What would be today the open problems of Causal Dynamical Triangulations?
CDT today has many open problems which should mean it is relatively easy to get started in. there is not much prior literature to read either
one important result would be to extend the work of Willem Westra to higher dimension.
Westra is a PhD student working with Loll who, with Loll help, has proved something in 1+1D which would like to be extended to 2+1D and ultimately if possible to 3+1D.
Now Loll has two graduate students working in this area, Willem Westra and also Stefan Zohren.
basically ANYTHING about including topology change in the path integral so that there can be brief microscopic wormholes in spacetime that are too small for us to be aware of them. so far there is very little on topology change and so virtually any result is interesting.
also the CDT people have made only a beginning with including matter, so there is a new term in the Lagrangian and some new rules for the Monte Carlo simulation. So far they have only done matter, i think, in 1+1D.
CDT is at a very young stage where they are trying out the model (in 2, 3, 4 dimensions) and experimenting with Monte Carlo simulations of 4D spacetime, and almost everything they do is a new result.
It should be clear from looking at the papers the past couple of years. Konopka a grad student at Perimeter just jumped in without a by-your-leave. he did not go to utrecht first and work with Loll, he just jumped in.
I think Ansari also did that. It is so much at ground floor that you dont need someone to hold your hand. This is how it looks to me.
However it also looks to me that the problem of realizing a Black Hole in CDT is very very difficult.
I am just reacting off the top of my head, just saw yr post. So I may change my mind later. Anyway thats a first response.
I think the question you asked needs to be asked in the context of who the researcher is.
for a masters or PhD grad student who wants a thesis problem it has to be a DO-able problem. Also for a newcomer who has been a string postdoc, say, and wants to try a new line of research out. It has to be an approachable problem
You might even get a suggestion free for the asking from Renate Loll!
She has been putting many grad students and postdocs on the CDT track recently. Maybe she has a lot of problems in mind and is generous about giving them out.
It might be worth asking. But of course NORMALLY someone like that saves the good do-able problems to give to his or her own students. She does not HAVE to reply.
the way that field is going there are all sorts of little results building up (first in dimension 2, then 3 then 4, first without matter then with some matter then with more matter0 which little by little are filling in the picture and increasingly indicating that it is a GOOD QUANTUM SPACETIME.
but this is a gradual accumulative process. not to do in one big jump
so the little do-able problems are important. Loll is getting a lot of useful important work out of her young people. everybody is helping push the wagon.
and then there is another sense in which you could be wondering what is a BIG open problem. what is that monumental one heroic landmark PROBLEM of CDT? not the small incremental results, but the big Sphinx-like question mark.
well somebody might say it would be putting matter fields into this limits of triangulations format and trying to reconstruct the standard model on that new quantum spacetime, but I cant think on that scale. i will tell you my personal CDT Sphinx:
it is What IS the CDT spacetime? For me that is the big OPEN PROBLEM that I wonder about. Because I do not think it looks like a differentiable manifold. But I think it is a real thing. there just is no usual familiar mathematical structure that we learn to use in grad school that corresponds to it. this is my suspicion.
It is a LIMIT (in a sense of limit that still must be defined) of SIMPLICIAL MANIFOLDS as the triangulation scale shrinks-------and even though any particular simplicial manifold could be replaced by a nice smooth differentiable manifold which the simplicial manifold is a triangulation of---even tho any one of the sequence could be replaced by something smooth----the limit (in this sense I am not sure how to define) is NOT smooth and in fact is not coordinatized and does not have a uniform dimensionality.
So for me the open problem is there is some other kind of continuum that we are not used to, that the Loll quantum triangulations converge to. And I feel a bit silly telling you this because I am not sure of it. But hey, you asked for open problems! here is one: define this other kind of continuum and find out what it is like to do mathematics on it!
but seriously, if you are a grad student then you should focus on what a good thesis advisor can fish up for you as a problem to work on. dont waste your early years on something that is too Sphinxy.
I think issues in CDT are not so much divergent as convergent. I realize this sounds naive, but isn't gauge invariance a bit problematical to begin with?
In what way do you mean? Fadeev and Popov showed how to quantize gauge theories and 't Hooft and Veltman showed how to renormalize them, Becchi, Rouet, Stora, and also Tyutin (BRST) explained the ghost particles that Fadeev and Popov had found as due to a native supersymmetry that gauge field theories enjoy. The ghosts (scalar particles with Fermi-Dirac statistics) are not threatening; they are always off-shell and actually help to preserve unitarity. Do you regard any of this as problematical?
another research problem for Alamino-----show the equivalence between the causal sets approach and the CDT path integral
it's going to take monkeying with causal sets
and redefining or reinventing it a little, but causal sets can stand it
because it already comes in several flavors and you still can't calculate
with it so why not monkey with it a little and get some kind of bridge theorem
Alamino asked for open problems, like he/she was a grad student wanting a research topic to think about. or maybe not, maybe wants a big open problem to think about on his/her own time. that's interesting. here is another:
CDT is different from the other nonperturbative backgroundindependent QG approaches because neither does it assume any discreteness nor does it find any evidence of spacetime discreteness so far, or any hint of a minimal length scale.
