# Is space discrete?

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Lee Smolin argues pretty convincingly that space is discrete at the Planck scale. He does so in the book Three Roads to Quantum Gravity, where he claims that each of the three roads (string theory, loop quantum gravity, and black hole thermodynamics) all paint a picture of discrete space. The string theory angle was an interesting one as to my knowledge string theory still used a continuous spatial background, but Smolin elucidated me as to the idea that strings are comprised of "string bits" which are discrete.

I was wondering if there are any theories which specifically require space to be continuous and are incompatible with a discrete approach.

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Yes, I believe space is discrete, don't see how it could be anything other.

String theory, with or without "string bits" is silly.

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I don’t understand the use of the word discrete enough I think. I mean individual organisms might be discrete biological units but I would qualify life as a discrete unit then also. Life though as evidenced by natural selection and ecology then must be a discrete unit of the earth:confused: As such the earth would be a discrete unit of the solar system, then again it would seem physical laws are not all that discrete unless you favor the idea that the universe itself is an aggregate of something. I think if the reality that the universe faces inflation post big bang that post heat death it should then shrink?

I guess I just don’t understand how you relate discreet to the reality that reality seems to be a whole as far as I can observer, as in we are made of star dust. Plus if the universe was discrete does that not mean you cant apply calculus to it?

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Well.

The universe/space appears continuous because of how we observe it, basically our eyes suck. But when you get down to the most fundamental level, thats where Planck's constant comes in, existence is discrete.

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Well.

The universe/space appears continuous because of how we observe it, basically our eyes suck. But when you get down to the most fundamental level, thats where Planck's constant comes in, existence is discrete.

So basically the forces described by the standard model happen to be what keeps the universe together? Then the big trick is basically evolving say how a solar system comes to be from that then?

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I was wondering if there are any theories which specifically require space to be continuous and are incompatible with a discrete approach.

As I'm sure you already know, a consequence of classical physics is that it assumes continuous space, so for any theory (I presume you mean a theory of gravity) to agree with QFT, the geometry of space needs to be discrete, that is volume, area et.c.

Continuous space is not really an observation flaw, which theCPE seems to be saying, (though I may have misread) but a mathematical one...infinitely divisible. For instance Maxwells equations rely on a continuous field, that is, it's not composed of quanta. This is fine with low frequency EM waves, but becomes a problem with higher frequencies. With regards to relativity, as gravity is a description of the geometry of space, at the Planck scale the geometry needs to be expressed in discrete units, otherwise you run into infinities, or indefinites. So in short, any theory that assumes continuous space, simply won't agree with QM...so isn't in anybodies best interest to pursue.

I think Bascule probably wanted an expert to answer, and somebody correct me if I missed anything with regards to the above.

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Continuous space is not really an observation flaw, which theCPE seems to be saying, (though I may have misread) but a mathematical one...

Its an observation flaw in that from our perspective space appears continuous with our naked eye and unsophisticated equipment.

I wasn't suggesting space must be discrete ONLY due to an observation flaw and there are no mathematical implications.

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I was wondering if there are any theories which specifically require space to be continuous and are incompatible with a discrete approach.

I'll try to say more about this later today, time permitting. As I see it the critical issue is can you handle gravity in a quantum theory without requiring some form of discreteness.

There are at least two lines of QG research that are getting a fair amount of attention which don't REQUIRE any kind of minimal length scale or any kind of discreteness.

discreteness can appear in various ways, so the discussion is inevitably a bit vague---what do you mean by discreteness, do you mean actually marbles and spiderwebs down there or do you just mean that the geometric spatial MEASUREMENTS we make (like area and volume) are necessarily quantized with only a discrete spectrum of values ..... I don't want to have to deal with semantic questions like that, so I remain vague

But there are at least these two QG approaches that don't have any discreteness in any way shape or manner-----you can let the scale parameter or the UV cutoff go to zero, and they don't blow up. Or they seem not to blow up.

One is Martin Reuter and Roberto Percacci Asymptotic Safety approach

the other is Renate Loll and Jan Ambjorn Triangulations approach.

You know how to use arxiv.org.

