Building SM Matter from Discrete Quantum Geometry

[marcus]

You wouldn't have a discrete structure and diffeomorphism invariance 'at the same time'; mathematically diffeomorphism invariance would reduce the smooth structure to a kind of discrete set set of equivalence classes of smooth spacetimes.

Even though I found the Achim Kempf paper (and his PIRSA talk) interesting and convincing at the time, still I am inclined to agree with Tom view. Kempf's idea was a conjecture--it could be that discrete and continuous representations are on equal footing.

But Tom is talking about how reality looks after diff-invariance is "factored out". It may be a deeper picture than Kempf's. I don't feel sure, but if I had to choose I think I would go with this "discrete set of equivalence classes" idea. Just my two cents.

atoms. orbitals. Fock space...these are fundamentally discrete, could geometric relations also be?

Kempf has some good points about information/sampling theory though

 

[atyy]

The other attractive view to take spin foams as really discrete is Barrett's http://arxiv.org/abs/1101.6078.

"This is done by generalising Sakharov’s idea of induced gravity ... For this idea to work, it is necessary for the spacetime geometry to exhibit discreteness at the Planck scale ..."

"Although the state sum models are discrete in nature, it is envisaged that an approximate continuum description should emerge at energies below the Planck scale. Therefore state sum models are constructed with this limit in mind - it guides the expectations of the physical content of the model."

"The wish-list of properties for a state sum model is

• It defines a diffeomorphism-invariant quantum field theory on each 4-manifold
• The state sum can be interpreted as a sum over geometries
• Each geometry is discrete on the Planck scale
• The coupling to matter fields can be defined
• Matter modes are cut off at the Planck scale
• The action can include a cosmological constant

Diffeomorphism invariance here actually means invariance under piecewise-linear homeomorphisms, but this is essentially equivalent. The piecewise-linear homeomorphisms are maps which are linear if the triangulations aresubdivided sufficiently and play the same role as diffeomorphisms in a theory with smooth manifolds. This invariance is seen in the Crane-Yetter model and also in the 3d gravity models, the Ponzano-Regge model and the Turaev-Viro model, the latter having a cosmological constant. The 3d gravity models can be interpreted as a sum over geometries, a feature which is carried over to the four-dimensional gravity models [BC, EPRL, FK], which however do not respect diffeomorphism invariance."

 

[czes]

"High-energy gamma rays should show marked differences in polarization from their lower-energy counterparts. Yet studying the difference in polarization between the two types of gamma rays, Philippe Laurent of CEA Saclay and his collaborators found no differences in polarization to the accuracy limits of the data.
Theories have suggested that the quantum nature of space should become apparent at the Planck scale: 10^-35 of a meter. But the Integral observations are 10,000 times more accurate than any previous measurements and show that if quantum graininess exists, it must occur at a level of at least 10^-48 m or smaller."
http://www.centauri-dreams.org/?p=18718

The above is true if the space-time would be built of the physical particles-antiparticles interacting with the photons.
I suppose the space-time is nothing more but a topology constructed by a relation between the non-material information like wave functions of the Schroendiger equation.
If the information of the wave-functions might be non-local they may create relations everywhere due their probability and create the space-time.
Another supposition is that each relation of the non-local wave-function encodes a Planck time dilation and it causes the sequence of the events and time flow.

I think, each theory has to include information conservation, superposition and non-locality. It suggests that even the smooth information has to be observed as a discrete after each relation.

 

[tom.stoer]

not every theory based on a "discrete structure of spacetime" does indeed predict Lorentz invariance violations, anomalous dispersion relations etc.; so not all these theories have been rules out, especially not LQG which can be formulated in a Lorentz-covariant manner and from which no Lorentz violation has been derived so far.

 

[Naty1]

CZES posts:

...if quantum graininess exists, it must occur at a level of at least 10^-48 m or smaller."

VERY interesting!

 

[czes]

not every theory based on a "discrete structure of spacetime" does indeed predict Lorentz invariance violations, anomalous dispersion relations etc.; so not all these theories have been rules out, especially not LQG which can be formulated in a Lorentz-covariant manner and from which no Lorentz violation has been derived so far.

Yes. I hope somebody will find the correct theory.

