Connectedness of Discrete Space-Time Models

[runner108]

Been enjoying Physics Forums and this is my first question so before I ask I should I applaud everyone for the helpful friendly atmosphere.

I'm trying to understand the concepts of continuity and connectedness. I have no set theory background nor physics backgrounds so I just read online for guidance so please keep answers to a relatively conceptual level.

I understand continuity as it applies to mathematics in regard to epsilon-delta. Also, as I understand it, continuity when it applies to topologies seems to be a basic tenet. Space-time seems to be described as a four-dimensional, smooth, connected Lorentzian manifold. On this type of a manifold it seems to be that connectedness and path-connectedness are the same. I understand that this allows for comparing frames of reference and movement of energy and particles. It seems to me in my very uneducated guess that this allows for differentiability (which at least in mathematics means you can figure out instantaneous rates of change). My question is, in discrete space-time models such as LQG, they say that it also talks of a differentiable manifold. On the other hand they say that according to LQG space is not continuous. My understanding is that if something is differentiable it is necessarily continuous. Certainly my lay understanding is leading my a-stray.. can anyone help me fill in the gaps to my thinking? MUCH APPRECIATION.

 

[marcus]

... My question is, in discrete space-time models such as LQG, they say that it also talks of a differentiable manifold. On the other hand they say that according to LQG space is not continuous...

Who is "they" who is supposed to be saying the LQG space is not continuous? :-D I don't think I have.

Quantum measurement of some geometric quantities, in LQG, has been shown to have a certain discreteness property.

If you have some concrete physical setup that defines an area, and you consider the operation of measuring that area (measurement operators, in a quantum theory are called "observables") then even though discreteness is not put into the definitions of LQG it is a theorem that area observables have discrete spectrum of outcomes. The operation of measurement can only produce a discrete set of possible values.

That does not say that space is grainy like sand. What it tells you about is measurement. The theory does not impose conditions on "space itself" whatever that is :biggrin". It is about geometric measurement. How geometry responds to measurement.
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You are right that in LQG space, and also 4D spacetime, are represented by smooth manifolds. Space is represented in the theory by a continuum.

The business about certain operators (like the area observables) having discrete spectrum is something that is derived, proven as a mathematical theorem later, after the Hilbert-space of quantum states of geometry has been constructed.
In a quantum theory you need to represent the quantum states of the system and the set of all states has a certain convenient structure named after David Hilbert. In LQG the (very large) set of the quantum states of the geometry of the universe is given that kind of convenient structure. States of geometry form a Hilbert-space, and measurements of geometry variables are then operators defined on that Hilbert space of states.

What answer you get when you measure depends, of course, on what the state of geometry is. Anyway the discreteness in LQG is a feature that arises in the context of measurement.
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In the quantum theory of light that people invented over 100 years ago, light can come in a continuous range of energies. You can have light of every possible wavelength. However if you measure the energy of light emitted by a certain type of atom, like a hydrogen atom, it appears to have "energy levels" and to be absorbing and emitting light only from a discrete menu of possible energies.
So it seems that energy is a continuous thing. It can be any number of electronvolts or joules that you want. But if you make a Hilbert-space picture of the quantum states of an atom, and start to consider measurements of energy to be made on that atom, then you run into discreteness----discreteness of the spectrum of outcomes of certain operators.

It's mildly paradoxical. Just how the quantum world is.

Geometry, when made into a quantum theory, can be mildly paradoxical in a similar way.
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When people do a sloppy popularization job on LQG, say at the Scientific American level, they can really confuse things by not making this clear. Magazine editors probably don't think that a popular audience can handle the idea that "space itself" could be smooth while geometric measurements nevertheless have discrete spectrum.

