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Current Essay Contest

Contest Partners: The Peter and Patricia Gruber Foundation and Scientific American

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What Is “Fundamental”
October 28, 2017 to January 22, 2018
Sponsored by the Fetzer Franklin Fund and The Peter & Patricia Gruber Foundation

Wandering Towards a Goal
How can mindless mathematical laws give rise to aims and intention?
December 2, 2016 to March 3, 2017
Contest Partner: The Peter and Patricia Gruber Fnd.

Trick or Truth: The Mysterious Connection Between Physics and Mathematics
Contest Partners: Nanotronics Imaging, The Peter and Patricia Gruber Foundation, and The John Templeton Foundation
Media Partner: Scientific American


How Should Humanity Steer the Future?
January 9, 2014 - August 31, 2014
Contest Partners: Jaan Tallinn, The Peter and Patricia Gruber Foundation, The John Templeton Foundation, and Scientific American

It From Bit or Bit From It
March 25 - June 28, 2013
Contest Partners: The Gruber Foundation, J. Templeton Foundation, and Scientific American

Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?
May 24 - August 31, 2012
Contest Partners: The Peter and Patricia Gruber Foundation, SubMeta, and Scientific American

Is Reality Digital or Analog?
November 2010 - February 2011
Contest Partners: The Peter and Patricia Gruber Foundation and Scientific American

What's Ultimately Possible in Physics?
May - October 2009
Contest Partners: Astrid and Bruce McWilliams

The Nature of Time
August - December 2008

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Author Yuri Danoyan+: on 4/24/11 at 19:46pm UTC, wrote It seems to me very interesting ...

Lawrence Crowell: on 4/24/11 at 1:25am UTC, wrote The argument involves quantum uncertainty between different coordinates. ...

Author Yuri Danoyan+: on 4/13/11 at 18:53pm UTC, wrote My opinion about supersymmetry see here ...

Lawrence Crowell: on 4/13/11 at 12:43pm UTC, wrote I ended this before concluding with the nature of F_4 and the 24 cell. The...

Lawrence Crowell: on 4/13/11 at 12:35pm UTC, wrote @ Yuri, There are three types of symmetries in the world. The first type...

Author Yuri Danoyan+: on 4/13/11 at 3:10am UTC, wrote Dear LC If Feynman, and earlier Stueckelberg, proposed an interpretation...

Lawrence Crowell: on 4/12/11 at 12:13pm UTC, wrote Tha naonymous above is me, the author. I sometimes find these web pages do...

Author Yuri Danoyan+: on 4/11/11 at 16:11pm UTC, wrote Dear Lawrence To my mind 24 Cell theory-redundant theory. I can imagine...


John Cox: "Zeeya, It is past time for an overhaul to get some protection against..." in The Sudoku Universe, Why...

Zeeya Merali: "Do we live in a giant Sudoku puzzle? Are real numbers really real? Does..." in The Sudoku Universe, Why...

Georgina Woodward: "Joe, your credo has very limited explanatory power and is therefore of very..." in Quantum Dream Time

Joe Fisher: "Dear Georgina, You wrote: “I agree that there have been surfaces prior..." in Quantum Dream Time

shery williams: "Office Setup is the full suite of Microsoft productivity software that..." in Are We Merging With Our...

shery williams: "McAfee offers world-class security solutions for Windows & Mac OS as well..." in Are We Merging With Our...

Yelena Hopper: "Its a decent approach to present a few thoughts in material science yet you..." in We Are All Connected

Yelena Hopper: "There is an existence in the wake of death. Spirits and souls are genuine...." in Wandering Towards a Goal:...

click titles to read articles

Quantum Dream Time
Defining a ‘quantum clock’ and a 'quantum ruler' could help those attempting to unify physics—and solve the mystery of vanishing time.

Our Place in the Multiverse
Calculating the odds that intelligent observers arise in parallel universes—and working out what they might see.

Sounding the Drums to Listen for Gravity’s Effect on Quantum Phenomena
A bench-top experiment could test the notion that gravity breaks delicate quantum superpositions.

Watching the Observers
Accounting for quantum fuzziness could help us measure space and time—and the cosmos—more accurately.

Bohemian Reality: Searching for a Quantum Connection to Consciousness
Is there are sweet spot where artificial intelligence systems could have the maximum amount of consciousness while retaining powerful quantum properties?

November 25, 2017

CATEGORY: Is Reality Digital or Analog? Essay Contest (2010-2011) [back]
TOPIC: Discrete Time and Kleinian Structures in Duality Between Spacetime and Particle Physics by Lawrence B Crowell [refresh]
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Author Lawrence B Crowell wrote on Jan. 12, 2011 @ 15:38 GMT
Essay Abstract

The interplay between continuous and discrete structures results in a duality between the moduli space for black hole types and $AdS_7$ spacetime. The $3$ and $4$ Q-bit structures of quantum black holes is equivalent to the conformal completion of $AdS$.

Author Bio

I did my graduate work at Purdue University and have since worked in affiliation with the AIAS and industry, which has been with spacecraft navigation and software. For the last several years I have been working to set up the problem which this paper is related. Quantum cosmology is a quantum error correction process.

Download Essay PDF File

Philip Gibbs wrote on Jan. 12, 2011 @ 22:07 GMT
Lawrence, it is good to see your essay has been posted. It is quite a technical paper and I want to understand it better, so I am likely to ask some questions once I have looked at it some more.

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Author Lawrence B Crowell wrote on Jan. 13, 2011 @ 00:17 GMT

Unfortunately this paper is a bit sketchy in some ways. The later part I illustrate the role of a G_2 holonomy. This Killing vectors, N for N-SUSY, satisfy a commuting condition except for one which is K_t, which is not covariantly constant. There is a U(1) fibration over the 6 dimensional space given by the coset condition G_2/SU(3), where G_2 can be seen as decomposed into SU(3) x S^6. Therefore the space cannot be factorized and the holonomy is equal to G_2, which is a property of G_2 manifolds. This constitutes a fixed point set, and a set of timelike geodesics. These geodesics are identified on the coset space for the gauge connection, which is the moduli space

For a given Killing vector k, a hypersurface defined by |k|^2 = 0 is the fixed point

set and degenerates if the surface gravity vanishes, The fixed points of all Killing vectors are non-degenerate which implys a periodic identification along the Killing direction to ensure the absence of conical singularities. This fact is well known from non-extreme black holes where the event horizon is a non-degenerate fixed point set for a timelike Killing vector. In the case here, the compactness of the Killing direction ensures that the Kaluza-Klein gauge group becomes U(1) and that the D6-branes, which are identified as the fixed point set.