 

[marcus]

not every theory based on a "discrete structure of spacetime" does indeed predict Lorentz invariance violations, anomalous dispersion relations etc.; so not all these theories have been rules out, especially not LQG which can be formulated in a Lorentz-covariant manner and from which no Lorentz violation has been derived so far.

That's certainly true. One can have geometric discreteness without "graininess"---in the naive sense of space made of permanently divided grains.

Angular momentum appears discrete when one measures but does not consist of permanent lumps. An atom's energy appears in discrete levels but energy itself is not divided into fixed lumps.

So geometric measurements of continuous media can have discrete spectra (angle, area, volume...). I guess one can regard this as analogous to the wave-particle duality. Perhaps these dualities are different faces of the same core fact about nature.

 

[marcus]

CZES posts:
"...if quantum graininess exists, it must occur at a level of at least 10^-48 m or smaller..."
VERY interesting!

I believe we discussed that particular finding earlier. It is unclear how to relate it to current LQG. It wasn't obvious what was meant by "quantum graininess" or what version of QG it might apply to.
Czes linked to a misleading popularization in a blog whose main focus seems to be space-travel. The actual scientific paper by Philippe Laurent et al did not directly mention LQG or draw any explicit connection. It is here:
http://arxiv.org/abs/1106.1068
Constraints on Lorentz Invariance Violation using INTEGRAL/IBIS observations of GRB041219A
The June 2011 paper is based on this earlier analysis by Götz et al
http://arxiv.org/abs/1103.3663

After the June 2011 paper appeared, Philippe Laurent was quoted in popular newsmedia making some loose unqualified claims that their paper applied to some unspecified version of string theory and some unspecified version of LQG. He may have had something definite in mind. LQG has evolved quite a bit over the past decade and if you go back into earlier versions say from the 1990s you can undoubtably find a lot of variety. Or even before 2007.
But he didn't give any footnotes or pointers to actual research literature so it was not clear what he was talking about.

 

[marcus]

It seems that what Czes was saying about "graininess" and last June's Philippe Laurent paper about a gammaray burst is not relevant to Loop/Spinfoam QG, so does not apply to what we are discussing in this thread.

But Atyy's post #37 (just a bit earlier) was interesting. BTW the quote from Barrett points out that (quantum states of) geometry can be discrete while the underlying manifold is continuous.

==quote Atyy https://www.physicsforums.com/showthread.php?p=3710407 ==
The other attractive view to take spin foams as really discrete is Barrett's http://arxiv.org/abs/1101.6078.

"This is done by generalising Sakharov’s idea of induced gravity ... For this idea to work, it is necessary for the spacetime geometry to exhibit discreteness at the Planck scale ..."

"Although the state sum models are discrete in nature, it is envisaged that an approximate continuum description should emerge at energies below the Planck scale. Therefore state sum models are constructed with this limit in mind - it guides the expectations of the physical content of the model."

"The wish-list of properties for a state sum model is

• It defines a diffeomorphism-invariant quantum field theory on each 4-manifold
• The state sum can be interpreted as a sum over geometries
• Each geometry is discrete on the Planck scale
• The coupling to matter fields can be defined
• Matter modes are cut off at the Planck scale
• The action can include a cosmological constant

Diffeomorphism invariance here actually means invariance under piecewise-linear homeomorphisms, but this is essentially equivalent. The piecewise-linear homeomorphisms are maps which are linear if the triangulations aresubdivided sufficiently and play the same role as diffeomorphisms in a theory with smooth manifolds. This invariance is seen in the Crane-Yetter model and also in the 3d gravity models, the Ponzano-Regge model and the Turaev-Viro model, the latter having a cosmological constant. The 3d gravity models can be interpreted as a sum over geometries, a feature which is carried over to the four-dimensional gravity models [BC, EPRL, FK], which however do not respect diffeomorphism invariance."

The red is a serious issue. Loop/Spinfoam QG should respect diffeo invariance. I am not convinced that what Barrett said in this case is right. The issue may now have been addressed by the Freidel Geiller Ziprick paper:

http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges.
27 pages

 

[tom.stoer]

Diffeomorphism invariance here actually means invariance under piecewise-linear homeomorphisms ...

