Both continuous and discrete in the same space at the same time (Kempf)

[marcus]

Kempf gave a talk on this. I'll find the PIRSA link. I remember watching the whole video and being impressed. It may be easier to understand than the paper because communicating a higher proportion of the person-to-person intuition---more beginner level. You can try it either way. Either watch the talk first like I did and then read the paper, or some other way. He's on to something nontrivial.

http://arxiv.org/abs/1010.4354
Spacetime could be simultaneously continuous and discrete in the same way that information can
Achim Kempf
(Submitted on 21 Oct 2010)
"There are competing schools of thought about the question of whether spacetime is fundamentally either continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the bandlimit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possesses an ultraviolet cutoff. Most recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e., of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale. Here, we develop these results further, and we here also consider the generalization to curved spacetimes, i.e., to Lorentzian manifolds."

 

[marcus]

Kempf also taught a Perimeter course on Quantum Field Theory for Cosmology
There are 20+ video lectures:
http://pirsa.org/C10003

But the lecture I am looking for is about spacetime being both continuous and discrete---when we talk about information it is a spurious distinction. Still looking

pirsa seems to be down, but I found a 18 September 2009 post of mine which gives the pirsa link along with other information
==quote 18 Sept 09 https://www.physicsforums.com/showthread.php?t=338459 ==
Geometry both discrete and continuous at once, like information--Kempf

It is possible for a geometry to be both discrete and continuous. We don't know if our universe's geometry is like that, but it could be. Video of a talk at Perimeter by Achim Kempf, describing this, was put online yesterday.

http://pirsa.org/09090005/
Spacetime can be simultaneously discrete and continuous, in the same way that information can.

It refers to this paper published in Physical Review Letters
http://arxiv.org/abs/0708.0062
On Information Theory, Spectral Geometry and Quantum Gravity
Achim Kempf, Robert Martin
4 pages
(Submitted on 1 Aug 2007)
"We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe."

and also to this paper Kempf recently posted on arxiv:

http://arxiv.org/abs/0908.3061
Information-theoretic natural ultraviolet cutoff for spacetime
Achim Kempf
4 pages
(Submitted on 21 Aug 2009)
"Fields in spacetime could be simultaneously discrete and continuous, in the same way that information can: it has been shown that the amplitudes, , that a field takes at a generic discrete set of points, , can be sufficient to reconstruct the field for all x, namely if there exists a certain type of natural ultraviolet (UV) cutoff in nature, and if the average spacing of the sample points is at the UV cutoff scale. Here, we generalize this information-theoretic framework to spacetimes themselves. We show that samples taken at a generic discrete set of points of a Euclidean-signature spacetime can allow one to reconstruct the shape of that spacetime everywhere, down to the cutoff scale. The resulting methods could be useful in various approaches to quantum gravity."
...
==endquote==

 

[marcus]

OK now here is a key bit of the new paper:
==quote Kempf new paper==
9. Summary and Outlook

We showed that spacetime could be simultaneously continuous and discrete, in the same way that information can. To this end, we considered the cutting off of the spectrum of the Laplacian (or d’Alembertian) at an eigenvalue close to the Planck scale. We found that, in this case, physical theories possesses equivalent continuous and discrete representations, and that external symmetries can be fully preserved. ...

... Further, it should be very interesting to go beyond the study of the kinematics and to investigate the dynamics. First steps were taken in [5], by considering the number, N , of sample points needed for reconstruction of fields on a compact Riemannian manifold with cutoff, and the reconstruction of the manifold itself. N is the number of eigenvalues of the Laplacian, in the simplest case of the Laplacian on scalar functions, which are below the cutoff. Interestingly, this number, N , has an expansion in terms of the curvature...