The G_2 is 14, which in complexification is a 28. This then provides a map between these orbits on the SO(4, 4)/SO(2, 2)xSO(2, 2) moduli space of the 4 qubit system, where the orbits of SO(4; 4) are the adjoint 28 for the black hole, with the holonomy of G_2. The holonomy is a geodesic on a timelike direction in the two moduli spaces. The difference is there is a triality condition here, which means the map between the two are 3 to 1, and the Z_3 is the cyclicity.

By taking the AdS_7/Z_3 this collapses this condition, where in addition this discrete condition defines the conformal completion on the AdS. This may be used to construct the boundary of the AdS, where the isometries of AdS are conformal fields on the boundary. These conformal fields are the T^6 coincident with the D6-brane. This is then a qubit discrete approach to the Maldecena result AdS ~ CFT.

Cheers LC

Eckard Blumschein replied on Jan. 21, 2011 @ 06:00 GMT
Dear Lawrence,

If I recall correctly you promised to include pictures in your essay that will show how to imagine something like spacetime foam. Did I got you wrong, or did you decide to hide your colorful results?


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Lawrence B. Crowell replied on Jan. 22, 2011 @ 12:35 GMT
I guess I am not seeing on this page here where I made such a promise. I might have said something about including a picture of a tessellated AdS spacetime. This is one of MC Escher’s prints, where this illustrates the lattice system on the AdS_2.

Cheers LC

attachments: AdS_circlelimit.JPG

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Eckard Blumschein replied on Jan. 24, 2011 @ 16:10 GMT
On Dec. 7, 2010 in 782#post_29212 you wrote:"Signatures of this structure lie around us in the universe, such as the images I attach. So fractal geometry is important, and in fact what I outline above is what my essay will entail.

Cheers LC

attachments: cosmic_filaments.JPG, cmb_popup.jpg "

"AdS circlelimit.JPG" is also a nice but naked figure without an explaining legend. I have humbly to admit to be a layman who not even heard of tessellated AdS. Let me try a wild guess: In't there a Tom Essel, and could dS stand for de Sitter? A? Hm, maybe Arahonov? Anyway, a certain part of your readers would certainly appreciate at least a list of abbreviations.


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Steve Dufourny wrote on Jan. 13, 2011 @ 12:47 GMT
Hi all,

Congratulations for this beautiful maths extrapolation and superimposings.

It's a mix of some theories of strings.

Probably it could be interesting if Mr Witten, Mr Baez and Mr Penrose come here for a discussion in transparence about some points of vue.That will be interesting it seems to me.

I invite them to see your beautiful mathematical play.and we shall discuss about the relevances and the errors.

The QM is logic and all its superimposings must be rational.

All road conducts to truth if and only if they are purely correlated with our universal constants.

Good luck.

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Author Lawrence B Crowell wrote on Jan. 13, 2011 @ 13:23 GMT
Ed Witten probably has a lot better things to do than peruse the FQXi essays. The guy is utterly brilliant. He writes very long papers, where he is one of the few people who write 60+ page papers I am willing to read. For the most part I prefer not to read papers over 25 pages in length. Of course one point in writing these essays is to put at least one ball into the basket so that I might get a bit of attention --- Warhol’s 15 minutes of fame.

Cheers LC

Steve Dufourny wrote on Jan. 13, 2011 @ 18:34 GMT
I translate my french a little,

Notes of Math overlook the confusion.

Such a sweet and serene symphony clearing the unlimited sharps.

Is it important to devote training practices of these notes in series without changement of speeds?

It would become, you will agree, though unsuccessfully to play the score universal score, without these so-called harmonic overlays.

Good luck for the contest, sincerely.


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Edwin Eugene Klingman wrote on Jan. 15, 2011 @ 03:14 GMT
Lawrence - I just noted your comment to Darth on my page and feel compelled to offer the following clarification. I am probably responsible for some of the confusion. If you are interpreting my version of the gravito-electro-magnetic field to be the same as Sweetser's GEM, then I have mislead you. I show Sweetser's diagrams because I believe they are relevant to understanding significant aspects of 'metric' vs 'potential' approaches to physics. I do NOT accept all of his approach to GEM. Part of the confusion is that I have been using the abbreviation 'GEM' for years before knowing about Sweetser, and neither he nor I have a monopoly on this term. It often refers to Maxwell's original invention--based on symmetry--of the gravito-electro-magnetic (GEM) equations analogous to his electro-magnetic field equations. I don't know a way around this confusion. I often refer to the 'Gene Man' theory, which is more specific, but also more self-referential, and less familiar to all.

My field equations (see my essay) are neither Maxwell's nor Sweetser's.

I regret the confusion.

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Author Lawrence B Crowell replied on Jan. 15, 2011 @ 13:23 GMT
The clear problem with GEM idea in general is these are an intermixing of internal and external symmetries in ways that are problematic.

The Einstein field equations are similar to Maxwell's, and in the post^3 Newtonian form are identical (modulo a factor of 2) to Maxwell’s equations. The post^2 form is similar to electric and magnetic statics. The magnetic field analogue is similar in some ways to magnetism in EM theory. This is usually what is meant by gravito-magnetic field,

Cheers LC

Edwin Eugene Klingman replied on Jan. 16, 2011 @ 02:00 GMT

The equations are similar in appearance, but electromagnetic fields are essentially linear, whereas the gravitomagnetic (C-) field is inherently non-linear, and is therefore a Yang-Mills equation, not a 'Maxwell'-type equation. This has extremely significant consequences.

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Steve Dufourny replied on Jan. 17, 2011 @ 11:14 GMT
Indeed, indeed,....but why they mix like that.

It's even not to imply confusions.

NUCLEAR CRYOGENICS???? magnetic polarizations..........YANG !!!

beta particles ...electrons....see the cobalt for example!!!The apparatus of Ambler,Hudson,Wu ...mesures that whith He and N2.

I don't see concrete mixing of our equations,physical.How is it possible to derivate or intergrate correctly if the real physicality isn't inserted with the biggest rationality.


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Author Lawrence B Crowell wrote on Jan. 16, 2011 @ 17:14 GMT
The difficulty remains if the field theory is abelian, as in U(1) QED, or if it is nonabelian in the YM sense. This comes from the problem of characterizing the symmetries of the S matrix if there is a mass. Review the Dirac equation and the mass-gap in the two solutions which separate by a gap p ~ m on the momentum light cone.

The S-matrix acts on shift the state or momentum state of a...

view entire post

Author Lawrence B Crowell wrote on Jan. 18, 2011 @ 02:38 GMT
I have attached a rough draft that is somewhat sketchy, on more mathematical detail with the physics here.

