Is Space-Time a Continuum or a Physical Object?

[TrickyDicky]

GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.

 

[phinds]

This is a somewhat contentious issue with no known resolution. There are numerous threads on this forum discussing and you might do a search to check them out, but what it boils down to is we don't know.

 

[marcus]

GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.

There are indications, are there not?, that geometry is also discrete when probed at small enough scale. What are these indications? You certainly know some. We can't be sure yet, but there are reasons to suspect geometric discreteness---what are some of them?

Jacobson's 1995 derivation of GR as a thermodynamic equation of state?
So what might be the geometric molecules of which GR is the EoS?
Finite entropy beyond bh horizon?
Continuum GR cannot be entirely right since it fails at extreme density?

 

[marcus]

I wonder if this old (1983) paper of Rafael Sorkin is relevant.
http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/31.padova.entropy.pdf
He argues for geometric discreteness simply based on the finiteness of bh entropy
==quote Sorkin intro==
The evidence is very strong that a black hole presents itself to the outside world as a thermodynamic system with entropy proportional to its surface (horizon) area. Yet the physical origin of this entropy is far from clear. In fact the formula S = k lg N , on which our general understanding of the Second Law is based, entails the absurdity S = ∞; for— unlike in flat space—a bound on the total energy does not suffice to bound the number of possible internal states. In particular the Oppenheimer-Snyder solutions [1] already provide an infinite number of possible internal configurations for a Schwarzschild exterior of fixed mass.

A related observation is that the internal dynamics of a black hole ought to be irrelevant to its exhibited entropy because—almost by definition—the exterior is an autonomous system for whose behavior one should be able to account without ever referring to internal black hole degrees of freedom. In particular one should be able to explain why it happens that a sum of two terms, one referring to exterior matter and the other only to the black hole geometry, tends always to increase.
==endquote==
==quote Sorkin conclusions==
On dimensional grounds it is easy to see that S will be ultra-violet infinite in the continuum limit l→0. ...
To obtain an entropy of the correct order of magnitude for a black hole, the cutoff l must be chosen approximately equal to the Planck length. Conversely, if S^ext really can be identified as the black hole entropy we obtain evidence of the physical necessity for such a cutoff to exist.
...
...
==endquote==

 

[Chronos]

Spacetime quantization is an active area of research, but, the results are inconclusive. In theory, extremely energetic gamma rays from distant sources should suffer measurable dispersion if spacetime is quantized. It is expected this would result in a time delay vs the arrival time of lower energy photons. This, of course, assumes that GRB gamma emissions at different frequencies occur at a single point instantaneously, which is obviously debatable. The Fermi LAT gamma ray telescope has been utilized to conduct such measurements. Data to date is inconclusive. One possibility is quantization occurs at sub-Planck scales. This is somewhat unpalatable to some scientists, but, I'm unconvinced the Planck scale is a brick wall as opposed to a human convenience - e.g., obviously matter exists at sub Planck mass scales. For further discussion see: http://arxiv.org/abs/1109.5191, Bounds on Spectral Dispersion from Fermi-detected Gamma Ray Bursts; and, http://arxiv.org/abs/1406.4568, Lorentz violation from gamma-ray bursts.

 

[Chronos]

I wonder if this old (1983) paper of Rafael Sorkin is relevant.
http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/31.padova.entropy.pdf
He argues for geometric discreteness simply based on the finiteness of bh entropy

Sten Odenwald seems to think so - http://www.astronomycafe.net/qadir/BackTo286.html

 

[marcus]

Note that Sorkin's approach to geometric discretness preserves Lorentz Invariance.
This is one of the first things proved in the Perimeter lectures on Causal Sets, that Dowker and Sorkin presented a year or two back.
Same with LQG, there is that paper by Rovelli and Speziale proving Lorentz invariance.

So we don't want to get geometric discreteness MIXED UP with the old idea of "graininess" which somehow leads to higher energy gamma arriving EARLIER or as other people imagined LATER than other light. That old idea never seems to go away, does it? : ^)
There is it cropping up in that "Astronomy Cafe" piece by Sten Odenwald! The old L.I.V. idea ("lorentz invariance violation")
We have to learn to make a sharp distinction.

 

[marcus]

T.D.
You might be interested in this January 2014 paper by Chirco et al. They take Jacobson's 1995 idea one step farther and identify what the microscopic DoF of which GR is the equation of state could actually be!
==quote page 1 of http://arxiv.org/pdf/1401.5262v1.pdf ==
...The microscopic degrees of freedom are those of the quantum gravitational field and the Einstein equations express only the classical limit of the dynamics. The entropy across the horizon measures the entanglement between adjacent spacetime regions. Its finiteness is evidence for the quantization of the gravitational field: this is analogous to the fact that the finiteness of the black-body electromagnetic entropy is evidence for the quantization of the electromagnetic field.

We show that the Jacobson result is consistent with this simpler and tighter scenario. The finiteness and the universality of the entanglement entropy across spacetime regions indicates ultraviolet quantum discreteness, as it did for Planck and Einstein at the beginning of the XX century.
==endquote==

Forgive me if I say something in probably the dumbest crudest way, by comparing geometry (as we come to understand it now) with LIGHT as people came to understand it early in XX century.

there is no need to imagine that light is "grainy" when it is just traveling along minding its own business. The quantization is observed when it INTERACTS with something material---when it is emitted, or absorbed, or scattered, etc.