With Loll it's easy because no other Lolls.

http://arxiv.org/find/grp_physics/1/au:+Loll/0/1/0/all/0/1

With Reuter it's harder because there are several M Reuter

He's the one at Mainz University. But actually it is not so hard, most of the M Reuter papers are by him and not by others with the same initial.

http://arxiv.org/find/grp_physics/1/au:+Reuter_M/0/1/0/all/0/1

I think it's cool that they have models that they can compute with, that they are constantly making better, that handle quantum spacetime geometry------and that don't blow up when you let the scale go to zero.

It means that the model says it is OK either way! If there is some discrete structure really down there, then fine, i'm just an effective theory that can be used to compute numbers about spacetime down as far as you want, until you begin to see new physics. That's called an effective theory. Or maybe there is no discrete reality down below planck scale, in which case Loll Triangulations or Reuter Asymptotic Safety could actually lead to FUNDAMENTAL theories. It's nice to have that flexibility of being useful in either case however it turns out.

the thing with Loll is she uses triangles (tetrahedra, or their analogs called simplexes) but then she lets the size of the triangles go to zero. so it is basically a continuum theory---the continuum is just getting weird down at very small scale because it begins to look like fractals and the idea of dimensionality gets weird. but it's still continuum.

the thing with Reuter is he and coworkers found a sweetspot in the gravity action where you can let the scale go to zero and it doesnt blow up----they found a place where it's safe to take the limit as scale --> 0. People used to think there wasn't any such spot. Fair amount of computer work went into finding this socalled "fixed point" in the gravity action flow. A young guy named Frank Saueressig helped him. People used to think gravity was "non-renormalizable" because of this tendency to blow up at small scale. Finding the sweet spot means it's actually renormalizable after all (just not in so-called perturbative case that people are accustomed to consider). So that's cool too. None of this stuff is finally decisive. It is all still in progress.

Loll and Reuter approaches may eventually converge. Both have a similar picture of fractally spacetime continuum below planck scale. both get the same result about weird dimensionality at very small scale (but come to this by radically different methods).

the jury on discreteness is still out. IMO we simply don't know what things look like smaller than planck. (or even remotely close to planck IMO). Very fascinating stuff.

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Simple: The space is infite, the universe is discrete.

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I was wondering if there are any theories which specifically require space to be continuous and are incompatible with a discrete approach.

Yes, e.g. relativity and quantum field theory. Note that while they are incompatible with a discrete approach (due to appearing differentials), they are not necessarily completely incompatible with discrete space - they could be effective theories for large scales ("large" meaning something like the size of a nucleus or larger). Bjorken, Drell make a related statement in the 1st chapter (chapter 11.1 ) of their book "Relativistic Quantum Fields":

11.1 Consequences of a description via local fields

Before we proceed investigating to which results the application of quantization on classical fields obeying wave equations leads, the consequences of such an approach shall be discussed. The first consequence is that we are lead to a theory with differential wave propagation. The wave-functions are steady functions of continuous parameters $\vec x$ and t and changes of the field at point x are determined by the properties of the field at infinitesimally close points x'.

For most fields (e.g. sounds waves) such a description is an idealisation only valid for wavelengths greater than a characteristical scale that is a measure of the structure of the medium [the waves propagate in]. For smaller wavelengths, serious changes have to be made to the theory.

The electromagnetic field is a remarkable exception. [ ... bla bla, you all probably know about light and aether ...]. There is no proof for the existance of an aether in which the electromagnetic wave propagates in. It is, however, a very far extrapolation of our current [~1963] experimental knowledge to assume that the wave propagation which was very successful in describing systems of atomic scale (~$10^{-8}$ cm) can be generalized to infinitely small scales (e.g. scales of the size of a nucleus (~$10^{-13}$ cm)

[a section about self-energies and renormalization possibly being a hint about the breakdown of the differential approach]

We could ask why local field theories [...] then are so widely accepted? There is several reasons for that, including that with such theories a great agreement between theory and experiment has been achieved, [...]. But the most striking one is simply the following: There exists [again, we are talking about ~1963 !] no convincing theory, that works without differential equations for the field.

Lose translation from german done by me, buying the book highly unrecommended (due to its age).

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As I see it the critical issue is can you handle gravity in a quantum theory without requiring some form of discreteness.

Yep, that's exactly what I was getting at. Sorry for leaving it vague in the OP.