Cheers LC

attachments: AdSqubit.pdf

Ray Munroe replied on Jan. 18, 2011 @ 20:41 GMT
Hi Lawrence,

WOW! A little bit of light reading for the next time I have trouble sleeping...

I like these qubits - they are simultaineously discrete kissing-sphere particle-like and continuous string wave-like.

Ceratinly, Philip's 4 qubit is an important sub-symmetry. My models require this for icosian symmetries.

Regarding your 3 qubit idea, Could we have a 5 qubit (the NS5-brane) decomposing into a (3+2) qubit?

Have Fun!

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Author Lawrence B Crowell replied on Jan. 19, 2011 @ 03:05 GMT
The 4 quantum bits with an SO(8) realization have an 24-cell realization. The 24-cell is represented by the B_4 ~ SO(9), D_4 ~ SO(8) and F_4. With a part of what I am doing in this paper, in particular within the attachment, is to work out a G_2 holonomy for these quantum bits. The G_2 is the automorphism of E_8 and F_4 the centralizer. I think by this means we can push this up to the 8-qubit entanglement problem. A part of what I have laid out is what I think are some step in this direction.

The 5-qubit problem I think is outstanding, though one can derive a combination of 4-quibits into a 5. Also within an E_8 system there are two dualities that exist with respect to the Steiner system (or Hamming distance). The standard one is [3, 5, 8] and the self-dual system is [4, 4, 8] that has a Hilbert space correspondence. The Steiner system has a dual between 3 and 5, so within the E_8 system the 3-quibit subsystem is equivalent to the 5-qubit subsystem. By this is would mean that the 3-qubit system would have 5 separable states plus 3-qubits in a W or GHZ state.

Cheers LC

Edwin Eugene Klingman replied on Jan. 29, 2011 @ 21:45 GMT

That's a lot of math to reach the strong conclusion that "there is only one electron, one proton (well really one up quark and one down quark), one photon and so forth in the entire universe, and what we see as multiple copies of them are a sort of illusion induced by a large scale decoherence and the subjective appearance of decoherent classes of histories."

Why hold back? Why not say something radical?

Have you figured out where consciousness [with or without free will] comes from in this scenario?

Also, I missed how W's and Z's fit into this scheme and the three generations. Are these included in the "and so forth"?

Edwin Eugene Klingman

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amrit wrote on Jan. 22, 2011 @ 22:19 GMT
spacetime is an empty term.

time is a numerical order of change in a 4D space

yours amrit

attachments: Is_Einstein_still_misunderstood.pdf

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Author Lawrence B Crowell wrote on Jan. 24, 2011 @ 03:19 GMT
The discrete nature of time might be compared to a numerical ordering. I carry this further to consider a discrete structure in general.

Cheers LC

Steve Dufourny wrote on Jan. 31, 2011 @ 10:56 GMT
In all honesty, do you think time travel is possible dear Lawrence?

I don't need points of vue but a simple answer.

Cheers Steve

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Author Lawrence B Crowell replied on Jan. 31, 2011 @ 13:37 GMT
I do not think classical closed time loops are possible. So I don’t think time travel per se is possible. In quantum gravity I think correlation between pre and post selected states have an ambiguity in time ordering. This is a subtle issue with what an event in spacetime means with holography. However, this does not translate into any ability to time travel.

Cheers LC

Steve Dufourny replied on Feb. 5, 2011 @ 12:22 GMT
Thank you dear Lawrence,



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Author Lawrence B Crowell wrote on Feb. 3, 2011 @ 13:43 GMT

I looked at Joy Christian’s paper. It looks dense. It appears that she is arguing the Hopf fibration across the 7 sphere is responsible for the Bell inequality violation with the GHZ state.

I think honestly I will wait for others to comment on this. My time is a bit limited. Another reason is that people who raise up objections to quantum non-locality have a consistent record of being wrong. In fact Joy Christian proposed a reason for the illusion of nonlocality back in 2006 and that was torn down. I frankly suspect something similar will happen here. I also suspect that if she is building up nonlocality from a geometry, then if there is no physics to the internal structure that can be measured this then all amounts to a sort of geometric quantization. In effect it just arrives at nonlocality by other means and then defines nonlocality as an “illusion.”

Cheers LC

Edwin Eugene Klingman replied on Feb. 3, 2011 @ 19:46 GMT

First, I've discovered that Joy is a man.

Second, I don't believe he is arguing that "the Hopf fibration across the 7 sphere is responsible for the Bell inequality violation with the GHZ state." What I believe he is saying is that Bell's mistake was in thinking that "correlations between the points of a real line have anything to do with the correlations between elements of reality", and it is "topologically impossible for any Bell type map to constitute a manifold of all possible measurement results."

This, as I understand it, is incompatible with the basic completeness criterion of EPR that "every element of physical reality must have a counterpart in the physical theory."

But "correlation between the EPR elements or reality are correlation between the respective points of two 2-spheres" and has "nothing whatsoever to do with the correlations between the points of two 0-spheres as Bell unjustifiably assumes." Bell's incomplete description of physical reality doesn't count all possible measurement results.

The significant result is this: Bell incorrectly found the value 2 while QM found 2*sqrt(2) and experiments show that Bell's value is violated but the QM value is never violated. What Joy Christian finds **in every case** is the value 2*sqrt(2) as the appropriate measure. Since all measurements always fall within this value, the correct inequality IS NEVER VIOLATED.

If he is correct, then all non-local, non-real, entanglement arguments [ie, all 'spooky' and 'weird' stuff] were based on Bell's incorrect value!

Of course these 'spooky' and 'weird' arguments have been going on for decades, they have subtly and not-so-subtly affected the minds of most physicists, even to the point that someone as bright as Florin remarked to you about "has the smell of local realism". Fortunately, Florin has now begun to study Joy's work and seems to have an open mind.

My interest, as I said, is so strong because my theory is based on local realism. You are correct that other challenges have failed, so we'll see.

I believe you are incorrect to think that Joy has arrived at non-locallity by other means. What he has done is demolished non-locallity. It may take some time for you and others to grasp this notion, but I believe that's what will be required. Of course I may be wrong.

I'm glad Florin has committed to studying this issue, as I believe it is the most important issue facing physics today.

Thanks for playing,

Edwin Eugene Klingman

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Author Lawrence B Crowell replied on Feb. 3, 2011 @ 22:45 GMT
I had a funny feeling Joy was a man, but … .