So I am not imagining that spacetime is a grainy material or "fabric". It can be as continuous as it wants ON ITS OWN TIME when I am not making some geometric measurement. Or not! I think it is meaningless to talk about how geometry is when there is no interaction. In any case, the discreteness comes in only when I interact with it, e.g. have some materially physically defined angle, or area, or volume. I have something I want to measure, and therefore there is an interaction of some kind.

Just quietly sitting there being the geometry, defining the geodesics and what the triangles would add up to IF you bothered to measure, that is neutral, it is the "default" It is not an interaction for light to be following the geodesics, because where else is it going to go?

The geometric quantumness only shows up when there are events. Am I being inconsistent, or too vague?

 

[bahamagreen]

Is it meaningful to ask if these two questions are the same?

1] Is the universe continuous?
2] Is spacetime continuous?

And... continuity of what, exactly?

Is it meaningful to distinguish asking about the continuity of what comprizes the universe or spacetime, or asking about the continuity of the "physical laws" (in the universe or spacetime)?

Also, doesn't the universality of local c assume or require some strict assumptions about continuity? Continuity is always a locally applied concept so even under GR continuity would hold "everywhere"... right?

 

[marcus]

Let's talk about the geometry , not about "space-time" as if it were some naive thing like a "fabric", or some weird sort of material substance.

Geometry means measurements (...metry) and it means geometric relationships, and geodesics, and between/not between, bigger area than, or not, intersecting, or not. Bigger or lesser angle.

if someone has to think of it as some kind of material thing, let it be a WEB of measurements/relationships.

"Continuous" would mean you can keep dividing a distance by two forever. No smallest MEASUREMENT.
"Discrete" means you can't. You can't measure below a given positive size.

I think there probably is a a smallest distance that you can measure, and if you try to measure a smaller distance then you might more or less randomly get a zero result, or the minimal eigenvalue. Have to go.
Back now. Yeah, the idea is that if you TRY to measure a width that is less than the minimal positive measurable distance "l" then Nature won't allow it and you will get a "fluctuation" between zero and the minimal "l".

We are not concerned with what somehow "really exists" down at that scale---it doesn't mean anything. Only measurements matter. We are concerned with how Nature responds to measurements. That's what geometry is. And why I'd say it is not continuous.

 

[td21]

Planck constant may be the smallest piece we can measure. But I still cannot comprehend what discreteness means in terms of space.
In my opinion, space is defined to be continuous (isn't it)? One may assign values to certain space-coordinates (to create the discreteness), but the coordinates itself is always continuous.

 

[dragoneyes001]

wouldn't extreme gravity be a factor in this. if the universe can be warped by "super" black holes wouldn't that allow for other forces to affect the shape and continuity of the whole?

 

[marcus]

For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
http://www.signalscience.net/files/Regge.pdf
That goes back to 1960 when Regge was at Princeton. His 1960 paper was called
General Relativity Without Coordinates
It is a famous paper that introduced a famous approach (Regge calculus) that succeeded in doing Einstein General Rel without using coordinates.
That link gives a fax of the original paper.

Since then there have been many ways developed to treat spacetime using discrete entities, like simplices, cell-complexes, or graphs, rather than using the usual continuum model (the differential manifold).

Just to take one example, there is so called "CDT" (causal dynamical triangulations) as developed by Renate Loll and Jan Ambjorn, with others.

I wouldn't say that a clear winner has emerged in the competition for "best discrete model of spacetime geometry" but it cannot be said that it is
"continuous by definition." : ^)
Causal Sets, as developed by Rafael Sorkin and Fay Dowker, with others. If anyone wants they can do a search by author names at arxiv.org and find a large collection of paper that appeared over the past 10 years, and some earlier.

 

[td21]

For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
http://www.signalscience.net/files/Regge.pdf
That goes back to 1960 when Regge was at Princeton. His 1960 paper was called
General Relativity Without Coordinates
It is a famous paper that introduced a famous approach (Regge calculus) that succeeded in doing Einstein General Rel without using coordinates.
That link gives a fax of the original paper.

Since then there have been many ways developed to treat spacetime using discrete entities, like simplices, cell-complexes, or graphs, rather than using the usual continuum model (the differential manifold).

Just to take one example, there is so called "CDT" (causal dynamical triangulations) as developed by Renate Loll and Jan Ambjorn, with others.

I wouldn't say that a clear winner has emerged in the competition for "best discrete model of spacetime geometry" but it cannot be said that it is
"continuous by definition." : ^)
Causal Sets, as developed by Rafael Sorkin and Fay Dowker, with others. If anyone wants they can do a search by author names at arxiv.org and find a large collection of paper that appeared over the past 10 years, and some earlier.

Thank you for introducing me to this wonderful paper. I said space is continuous by definition because I don't know what the stuff in between would be called if space is discrete.

 

[TrickyDicky]

Ideal measurements are obviously discrete, then one could argue that in practice no ideal measurements can be made.
I think we could all start by agreeing that nature is not scale-invariant, and what that implies.
For many experts this simple fact discards both a fractal and a continuous universe, solving the issue in favor of discreteness.

 

[phinds]

Ideal measurements are obviously discrete ...

But it seems to me that ideal measurements measure what IS. If what IS is not continuous then it would not measure as continuous. a simple example would be making measurements of the distance between a series of fence posts that are 10 feet apart. It just doesn't make sense to talk about a measurement of 23.78... feet in that realm.

 

[Torbjorn_L]

GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.

I don't think that is correct.

