key quote from Larsson's most recent post on SPR:
----quote from Thomas Larsson---
Thus, I believe that it is a fair chance that AJL have indeed succeeded in quantizing gravity.
They do so not by assuming a lot of experimentally unconfirmed new physics, but rather by strictly implementing the time-honored principles of old physics, especially causality.
Larsson makes important points in this post.
I defer to his judgement and generally agree, but have a couple of
comments to make in the context of this thread.
Here is the text of his post, which was in reply to Baez.
-----Larsson post, for possible comment-----
(John Baez) wrote in message news:<email@example.com>...
> In article <firstname.lastname@example.org>,
> Thomas Larsson <email@example.com> wrote:
> >firstname.lastname@example.org (John Baez) wrote in message
> >> Given all this, I'm delighted to see some real progress on getting 4d
> >> spacetime to emerge from nonperturbative quantum gravity:
> >> 3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Emergence of a 4d world
> >> from causal quantum gravity, available as hep-th/0404156.
> >This is pretty exciting.
> I'm glad you think so! I sure do!
Maybe I was overreacting. It was becoming boring to be negative all the time, so when I realized that somebody had made tangible progress towards some kind of quantum gravity, I got carried away.
Anyway, I would like to discuss to what extent AJL really have succeed in constructing a model of QG in 4D. As I see it, there are three things that could go wrong: that the model isn't quantum, that it isn't gravity, or that the measure is wrong.
1. Is the AJL model really quantum? Some time ago, Urs Schreiber argued that LQG, or at least the LQG string, fails to be a true quantum theory, and I tend to agree. However, the AJL model can be viewed as a statistical lattice model, and if such a model has a good continuum limit, it is AFAIK always described by some kind of QFT. What else could it be?
2. Is the AJL model really gravity? The action is a rather straightforward discretization of the Einstein action with a cosmological term:
\int R => sum over (d-2)-simplices
\int det g = volume => sum over d-simplices.
What is perhaps somewhat unusual is that all edges have the same length, which is different from Regge calculus. Nevertheless, I don't think that this really matters, but one could check if the results look different if you allow for variable edge lengths.
3. Is the measure right? Here is the place where AJL differ significantly from previous simulations. AFAIU, the crux is that AJL insist on a strict form of causality: they exclude spacetimes where the metric is singular, even at isolated points. This may seem like an innocent restriction, but it rules out things like topology change and baby universes, which require that the metric be singular somewhere.
It is not obvious to me whether one should insist on such a strong form of causality or not, but this assumption leads at least to better results, e.g. a reasonably smooth 4D spacetime. Thus, I believe that it is a fair chance that AJL have indeed succeeded in quantizing gravity. They do so not by assuming a lot of experimentally unconfirmed new physics, but rather by strictly implementing the time-honored principles of old physics, especially causality. That is cool.
Notice that he says
"What is perhaps somewhat unusual is that all edges have the same length, which is different from Regge calculus."
This is the "dynamical triangulation" approach which has been extensively pursued since around 1985. In the 1990s it has seemingly replaced Regge calculus as the main focus of attention, or so is my impression. Here is a very good historical account from 1992 by Ambjorn Jurkiewicz and Kristjansen
"Quantum gravity, dynamical triangulations and higher derivative regularization"
I have put keywords "dynamical triangulation" in arxiv search and come up with 155 papers mostly since 1995----this includes some search-engine mistakes, not all are dynamical triangulation quantum gravity.
Surveys of quantum gravity typically list DT along with Regge approach on equal footing in the "Discrete Approaches" category. For example in Rovelli's
1998 survey "Strings Loops and Others" (gr-qc/9803024) plenary talk given at the GR15 conference, the approaches are listed:
string, loop, Regge, dynamical triangulation, Ponzano-Regge, euclidean quantum gravity (a Hawking favorite),....,etc,....
the history makes no difference to Larsson's
excellent and well-qualified points but I want to know it anyway
DT has been there all along, since 1985 work by David and by Ambjorn, and maybe earlier. But I at least simply did not notice! There is a lot of work, a lot of computer simulations, review papers, interesting graphix including spacetime animation. We should have this stuff assembled and be aware of it. Here are some 1985 papers that I think are DT
 F. David, Nucl. Phys. B 257 (1985) 45.
 J. Ambjørn, B. Durhuus and J. Froehlich, Nucl. Phys. B 257 (1985) 433.
 F. David, Nucl. Phys. B257 (1985) 543.
 V. A. Kazakov, I. K. Kostov and A. A. Migdal, Phys. Lett. 157B (1985) 295.
these are from the good survey by Ambjorn, Jurkiewicz, Kristjansen
Renate Loll also has a 2003 introduction
"A discrete history of the Lorentzian path integral"
this is of course DT, but also more----it is foliated
and this goes to another point Larsson made, his point 3, about
the "strict causality"
This has been the variant of the DT approach prevalent since 1998.
Loll gives somewhat of the history of this "Lorentzian" or "causal" DT.
Lots to discuss here