My sense is this. I am pretty sure there is a fly in the ointment somewhere with this. I have a hard time thinking that quantum nonlocality is wrong. Of course maybe Joy has found a loose thread in the quantum edifice and has started to pull at it so it might unravel. My immediate suspicion is that just as his last 2007 attempt to overthrow nonlocality failed to make the grade the same will happen here. If I have the time to dig into this paper I will do so, but it is pretty dense and clearly requires a bit of time to digest. I am currently reading papers on the latest proof of a modular function on the partition of the integers. This is the latest hot breakthrough in mathematics (number theory) and this clearly has implications for the statistics of string modes which compose black holes. So right now that takes a higher precedence. Maybe Florin has more time to work through this.

My bet is this will not fly.

Cheers LC

Suzi Q Pea replied on Feb. 26, 2011 @ 04:27 GMT
I respectfully beg to differ with L.C. Fresher minds are doubting the validity of nonlocality, but we dare not speak up lest our self-satisfied professors not toss us the occasional publishing bone. It is no longer acceptable to question the dogma of the "True Faith" of the Powerful and Academically Mighty. LC, understandably, wants to keep it that way.

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T H Ray wrote on Feb. 11, 2011 @ 12:45 GMT
Hi Lawrence,

It's taken me a while, because as usual you send me back to the books and I haven't had much opportunity to catch up. In particular, I want to read Goyal's papers.

Also as usual, though, you frame your questions in precise mathematical terms, which I appreciate. Always a "10," in my estimation.

I'm reticent to get into a technical exchange that I can't finish, though from our previous dialogue I expect we are still in accord over which mathematics to start with ... algebraic topology in the complex Hilbert space, commutative structures and analytic continuation. As you say, " ... the discrete binary aspect of the universe is equivalent to the continuous structure of the universe" (in a continuous exchange of curves for discrete points).

Best wishes in the contest.


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Author Lawrence B Crowell replied on Feb. 12, 2011 @ 14:03 GMT
There are further or deeper structures involved here as I have recently found. The paper by Phillip Gibbs complements my paper. He illustrates how there are elliptic curve realizations with the hyperdeterminant for n-partite entanglements, which have a correspondence with black hole types. I have found that a generating function for strings on black holes has a combinatorial structure which is one exact and secondly has elliptic curve structure. This is by pursuing the problem from a completely different perspective. This also seems to lead to zeta functions through modular structures. The generating function is a modular form of a type with a group action given by the Calabi-Yau form. The application of a particular type of operator on this generating function defines a non-holonomic form which is an integer partition of states on a black hole.

Cheers LC

Georgina Woodward wrote on Feb. 13, 2011 @ 10:26 GMT
Dear Lawrence,

I am of course impressed by your knowledge of physics theory and mathematics. I would not like you to think my question disrespectful of that. I will admit I have not downloaded your essay as I doubt very much that as a non specialist I could even begin to understand it. I am quite familiar with your style of communication of your own ideas from FQXi blog threads. My question to you is why do -you- consider your interpretation of the competition question, answered with discussion of moduli space for black hole types and $Ads-7$ space-time, to be foundational?

Kind regards, Georgina

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Author Lawrence B Crowell replied on Feb. 13, 2011 @ 18:35 GMT
If you read my paper you should also read Phillip Gibbs’ paper. Our papers complement each other in some ways. Another interesting source to read is

On the Black-Hole/Qubit Correspondence

L. Borsten, M. J. Duff, A. Marrani and W. Rubens

which illustrates how n-partite entanglements are equivalent to black hole types. What I have done...

view entire post

Georgina Woodward replied on Feb. 13, 2011 @ 19:35 GMT

thank you for taking the time to reply. For me your sentence "I would say that what I have done is proximal to the foundations of the universe because it involves some important structures found by some interesting research of late." does explain to me why you have presented this particular material. I can see that you consider this recent research groundbreaking stuff, which is one of the criteria FQXi are looking for.

I might download your essay just for the challenge of seeing what I can decipher. It is undoubtedly too technical for me to enjoy but your high position indicates that there may be sufficient readers who have not found it so and regard it as fulfilling the competition evaluation criteria well.The very best of luck to you. Georgina.

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Author Lawrence B Crowell wrote on Feb. 16, 2011 @ 22:37 GMT
This is a bit of an update. Things have been very quiet of late. The two papers in the FQXI contest which most closely correspond to my work ARe Plillip Gibbs' and Jarmo Matti Mäkelä's. The correlation between Phil's paper and mine is fairly clear, as both invovle quantum bits in n-partite entanglements. My paper does though invoke a discrete structure when applied to the AdS spacetime. Mäkelä's paper came about at the right time, for his program, outlined in the narrative with Newton, is an accounting of states on a black hole. This has come concurrent with the recent proof of an exact formula for the partitions of the integers. The Eisenstein construction is due to a coset construction with a discrete group from the Calabi-Yau form. This tightens up the ansatz I invoke on the Z_3. So this is an ongoing process at this time.

Cheers LC

Andrew Beckwith wrote on Feb. 24, 2011 @ 20:17 GMT

The connection between light cones and quantum physics is drawn tighter with the discrete structure.

Discrete structures are more appropriate for quantum information. In what follows the entanglement types

of 3 or 4 quantum bit system is equivalent to black hole types, which is extended to the AdS spacetime

as well. The Taub-NUT spacetime is essentially just a black hole with the meaning of radius and time

reversed in the metric elements.

end of quote

Lawrence, I would like to know what precisely you mean by the "connection between the light cones, and quantum physics". can you e mail me with more details?

Andrew Beckwith, e mail of

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Author Lawrence B Crowell replied on Feb. 25, 2011 @ 00:31 GMT
The discrete Klein group structure defines a light cone structure and a Heisenberg group. This involves a bit of mathematics, which went beyond the scope of my paper here --- if you get too mathematical you do not do so well.

The AdS_{n+1} group of isometries O(n,2) contains a Mobius subgroup, or modular transformations, so that this discrete group does not necessarily act effectively on AdS_{n+1}. This means that the discrete group Γ is not necessarily convergent on the boundary space M_n. Such a convergence means there exists a sequence {g_i} \in Γ which admits a “north-south” dynamics of poles p^{+/-} on a sphere, which in the hyperbolic case defines the past and future portions of a light cone. The limit set of a discrete group is a closed Γ -invariant subset that defines a Λ _ Γ \subset M_n so the complement \Omega_ Γ acts properly on M_n. This Γ -invariant closed subset of Λ _ Γ \subset L_n is the space of lightlike geodesic in M_n.. This has some interesting properties. The action of Γ on Λ _ Γ U AdS_{n+1} (U = union) is contained in M_n. The open set Λ _ Γ is the maximal set that the Γ acts properly on Ω_ Γ UAdS_{n+1}. The other is the discrete group Γ is Zariski dense in O(n,~2).