... Up to corrections that are of higher order in the Planck scale, this is of the form of the euclidean Einstein action,... Here, before renormalization, c₀ is related to the (unrenormalized) cosmological constant Λ via Λ = c₀ /16πc₁. When there is no curvature, c₀ expresses the density of degrees of freedom in the theory...
...
Curvature then is acquiring a new interpretation. In addition to expressing the nontriviality of parallel transport, curvature is here seen to be the local modulation of the density of degrees of freedom in the theory. It should therefore be possible to re-express also the variational principle as an extremization with respect to the density of degrees of freedom, or their overall number in a finite volume.

In effect, we are diagonalizing the Einstein action by expressing it in the Laplacian’s eigenbasis.

A key challenge will be to express and investigate also a bandlimited formulation of interacting theories of the standard model in this basis. This will introduce the Dirac operator and generalized Laplacians on tensors, which nicely fits with our observation in Sec.8 that the spectra of such operators are in any case needed to complete the information theoretic description of spacetimes.
==endquote==

 

[Boy@n]

Hoes does the PF community accept this view of space-time being both continuous and discrete at the same time, as information is?

This thread is almost a year old, so, anything new? Has this view become a mainstream in serious science?

 

[marcus]

Hoes does the PF community accept this view of space-time being both continuous and discrete at the same time, as information is?

This thread is almost a year old, so, anything new? Has this view become a mainstream in serious science?

The simple short answer is no. The paper may be good. It may also be part of a developing trend involving other authors moving in a parallel direction but unaware of Kempf and not citing him. He is focusing on information and allowing the degrees of freedom to increase.

The physics is allowed to depend on how detailed you look at the system, how deep into detail you go, how many d.o.f. your model of the system includes. OTHER PEOPLE MAY be paralleling this general trend, but not be aware of Kempf.

There is this big development in LQG BH entropy where you consider a variable N the number of horizon "degrees of freedom" technically called "punctures". Where the spin network which models geometry comes out (like "hair") through the BH horizon. Each leg or link of the network contributes a bit of area and a bit of curvature to the horizon. The more hair to the picture the more finely the area etc is divided up.
http://arxiv.org/abs/1107.1320 (BH entropy paper by Ghosh and Perez)
http://arxiv.org/abs/1107.4605 (BH entropy paper by Mitra)

The people writing those LQG BH papers do not seem aware of Kempf. But to me it is like a school of fish moving in parallel without being completely collectively aware.

So the answer is NO. I DON'T SEE HARD REALWORLD OBJECTIVE INDICES SAYING THAT KEMPF IS INFLUENCING OTHER RESEARCHERS. But I think he is bright and creative and "on to something". So I don't dismiss or consider him fringe. However I have no idea what other PF people think. Which again may not matter much---it is not like we constitute a representative sample, objectively just a bunch of individual viewpoints. No one really in position to pontificate.

Thanks for bringing this up, let's keep an occasional eye on his output
http://arxiv.org/find/grp_physics/1/au:+Kempf_a/0/1/0/all/0/1
and on what cites his papers get. (over on the right of the arxiv page of a paper you see a link for "cited by" so you can see what other papers by whom referred back to the it)

 

[Naty1]

There have been suggestions for many years that everything around us in information based...that "reality" IS information.

This goes from entropy and energy...the stuff of thermodynamics...as a subset set of Shannon information theory (where for example nature disippates information as it increase entropy) to the holographic principle where the information within a horizon (a boundary) is related to the AREA of the enclosed sufrace rather the entire volume...even explains some black hole information loss stuff that Susskind used to overcome Hawking's claim that information was lost in blacl holes.

An interesting introductory book to these concepts is Charles Seife's DECODING THE UNIVERSE...conceptual, not mathematical.

 

[Boy@n]

http://en.wikipedia.org/wiki/Stephen_Wolfram" ...

I'd guess that this (and probably not well accepted in science arena?) supports both ideas - existence being based on information and that space-time is discrete and continuous at the same time, as information is?

 

[marcus]

There have been suggestions for many years...
...that "reality" IS information.
...

True, there have been those suggestions for many years. It is a school of thought, almost, verging on the philosophical.

But I don't see Kempf as part of that. He is not saying that spacetime IS information. Or that it IS a network of automata, or anything resembling that.

