Hi selfAdjoint
It seems to me pretty clear that AJL are making progress and Lubos comment could have some interesting angles on it---hostile or dismissive tho the attempt be. So we might do well just to copy it here so all can take a gander at it.
I omit whatever looks like a general condemnation of "discrete gravity people", and which we have heard several times already, and include only what seems aimed at the AJL paper specifically.
There may be mistakes here so read at your own risk (don't necessarily take it for gospel :) )
---quote Lubos SP.strings comment on AJL paper---
*
http://www.arxiv.org/abs/hep-th/0411152 - Triangulated gravity
These colleagues first repeat a lot of the commercials about "Causal
Dynamical Triangulations" that they've already written in many previous
papers. The starting points are very obvious and sort of naive: try to
define the path integral of quantum gravity in a discretized form. (It's
like spin foams in loop quantum gravity, but you don't necessarily require
that the details will agree.) OK, so how can you discretize a geometry?
You triangulate it into simplices, and you imagine that every simplex has
a region of flat Minkowski spacetime in it.
(That's not like loop quantum gravity - the latter assumes that there is
no geometry "inside" the spin foam simplices - the geometry is
concentrated at the singular points and edges of the spin foam.)
Then you write down the Einstein-Hilbert action many times and you
emphasize that it is discretized. There are many other differences from
loop quantum gravity: while the minimal positive distance in loop quantum
gravity is sort of Planckian, in the present case they want to send the
size of the simplices to zero and the regulator should be unphysical. Of
course that if you do it, you formally get quantized general relativity
with all of its problems: as soon as the resolution becomes strongly
subPlanckian, the fluctuation of the metric tensor becomes large. The path
integral will be dominated by heavily fluctuating configurations where the
topology changes a lot and where the causal relations are totally obscured
- and the results of these path integrals will be non-renormalizably
divergent - at least if you expand them perturbatively. But this is simply
what a correct, authentic quantization of pure gravity gives you.
These authors are doing something different in one essential aspect. They
don't want to sum over all configurations, all metrics - the objects that
you encounter in the foamy GR path integral above. They don't do it
because they sort of know that pure GR at subPlanckian distances is
rubbish. Instead, they truncate the path integral to contain "nice and
smooth" configurations only. The allowed configurations they include must
be not only nice, but they must have the trivial causal diagram as well as
a fixed topology - namely S3 x R in their main example. Well, if you
restrict your path integral to configurations that look nice, it's not
surprising that your final pictures will look nice and similar to flat
space, too. But it by no means implies that you have found a physical
theory.
Any path integral that more or less works simply must be dominated by
configurations that are non-differentiable almost everywhere, by the very
nature of functional integration and by the uncertainty principle. One can
often show that the path integral localizes, but that's just a result of
theorems and calculations. One cannot define the path integral to include
smooth and causal histories only. Such a definition simply violates the
uncertainty principle as well as locality, if you make some global
constraints on the way how your 3-geometry can look like. Consequently, it
also violates general covariance, and you won't decouple the unphysical
polarizations. If you also make global constraints about the allowed
shapes as functions of time that cannot be derived from local constraints,
you will also violate unitarity.
...
...
[edit: the rest is not about AJL specifically and is stuff we've heard before]
...
...
At any rate, they show that these strange rules of the game admit some
big-bang big-crunch cosmological solution described by some collective
coordinates (a nice picture animates in front of your eyes), and they
construct or propose a wave function of the Universe that depends on the
observable representing the "3-volume of the Universe".
posted by Lumo at 8:17 PM
---end quote from Lubos---
---quote Robert Helling's reply (Lubos riposte later)---
- Triangulated gravity
>
> These colleagues first repeat a lot of the commercials about "Causal
> Dynamical Triangulations" that they've already written in many previous
> papers.
Once again, Lubos is much faster than me and I make my comments
without having ready anything of the papers than the abstract. And I
agree, when I saw this paper on the arxive, my reaction was "another
one of those. how for into the abstract do I have to read to find the
new stuff?" as was yours.
However, again once again, I am a bit less critical than you are. OK,
it seems they beat the publicity drum a lot but I think this is fair
if you are a small group that wants to be noticed in the stringy
atmosphere of hep-th. And I should mention that Jan Ambjorn has worked
on many different things including matrix models (the old ones),
lattice theories and string theory.
So let me try to say a couple of words in defence of their approach:
This stuff obviously has its background in the matrix model
literature and the realization of 2d gravity in terms of dynamical
triangulations (dual to matrices) was one of the successes of the
80s. But you are right, the Euclidean path integral is not only
dominated by but also seems to localise in non-smooth geometries.
So they try to cure this problem by changing the rules of their path
integral.
[LM#1]
It is probably fair to divide geometry into different levels of
structure. One possible distinction is
0) differentiable structure
1) topology
2) causal structure
3) conformal structure
4) metric structure
It is up to discussion at which of these levels you start varying in
your path integral and which parts you keep fixed.
[LM#2]
I guess, Lubos wants to vary 1-4 while keeping 0 fixed, Ambjorn and
friends only vary 3-4 and I had long discussions with Hendryk Pfeiffer in
which he tried to convince me that one should vary all 0--4.
[LM#3]
Nobody has done a really convincing 'sum over geometries' yet, so I
think it should be allowed to try all these approaches.
[LM#4]
What Ambjorn etal find is that again in 2d you can solve this model
exactly (ie compute the partition function with sources) and it agrees
with expectations (whatever those would be). Second the typical
configurations look much smoother (something they haven't put in, they
only demand causality and global topology) than in the Euclidean case.