The lightlike geodesics in M_n are copies of RP^1, which at a given point p define a set that is the lightcone C(p). The point p is the projective action of π(v) for v a vector in a local patch R^{n,2} and so C(p) is then π(P∩C^{n,2}), for P normal to v, and C^{n,2} the region on R^{n,2} where the interval vanishes.

The space of lightlike geodesics is a set of invariants and then due to a stabilizer on O(n,2), so the space of lightlike curves L_n is identified with the quotient O(n,2)/P, where P is a subgroup defined the quotient between a subgroup with a Zariski topology, or a Borel subgroup, and the main group G = O(n,~2). This quotient G/P is a projective algebraic variety, or flag manifold and P is a parabolic subgroup. The natural embedding of a group H --> G composed with the projective variety G --> G/P is an isomorphism between the H and G/P. This is then a semi-direct product G = PxH. For the G any GL(n) the parabolic group is a subgroup of upper triangular matrices. An example of such a matrix with real valued elements is the Heisenberg group of 3x3 matrices.

Cheers LC

Sreenath B N wrote on Feb. 27, 2011 @ 16:58 GMT
Dear Author Lawrence B Crowell,

Iam really astonished by your indepth knowledge of quantum-mechanics and Black-Holes.In your concluding remarks,you have expressed your inability to reconcile the digital and analog nature of reality.But,according to me,such a thing is possible.For this you, please,go thro' my essay and also express your views.

Good luck and best wishes.

Sreenath B N.

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Author Lawrence B Crowell replied on Feb. 27, 2011 @ 18:25 GMT

I have been working on discrete quotient systems. These structures ultimately involve explicit density of string states calculations, such as the stringy entropy of a black hole. I was less motivated here to try to conclude that nature is primarily continuous or discrete. I will say that I think there is some sort of complementary principle, where continuous structures are epistemic and discrete are ontological.

What is interesting to note in reading some of these papers is how people can argue in an adroit manner both sides of the dichotomy: nature is digital or nature is analogue. The relationship between the two borders on the metaphysical, for we have little physical idea of what we mean by ontology and epistemology. Quantum mechanics by Bell’s theorem on inequality violations tells us that the universe fundamentally is not realistic in a classical sense. So wave functions are considered to be epistemic, and they do not have exactly the same ontology of a particle --- in fact no ontology. The measurement produces a particle which we register as a discrete “click.” We can point to that and say “it exists.” However, to interpret all that discrete stuff we need to think according to all that continuous stuff. Noether currents are not conserved in discrete structure, only in continuous ones. So we are left with a question on existentialism, where we ponder whether something which is purely epistemic can have the same existential categorical status of something ontological.

I am slowly working my way through as many of these papers as I can read. This month I have been horribly sick with pneumonia, which if it does not kill you then at times you wish you were dead. So I will pull your paper up and try to read it today.

Cheers LC

Georgina Parry replied on Feb. 28, 2011 @ 09:52 GMT

take care of yourself and get well soon.

Best wishes, Georgina.

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Author Lawrence B Crowell wrote on Mar. 1, 2011 @ 02:35 GMT
This is fairly serious. It is not just a matter of enduring a cold, but almost a battle in a way. It is get down and serious. Some days are better than others. Today was not too bad, yesterday worse and most of last week and the week before really rough. I started to read Sreenath’s paper yesterday, but I ended up having to sleep early.


Steve Dufourny replied on Mar. 4, 2011 @ 10:19 GMT
Hope you are well dear maverick of maths.



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Member Tejinder Pal Singh wrote on Mar. 6, 2011 @ 13:17 GMT
Dear Lawrence,

Yours is an interesting and noteworthy essay, especially for its mastery over AdS. I learnt about the useful work of Philip Goyal, and am happy to observe some commonality with what I have said about discreteness.

Good luck to you in the contest,


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Author Lawrence B Crowell replied on Mar. 7, 2011 @ 01:17 GMT
Thanks for the encouragement. I have made some progress on this since I submitted this paper. However, I have also been terribly ill the last month, so things have been difficult. A bit of this is based on Goyal’s papers, and lecture from the PI he gave recently. This business leads to some general theories I think on the discrete structure of sequences and information in a general context. ...

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Author Lawrence B Crowell replied on Mar. 8, 2011 @ 00:31 GMT
I read your paper through in detail last night and started writing this. I wrote further today, so this got a bit long. I even looked at Adler huge paper, though being 175 pages in length I of course could not read the whole. The A-M matrices, traceless diagonal elements of i = sqrt{-1} and -i, forms is related to the Kahler matrix. This is a line bundle form of the symplectic matrix. To...

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Peter Jackson wrote on Mar. 7, 2011 @ 12:44 GMT

I hope you're recovering. My trepidation in tackling you essay proved misplaced and I don't know why it's languishing. I suppose it may be that your conclusion; "The model here indicates a curious relationship between continuity and discreteness in nature." is no surprise, but I was interested in your take on the wide range of aspects and views. Like Edwin, we speak quite different languages, but I'm encouraged you may be able to do the same as me, and I anyway see disparity is a potential strength in problem solving.

Some thoughts that did occur, have you explored the possibly interesting parallels betwen Goyals work and the Huygens Construction extending via Fourier transform to Ewald-Oseen extinction, giving something very similar to the path integral, but most importantly opening a vista into classical space time and SR to be explored.

Linked to this is the fact that light cones are, in current cosmology, far from smooth. Einstein lensing and gravity well caustics produce large surface perturbations. Acceptance of this seems to show some 'causality' concerns can be eased.

Black Holes. I've derived that the 'folding in' results in toroids, or Tokamaks, which indeed may extend, for instance, from our active galactic nuclii to the EM field of the whole galaxy, and indeed at a greater scale (not discussed here). Have you yet considered the event horizon in this morphology? There is photographic evidence (see the Chandr IR Crab nebula 'Neutron Star' core. I believe I've come across where all the missing Lithium 7 went, focused at the Tokamak (as we wish to do in fusion reactors). Astonishingly, if this is correct, I think I may have an unrecognised smbh in the HH34 Photo in my essay. Look for the macro lensing.

I hope you'll read my essay with an open mind. I'd like a scientific falsification of the proposition, which may otherwise show a fundamental quantum mechanism driving SR, also consistent with GR.

In that vein 'd like to pose a question first. You're watching a train passing by, if the light signals reaching you, from a pulse through the fluorescent tube, is scattered sequentially by the gas molecules (by em waves doing 'c' in the tube), and the light reaches you at 'c'. Would anything actually be doing c plus v anywhere so need to contract? If perhaps the answer is 'No', SR may be better explainable within the postulates and without paradox! And there are other interesting implications, which also do not exactly follow current assumptions. (Please assume I do know what those are).