this is a huge difference from, for instance, causal set (which assumes discreteness from get-go) or Loop (which emphasizes results about discrete area and volume spectra and is sometimes based on spin networks which are kind of discretish)
this would be a good thing to investigate.
there is this famous passage in "Spectral Dimension of the Universe" where Loll says that their research has revealed no indication of spacetime discreteness or minimal length scale and she contrasts that with other nonpert. QG.
as you go down smaller and smaller scale in CDT you get that space and spacetime both get more and more wrinkly and fractally and chaotic and frantically nonclassical, but it is always a topological CONTINUUM that gets wrinkly
what you are approximating with, and going to the limit with, is always a triangulated continuum---it never becomes discrete.
but what about the limit? well I suspect that the limit is neither discrete nor indiscrete. it is something you can CALCULATE about, because you do the calculations on the triangulated manifolds that are approaching it as limit. every calculation is done on nice things and implicitly goes to the limit as the mesh gets finer
so you have a way of all the calculation you want and all the quantum expectation values, but you cant say what spacetime IS without some new mathematics. what it IS is a new kind of structure which is defined by going to the limit with triangulated manifolds.
and that, I expect, is neither discrete nor indiscrete----but of course neither are women, or lots of things, probably even Mozart wasnt.
so if you are a grad student and want a DO-able problem then you should write to Loll or one of the others and get some ideas, prefereably involving just the 2D case where I suspect there is still plenty to investigate (like including some mickey mouse matterfields in the 2D case)
but if you already have a life, and you want a blue sky wide open PROBLEM to think about, then there is this curious fact that CDT is different from all the rest because of the fact that, no matter how fine you make the triangle scale (which is going to zero) you never find evidence of a minimal length
First, let me thank you for the answers and the advices, they are exaclty what I was looking for. And, by the way, you're right: I'm a grad student and I looking for problems in CDT. In truth, I will finish my PhD in Statistical Physics next month and as I always liked quantum gravity related matters, I saw in CDT an area where I could merge both knowledges and I trying to enter in the field.
Maybe write to Dario Benedetti. he was at Rome "La Sapienza" for the Laurea and then went to Utrecht for PhD. seems like a friendly person from website and might have suggestionss
he and Loll have a paper in the works. I saw a reference to it but so far there is no preprint posted
Here is another possible contact:
his CV looks like statistical physics, he is still at Rome, but he is co-authoring a CDT paper with Loll and Benedetti
here is an exerpt from Zamponi CV:
<<Research interests (keywords)
Nonequilibrium statistical mechanics; chaotic dynamical systems; large deviations; entropy production; fluctuation theorem. Numerical simulation of classical many body systems; molecular dynamics; sheared fluids. Structural glasses; configurational entropy; fragility. Langevin dynamics; fluctuation theorem for stochastic systems; dynamics of disordered mean-field models. Topological properties of the potential energy surface in classical many-body systems. >>
so somehow even though Zamponi is still finishing a PhD in Rome in statistical physics, with supervisors Prof. Giancarlo Ruocco and Prof. Giorgio Parisi (his CV says) he is ALSO collaborating on a CDT paper with Loll and Benedetti.
This paper, which has not appeared in preprint, is reference  in
the recent CDT black holes paper by Loll and Dittrich
I dont know what it is about or what the connection (if any) with black holes is. it is just a sign that someone in Rome in statistical physics can also get a foot into CDT somehow
maybe Zamponi or Benedetti would say how to do it
I took a look at the paper by Dittrich and Loll, although it is a little long and I will take some time to read it. This is the abstract:
We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations.
Black holes are indeed the favorite matter when a statistical physicist starts to work with gravity due to the amount of statistical physics and themrodynamics involved. Maybe it is an interesting starting point, or at least worth to take a look.
I´ve just sent an email to Renate Loll and I will try to contact the other persons you indicated.
Thanks for the help.
my pleasure, hope something interesting works out
...sounds like swirling surface tension and scaled bubble connections to me depending on whether the system is macro or micro given that bubbles always connect at triangles
sorry if that doesn't make sense but i'm no expert, i do however have this picture in my head...
yeah yeah I know :bubblehead :tongue2:
Thanks SA, that was pretty deep. I have studied QFT, but am not very confident in my knowledge [which suggests I would be better off listening than posting]. I follow it pretty much to here:
I went through that really excellent link and didn't see anything that suggested the author(s) found quantum gauge theories problematical. Could you specify?
BTW, I do recommend that link to anyone reading this who wants more understanding of gauge theories.
Agreed, nothing particularly problematic is suggested by that source, SA. I should clarify. It is my understanding quantizations cannot be done without gauge fixing, and is very complicated when trying to do so non-perturbatively. Some examples:
Gribov Problem for Gauge Theories: a Pedagogical Introduction
Authors: Giampiero Esposito, Diego N. Pelliccia, Francesco Zaccaria
Gauge fixing in Causal Dynamical Triangulations
Authors: Fotini Markopoulou, Lee Smolin
Separate names with a comma.