He is saying (I think) that spacetime geometry could have a certain restricted SIMILARITY with information--a limited analogy.

He reminds us of Shannon's analysis of continuous analog signal transmission which measures the content of the signal in a finite number of discrete bits suffcient to reconstruct the signal. The message sent along the telephone wire can be equally well regarded as continuous waveform or in terms of discrete bits.
Particularly if there are only a finite number of harmonic components.

He suggests that the geometry we live in could be similar to the signal in the limited sense that it also might be equally well described with a continuous model and a discrete model. He suggests the two descriptions can be reconciled.

So when you read the paper you see him beginning to talk about this field N that is defined everywhere on spacetime and describes the density of degrees of freedom (i.e. "bits") which it would take to describe the geometry.

http://en.wikipedia.org/wiki/Stephen_Wolfram" ...
...

Personally I don't see much in common between Kempf and Stephan Wolram, but other people might draw some connection.

It seems to me that Wolram is suggesting that the world really consists of an array of automata. I've always found this idea unpersuasive, even kind of naive and boring.
It seems very different from Kempf, who allows spacetime to be a CONTINUUM (but also finitely describable).

Kempf's picture allows spacetime to really be a smooth continuum, not broken up into little cells or grains. In his picture it is a continuum with a subtle sophisticated difference: it is describable in terms of a discrete set of "degrees of freedom" (d.o.f. just means descriptors, the numbers that it would take to reconstruct the thing.)

Perhaps I am being unfair to Wolram (I have not read his book) but I think Kempf's picture is more beautiful because he allows the geometry of a region of spacetime to be BOTH CONTINUOUS and DISCRETE in the sense that the continuum can be described by a finite set of descriptors.

I see a trend in QG (quantum geometry) that goes in parallel with Kempf. Bianca Dittrich (at the Albert Einstein Institute near Berlin) has been working on the problem of what does diffeomorphism-invariance of the Einstein GR continuum look like if the continuum is describable by a discrete triangulation---a finite set of d.o.f.
Diff-invariance is an essential feature of the GR spacetime geometry. She has found a way that seems to perfectly reconcile the property of diff-invariance with the ability to represent spacetime discretely with building blocks. It is a kind of synthesis between the discrete and the continuous. Her concept of discretizable diff-invariance has been called "Ditt-invariance" as a concise pointer to her name Dittrich. (I don't know how she feels about that, she strikes me as a modest retiring person. What does she think of her nick-name being used in some mathematics terminology?)

I see more in common between Kempf and researchers like Dittrich (and also Ghosh, Perez, Mitra's recent work on Black Hole entropy) than I see connection with Wolram.

It may be much too technical to be useful in this context but here is a recent paper about "Ditt-invariance" http://arxiv.org/abs/1107.2310
I hope the connection I am drawing is not too far-fetched. It is basically an intuitive one.

 

[Boy@n]

Nice and informative post, Marcus, thanks! I am just a layman and could more or less understand it, but I won't if it gets more technical.

May I ask you something, which seems like a too simple question to be even asked and I guess that's why no one asks it.

How is motion even possible in theories which say that space-time is discrete?

Just imagine your hand moving from point A to point B. Now, if you record that with a Perfect video camera, how many frames per second would you record? Would there be infinite amount of frames/steps? But then, if infinite, you could never finish moving your hand, or to say, you could never move it at all.

If not infinite, then I'd say that there have to be as many frames (or steps) as there are there Planck length fitting that distance your hand made (say 10cm/10^-37cm), which is a huge number. (Actually, it would be even more, since at quantum level a particle is not a single-pointing-particle but like a wave, a set of particles, with particles being distributed over space and time, as I understand it.)

I’ve made a https://www.physicsforums.com/showthread.php?t=515859", where I didn’t focus so much on space-time being discrete or continues, but on the "fact", that human perception of reality is, so to say, very “slow” compared to true “speed of reality” (e.g. if we reviewed our perfect recording of a simple hand moving on a TV set in slow motion, say, one second for every frame, it would take us more time to finish watching it than the age of Universe^2).