There are at least two lines of QG research that are getting a fair amount of attention which don't REQUIRE any kind of minimal length scale or any kind of discreteness.

discreteness can appear in various ways, so the discussion is inevitably a bit vague---what do you mean by discreteness, do you mean actually marbles and spiderwebs down there or do you just mean that the geometric spatial MEASUREMENTS we make (like area and volume) are necessarily quantized with only a discrete spectrum of values ..... I don't want to have to deal with semantic questions like that, so I remain vague

I do really like what Smolin keeps describing as an "evolving network of relationships." I guess originally when he was developing LQG it was a fixed lattice structure which had a number of problems, so he moved to Penrose's spin networks.

One is Martin Reuter and Roberto Percacci Asymptotic Safety approach the other is Renate Loll and Jan Ambjorn Triangulations approach.

You know how to use arxiv.org.

It's easy to use arxiv.org to find titles and abstracts that sound interesting to me but...

With Loll it's easy because no other Lolls.

http://arxiv.org/find/grp_physics/1/au:+Loll/0/1/0/all/0/1

With Reuter it's harder because there are several M Reuter

He's the one at Mainz University. But actually it is not so hard, most of the M Reuter papers are by him and not by others with the same initial.

http://arxiv.org/find/grp_physics/1/au:+Reuter_M/0/1/0/all/0/1

I'm afraid those are both over my head

I think it's cool that they have models that they can compute with, that they are constantly making better, that handle quantum spacetime geometry------and that don't blow up when you let the scale go to zero.

It means that the model says it is OK either way!

Now that is interesting...

Loll and Reuter approaches may eventually converge. Both have a similar picture of fractally spacetime continuum below planck scale. both get the same result about weird dimensionality at very small scale (but come to this by radically different methods).

The use of fractals is about the only way I can see continuous space making sense. Smolin's argument seemed to be that continuous space would necessitate infinite information while black hole thermodynamics essentially entailed black holes to contain finite information. With something like a fractal it would seem you could have both...

the jury on discreteness is still out. IMO we simply don't know what things look like smaller than planck. (or even remotely close to planck IMO). Very fascinating stuff.

What's your personal take on the whole thing?

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I think a related question when asking if space if discrete or not, is if the answer is conditional on the observer or not, because a question is formulated in the context of something.

Ie. maybe it's related to "observational resolution".

This is my personal take on the problem, that the key is not only in space itself, but as much as in the observer relating to it - is the observer discrete?

/Fredrik

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What's your personal take on the whole thing?

Bascule, I saw your question yesterday evening and started thinking about it.

I haven't got a ready answer. I'll try to respond when I get my thoughts in order.

My basic attitude is that it's like watching 4 or 5 mountainclimber teams all trying to climb the same mountain.

Or maybe you are getting periodic radio reports from each climbing party but the whole mountain is shrouded in fog and the teams can't see each other and cant see the whole mountain.

Some of the teams are probably going to merge----discover that they are both on the same approach.

I guess there are more than 4 or 5. There's one that hasn't reported in over a year----not on arxiv anyway. So you wonder how they'r doing.

You can't see the mountain because of the fog, so you can only try to guess the map of it by what you learn from the reports radioed by the climbers (and they can't see all that well either)

So to a certain extent I dont pick favorites. I don't bet on winners.

============

Of course I am somewhat selective. i stopped being interested in string approach a long time ago because they don't quantize geometry---they tend to depend on a fixed prior choice of background geometry instead of letting the geometry be undetermined or dynamic---and the artificiality of extra dimensions, and the landscape of 10^500 versions of physics---it just seems to have stalled back in 2003, if not earlier. I watch a broad range of research lines, but ignore some that don't seem interesting.

So I watch a select handful of approaches, and try to gauge their progress. Among those I don't choose favorites, and I also don't exclude new arrivals. Like those two Brazilians who just showed up: Pereira and Aldrovandi.

They might be a good example to talk about. P and A have an approach they call de Sitter General Relativity where they can CALCULATE (from basic numbers like the observed density of matter in the universe) what the apparent dark energy factor should be and what delay MAGIC should observe in gammaray flares, from a given distance.

In other words, with classic GR you have to put Lambda in by hand, and its value is mysterious. It is a small positive energy density that has no conventional explanation---to account for the observed acceleration.

classic GR is Poincare GR and they say to modify it by using the deSitter group instead of the Poincare group. And they do that and get a slightly different Einstein Field Equation and a slightly different Equivalence Principle. And then they turn out to be able to calculate stuff that you otherwise couldn't.