I know that Donaldson et al. showed that there are excpetions to the equivalence "diffeomorphisms = PL homeomorphisms" (exotic smoothness); could this play a role here?

 

[marcus]

I know that Donaldson et al. showed that there are excpetions to the equivalence "diffeomorphisms = PL homeomorphisms" (exotic smoothness); could this play a role here?

There is always that intriguing possibility. I remember one of the people who participated in the prototype "Loops 2004" conference at Marseille was Hendrick Pfeiffer. He was writing about exotic differential structures then. I'll get the link in case it is of interest. He probably refers to the work of Donaldson.

"Loops 2004" was called something else but it got the series of conferences started. Anyway there was already that idea of quantum gravity having something to do with different PL structures, or not PL but smooth, that can live on the same topology. Or so I recall. I should get the link to make sure.

Here it is. I recall being impressed. But I cannot confidently answer your question could it play a role.
http://arxiv.org/abs/gr-qc/0404088
Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
(Submitted on 21 Apr 2004)
The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space-time geometries, gives rise to a formal invariant of smooth manifolds. This is an opportunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d=3+1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d=3+1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d<=5+1, the classification results provide us with triangulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than absence of gravitons that renders pure gravity in d=2+1 a `topological' theory.
41 pages

Yes, he does cite a 1990 paper of Donaldson. And also incidentally a 2002 paper by Torsten A-M, whom we know from discussions here!

 

[tom.stoer]

I can't find the thread but sometimes ago I posted the idea that exactly these inequivalent smooth structures in dim=4 singles out dim=4. Suppose you could write down a PI summing over all non-diffeomorphic smooth manifolds in all dimensions. Then the probability for a 4-dim. manifold in the PI is 1, b/c there are uncountably many inequivalent 4-dim. smooth manifolds, whereas for all other dimensions there are only countably many. So dim=4 is explained exactly by diff. inv.

 

[marcus]

I can't find the thread but sometimes ago I posted the idea that exactly these inequivalent smooth structures in dim=4 singles out dim=4. Suppose you could write down a PI summing over all non-diffeomorphic smooth manifolds in all dimensions. Then the probability for a 4-dim. manifold in the PI is 1, b/c there are uncountably many inequivalent 4-dim. smooth manifolds, whereas for all other dimensions there are only countably many. So dim=4 is explained exactly by diff. inv.

I believe I saw that! You and Hendryk Pfeiffer were thinking somewhat along similar lines. After reading your post I went back and colored part of the Pfeiffer abstract blue, so if anyone else is reading they can quickly see where the similarity of ideas is. There's a suggestion of why the world has four dimensions. There just could be something to the idea.

I am interested by it and was impressed by Pfeiffer's paper back in 2004 but I don't understand it well enough to make a guess as to whether it is a good idea or not.

Actually I have the same problem with Louis Crane's paper here. I would like to hear other people's opinions because I am not able to confidently evaluate.

You know how some people do "numerology" and find amazing coincidences of numbers. Could it be that Louis Crane has fallen prey to "groupology" and has found an accidental appearance of the alternating group on 4 letters, namely A₄. And could he be overinterpreting or overreacting to this?

On the other hand could it be that the paper is an important opening to new territory that deserves to be explored? (whether or not eventually leading to success.)

He says he is going to follow it up with a longer paper. At the bottom of page 3 he says "A more detailed study of the structure will follow."

That is right after he states a conjecture.
==quote==
CONJECTURE: The EPRL model coupled to the tetron field gives a unified model which breaks to give the standard model.
==endquote==
It's a bold interesting conjecture and I have to take something like that seriously, as potentially important, unless there is a good reason to dismiss it.

 

[czes]

After the June 2011 paper appeared, Philippe Laurent was quoted in popular newsmedia making some loose unqualified claims that their paper applied to some unspecified version of string theory and some unspecified version of LQG. He may have had something definite in mind. LQG has evolved quite a bit over the past decade and if you go back into earlier versions say from the 1990s you can undoubtably find a lot of variety. Or even before 2007.
But he didn't give any footnotes or pointers to actual research literature so it was not clear what he was talking about.

Thank you Marcus for remark. I have had some troubles with Philippe Laurent article. He did want to provoke a discussion about the essence of the space probably.