GR describes spacetime as smooth, but you can insert singularities like particles from quantum field theories in semiclassical physics or perhaps idealized black holes (I'm not well versed in GR).

- If you ask if there is a problem, evidently not. GR is known to be an effective, quantizable theory, so it both allows and has particles (at low energies, at least). Same goes with string theory, discrete strings on smooth branes.

- If you ask if space (or time) is discrete, it is an open question.

As Chronos notes observations arguably seem to imply relativity (so smoothness) is valid below scales where presumed discreteness ought to have kicked in.

Sorkin's approach to geometric discretness preserves Lorentz Invariance

I would like to see a reference. All previous examples failed to do that, as would be expected. All discrete approaches to gravity also fails to have a dynamics (no harmonic oscillators) mainly due to that they fail to have an energy (no lower energy bound), so the attempt to use thermodynamics should be erroneous.

Sorkin's other paper is simply claiming to put harmonic oscillators of quantum field theory (say) on a lattice, but it could as well be a particle gas of some well defined density. It is a bait-and-switch, "it is a tale. Told by an idiot, full of sound and fury, Signifying nothing."*

For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.

That looks like another bait-and-switch. The Regge paper makes its skeleton spaces into differentiable manifolds by smoothing them.

Sure, geometry (of GR, say) can be expressed without coordinates. So what? String theory use that, doesn't it, when it puts up a GR scaffold solution and then remove it (with or without coordinates) as I take the procedure is?

*I'm not saying that Sorkin is an idiot, maybe I am. And he is technically proficient in his field, which isn't mine. But the intent of the paper looks superficially like idiocy.

 

[martinbn]

For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
http://www.signalscience.net/files/Regge.pdf
That goes back to 1960 when Regge was at Princeton. His 1960 paper was called
General Relativity Without Coordinates
It is a famous paper that introduced a famous approach (Regge calculus) that succeeded in doing Einstein General Rel without using coordinates.
That link gives a fax of the original paper.

Since then there have been many ways developed to treat spacetime using discrete entities, like simplices, cell-complexes, or graphs, rather than using the usual continuum model (the differential manifold).

Just to take one example, there is so called "CDT" (causal dynamical triangulations) as developed by Renate Loll and Jan Ambjorn, with others.

I wouldn't say that a clear winner has emerged in the competition for "best discrete model of spacetime geometry" but it cannot be said that it is
"continuous by definition." : ^)
Causal Sets, as developed by Rafael Sorkin and Fay Dowker, with others. If anyone wants they can do a search by author names at arxiv.org and find a large collection of paper that appeared over the past 10 years, and some earlier.

http://sigrav.na.infn.it/tullio-regge/

 

[TrickyDicky]

I don't think that is correct.

GR describes spacetime as smooth, but you can insert singularities like particles from quantum field theories in semiclassical physics or perhaps idealized black holes (I'm not well versed in GR).

GR is known to be an effective, quantizable theory...

Wow, so to you GR is quantizable and discrete. I think this indicates your admitting not being well versed in GR is somewhat an ironic understatement.

 

[TrickyDicky]

Is it meaningful to ask if these two questions are the same?

1] Is the universe continuous?
2] Is spacetime continuous?

And... continuity of what, exactly?

Is it meaningful to distinguish asking about the continuity of what comprizes the universe or spacetime, or asking about the continuity of the "physical laws" (in the universe or spacetime)?

These are important points, to the first I would reply that universe is not necessarily the same as spacetime, for instance if we make the distinction between spacetime as something different from its content.
The second is also key, continuity of what? Depending on whether one is essentialist, substantialist, ... Etc will respond differently.
The thirs leads us to discuss about how to define physical laws, and relate it to whatever it is that we consider discrete or continuous.

But it seems to me that ideal measurements measure what IS. If what IS is not continuous then it would not measure as continuous. a simple example would be making measurements of the distance between a series of fence posts that are 10 feet apart. It just doesn't make sense to talk about a measurement of 23.78... feet in that realm.

Since in reality there are no ideal measurents by definition we find that physical measurents must introduce some sort of scaling and coarse graining that deviates from what otherwise would be a scale independent process, this is usually dealt with when studying the renormalization group.

 

[PeterDonis]

I don't know what the stuff in between would be called if space is discrete.

If space is discrete, there isn't any "stuff in between". "Discrete" doesn't mean a bunch of discrete lattice points superimposed on a continuous background. It means discrete, period.

 

[George Jones]

Presumably, "discrete" means that, a set, spacetime is countable.

 

[phinds]

Since in reality there are no ideal measurements by definition we find that physical measurents must introduce some sort of scaling and coarse graining that deviates from what otherwise would be a scale independent process, this is usually dealt with when studying the renormalization group.

No argument that there is a point below which we cannot make (and likely never will be able to make) actual measurements but I don't get how that relates to whether or not what we are measuring is continuous, based on theory. If I have a meter stick that will only measure down to a millimeter, that does NOT constrain my height to be an integer multiple of millimeters.

 

[marcus]

Presumably, "discrete" means that, a set, spacetime is countable.

George, I have a different perspective. I think of QG as representing GEOMETRY. A theory would not necessarily contain a set that represents the "points of space-time". A theory should have a representation of quantum states of geometry, and perhaps transition or boundary amplitudes. But it might not have a set which are imagined to be the points of space-time. So "countable" might not refer to any thing.
However in the case of Causal Sets. Your presumption DOES apply, I think.
I want to recap a bit here, to try to make my POV clearer.
A discrete GEOMETRY would I think be one where the geometric measurements are discrete in some sense. E.g. the observables, the operators have discrete spectrum. Where states of geometry are determined by a finite (or countable if you want) number of geometric measurements or conditions e.g on deficit angles, areas, volumes...

GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.

There are indications, are there not?, that geometry is also discrete when probed at small enough scale. What are these indications? You certainly know some. We can't be sure yet, but there are reasons to suspect geometric discreteness---what are some of them?

Jacobson's 1995 derivation of GR as a thermodynamic equation of state?
So what might be the geometric molecules of which GR is the EoS?
Finite entropy beyond bh horizon?
Continuum GR cannot be entirely right since it fails at extreme density?

Note that Sorkin's approach to geometric discretness preserves Lorentz Invariance.
This is one of the first things proved in the Perimeter lectures on Causal Sets, that Dowker and Sorkin presented a year or two back.
Same with LQG, there is that paper by Rovelli and Speziale proving Lorentz invariance.

So we don't want to get geometric discreteness MIXED UP with the old idea of "graininess" which somehow leads to higher energy gamma arriving EARLIER or as other people imagined LATER than other light. That old idea never seems to go away, does it? : ^)
There is it cropping up in that "Astronomy Cafe" piece by Sten Odenwald! The old L.I.V. idea ("lorentz invariance violation")
We have to learn to make a sharp distinction.

If anyone wants the Perimeter lectures on Causal Sets (e.g. where they show the sense in which Causets preserves Lorentz invariance) just google "pirsa Dowker" or "pirsa Sorkin". Pirsa is the online archive of videos of Perimeter Institute seminars and lectures.

I mentioned the Rovelli Speziale paper so should fetch a link for that. I'll google "rovelli lorentz". Yes, the first hit is:
http://arxiv.org/abs/1012.1739
Lorentz covariance of loop quantum gravity
Carlo Rovelli, Simone Speziale
(Submitted on 8 Dec 2010)
The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional SU(2)-spin-network Hilbert space can be mapped to a space K of SL(2,C) functions, where Lorentz covariance is manifest. K can be described in terms of a certain subset of the "projected" spin networks studied by Livine, Alexandrov and Dupuis. It is formed by SL(2,C) functions completely determined by their restriction on SU(2). These are square-integrable in the SU(2) scalar product, but not in the SL(2,C) one. Thus, SU(2)-spin-network states can be represented by Lorentz-covariant SL(2,C) functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally SL(2,C)-invariant in the bulk, and yields states that are preciseley in K on the boundary. This clarifies how the SL(2,C) spinfoam formalism yields an SU(2) theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.
6 pages, 1 figure.

But there is a much more basic point to make! We do not know what Nature is "made of ", what we know, and what we are interested in is "how she responds to measurement"
Measurements are interactions. Theories are supposed to model and explain interactions, they have to fit the world in that sense.
If there is discreteness, it lies in the interactions.
So, in particular, in the quantum observables of the theory. The geometric observables should have discrete spectra. this can be fully compatible with Lorentz invariance.

 

[PeterDonis]

Presumably, "discrete" means that, a set, spacetime is countable.

That's how I was interpreting it, yes.

I think it's worth noting that the converse is not necessarily true: that is, points of spacetime being a countable set does not necessarily mean spacetime must be discrete. The rational numbers are a countable set, but they are also continuous (at least with the standard ordering on them).

 

[marcus]

Presumably, "discrete" means that, a set, spacetime is countable.

As I'm sure you realize, that presumption does cause some semantic inconvenience because then for example LQG is not a "discrete" theory, in that sense of "discrete".

I am not sure that Causal Dynamical Triangulations (CDT) another approach to QG would be "discrete" in that sense, either. You start with a space-time continuum and you triangulate it---divide it up into simplices. Both the continuum and the individual simplices of course have uncountable sets of space-time points.
But CDT is often spoken of as a discrete QG approach because although the space-time is a continuum (and of course uncountable) the geometry of that continuum is being described discretely with information about a finite number of simplices.

Information such as the numbers of simplices of one sort or another adjacent to each other.

Something similar occurs in the version of LQG where you start with a spacetime continuum and you embed graphs into it which carry information about the geometry. The graph or network carries information about the geometry of a particular slice of the manifold. Or there may be an embedded 2-cell complex (analogous to a graph) which carries information about the geometry of a region.
This version was the standard for many years and is still preferred by influential people like Ashtekar and Lewandowski. It's probably still the majority version. Spacetime, to the extent that it is represented in LQG as something other than the geometry living on it, is of course uncountable.

One can also develop the theory in a more abstract way in which there is no embedding in an underlying manifold. In that case there are no "space-time points" in the theory, there is only the geometry itself: quantum states of geometry combinatorially represented. The catch there is that when you want to show something about the continuum limit you have to re-introduce the continuum.

 

[TrickyDicky]

No argument that there is a point below which we cannot make (and likely never will be able to make) actual measurements but I don't get how that relates to whether or not what we are measuring is continuous, based on theory. If I have a meter stick that will only measure down to a millimeter, that does NOT constrain my height to be an integer multiple of millimeters.

Of course, I'm not equating discrete measurements in that sense to a discrete universe. I'm bringing in the theme about scale independence that I find perhaps more specific mathematically than the wider concepts of discrete and continuous.

 

[marcus]

Of course, I'm not equating discrete measurements in that sense to a discrete universe. I'm bringing in the theme about scale independence that I find perhaps more specific mathematically than the wider concepts of discrete and continuous.

