Discrete space <-> graph theory

[birulami]

My complete layman's question. In two of Smolin's books as well as in popular science journals I read that there is the idea of a discrete space, i.e. space would not be completely continuous but rather have "smallest pieces".

I wonder if this means that space can be modeled as a graph (the one with nodes and edges, not the plot). The smallest pieces of space would be the nodes of the graph. But what is the neighborhood relation? How many neighbors would every node have? I guess it is not just 6 neighbors like stacked cubes.

Harald.

 

[Jim Kata]

Yes, it can. A spin network is a graph

 

[marcus]

Hi Jim and birulami,
there are some deep questions lurking here which I won't try to answer
(what is space, is it just an illusion that emerges at macroscopic scale from something more basic, or does it really exist and if so what should be the mathematical model of it, what are its fundamental degrees of freedom, is space different from its geometry or does it consist merely of geometric relationships, or causal relationships between events, and what are events?...and so on)

I just have a simple comment about ordinary LQG. In standard LQG as it developed in the 1990s spacetime is represented by a smooth manifold and space is represented by a smooth manifold of one lower dimension.

it does not look like something made out of little sticks. It is mathematically represented by a smooth continuum.

whatever is written in popular books that contradicts this does not matter, it is just trying to get ideas across to lay public.

In standard LQG spin networks serve as quantum states of geometry which seems like a different idea from space. Maybe you like to equate them but I feel there is a distinction between space and the state of spatial geometry which the observer sees.
The mathematical stand-in for space, the smooth manifold, the continuum, has a big configuration space of different classical geometries represented by smooth connections on the manifold. Spinnetworks are functions defined on that configuration space (analogous to the wavefunction of a particle as might be defined on the line).

In GR if you want to talk about an area or a volume you need events which practically requires MATTER. An area has to be the area of some definite piece of metal or some particular desktop. A volume has to be the volume of a particular something defined by events. Space by itself has no meaning. (this is because of general covariance, socalled diffeomorphism invariance, was pointed out by Einstein as early as 1916).

So LQG carries on this GR tradition and when it talks about measuring areas or volumes that is of SOMETHING----and you get discrete spectrum.

Of course this doesn't mean that space is made of little Planckscale sticks. It is a fact about MEASUREMENT in the context of that theory, in particular it is about GEOMETRIC meaurement, typically involving some matter to determine events, like a dustgrain or a desktop.

To me it is somewhat mysterious that the spectrum of area operators in usual LQG turns out to be discrete. But it is one of those things like the Heisenberg principle that has to do with measurement. I don't know of any satisfactory classical or hiddenvariable mechanistic reason for it. In usual LQG, discreteness is NOT put in and you start with a smooth differentiable manifold that you construct everything on, but after several steps you get classical (smooth) states of geometry, and then you get quantum states of geometry, and then you get geometric measurement operators, and then these operators turn out to have discrete spectrum.

If you read in a popular book that this means space is made of atoms, it's probably not meant to be taken too seriously. It is just someone doing the best they can to convey the idea of a quantum operator having discrete spectrum. We already saw this with the hydrogen atom.

Maybe this is obvious to you, or maybe you disagree. I don't know how you approach the subject.
====================

there are other background independent QG approaches that don't have this discrete spectrum geometric operators thing. I'm not sure how this is going to play out. here I just discussed what is usually understood by LQG but there is active research in several other approaches.

In particular people are especially excited by the Asymptotic Safety approach of Reuter and Percacci and their groups. They use the term "nonperturbatively renormalizable".
In their approach there is no discreteness that I can see at all. Although Percacci says that it would be compatible with some kind of discreteness at Planck scale. They just use an ordinary smooth differentiable manifold and an ordinary Einstein metric to describe the geometry.

IMO anyone interested in QG should read Percacci's paper called Asymptotic Safety. It just came out and is on arxiv. Let me know if you have any difficulty finding it.
He discusses the discreteness issue in the Q/A section at the end.

 

[marcus]

It seems to me that the UV divergences in a quantum theory of gravity could be canceled by just requiring the distance between any two points be finite.

Jim you said this in another thread, and it seems part of the same train of thought that we have here.
Are you sure that the quantum theory of gravity must have UV divergences? It seems that recent work of Reuter Percacci and others provide rather persuasive evidence challenging that assumption.

At least that's what they are saying, and the work has attracted considerable attention---Reuter was an invited plenary speaker at both Loops 05 and Loops 07.
Satz, a UK grad student, has an informative blog about some lectures he gave earlier this year, which probably has links to papers if you don't know the Asymptotic Safety work already. Here is Satz blog, ask if you want more
http://realityconditions.blogspot.com/2007/04/report-on-quantum-gravity-school_10.html