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Another active and prominent field of quantum geometry is called CDT (causal dynamical triangulations) initiated by Renate Loll's group. In my signature at the end of the post there is a link to a Loll article on CDT that appeared in the Scientific American. The remarkable thing about this article is that it is NOT as sloppy and misleading as popularizations at that level often are. Loll is an unusually good science writer as well as a brilliant theoretical physicist. And also she has been very lucky---she and Ambjorn took a long shot gamble on a new approach back in 1998 and it has prospered.
CDT is unlike LQG in many ways, but the article is still a good introduction to quantum gravity---or, since gravity = geometry---one could equally well say an introduction to quantum geometry.

Loll's approach is also based on a continuum spacetime manifold. However at each stage of approximation the geometry is represented by dividing the continuum up into little triangle-type building blocks. The quantum state of geometry is describe by how the millions of tetrahedra-type blocks fit together. But then the size of the blocks is allowed to go to zero.

In other words, "space itself" (whatever that is) is not imagined to consist of little blocks. And nevertheless a discrete method is used to describe and analyze the quantum states of geometry.

Results are derived, some of them similar to what one can prove in the LQG context, and some different.

Loll's SciAm article is a good introduction to the whole field of QG and I don't know anything that is as good that introduces QG by way of Loop. That does not mean that Loll's CDT is better than Ashtekar and Rovelli's LQG . It may say something about how hard or easy the theories are to explain at a popular level though.

 

[runner108]

When I referred to 'they' I referred to most articles I read that said something like:

"some people say space is continuous, but new theories suggest it is discrete"

The use of the word 'but' always seemed to contradict the 'continuous' quality.

Perhaps a more accurate statement would be that 'space is quantized' in QLG? Thanks for the reply!

 

[runner108]

I guess a simpler question would be, are their any 'dis-continuous' space models that I can look up to get an understanding of what what a counter argument to 'continuous space' might be?

 

[marcus]

I guess a simpler question would be, are their any 'dis-continuous' space models that I can look up to get an understanding of what what a counter argument to 'continuous space' might be?

Well there is the "causal sets" approach to QG. There is probably a Wikipedia about it. The main inventor is Raphael Sorkin.

And there is something called "quantum graphity" being developed by Fotini Markopoulou. That doesn't divide space up into little bits. But it does the next closest thing and uses graphs (vertices joined by links).

There are various other discrete approaches where "space itself" is represented by a vast heap of little bitty elements. But extreme cases of that tend to be marginal. My impression is that "causal sets" was worked on in the 1990s and declined in research activity afterwards, especially after, say, 2004-2005. Sorkin is brilliant and causal sets is mathematically interesting, but right now it looks slightly passé.

If you like discreteness aesthetically then I repeat my suggestion that you look at Loll's CDT article in the SciAm. In her picture a FRACTAL quality of space emerges at very small scale. It is very beautiful. The article is well illustrated.
Who cares if "space itself" is not a pile of marbles or grains of sand? In CDT you get to picture it as a swarm of little triangular blocks that spontaneously fit themselves together and form a dynamically evolving shape.

Like the shape of a flock of birds. In her mathematical model the overall shape of space can arise from microscopic geometric interactions. It is just that you are not limited to one size of sandgrain. You can consistently allow the size of the grains to go to zero.

(It's important to be able to do that, for one thing just so that special relativity carries over to the quantum picture.)

Check it out. Loll's article should satisfy any normal person's appetite for discreteness without actually introducing a 'minimal-size grain'.

 

[runner108]

Exactly what I was looking for. Thank you Marcus. I will take some time to digest what you said.

 

[Naty1]

You are getting into an interesting and sophisticated area of mathematics and physics,
A few interesting blurbs from Wikipedia:

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus.

Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, and holomorphic

http://en.wikipedia.org/wiki/Differentiable_manifold

If you have acces to Roger Penrose's, THE ROAD TO REALITY,2004, in chapter 6, Real Number Calculus, he shows some interesting functions and discusses differentiability, smoothness, curvature and so forth...and also gets into complex differentiability z = a + bi...it's a good read to get a "feel" for different mathematical constraints and freedoms.