Now this needs very much to be TESTED, by more observations by MAGIC telescope, to see if their prediction of the delay is right when you look at different flares that have traveled different distances. They got the 4 minute delay on that one flare, but would they be right the next time. And that was just one flare so the data may be wrong in the first place.

So P and A version of GR needs to be tested. But it raises this issue: if we are going to unify QM and GR then it will presumably work better if it is the right version of GR, and maybe classic GR is the wrong one.

Maybe a successful quantum version of General Relativity will turn out to be a quantum version of DE SITTER General Relativity.

that is one detail of the picture. Their paper just appeared and illustrates why it is hard to pick winners. But if I were going to list favorites, here is what I'd say at this point.

Basically I like Rovelli's new spinfoam model (big advance this year)

and Ashtekar's quantum cosmology (how it treats the big bang)

and Loll's triangulation approach (because they run little universes in the computer)

and Reuter aymptotic safety.

I think that as people begin to get quantum geometry/gravity right they will see how to include the standard particles in the picture as aspects of geometry. Smolin and a bunch of people are currently trying out a way to do that.

I think that cosmology is the proving ground for QG. That is coming out clearly in the work of Reuter and Ashtekar and I expect that Rovelli spinfoam will be put to cosmological test. Each one of these names stands for half a dozen others as well.

A strange paper like that about deSitter GR could appear and if observations confirm its predictions that could have a profound effect on the approaches that I mentioned by changing the version of GR that they are trying to quantize. that would be a surprise shock, and shocks like that are possible

that's all that occurs to me for now, maybe I'll have some more definite take on the QG scene later.

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Yes, I believe space is discrete, don't see how it could be anything other.

String theory, with or without "string bits" is silly.

Yes, me too. Long time ago space (well they said matter) was thought to be continuous, but nowdays physics tells about it's discrete feature!

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Smolin's argument seemed to be that continuous space would necessitate infinite information

This is the flavour of reasoning I like.

I have tried to, but I fail to imagine how a general observer can relate to a continuum. I've also tried to connect logic reasoning to a continuum, and it seems the only logical and rational way to introduce the continuum is as a limiting case of discrete models. In effect this is what we do when we introduce real numbers in mathematics. When we are dealing with continuum models I think we are dealing with limiting cases. And can there be interesting physics taking place where this limiting case is invalid? I think so.

/Fredrik

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I did want to interject Lee Smolin's take on this. A consistent theme of his book Three Roads to Quantum Gravity is that physicists are presently in a similar state to the observation of atoms regarding the discreteness of space. That is to say: many have discovered the prerequisite knowledge which has lead them towards the impression that space is discrete.

Here's specifically what he had to say on the matter:

...we looked at three different approaches to quantum gravity: black hole thermodynamics, loop quantum gravity, and string theory. While each takes a different starting point, they all agree that when viewed on the Planck scale, space and time cannot be continuous. For seemingly different reasons, the end of each of these three roads one reaches the conclusion that the old picture according to which space and time are continuous must be abandoned. On the Planck scale, space appears to be composed of fundamental discrete units.

Rather than being continuous, Smolin argues that space is comprised of an evolving network of relationships.

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another comment on this

Smolin elucidated me as to the idea that strings are comprised of "string bits" which are discrete.

I'm not a string advocate at all and won't be one, but it has occured to me as well that the simplest possible extension to a boolean record, as the memory size increases is a structure defined by the history or the flipping states, which in in the continuum limit defines the probability. Not consider that this probability is changing, and is thus unstable. One can easily imagine this as the probability range 0-1 defining the coordinate along a string with unit length, and the uncertainty of the value at each position can defined a new dimension - so as to see this as a one dimensional string, swinging into further dimensions, and the dimensionality of the external space is related to the "complexity of the uncertainty".

This way, the origin of a string is understandable, and IMO the string dynamics can be deduced from more first principles. Ie. there is no need to postulate existence of elementary strings.

I have not read smolins original suggestions of what he means with string bits, but if this has any relation to it, then the string bits are really more to be seen as information quanta. And the string is more like the simplest possible imaginable "pattern". And this patterns can further be excited and so on. This has the potential to make percet sense.

But one would only wonder what took them so long. I guess it's because this is supposedly a philosophical question, like everything else there are no clean and obvious routes to find the answers to

/Fredrik

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Lee Smolin argues pretty convincingly that space is discrete at the Planck scale.