Very best wishes.


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Author Lawrence B Crowell replied on Mar. 7, 2011 @ 16:54 GMT
My paper is a presentation on quotients and cosets of spaces and moduli which are of a continuous and discrete nature. I did not write this paper with an idea of trying to say nature is either continuous or discrete. There are clearly aspects of both, even if the continuous aspects are more epistemic than ontic, where discrete structures are measured. Yet the discrete measurements give rise to signatures of continuous structures, such as how the 2-slit experiment for single photons gives a discrete pattern which indicates a continuous or wave nature. I tend not to assign any existential preference to things which are epistemic or ontic, but simply enjoy what seems to be the interplay between the two.

I am a bit disappointed in the downward drift of my essay of late. I am going to say that there are 8 essays ahead of mine which are pure balderdash and rubbish. These papers are complete nonsense, and yet they are ahead of mine, where even if I am not right at least I crafted the work with mathematics and calculations. In fact I will say that a couple of these nonsense papers are in the top 10. In addition to these 8 rubbish papers there are about an equal number of papers which are of dubious reasoning, or they are “questionable.” As I look at papers beneath mine on the community rating there are about a half dozen or so which are far too low to ever get into the top 35 which are actually fairly good, and clearly superior to a number near the top. I hesitate to give out author’s names here, but it is clear if you give some of these papers a reading that the value of the “community ranking” is somewhat off base. I can only hope that before the close date there is some assessment by the FQXi members which adjust things some. I look upon this with some disappointment, where in the last week I have seen my paper passed up by a succession of papers, where most are of questionable value.

The issue of the speed of light, where your paper connects ideas of indices of refraction and the like, is that in vacuum the speed of light is an invariant. There is no communication of signals faster than light. There is the velocity of a phase front, a region of constant action v_p = E/p = ω/k. There is the group velocity v_g = ∂ω/∂k, which is

v_g = v_p - λ∂v_p/∂λ

This dispersion does permit the envelope of a wave to move faster than light, but this does not communicate information.

Cheers LC

Constantinos Ragazas wrote on Mar. 7, 2011 @ 19:26 GMT
Dear Lawrence,

Hope you are now feeling better. I couldn't let this contest pass without asking that you 'review' a key result to be found in my essay:

"Planck's Law is an exact mathematical tautology that describes the interaction of measurement"

The proof is simple and elegant. It uses only continuous methods and does not use 'energy quanta' or statistical physics. Furthermore, I argue that this mathematical tautology that is Planck's Law explains why the blackbody spectrum obtained experimentally is indistinguishable from that obtained theoretically from Planck's Law.

In my essay also you will find results pertaining to entropy and The Second Law of Thermodynamics. This you may recall from the number of exchanges we had on this topic last summer. Hope you consider my essay and support my efforts to have these results 'peer reviewed' by the 'judging panel'.

All the best,

Constantinos Ragazas

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Author Lawrence B Crowell replied on Mar. 8, 2011 @ 02:57 GMT
I will have to get back to you on this. I gave your paper a first look. Yes I do remember this sort of idea last summer. I will have to make an assessment of this and write back in the next day or two.

Cheers LC

Author Lawrence B Crowell replied on Mar. 10, 2011 @ 22:35 GMT
In reading your paper I find one curious question. This is I think the same as the question I raised last summer. The equation δt/ħ = 1/kT defines a scale of fluctuation, here with a Euclideanized time. This is a scale of time where the observable uncertainty or disorder of a quantum system is equivalent to thermal fluctuations at some temperature. However, in much of what you do it appears to be used as a variable. It is used as a time in various integrations and as units in a time line. This step actually requires some subtle justification.

Cheers LC

Constantinos Ragazas replied on Mar. 11, 2011 @ 00:20 GMT
Dear Lawrence,

In “A World Without Quanta” there are no “scales of fluctuation” or “disorders of a quantum system”. All the simple logic and mathematical derivations in my essay become clear and convincing if viewed without using the prism of current theory. I am not a physicist! I have no idea what you are talking about. But we can have a good conversation on the results in my essay if you keep to the language and terms I use in it.

The time variable t is a continuous variable, but the equation you point to δt/ħ = 1/kT does not appear anywhere in my essay in that form. The closest to it, I think, is Δt = h/kT. If that is what you are referring to than I can explain that this duration of time is for an 'accumulation of energy' h to occur. This is a result shown in the essay.

You write,

“... in much of what you do it appears to be used as a variable...”

If the 'it' is time t, then yes. It is a continuous variable. The view in all of this is of a 'continuous Universe'. The amazing think is that it is possible to have such a naïve view, and still explain and derive basic results in physics. That's all I can do! Perhaps you and others can do much more. Take it as 'food for thought' and see if it can nourish your physics.

The key result in the essay is “Planck's Law is an exact mathematical tautology that describes the interaction of measurement”. The mathematical derivation is simple and elegant. It does not use 'energy quanta' or statistics. Furthermore I argue that it can fully explain why the experimental blackbody spectrum is indistinguishable from theory. Please comment on that!

As a further enticement, I am about to post a paper that proves the following proposition using and extending the same ideas in my essay:

“If the speed of light is constant, then light is a wave”

Please help me get this essay to the 'church'!

Best regards,


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James Lee Hoover wrote on Mar. 7, 2011 @ 20:24 GMT
This means there should be considerable work available for philosophers long in the future.


Well done and quite esoteric.

This means you put the question in the realm of philosophy. This is probably more objective than my stab at analogue, but I'll stick with my guess.

Jim Hoover

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Author Lawrence B Crowell replied on Mar. 8, 2011 @ 01:25 GMT
I got a bit crimped for space towards the end. The philosophy part got clipped. Discrete things tend to be things which count as data points, a spot on a photoplate in a 2-slit experiment, the click of a photomultiplier tube that registers a count, a Raman spectrogram and so forth. These things are ontological, the reduced state of a particle, or the energy level of an atom and so forth. The continuous stuff are all the fields and waves and the like, which we sort of infer by a collection of the discrete stuff --- the data. The continuous stuff is frankly what makes the discrete stuff happen, but largely these are not directly measurable, or ontological. These tend to be more epistemological.

Physics then has this duality between the two. So this relationship is in a way existential, for it involves the question of what is existence and it seems to somehow involve the nature of an observer. In that way there may be some future work for philosophers, though this will have rather indirect impact on actual physics.

