The naturalness of the zeon length scale

[marcus]

Wabbit had an interesting comment in the "Aeon to Zeon" cosmology thread. Here's an excerpt, for more follow the link.

One stray comment here inspired by the zeon and related natural GR unit of length ## \Lambda^{-1/2} ##
...
So the largest possible measurable distance is ## D= \frac{1}{\sqrt{2\pi\Lambda}} ## ~ one zeon.
Or more directly, the maximum area of the boundary of the observable universe, and hence the maximum area of any sphere, is ## A_{max}=\pi D^2=\frac{1}{2\Lambda}## ~ one square zeon.
And the maximum volume of any observable spatial region at fixed comoving time is some constant times ##\Lambda^{-3/2}## ~ one cubic zeon.

This expresses in the flat case the zeon as kind of counterpart to the Planck length, or the inverse of the cosmological constant as the maximum area in units of the minimum area, the Planck area - which I also find interesting from an information viewpoint.

So it seems the zeon (or the zeon squared) may have something fundamental about it :)

I replied saying I'd start a separate thread.

The naturalness of the zeon length.
This is interesting, and it could be a separate topic
http://inspirehep.net/record/899089?ln=en
http://inspirehep.net/record/899089/citations
http://inspirehep.net/search?ln=en&p=refersto:recid:899089
"Smallest measurable angle" call it a "planckian" angle, might be a Planck length or Planck area held out at the "fundamental large distance". Or perhaps the distance scale is a consequence of the limitation on angle measurement instead of the other way round.
I will try to start a separate thread. Maybe it should be in BtSM forum...

 

[wabbit]

Thanks for the link to the Bianchi-Rovelli paper. There's something I m missing in the angular argument. Yes the angular resolution is bounded by ## l_P\sqrt{\Lambda} ## . But I would expect local angular resolution to be bounded far more strictly than that, by the angular resolution of the receiving small sphere, at ## l_P/R\ll l_P\sqrt{\Lambda}## . so the local rotation group should be far more fuzzy than the one quoted, no ?

 

[wabbit]

On the information side, there seem to be two bounds

One is from the area of the horizon in Planck units ## ~ 1/(\Lambda l_P^2) ## , but I am not sure what this gives - is it a bound on the amount of information any system can have on any other system ?

The second bound, assuming a bit of information requires a Planck volume, is given by the volume of the observable universe ## ~ 1/(\Lambda^{3/2}l_P^3) ##. This seems to give a limit on "external" information storage, or the total amount of information potentially accessible to a given system.

But are all this upper bounds, on length or information (or even mass, since mass=length) fundamental (i.e. is lambda a fundamental constant ? ) or merely accidental (i.e. lambda happens to have this value in our universe but it could just as well be different ? )

 

[marcus]

I expect you are right. The other is more a theoretical limit to accuracy when you have all the time and distance in the world to measure the angle.
I'm not a good person to ask though, not learnéd in such matters. I recall that Seth Lloyd has one or more papers about the quantization of angle.
Basically my intuitive sense is "this is fertile ground".
there is nothing firmly established yet, only signs of germination

At the moment the aspect that to me is most exciting (and it needn't be to you) is this:

• a good way to do quantum gravity (i.e. quantum spacetime geometry) is to use simplexes. little chunks of spacetime geometry.
• in that approach, a good way to include the Λ is to use chunks with a small constant curvature instead of chunks cut out of flat
• if you do that the it seems that the phase space of geometry enjoys a natural compactness and there is a kind of quantization of time, states in phase space make little discrete transitions, geometry is changing by little quantum transitions. the Hilbert space of states of geometry actually becomes finite dimensional. Of course the dimension is extremely large, but it is finite, which is something.

This is just my (uninformed) impression of some recent research.
I suspect that this embodiment of the Λ is actually closely related to what we are talking about, this other embodiment of Λ as a natural tendency to for distances to expand at a residual asymptotic rate after whatever momentum has died down
For the second point, if anyone's curious, google "haggard han kaminski riello"
For the third point, try googling "compact phase discrete" since it involves compact phase space of geometry, and discreteness of time.

 

[wabbit]

Yep, things seem to be popping up all over the place nowadays in QG from different perspectives and links between them, very interesting to watch that game : )

I like simplexes but not putting lambda by hand in the construction, I feel there should be some reason for lambda to be what it is.

An intriguing alternative I saw searching for papers by Seth Lloyd as you suggested, is that oultined in http://arxiv.org/abs/quant-ph/0501135 : (possibly distorted reading) inflation can arise at any scale and its rate comes from the scale at which it arises. It happens naturally at an initial singularity (something that both LQC and condensed matter models also find), is then suppressed by the emergence of matter from quantum fluctuation (I wasn't aware of such a mechanism, but this is needed if inflation is not an artificial construct) , and can reappear later at a much slower rate - this paper doesn't seem to have much of a progeny, but it's also an interesting possible qualitative description. In that picture lambda is not only accidental, it can also take different values in different regions - which is another way of being natural, sidestepping the "why this particular value" question. Still, a natural value would be more economical and explanatory than a random value.

I don't get the MERA papers but my impression is that they hint at a natural emergence of curvature from the interplay of information and scale - this would be close to the kind of inevitable process that could provide a natural value.

 

[marcus]

Wabbit, I'm glad you found something that interested you by Seth Lloyd. I'm embarrassed to say I had a memory glitch and said Seth Lloyd when I meant Seth Major. I didn't have time to check the reference or to see whether SM's papers were something I should recommend, I just recalled some Seth had been talking about quantizing angle. No relation, that I know of in this work, to Λ.
http://arxiv.org/abs/gr-qc/9905019
Operators for quantized directions
Seth A. Major
(Submitted on 6 May 1999 (v1), last revised 19 Oct 1999 (this version, v2))
Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two ``bundles'' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.
20 pages, 23 figures

http://arxiv.org/abs/1005.5460
Shape in an Atom of Space: Exploring quantum geometry phenomenology
Seth A. Major
(Submitted on 29 May 2010 (v1), last revised 1 Jun 2010 (this version, v2))
A phenomenology for the deep spatial geometry of loop quantum gravity is introduced. In the context of a simple model, an atom of space, it is shown how purely combinatorial structures can affect observations. The angle operator is used to develop a model of angular corrections to local, continuum flat-space 3-geometries. The physical effects involve neither breaking of local Lorentz invariance nor Planck scale suppression, but rather reply on only the combinatorics of SU(2) recoupling. Bhabha scattering is discussed as an example of how the effects might be observationally accessible.
Comments: 14 pages, 7 figures; v2

http://arxiv.org/abs/1112.4366
Quantum Geometry Phenomenology: Angle and Semiclassical States
Seth A. Major
(Submitted on 19 Dec 2011)
The phenomenology for the deep spatial geometry of loop quantum gravity is discussed. In the context of a simple model of an atom of space, it is shown how purely combinatorial structures can affect observations. The angle operator is used to develop a model of angular corrections to local, continuum flat-space 3-geometries. The physical effects involve neither breaking of local Lorentz invariance nor Planck scale suppression, but rather reply on only the combinatorics of SU(2) recouping theory. Bhabha scattering is discussed as an example of how the effects might be observationally accessible.
5 pages