Out of curiosity, is Smolin actually arguing this or presenting it as a possibility? As Atheist points out, GR and QFT make predictions on arbitrary length scales. Martin reports on LQG research producing models that might mature into effective field theories. If Smolin is making a positive argument for a fundamental quantization of topology, I'd like to know the outline of his case.

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... Martin reports on LQG research producing models that might mature into effective field theories...

I'm not always as clear or consistent as I should be. This is just to clarify. Non-string QG is a broad class of approaches and most (or a substantial representative sample) showed up at LOOPS '07 this summer, so you could consider them part of the LQG community. But the language gets confusing because these are different (and even to an extent competing) approaches. I wasn't there but I've studied reports from the conference fairly closely and have listened to a number of the talks, which are online.

http://www.matmor.unam.mx/eventos/loops07/plen_abs.html'>http://www.matmor.unam.mx/eventos/loops07/plen_abs.html

Just for starters, one invited speaker was an effective field theory guy, Donoghue, that was the title of his talk.

===

John F. Donoghue: Effective field theory of general relativity · slides (pdf) · audio (mp3)

I will review the foundations and recent progress in the use of effective field theory to elucidate the quantum predictions of general relativity.

===

One who made a fairly big splash was Martin Reuter, whose Asymptotic Safety approach would lead to either an effective (or possibly he may hope a fundamental) field theory. Reuter's approach is definitely non-discrete.

Another plenary speaker was Jan Ambjorn, who triangulations approach is supportive of Reuter's----the triangulation scale goes to zero---and could also be seen as leading to an effective or fundamental field theory depending on how things work out.

So subjectively in terms of visibility and impact (these guys especially Reuter got a lot of attention) the NON-DISCRETE approaches seem roughly 50 percent of the show, theory-wise.

But I don't want to give the impression that LQG proper is part of that. The Loop community is rather open, so they include approaches different from their own.

Conventional LQG going back to the 1990s and the other closely related stuff that it has spawned has a definite discrete character (because the area and volume operators turn out to have discrete spectrum).

So I would definitely agree with saying "non-string QG research producing models that might mature into effective field theories.."

or "Loop-community research producing models that might mature into effective field theories.."

But I shy away from saying " LQG research producing models that might mature into effective field theories.."

because of the semantic confusion. People could get exactly the wrong idea. Because LQG proper doesn't seem to me to be going in that direction.

The trouble is, people use the term LQG as a general catch-all for the competition to string, and also as a specific term meaning something definite. This almost always leads to confusion.

If you want an overview, here is the conference website

http://www.matmor.unam.mx/eventos/loops07/

Maybe about a third of the emphasis now is computer modeling, phenomenology, and application to cosmology and black holes.

Maybe about a third is non-discrete theory ( triangulations, asymptotic safety...)

And roughly a third is discrete (LQG, spinfoam, causal sets)

Another reality check will be the conference around 1 July 2008 in the UK

called "QG2 - 2008" standing for Quantum Geometry and Quantum Gravity.

The website has not yet posted the list of plenary speakers, registration just opened.

http://www.maths.nottingham.ac.uk/conferences/qgsquared-2008/

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Conventional LQG going back to the 1990s and the other closely related stuff that it has spawned has a definite discrete character (because the area and volume operators turn out to have discrete spectrum).

Just out of curiousity, I understand that the area and volume operators are constructed from surface integrals with discrete spectra, but could you summarize what Reuter and Loll are doing differently?

But I shy away from saying " LQG research producing models that might mature into effective field theories.."

because of the semantic confusion. People could get exactly the wrong idea. Because LQG proper doesn't seem to me to be going in that direction.

I don't mean to suggest that's where LQG is going as a field, but wouldn't you consider Loll and Reuter leading voices in that area? I haven't read the papers yet, but my impression from your report is that Asymptotic Safety and triangulation are LQG products and alternatives to rank-3 quantization. If I missed something, I think I'll need time to digest the lit.

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Just out of curiousity, I understand that the area and volume operators are constructed from surface integrals with discrete spectra, but could you summarize what Reuter and Loll are doing differently?

the most efficient thing would be for you to glance at Loll's paper. It is relatively easy reading, illustrated, pedagogical. Loll and Ambjorn call their approach Causal Dynamical Triangulations (CDT). I say "triangulations" for short, which is sloppy. I should say CDT but i like to avoid acronyms.

http://arxiv.org/abs/0711.0273

On the face of it, CDT is totally different from Reuter and both of them are totally different from LQG.