That seems like something interesting to try. Maybe if someone comes in with a new QG theory we could ask them
"Is there a minimum geometric quantity in your theory?"

Like, for example, "Is there a minimum measurable area?" If there is, then it would seem that their theory is NOT scale invariant.

I remember reading a CDT paper by Renate Loll some years back in which she proved that the simplices of CDT had to be at least a certain size. I forget how she proved it, I'm sorry to say. It wasn't put in at the start, the size arose somehow. That would indicate that CDT is not scale invariant. But there could be some disagreement about that.

What about AsymSafe QG? Do you know, or does anybody know, a reason why that would not be scale invariant? One would think, since it is such a close imitation of GR, that it would be scale invariant. But the AsymSafe people claim that it experiences spontaneous dimensional reduction at very small scale. The space-time dimension dwindles down from 4 to around 2.

I wish I knew more about this. It seems like something interesting to consider.

With LQG it is straightforward. LQG has a length scale. It is clearly not scale invariant!

there is also a minimum positive area that one can measure.

 

[TrickyDicky]

That seems like something interesting to try. Maybe if someone comes in with ;a new QG theory we could ask them
"Is there a minimum geometric quantity in your theory?"

Like, for example, "Is there a minimum measurable area?" If there is, then it would seem that their theory is NOT scale invariant.

I remember reading a CDT paper by Renate Loll some years back in which she proved that the simplices of CDT had to be at least a certain size. I forget how she proved it, I'm sorry to say. It wasn't put in at the start, the size arose somehow. That would indicate that CDT is not scale invariant. But there could be some disagreement about that.

What about AsymSafe QG? Do you know, or does anybody know, a reason why that would not be scale invariant? One would think, since it is such a close imitation of GR, that it would be scale invariant. But the AsymSafe people claim that it experiences spontaneous dimensional reduction at very small scale. The space-time dimension dwindles down from 4 to around 2.

I wish I knew more about this. It seems like something interesting to consider.

With LQG it is straightforward. LQG has a length scale. It is clearly not scale invariant!

there is also a minimum positive area that one can measure.

I believe that discrete models are by definition not scale invariant, thus all the examples you cite, but I'm not so sure the opposite is true.

 

[marcus]

I believe that discrete models are by definition not scale invariant, thus all the examples you cite, but I'm not so sure the opposite is true.

What does it matter? I mean that seriously. Maybe "discrete" is a bad term, the word having been used so indiscriminately/inconsistently over the years that is has gotten dull.

Maybe "scale invariant" (or some functional equivalent thereof) would simply be more useful as a category.

If we go with George's usage, which may be professionally accepted standard, then we just have to remember that LQG is not a discrete quantum gravity approach. Spacetime is either not represented or (more often) it is represented by a smooth manifold, an ordinary continuum.
What exhibits quantum discreteness is the web of geometric relations inhabiting the continuum (not "points" or "grains", which are not imagined to exist).

Everything is OK then as long as we remember not to say it is a discrete QG theory. And then one can add that the area operator has discrete spectrum, so that for example there is a smallest positive EIGENVALUE of the LQG area operator: in other words a smallest observable area.

And this, as that 2002 paper showed, is consistent with Lorentz invariance. Glad you started this conversation, TD. Nice to review and clarify terminology. Do you have some idea of further constructive directions the discussion could go?

BTW it occurs to me that one could INVENT a new category for QG theories which have operators with discrete spectrum that correspond to geometric observables

 

[marcus]

TD, given that "discrete" seems semantically problematic what to do? You've suggested one way to turn. Another might be to say "the way to handle semantic problems is to have fun with them : ^)
google translator says the Greek word
χασμα
means "gap"

The LQG researchers refer to LQG having a smallest nonzero area eigenvalue as "THE AREA GAP".

So how about calling QG theories that have a smallest geometric eigenvalue like that by the adjective "METROCHASMIC"?

Or would "GEOCHASMIC" be better?

It just means "has gappy geometric observables" and of course in particular smallest nonzero eigenvalues for some of them.

 

[martinbn]

I think it's worth noting that the converse is not necessarily true: that is, points of spacetime being a countable set does not necessarily mean spacetime must be discrete. The rational numbers are a countable set, but they are also continuous (at least with the standard ordering on them).

That's not how I understand continuous. The rational numbers are not continuous because they are not complete, Cauchy sequences of rational numbers need not be convergent. The real numbers are defined exactly because the rationals are not continuous.

 

[rrogers]

Thank you for introducing me to this wonderful paper. I said space is continuous by definition because I don't know what the stuff in between would be called if space is discrete.

Just say that when one interprets the GR equation symbols as calculus limits on the smaller scales; the equations are stochastic. In other words the classical GR equations are summaries/extreme cases of some stochastic equations where the independent variables happen to be random at a small scale.
Having said that I haven't given up on the geometry at larger scales. My reasoning is that if the QM equations twist and curve like they are on a surface/substrate then a geometric approach or interpretation is entirely reasonable. My case in point is the causal cone of light bending when passing by a massive object. The QM equations have to have "geometric" modifications to explain that.
As far as continuity/discreteness goes: I haven't given up hope that someone will think of a way to experimentally establish whether the "real" universe is Aleph 0,1, or even 1/2 or 2 (which is my favorite). The fact that on the smallest scale the equations turn stochastic doesn't determine whether the the large scale effects aren't Aleph 1 (or higher) and isomorphic to the equations describing curved surfaces.