Cheers LC

Peter Jackson wrote on Mar. 7, 2011 @ 21:41 GMT

Thanks for your response. I agree with your comment; "The issue of the speed of light, where your paper connects ideas of indices of refraction and the like, is that in vacuum the speed of light is an invariant. There is no communication of signals faster than light. There is the velocity of a phase front, a region of constant action v_p = E/p = ω/k. There is the group velocity v_g = ∂ω/∂k, which is

v_g = v_p - λ∂v_p/∂λ

This dispersion does permit the envelope of a wave to move faster than light, but this does not communicate information."

However it shows you've missed the logic and the mechanism. I indeed propose and derive that nothing goes faster than light anywhere, yet it may 'apparently' in the same way a runner may appear to be running at 50mph if you view him from the inertial frame of your moving car. It shows any number of invalid frames exist and only one valid one, which is the same one as the observed phenomena.

It explains how SR can work with QM; the received speed of light in a vacuum is indeed always 'c', but the vacuum is itself divided into 'discrete fields' in relative motion by diffractive plasma boundary zones. These ensure the speed of light is always 'c' LOCALLY.

I appreciate your view on other essays, but if you haven't understood the conceptual basis of any such a view is obviously invalid! I'm not criticising as it's beyond most brains conceptual power to visualise and manipulate the number of moving variables needed. Only about 1 in 5 seem to be achieving it so far, but it's gradually increasing.

You should get there if you truly try, but as Einstein and Bragg said, it's finding new ways at looking at and understanding 'what nature has revealed to us' that is important in physics. I could derive it via trains, buses or light boxes if necessary. (Take the sides away and when the mirrors move the pulse moves off into space, as one might expect, but all fully SR postulate compliant!)

Do let me know how you get on.

Best wishes and best of luck.


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Author Lawrence B Crowell replied on Mar. 8, 2011 @ 01:06 GMT
The idea of there being 'discrete fields' in relative motion by diffractive plasma boundary zones, is to me a bit odd. The only thing I can maybe connect this with is a region of vacuum polarization or virtual particles in QFT. This as a virtual cloud of particles is similar in a way to a plasma. The polarization vector, say in QED, does determine an index of refraction. You invoke some astrophysical data on bow shocks and the like, which are very diffuse plasmas.

I am also a bit uncertain by what is meant by apparent motion. Special relativity unequivocally tells us the speed of light is the same in all frames. There is no special frame or a valid frame amongst lots of invalid frames. Using the case of a runner consider that runner is moving at 10m/s and you are riding a bicycle moving at 8 m/s in either direction along the runner’s motion. In a Galilean transformation you would see that runner move at 2 m/s or 18 m/s, depending on orientation. However, if that runner represents a photon you would find the runner moving at 10m/s --- either direction. There is no frame where you would observe the runner moving with some other velocity. This is what the Lorentz transformations give you.

There are some funny things which I have been trying to figure out about your paper, and I think these are couple of things which I do not entirely see.

Cheers LC

Peter Jackson wrote on Mar. 8, 2011 @ 13:20 GMT

Good, You are starting to grasp an initially very tricky set of variables. The fact it seems a bit odd shows you've glimpsed the important adjustment.

I agree SR simply tells us 'c' is constant IN all frames. I suggest another way of conceiving that; If you're not ON that bus going past you're not IN the same frame. (after all - all light you receive FROM that or any frame you receive at 'c' in your frame, so nothing breaks the rule!!)!

200 other people walking, running, driving past or flying over the bus on every possible vector will ALSO receive the light at 'c' IN their frame. What is more, that light (scattered from particles inside the bus) will only ever have DONE 'c' anywhere LOCALLY.

i.e. It does 'c' (or c/n) with respect to (wrt) the bus Inside the bus and within the glass (n=1.5) of the windows, then c/n through the air (n=1.0003), then c/n through the FINE STRUCTURE free electron plasma of your eye or glass instrument lense, then c/n through your eye lense (n=1.38).

Yes, you're spot on about plasma, and the density and frequency correlate to speed. (Yes - through the 'condensate,' 'combined' or 'C' field!- See Edwin Klingman's essay for some sums, which I think Ray now agrees with).

If you didn't feel uncomfortable it wouldn't be the NEW door onto reality the discrete field concept represents. NO absolute background frame, but a background no less. It's a new level of dynamic understanding of SR you only really see the value of when you start to apply it. I pointed it at some Galaxies, and 2 weeks later has a paper accepted for Peer Review. It also takes us a giant step towards GR, effectively simply condensing the mass with motion for equivalence.

As with Edwin, we think very differently Lawrence. That is a fantastic advantage the human race has! - if we USE it. I've just done the initial dynamic conception bit, your skills will be needed, (once you fully understand the conceptual basis), to take it on and make the necessary adjustments to both it and the jigsaw puzzle bits of current physics so it ALL fits together seamlessly at last.

You may now be ready for a pint of beer, and a description of how this can be explained to a barmaid... etc..! I hesitate, but I'm now starting to trust you; (You may also find links to many earlier papers there).

Have fun as Ray would say.


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Cristi Stoica wrote on Mar. 12, 2011 @ 05:56 GMT
Dear Lawrence,

interesting essay, interesting connections you find between entanglement for black holes and AdS spacetime.

Best regards,


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Author Lawrence B Crowell replied on Mar. 12, 2011 @ 13:05 GMT

thanks. The three-fold discrete structure I argued for above turns out to have very rich and deep properties. This turns out to be related to the Calabi-Yau form and the three terms in the Hodge diamond h_{1,1}, h_{2,0) ~ h_{0,2} and h_{3,0} ~ h_{0,3}. The three fold structure is then an aspect of the three-form of G_2 and N = 2 supersymmetry. The G_2 is the centralizer of F_4 in the exceptional E_8. This 3-folding is then an F_4 restriction to the E_6 on the manifold as a form of twistor space. Also the three fold structure defines modular forms, or the Picard modular function with some number theoretic connections with zeta functions.

So it is turning out there is a much deeper structure to this than what I thought at the time I wrote the essay. I am hoping that this FQXi contest might be a way of promoting this further in the future.

Cheers LC

Steve Dufourny replied on Mar. 13, 2011 @ 15:02 GMT
Hi ,

The zeta function, it's interesting that, let's assume a Sum where n=o and infinity with fourier....f(x)=pi²/3+4Sum(-1)exp n/n² cos nx .....where are these connections please with pi exp 4 if we substitute and set correctly.Now let's assume also a good approximation where the thermodynamics are as always proportional and relevant.Debye will agree at my humble knowledge.Now let's see the link with pi ...if the real serie of the zeta function is inserted ...1+1/x+1/y.....= pi exp4/z...