So, how is motion explained in discrete theory, how in continuous, and how when we view space-time as being both at once?

Now, if space-time is continuous then it would just mean that while we can still make a video recording in discrete steps (no matter how many, but never infinite though), we can simply never make a "Perfect" record, meaning one, in which we'd truly record reality in the way it happened.

And that's why continuous space-time makes much more sense to me than the discrete one. (While of course you can always “make it” discrete for practical and mathematical purposes.)

 

[marcus]

How is motion even possible in theories which say that space-time is discrete?

One thing to notice is that different things are meant by that in different theories...
so you need different answers depending on what theory.

LQG was developed using a continuum model of spacetime (a smooth region with undetermined "shape") on which you put a discrete geometry.

The discreteness comes in when you make geometric measurements---surface areas, volumes, angles, lengths. To establish the geometric state of this continuum.

It's like the energy levels of a hydrogen atom. When you measure the energy you always get a choice from among some discrete possibles. The quantum transition is a flow of probability between state A and state B. They are discrete levels but it can start to be more probable that she is in state A instead of state B. there is no state in between, but you see a kind of blurry change and finally you realize she is fully completely in state A and not at all in B.
It has to do with MEASUREMENT. The quantum operators describing measurement have only a discrete range of outcomes. (called their "spectrum") There may be a continuum in between but you can never get your hands on it because when you measure the atom, or the girl, is always either in state A or state
B.

Spin is like that too, when you measure you only can get discrete outcomes which also depend on the orientation of your measuring device. Of course there are a continuum of directions in space, but when you measure the system acts as if its spin axis direction is discrete.

So for example in LQG there is a very subtle difference---space is not imagined as made up of little grains---it is not some tinkertoy or lego-world. But when you make geometrical measurements you encounter the same quantum experience you do with a hydrogen atom or with spin. Measurement means we are talking about quantum states, and operators with discrete spectrums. So the quantum states of geometry, in LQG, are networks of area and volume measurements (by which the experimenter checks that the geometry is in a certain state) and these network states can evolve probabilistically into different network states.

(in the blurry way that an an atom evolves between levels, or the girl shifts from state B to state A without you knowing exactly when and there being nothing measurable in between)

It seems very generally that is how the world actually is. At least it is plausible that geometry is like other things----discrete when you measure, to check the state---but not composed of a fixed set of little grains or toy blocks.

The challenge is to develop the mathematical tools to describe this. One set of tools are the socalled spin-networks and spin-foams of LQG---proposed quantum states of geometry and evolution paths of geometry.

And you asked about motion. In a quantum theory it may be impossible or inconvenient to describe motion by a smooth continuous trajectory. One immediately confronts the limitation of what we can know, the finiteness of our ability to measure, the discreteness of measurement. We actually may only know that a particle has moved along a finite sequence of markers----passed through a certain finite sequence of slits or doorways. We may only be able to speculate or imagine what it was doing in between, along the way. It may be necessary to accept the limitations of our knowledge and not pretend to more than we actually know, not to presume certainty in the face of nature's essential uncertainty (at very small scale).

Maybe the world is a continuum with which we interact only discretely and which we cannot know with full certainty. Therefore to be predictively accurate our models must incorporate these features.

I think people think about this in different ways, maybe someone else will show you a different way to think about it.

 

[Naty1]

Always had a lot of intuitive appeal to me.

I think Kempf's picture is more beautiful because he allows the geometry of a region of spacetime to be BOTH CONTINUOUS and DISCRETE in the sense that the continuum can be described by a finite set of descriptors.

Boy:

But then, if infinite, you could never finish moving your hand, or to say, you could never move it at all.

That question stumped mathematicians for some 2,00 years.. Don't have any reference link, but some Greek's version was "If there are an infinite number of points to cross a river I'd never get to the otherside"...

Somebody PROVED that is not accurate...calculus I would guess.