Reuter calls his approach QEG (quantum einstein gravity) and sometimes to make it clear he says "Asymptotically safe QEG".

They are all part of the same community. Grad students and postdocs cross lines and circulate amongst approaches, everybody shows up at the annual conference. The community has no name. I call it non-string QG------one could say background independent/nonperturbative QG----or just "Loop community" because LQG is the oldest largest part. Most of the nominally LQG people are working on spinfoam rather than the older 1990s type LQG. Frustrating thing is it's hard to generalize. Whatever you think you know about LQG is probably not true about CDT, or QEG, or Spinfoam.

Reuter QEG and Loll CDT both date from around 1998---when first paper was published in each case. Neither of them have area or volume operators, as yet. (at least that I know of). Until 2007 the Spinfoam approach did not have area and volume operators. Just this year they say they linked up with LQG and got the same spectra. I'm waiting to hear more about that.

I'll try to indicate the main differences amongst the leading approaches

Everybody starts with a diff. manifold. they want some d.o.f. that describes the geometry of that manifold. the space of geometries is their configuration space, and they want quantum states to be defined thereon.

1. Loll CDT describes geometries by TRIANGULATIONS of the manifold. It is a PATH INTEGRAL approach where you can calculate an amplitude for each triangulation of spacetime. you add up the amplitudes of all the ways of evolving from initial to final. (this is one you can read about, Loll writes clearly). At each scale CDT can calculate amplitudes and expectation values etc. and then they let the scale of the triangulation go to zero. A lot of their work is numerical (in computer).

2. Reuter QEG describes geometries by METRICS the way Einstein himself did (that is why he calls his approach "quantum einstein gravity"). His thing is that everybody thought this would not work but in 1998 he got some results suggesting that it would work after all and evidence has been piling up since then. Key words: UV fixed point in the renormalization group flow. Solutions in QEG are scale dependent. The constants like G and Lambda that go into the action run with scale. Gravity at early universe energy is closer to the UV fixed point and behaves different from how it does now. A young string theorist named Frank Saueressig has switched over to this approach and just brought out a paper on it this week. Look up Saueressig on arxiv. Nice to get a young newcomer perspective.

But there is no QEG paper I can think of to recommend that is as easy to read as Loll's CDT latest. If you want an intro QEG, tell me and I will think about it.

3. Conventional LQG describes spatial geometries by CONNECTIONS on a smooth spatial slice---a connection is like a parallel transport machine. It is an alternative way to describe geometry without using a metric or a triangulation. Then conventional LQG needs a constraint operator to check to make sure that the instantaneous spatial geometry state could have evolved up to that point and will be able to evolve onwards past that point.

It is the only one of the approaches that is not explicitly 4D.

4. Spinfoam is a PATH INTEGRAL approach in some way like CDT, the spacetime is a way geometry could evolve from initial to final state. But the spacetime geometry ( the path thru the space of geometries) is not described by a triangulation. It is described by something analogous to a labeled graph but made of 2D polygons---mathematicians call it a "two-complex", but QG people call it a foam, or rather, since it is labeled by spin numbers, they call it a spin-foam. A quantum state of spacetime geometry is descibed by a spin-labeled two-complex, aka a spinfoam

1. in one approach the basic thing is a triangulation

2. in another, the basic thing is a metric

3. in another it is a connection (spin networks are functions defined on connections)

4. in another it is a spinfoam (stand-alone, not defined on anything)

Some not all of these approaches are path-integral type---fully 4D.

=================

I'm sorry I can't give a more clear understandable overview but the main point I'm trying to make is how DIFFERENT the various approaches are.

they even start off with geometry described by different degrees of freedom. so they are different at the root.

the most interesting part of the story is probably the convergence.

CDT and QEG have found similar results at very small scale (we heard about in 2005) still not explained

This year there was a convergence of oldstyle LQG and Spinfoam where they apparently got agreement on several things. (A path integral and a hamiltonian approach turning out to be compatible.)

I know you are busy so I don't want you to waste your time trying to read a lot of papers. I would suggest that anyone could glance at the intro and conclusions of Loll's paper "Quantum Gravity on Your Desktop". but not to get bogged down in any of it.

then if anyone wants an efficient introduction to Reuter Asymptotic Safe QEG, tell me and I will figure out an optimal paper to look at. Might be the young guy Saueressig. Might be Reuter, might be Percacci.

