQG approach.

 

[Demystifier]

Marcus gave a detailed answer, but perhaps a shorter answer that more directly emphasizes the main point would be more instructive.

All fundamental classical field theories are relativistic covariant, treating time and space on the same footing. The simplest way to achieve this is to write a Lagrangian that transforms as a scalar under spacetime transformations. Still, from any Lagrangian one can also construct the corresponding Hamiltonian. The Hamiltonian, by construction, treats time on a different footing than space. Nevertheless, in any classical theory, the Lagrangian formulation is fully equivalent to the Hamiltonian formulation. In the case of gravity it is not so trivial to show explicitly (it is a mess, as marcus said), but for simpler theories (e.g. scalar field) it is rather trivial. In particular, it means that the Hamiltonian formulation is also relativistic covariant, although it is not so obvious and manifest. Given this relativistic covariance of the Hamiltonian formulation at the classical level, it is reasonable to expect that the covariance will be restored at the quantum level as well, at least in a classical approximation of the quantum theory.

In even simpler terms, we expect spacetime covariance of LQG because the starting point in construction of LQG was a classical Hamiltonian formulation for which we already knew that it was spacetime covariant.

 

[Micha]

Demystifier,
I am a bit familiar with the Langrangian and the Hamiltonian method
and their application in field theory, so although I did not mention, I was able to get this point out of Marcus answer. But it is good to have made this point explicit.
To put it in my own word, it seems, that while LQG at least in its original form gives us a very good intuitive picture in the nature of space, it does (for now) not allow to understand the symmetry between space and time in GR in an intuitive way. But the good news is, it is formally respecting it by being covariant.
So one is looking for a path integral form of LQG like spin foam models to
learn more about time evolution.

I have found this article from Rovelli.
http://relativity.livingreviews.org/Articles/lrr-1998-1/ [Broken]
Chapter 6.10 has the promising title:
Unfreezing the frozen time formalism: the covariant form of loop quantum gravity
"In conventional QFT each term of a Feynman sum corresponds naturally to a certain Feynman diagram, namely a set of lines in spacetime meeting at vertices (branching points). A similar natural structure of the terms appears in quantum gravity, but surprisingly the diagrams are now given by surfaces
is spacetime that branch at vertices. Thus, one has a formulation of quantum gravity as a sum over surfaces in spacetime. Reisenberger [158] and Baez [30] have argued in the past that such a formulation should exist, and Iwasaki has developed a similar construction in 2+1 dimensions. Intuitively, the time evolution of a spin network in spacetime is given by a colored surface."
I must admit that I am not able to make sense of a "sum over surfaces".
Can anybody help? :-)