 

[PeterDonis]

The rational numbers are not continuous because they are not complete

There are different definitions of "continuous". The one I was using is that "continuity" is the property that, between any two rational numbers, there is always another rational number; there is no such thing as a "minimum" interval between two rational numbers. So if possible measurement values were represented by rational numbers, there would be no minimum possible measurement value.

You're right that this definition of "continuous" does not guarantee other properties that might be desirable, such as convergence of all Cauchy sequences.

 

[Torbjorn_L]

Wow, so to you GR is quantizable and discrete.

Yes and no respectively.

• For the first, obviously to any physicist GR can be expressed as an effective [quantum] field theory at low energies and large scales:

"It’s often said that it is difficult to reconcile quantum mechanics (quantum field theory) and general relativity. That is wrong. We have what is, for many purposes, a perfectly good effective field theory description of quantum gravity. It is governed by a Lagrangian

(1) S = ∫d4x−g−−−√⎛⎝M2plR+c1R2+c2R2μν+c3M2plR3+…+L_matter⎞⎠

This is a theory with an infinite number of coupling constants (the ci and, all-importantly, the couplings in L_matter). Nonetheless, at low energies, i.e., for ε ≡ E²M²_pl≪1, we have a controllable expansion in powers of ε. To any finite order in that expansion, only a finite number of couplings contribute to the amplitude for some physical process. We have a finite number of experiments to do, to measure the values of those couplings. After that, everything else is a prediction.

In other words, as an effective field theory, gravity is no worse, nor better, than any other of the effective field theories we know and love.

The trouble is that all hell breaks loose for ε∼1. Then all of these infinite number of coupling become equally important, and we lose control, both computationally and conceptually."

[ https://golem.ph.utexas.edu/~distler/blog/archives/000639.html ]

That gravitons are discrete doesn't mean that the field (or spacetime) is.

• As for the latter, I was careful to point out that GR describes spacetime as smooth!

[And after listening to Nima Arkani-Hamid's latest lecture, on today's physics and how constrained it is, over the weekend I have come to appreciate this even more. Spacetime is likely smooth, because the universe is large.

It goes like this:

Nima described how we can extract 3D and 1/r^2 forces from knowing about relativity and QM, as well as that only spin 0, 1/2, 1, 3/2 and 2 particles will be seen. The 3/2 slot is the only remaining unrealized degree of freedom in the semi-classical physics, and it is covered by supersummetry.

_If_ it is taken, according to Nima this correspond to curled up "quantum" dimensions of length 1. Since they share (entangle and decohere, I guess) quantum fluctuations, they regulate the vacuum energy to open up for a large, non-planck scale, universe. (See Chronos for some arguable observations consistent with this.)]

 

[torsten]

For me, it is a dogma that the spacetime in quantum gravity has to be discrete. As far as I know, there is no experiment showing the discreteness. It is interesting that a smooth manifold has much to do with discrete structures.
I will only mention a few one: (for more details, see my essay at FQXi http://fqxi.org/data/essay-contest-files/AsselmeyerMalu_FQXIessay201_1.pdf and http://fqxi.org/data/essay-contest-files/AsselmeyerMalu_AsselmeyerMa.pdf)
1. A smooth manifold is determined by a discrete set of information (the handlebody structure)
2. Areas and volumes can be discrete in hyperbolic 3-manifolds (smooth) by using Mostow rigidity.
3.The smooth structure of a 4-manifold can be very wild and is determined by an infinite discrete structure, a tree.

These are only a few points. Ok, currently our theory does not include gravity but we are working on this and it looks promising.

 

[marcus]

... It is interesting that a smooth manifold has much to do with discrete structures.
...

That's a good observation! And you mention some interesting examples of discrete structure. We should emphasize the distinction between structure (primarily geometry and topology in this case) and any underlying point-set which the theory might have to represent "spacetime".

I suppose that there can be a theory of geometry in which there are physical objects: tables, chairs, cell-phones, exploding fire-crackers, fried chicken, clocks etc.
and in which there are angles, distances, areas, volumes describing and relating these physical objects. And there might be no underlying point-set in the theory. The only things like coordinates in the theory would then be measurements of relative positions and angles etc as they correlate with the clocks.

A point-set representing "spacetime" might, in other words, be excess baggage.
I don't know of any experiment showing the physical existence of points of space. All I know about are measurements of relative position etc...that is, of geometric relations between things, events IOW.

If the geometric structure is, as you suggest, discrete, then I would expect the results of those measurements (the quantum observables) to have discrete spectra. Just as so many other quantum observables have discrete spectra.

That might or might not be the case. However it could be the case that the observables have discrete outcomes without our being required to imagine any underlying spacetime point-set (discrete or smooth or what-you-will). IOW the point-set is excess baggage. One may then only introduce it into the mathematical formalism as a convenience if that turns out to be useful.

 

[julcab12]

.. There has been a lot of formulation going on the structure of spacetime -- gravity as thermodynamics, LQG, Casual sets, Casual dynamical sets and holography. Each has it's own special way of representing and expressing reality. We do not know what kind of reality might be happening beyond. Or such ultimate fundamental 'thing' exist. Hence, Discrete or continuous. But we can offer clues or at least a sense of directional path that nature is trying to tell us(hypothetically). So, we construct a model that agrees with reality in a way that is effective or serves as a reflection. We introduced a vision that became the backbone on how to create a model of reality. Newton has that first picture of spacetime (space, time, particle moving in empty space/stationary/rigid Newton space) then Faraday and Maxwell adds fields/electromagnetism(that vibrate and expressed dynamics of deformity) to the picture. Later on. Einstein added another field (gravitational field--well a dynamic type of Newtonian space) but he didn't stop there! he look at the picture and integrate a concept of relativity of spacetime as a single block as part of the whole picture of the field. Then quantum mechanics--Dirac/Heisenberg etc kicks in gave a picture of particle as quantum objects/probabilistic that also resembles a field which is discrete, Despite the fact that the equations has a naturally occurring continuous dynamic as part of the formulation but not the structure itself.