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If Smolin is making a positive argument for a fundamental quantization of topology, I'd like to know the outline of his case.

Space as a discrete, evolving network of relationships is the central theme of his book Three Roads to Quantum Gravity.

His case involves looking at specific details of three different approaches to quantum gravity and extracting commonalities that would lead one to believe that, at the Planck scale, space is discrete. With loop quantum gravity the case is rather straightforward as space is modeled in terms of spin networks (which I believe is the same thing as spin foam... maybe Martin can explain) which are by their very nature discrete.

He devotes a chapter to another of the three roads, black hole thermodynamics, arguing why a correlation between information and a the area of a black hole's event horizon (rather than its volume) implies discrete space.

The case with string theory is rather roundabout as he starts off describing string theory within a continuous background. He introduces discrete entities into string theory called string bits which he explains are somehow necessary, but as I've never heard of string bits outside of his book I'm guessing most string theorists aren't going to agree that string theory necessitates discrete space.

Have a look at the bit of the book I quoted earlier if you want to judge his convictions regarding his intent to express that space is discrete.

His writing likens the idea that physicists are waking up to the idea that space is discrete in the same way that chemists and physicists in the 19th century started to believe in atomic structure prior to its actual observation.

He's not beating around the bush here. He's flat out saying that to the best of our knowledge space is discrete.

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He introduces discrete entities into string theory called string bits which he explains are somehow necessary, but as I've never heard of string bits outside of his book I'm guessing most string theorists aren't going to agree that string theory necessitates discrete space.

I suspect that if the strings are decomposed into smaller units, then it means strings aren't fundamental anymore - at least relative to the decomposition. However like in any axiomatic system approach, one can easily imagine that what is an axiom in one system is a theorem in another one, as long as they are consistent. Perhaps in the elementary string thinking the "bits" aren't given any ontological status of "physical bits", but merely bits that occured due to choice of description.

This relates to some other opinions if it was here or in anoher place, that people take on different views on models. If you consider the mathematical model just as as human model, one could mathematically imagine continous things, but there the continuum has no physical basis. It's just one of out probably many possible mathematical models.

But I have problems with such thinking - to me it is to not take the model seriously. I think the modelling should be kept in contact with reality and map onto the physical reality to the maximum possible extent. This is also related to the use of symmetries in physics. On one hand unbroken symmetries are trivial, and thus representing a complete redundancy. Broken symmetries are otoh the interesting ones, and it seems the breaking of symmetries is a key focus.

This is in particular relevant when you consider that ideally the laws of physics must exists also to others than humans. Therefore I think that ideally the models should some way or the other be representable as physical structures. I share part of this with tegemarks thinking. IMO the point is not to say that physics is mathematics, but rather to take the mathematical models seriously and find the minimal model that exactly maps onto reality. In this way, consider a particle. Then it seems the concept of efficiency of representation becomes important. What are the limits a small system can physically related to? How does two small systems actually communicate? They can hardly afford the luxurous redundant setting of human science? Yet, don't we expect that the same logic applies to a small subatomic system as to a human - in principle? when searching for a unified theory scalable over complexity?

Regardless of what one thinks of string theory I still think that it is an interesting link, because it can provide a link to other approaches and provide a sensible way to connect the approaches and perhaps find common denominators.

Another way I see a discreteness as the most natural starting point goes back to the philosophy of science and logic. In logic it seems the simplest possible thing is a statement that is true or false, a boolean statement. Using this, together with axioms, we have been able to construct impressive theories and framework for continous mathematics, but they can be thought to sort of all boil down to, or be built on, discrete logic from which relations and systems are built.

/Fredrik

I hope this isn't considered off topic, but I see it as closely related.

A very similary question is, while respeting measurement ideals: how can we RELATE to a infinitely resolved (continuum) probability? How? the analysis with frequentist interpretations shows several issues, one of them beeing the memory requirement. The memory directly limits the resolution possible of the probability/microstructure.

If we distinguish between the continuum of the formalism and the physical probabilities then the formalism contains nonphysical redundancies. If you don't care about this, fine, but if we look for the minimum representation the redundant degrees of freedom are wasting our resources.

Even the first principle considerations suggest that our probability spaces are discrete, and thus considering them microstructures forming a physical basis for encoding is better, but they make up a discrete probability space.