[LM#5]
Of course, in 2D gravity is not typical, all the dynamics is in the
cosmological constant and its conjugate variable, the volume,
respectively. And in higher dimensions it is not possible to solve the
problem analytically, you can only run in on your computer.
[LM#6]
Another success that they claim is that they break the 'c=1 barrier'. OK,
I have no idea what that really is because I try to stay away from all
this old matrix model technology but Matthias Staudacher, who was around
in those days, says this is quite non-trivial: In these models, you do not
have to restrict yourself to pure gravity, you can couple matter to it:
For example you can add an Ising spin degree of freedom to all your
triangles and sum over it as well.
As, you say, in all these models you have to take the continuum limit
and then you get a conformal field theory. In the old days, it was
observed that whatever you did matter wise or matrix wise, you could
only get models with central charge <1. But with causal triangulations
coupled to matter you can break this barrier.
Finally these people claim their model has a well behaved continuum
limit and I see no reason to doubt it.
[LM#7]
But in the end this is only gravity if you end up in the correct
universality class. That is, all your weird rules you make up to construct
your discretised space-times correspond only to irrelevant operators that
go away in this limit. And to show this is of course the hard part.
[LM#8]
Robert
---end quote---
LM#1:[Moderator's note: Well, I understand. That's what I criticize.
Every path integral in a quantum theory is dominated by
non-differentiable configurations because this is necessary
for the uncertainty principle. A classical configuration has
sharp, well-defined values of the fields like X(t) or PHI(x,t)
or g_{12}(x,t), and by the uncertainty principle, the uncertainty
of the canonical momentum must therefore be infinite, which is
reflected in the path integral by the fact that the |derivative|
of the field is typically infinite, i.e. the non-differentiable
configurations dominate. Do you agree that you could not get
quantum mechanics if your path integral only summed over
differentiable paths? If you succeeded to define this "truncated"
integral in quantum mechanics, it would violate unitarity
and the rule U(t1,t2)U(t2,t3)=U(t1,t3) because the different
intervals would disagree "how much differentiable" the functions
must be. LM]
LM#2:[Moderator's note: It is fair to divide geometries, but it is never
fair to "cut" some configurations from a path integral, I think.
We've had a recent debate on sci.physics.strings about the overcritical
electric field which was exactly about this issue - did you agree with
our conclusion that you can't ever omit "unwanted" configurations? LM]
LM#3:[Moderator's note: It is fair to divide geometries, but it is never
fair to "cut" some configurations from a path integral, I think.
We've had a recent debate on sci.physics.strings about the overcritical
electric field which was exactly about this issue - did you agree with
our conclusion that you can't ever omit "unwanted" configurations? LM]
LM#4:[Moderator's note: Nobody has found a really convincing luminiferous
aether theory, so should all of us divide to different approaches how to
construct aether? Actually I think that these two questions are more
similar, even in details, than you might think. ;-) LM]
LM#5:[Moderator's note: I am not getting this point at all. What's exactly
the difference between the input and output? Typical configurations
in the gravity path integral have strongly oscillating topology, both
in the Minkowski and the Euclidean case, and in the Minkowski case,
they have also a highly nontrivial and chaotic causal diagram.
If you unphysically cut the "ugly" configurations, of course, you will
end up with the "nice" ones, and because you made more constraints
about the allowed configurations in the Minkowski case, you will
get even nicer configurations than in the Euclidean space at the end. But
that's not a result, that's your assumption. And it's an assumption
that contradicts quantum mechanics. LM]
LM#6:
[Moderator's note: Right, 2D and 3D gravity don't really have gravitons
as local degrees of freedom. All of us know how to compute 2D gravity
as a path integral over "nice topologies" of two-dimensional spacetime:
it's called the stringy worldsheet. But the conformal structure on
the worldsheet is only "nice" because *any* configuration in 2D
can be mapped to the "standard ones" by diff x Weyl transformations.
Analogous things hold for 2D string theory - one really wants to
calculate the path integral over the scalar fields in spacetime
and their effects. Moreover, my arguments above that talk about the
uncertainty principle for g_{12} and its time derivative can break down
in d<4 because there is no such a physical degree of freedom. LM]
LM#7:[Moderator's note: in the case of the present paper, I don't have
difficulties with the word "continuum limit" but rather with the word
"model". You can define some set of rules that gives you a *classical*
theory in some limit, but it by no means implies that your rules,
before you take the limit, define a meaningful quantum theory, does it?
For example, you should always ask whether your rules can lead to a
unitary S-matrix, which path integrals should, and the answer will
be NO in the 4D case, I think. LM]
LM#8:[Modeator's note: That may be a different way to say the same thing.
You're simply not sure whether the "restricted path integral" has
anything whatsoever in common with the real path integral. LM]
----end quotes---
selfAdjoint said:
After reading the paper, I was afraid that Lubos would pick up and criticize their restriction on the path integrals, that they be nice and causal. He was criticising the subsetting of path integrals by LQG theorists yesterday, on the grounds that to be valid, path integration has to include everything, and hence be non-differentiable almost everywhere, as well as non-physical(FTL, etc.). Well I was right, he has included just that criticism in his latest review of hep-th papers on sps:
http://groups.google.com/groups?hl=...411162202230.21626-100000@feynman.harvard.edu. I don't know what to make of this; his strictures seem valid to me, but I would really like to see a response by a quantum gravity pro.