Ok. We know that the electromagnetic field has discreteness-- flickering particles and all it's weirdiness. It's is also natural that the gravitational field/spacetime 'might' have an minimal structure - granular -- LQG guys! I don't know if the direction is true but i know this guy Rovelli follows that same effective path or picture.

http://www.nature.com/news/theoretical-physics-the-origins-of-space-and-time-1.13613#reality

 

[PeroK]

There are different definitions of "continuous". The one I was using is that "continuity" is the property that, between any two rational numbers, there is always another rational number; there is no such thing as a "minimum" interval between two rational numbers. So if possible measurement values were represented by rational numbers, there would be no minimum possible measurement value.

You're right that this definition of "continuous" does not guarantee other properties that might be desirable, such as convergence of all Cauchy sequences.

Re the question of measurements: of length, say. It might be that there is a minimum measurable length and, perhaps, all measurements would be an integer multiple of that.

Or, perhaps any rational number is a possible measurement.

But, it's not clear to me how you could get an irrational number as a direct measurement. You might infer it (circumference of a circle from a rational radius). How could an irrational number be the direct result of a measurement? Is it possible?

Or, alternatively, could you define the length of something as the limit of a sequence of hypothetically more and more accurate measurements? Then lengths would typically be irrational.

 

[marcus]

.. There has been a lot of formulation going on the structure of spacetime -- gravity as thermodynamics, LQG, Casual sets, Casual dynamical sets and holography. ..

http://www.nature.com/news/theoretical-physics-the-origins-of-space-and-time-1.13613#reality

Julcab, thanks for the NATURE NEWS article! It covers a lot of different approaches to the quantum theory of space-time geometry currently being worked on, and it's written in popular wide-audience style. Zeeya Merali is a talented journalist, I've seen other articles by her and they're not bad.

But we should try to do is be clear about the terminology. discrete and continuous are basic terms out of point-set topology. A discrete (topological) space means something definite. It means something definite for a function f: X --> Y to be continuous. One normally does not apply the term "continuous" to sets or topological spaces. The expression "continuous space" is not used because it does not mean anything.

So if you look at Torsten's post, just now, you see he does not use the word "continuous" applied to spaces. He talks about "smooth" manifolds, a well-defined concept out of differential geometry. It's a condition on the coordinate mappings, smooth means that certain *functions* are infinitely differentiable---their derivatives are defined to all orders.

So discrete and continuous are not opposites and they apply to different classes of objects. There is the term "connected" in point-set topology. Sometimes when people say "continuous" they really mean connected. (can't be broken into two disjoint open sets)
The set of real numbers < 0 or >1 is not connected. It consists of the real line with usual topology, with the closed interval [0, 1] removed.
R \ [0,1] is not connected, however notice that if you pick any two points in it you can always find a point between them. Infinitely many points, in fact.
And you can make it even more disconnected and remove another closed interval.
If you pick any two points in R \ [0,1] \ [2,3] you can always find infinitely many points between them.

The rational numbers with the usual topology (which they get as a subspace of the reals) are not connected (easy to divide them into two disjoint open sets) but they are also not discrete. (any point is an accumulation point)
It does not make sense to call the rationals "continuous" because a "continuous space" or "continuous set" is not meaningful. It doesn't have a commonly recognized meaning in mathematics.

 

[marcus]

...
Or, alternatively, could you define the length of something as the limit of a sequence of hypothetically more and more accurate measurements? Then lengths would typically be irrational.

Maybe not. For example if Loop QG or something like it is right, the area observable has discrete spectrum. There's a practical problem of how you prepare many copies of a system that instantiates the same physical area. But what they are saying is if you could measure very small physical areas you would always be getting answers from the same discrete set. There is a minimum positive area measurement but the larger areas you can get, though they are discrete, are not simply multiples of it.

 

[torsten]

That's a good observation! And you mention some interesting examples of discrete structure. We should emphasize the distinction between structure (primarily geometry and topology in this case) and any underlying point-set which the theory might have to represent "spacetime".

I suppose that there can be a theory of geometry in which there are physical objects: tables, chairs, cell-phones, exploding fire-crackers, fried chicken, clocks etc.
and in which there are angles, distances, areas, volumes describing and relating these physical objects. And there might be no underlying point-set in the theory. The only things like coordinates in the theory would then be measurements of relative positions and angles etc as they correlate with the clocks.

A point-set representing "spacetime" might, in other words, be excess baggage.
I don't know of any experiment showing the physical existence of points of space. All I know about are measurements of relative position etc...that is, of geometric relations between things, events IOW.

If the geometric structure is, as you suggest, discrete, then I would expect the results of those measurements (the quantum observables) to have discrete spectra. Just as so many other quantum observables have discrete spectra.

That might or might not be the case. However it could be the case that the observables have discrete outcomes without our being required to imagine any underlying spacetime point-set (discrete or smooth or what-you-will). IOW the point-set is excess baggage. One may then only introduce it into the mathematical formalism as a convenience if that turns out to be useful.