If someone OTOH really believes in infinite information inside bounded structures, then I have serious difficulties to understand how than can be turned into something constructive and connect to logic. It doesn't seem to make sense, not to mention how you would be able to pull time out of processing of infinite sets.

/Fredrik

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...case is rather straightforward as space is modeled in terms of spin networks (which I believe is the same thing as spin foam... maybe Martin can explain) which are by their very nature discrete.

...

He's not beating around the bush here. He's flat out saying that to the best of our knowledge space is discrete.

this is a really interesting issue.

to keep it in perspective though, 3-Roads was a popular book written IIRC around 2001, published 2002----I don't know what tone of voice Smolin would take now if he were to make a popular speculative book around that issue. the idea of an Atomism revolution regarding spacetime geometry is a great and inspiring vision, but I just don't know what his current attitude would be.

also to keep it in perspective, when Smolin speaks as a scientist to other scientists he, in effect HAS NO CONVICTIONS*. Every idea is a theory or model which one develops as best one can and then TESTS. His only belief that I know of is in empiricism. It is worth developing an idea even if it is wrong because if you work hard on it and bring it to the point of making unambiguous testable (falsifiable) predictions and then Nature says it is wrong then you have learned something.

I've heard the empiricist message loud and clear and repeatedly from him. But in professional writing I do not recall ever hearing him express a belief about spacetime Atomism. I have heard him express a desire to TEST spacetime discreteness and often times a HOPE that it can be tested with orbital gammaray instruments like GLAST.

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He would probably like to see some dispersion in arrival time of different parts of a gammaray burst---some evidence of energydependence of speed of light. But that would not be decisive. Maybe in 2001 it seemed so.

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A spinnetwork is a GRAPH made of vertices and edges which embeds in 3D space and helps describe the quantum states of 3D space geometry. (the graph is labeled)

A spinfoam is analogous to a graph but with one extra dimension---it is made of vertices, edges, and faces (bits of surface)----and it embeds into 4D spacetime. A spinfoam illustrates how one spinnetwork can evolve into another. Think of dragging a network thru time and the edges leave blurs which are the faces of the foam. (technically the foam is labeled)

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He's flat out saying that to the best of our knowledge space is discrete.

Have to be careful though. the picture has changed enormously since 2001. Back in 2001 a man says "our best knowledge of microscopic spacetime geometry, our most advanced models anyway, are string, loop, and black hole thermo-----and I see evidence of discreteness in all three."

OK, fair enough. He was talking about vintage 1990s Loop Quantum Gravity.

There has been profound change in the nonstring QG community in the past 5 or 6 years.

I wouldn't say "to the best of our knowledge" in any case because we don't know very much until we have testable theories and test them by actual observation, but I would make sense of that phrase by saying "according to our most developed models".

But in 2007 "most developed models" means something different from string and vintage-1990s LQG.

And NOW our most developed models don't give me any feeling of unanimity regards the Geometric Atomism issue.

Don't get me wrong. I don't DISbelieve it either. I'm just waiting.

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*no convictions about what's the right theory. one should have moral convictions. uphold the empiricist ethic essential to the continuing existence of the science community etc. Smolin's recent book Trouble looks objectively at the menu of theories and says they could ALL be wrong. But it is very strong on the MORAL message. Different approaches should be tried. don't let a clique monopolize. Avoid groupthink. Insist that every theory be empirically testable. Give students choice and encourage independence... The book is passionate but not about pushing one particular model of nature, whether atomist or fractal.

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BTW the two most prominent non-Atomist approaches (Reuter's and Loll's) are both very different but curiously both arrived at a fractal-like picture of microscopic spacetime. Sounds crazy and I guess either or both approaches could get thrown out. there is no predicting the future of research. but for the moment, Atomism is not the only cool idea that lurks in the head of Mankind.

and I seriously doubt it is the only cool idea that lurks in the head of Smolin in particular.

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Bascule, did you TRY to read Loll's http://arxiv.org/abs/0711.0273 ?

You must have been sampling something else. I don't think you tried that one, the "Quantum Gravity on Your Desktop" paper.

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Bascule, did you TRY to read Loll's http://arxiv.org/abs/0711.0273 ?

You must have been sampling something else. I don't think you tried that one, the "Quantum Gravity on Your Desktop" paper.

Okay, I scrolled down to page 2 and I see pictures. That's encouraging.

I'll try to read it at some point here.

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