In principle I agree with: the discrete spektra is the most important point, see the smooth Schrödinger equation.
I also agree that the point is not fundamental in GR. If I remember correctly, it is the famous hole argument of Einstein
(see http://plato.stanford.edu/entries/spacetime-holearg/). It was resolved by assuming that matter fully (and only) generates gravity. But one can go a step further: what if matter is itself a part of spacetime (as a special 3-manifold)? Then there is only spacetime and spacetime is not only excess baggage.

 

[PeterDonis]

It might be that there is a minimum measurable length and, perhaps, all measurements would be an integer multiple of that.

I agree this is possible; but if it is true, then the measurement results do not form a continuous set, so if we have two length measurements that differ by the smallest measurable amount, then it makes no sense to ask what is "between" the measured points. That was the point of my original post along these lines.

 

[member 11137]

Continuous or not, it seems to be a question of scala and of motion. Just think about the propeller of a plane. If it doesn't move, you see for example three or four wings. When it turns, you just see a kind of circle. All this is inspiring a link between a stroboscopic principle and the fact that an object seems to be continuous or not...

 

[jambaugh]

I believe it is a mistake to even ask the question, or posit an answer to questions about the nature of space-time as if it were a real object. This mistake is natural given the beauty and elegance of the geometric representation of Einstein's GTR. But remember it is only a description. The equivalence principle led Einstein to equate a dynamical force of gravity to a curved space-time geometry in which there is no dynamical force, only geodesic motion. Note that this "equating" is a two way street. We cannot observe geometry. We can only observe a relative relationship between geometry and a dynamical gravitational force (in which inertial and gravitational mass are equivalent). We do not observe space-time geometry, or space-time anything. The fact that we can excise the dynamical force of gravity by suitable choice of geometry makes it easy to say "There is no force of gravity, only geometry" but that would not be scientifically correct. We should rather say "The force of gravity is only defined relative to a choice of geometry both of which are relative." One might as easily say there is no geometry but that makes formulating theories harder.

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct (a useful one and perhaps a uniquely necessary one but none-the-less...) and as such should not be argued about as if it were a physical object. Is it continuous? I find using a continuous one more useful for physics. You are free to choose a discrete one if you like.

(Side note: I think this "error" is also leading many down a dead end channel to attempt to "quantize space-time" in various quantum gravitation approaches. I'm just not smart enough to back this opinion up with a good journal article. ;))

 

[julcab12]

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct

...I'm just curious thought. Even visual information -- mental construct. Almost/all of our experiences are interactions fundamentally. Objects are also interactions and events. Spacetime is not just description. Whatever it is. It should have some structure.

http://iai.tv/video/spacetime-and-the-structure-of-reality

 

[PeterDonis]

We should rather say "The force of gravity is only defined relative to a choice of geometry both of which are relative."

I'm not sure this is quite right. First, GR does not define a "force of gravity relative to a choice of geometry". It says "gravity" is not a force at all. In other words, GR adopts a definition of "force" that is physically different (and, IMO, more reasonable) than the definition in Newtonian theory: a force in GR is something that causes proper acceleration, i.e., something that is felt as weight. An object moving solely under gravity is weightless, feeling no force; so GR says gravity is not a force. None of this says anything about "geometry"; it's all about what, physically, we should denote by the term "force".

Also, there is only one choice of geometry that eliminates gravity as a force, i.e., that respects the physical definition of "force" that GR uses. However, there is nothing requiring us to call the thing we are talking about "geometry" or "spacetime geometry". That name is chosen because the math we use is the math of Riemannian geometry (more precisely, pseudo-Riemannian geometry), but that doesn't mean the physical interpretation has to be the same as it is in ordinary geometry.

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct (a useful one and perhaps a uniquely necessary one but none-the-less...) and as such should not be argued about as if it were a physical object.

Even in the context of classical GR, I don't think this is quite right either. The whole point of GR is that the geometry of spacetime (or whatever word you want to use to denote that thing) is dynamical; it interacts with matter and energy. IMO that makes it a physical object, not just a mental construct.

In quantum gravity, spacetime has to be a physical object, because it is built out of simpler physical objects, just like atoms are built out of simpler objects. But that's going beyond GR.

 

[jambaugh]

...The whole point of GR is that the geometry of spacetime (or whatever word you want to use to denote that thing) is dynamical; it interacts with matter and energy. IMO that makes it a physical object, not just a mental construct.

That is the *model* we use in describing GR but it is not quite what the theory states. The theory states that the dynamical behavior of physical object will be such that... (fill in here with what you said). " But you cannot observe space-time itself in order to say if it does or does not dynamically interact with anything. It is just like the aether of the aetheric forms of SR where clocks are "really slowed by their motion through the aether" and measuring rods "are really shortened by their motion through the aether" and so the aether "must be physical".

Recall that in non-relativistic physics x is an observable but t is not... it is a parameter. Whether you ask if t is "really continuous" is meaningless in that theory. When we relativize we don't make t more "space-like" we make x more "time-like" in that it too ceases to be meaningful as an observable, instead becoming a parametric quantity.

Space-time is parametric. The geometry we overlay let's us distinguish between "free geodesic motion" and "motion under the influence of a dynamical force" but that dividing line is arbitrary and relative. Choosing the geometry is like selecting the gauge condition. The constraint surface of the gauge condition isn't physically real but it is physically relevant to the predictions of how a physical object behaves.