The properties are thermodynamical ....and thus we understand why it's essential to have REAL AND RATIONAL variables for intergrations, substitutions and settings..........pi exp4 /90.....if we consider -pi smaller or equatl to x and x smaller or equal to pi....where are these connections without complexs and imaginaries, x can have so many settings ....the thermo is rational.Could you please resume the link.



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Author Lawrence B Crowell replied on Mar. 13, 2011 @ 22:18 GMT
The Riemann zeta function is

ζ(s) = sum_{n=1}^∞ n^{-s},

which is a rather simple looking formula, but this has some mysterious properties. The function is zero at s = -2, -4, -6, … , and zeros at s = 1/2 + it, where the value of t gives the prime distribution.


T H Ray wrote on Mar. 14, 2011 @ 16:07 GMT

You know that I read and rated your paper (highest) over a month ago. Having just gotten around to Phil Gibbs's excellent essay, though, and knowing that our results in general almost converge (albeit by different methods), I felt compelled to leave a comment.

It occured to me, in the context of quantum error connection coupled to quantum least action, that any measurement by any observer at any instant of "the present" (which is the only kind of instant that quantum mechanics accommodates, since all quantum action is instantaneous)is the lowest energy state possible. That makes the estimated 10^500 vacuua of string theory seem pretty small. I had made the same conclusion, without rigorous mathematical support, in my ICCS 2007 paper



in my

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Author Lawrence B Crowell replied on Mar. 14, 2011 @ 17:27 GMT
The 10^{500} vacua are an estimate from the number of Dp-brane wrapping conditions and T-duality. The question of course we might ask is whether most of these are “real,” where by real I mean they define a spacetime cosmology that has some classical content. I suspect most of these are really quantum amplitudes which are small corrections to the grand path integral of the universe.

The one point of my essay is that with an equivalency between the horizon of a black hole in an AdS spacetime and the field theoretic information on the boundary there is a 3-cycle which is associated with the compactification of fields in the Calabi-Yau manifolds. I have been doing more calculations and work on this and it is getting into some very deep territory. In particular the coset construction leads to Eisenstein series which give an integer counting of quantum bits. This turns out to have strong relationships with the integer partition recently proven by Ono and his group Brunier, Folsom, and Kent. This then accounts for the number of qubits on the horizon of a black hole and the boundary of an AdS spacetime.

This of course touches on the whole multi-verse construction. This is an interesting thing to play with. It also suggests something is funny with our scientific paradigm. We normally have in science the idea that we can prepare experiments with various initial or boundary conditions. The theory then predicts some constant process which is transformed by those initial or boundary conditions, but not removed by them. With the universe we have in effect one grand scattering experiment. We also did not set it up. As such we have no ability to “prepare” systems to test theories, and further we have an ambiguity over what represents initial or boundary conditions and what represents fundamental principles. The current issue over why the universe started out with such low entropy is potentially of this nature.

The number of branchings which occur with measurements or state reductions involves the many world interpretation (MWI), which may be related to the multi-verse concept. With eigenbranching of the universe the numbers do become absolutely bewildering. Tegmark has written considerably on this, and in the context of a sort of platonia concept of universes or cosmologies which obey all possible mathematical structures. I don’t know exactly what to think about that idea. In general my approach to things is a bit more conservative.

Cheers LC

T H Ray replied on Mar. 18, 2011 @ 16:16 GMT
Hi Lawrence,

Anxious to get back to science after all the hoo-hah of late.

Max Tegmark has a following, although my philosophy will never completely converge with his, as long as I am convinced that finite language is inadequate to contain infinite meaning.

Against that meaning, 10^500 vacua is small. When we start asking which solutions are "real" we trap ourselves in the measurement problem. We assume that the nonlocal wave function is a calculational artifact. My view, however, is similar to Ken Wharton's -- taking the universe as a quantum system. The conclusion in my time barrier preprint, that the 4 dimension horizon is identical to the 10 dimension limit, identifies by precise numerical calculation a boundary condition that preserves the Euclidean R^4 of reversible time trajectory, while showing Jacobson-Verlinde type gravitic entropy over n-dimension Euclidean manifolds. I give time a specifically physical definition independent of spacetime geometry, allowing analytic continuation.

If there is a formal finite derivation of string theory vaccum solutions, 10^n, then no matter the value, my theory accommodates any finite range in the domain of 10 dimension physics.

All best,


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Anonymous replied on Mar. 18, 2011 @ 18:55 GMT
The existence of four dimensions is becoming clearer. The gravitational constant is G = 6.67x10^{-11}m^3/(kg-s^2). In naturalized units where mass has a unit of reciprocal length this is an area. In holography the quantum field information content in spacetime is defined by fields on a boundary or horizon. In spacetime the reduction is on space which reduces the horizon area to two dimensions....

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Anonymous wrote on Mar. 14, 2011 @ 21:31 GMT

The general trend here is not looking too good. I seem to be sinking. At this rate I will be below the 35 by end of the day tomorrow. I am not sure what you think, but there are about 5 essays ahead of mine which are pure bilge water, and about 5 that are highly questionable. There are a number below mine which are better than some above. So I am somewhat unhappy at the turn of events lately. My essay in the last month has been up as high as 16 and last week as low as 33, then back up to 20 by midweek. I can only hope this current downward trend is arrested.

I scored yours a while back, where I gave it an 8. The two points off are due to what I see as some incomplete concepts with respect to the “warping.” Ch. Corda has some interesting ideas about intrinsic curvature, where I have worked some plausible argument that this is due to the Heisenberg uncertainly principle. It sounds as if buried beneath what you are doing is something of that nature.

The sort of moduli space constructions of SLOCC state constructions Gibbs writes about, where our papers in some way complement each other, can be extended to E_{7(7}}/SO(8) type of moduli space. The E_8 decomposes into E_7 to E_6 and then E_6. The F_4 group has G_2 as its centralizer. The Calabi Yau construction which can exist is one then based on the invariance of F_4 with an h_{1,1} Hodge diamond with respect to the 3-form of G_2. The G_2 is then a dual of the Calabi-Yau 3-form. My essay touches on this with the G_2 - -> SU(3)xS^6, which gives a triality.

Cheers LC

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Constantinos Ragazas replied on Mar. 15, 2011 @ 02:00 GMT
Dear Lawrence,

Hope my rating of your essay helps a lot.


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Steve Dufourny replied on Mar. 15, 2011 @ 11:29 GMT
Me I have given 8 to you 3 , Christi,Constantin,Lawrence,...Ray also.

Dear all , have you already thought about the finite serie of primes.....



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Author Lawrence B Crowell wrote on Mar. 14, 2011 @ 23:20 GMT
The above is mine. I wonder why this logs me off automatically.


Donatello Dolce wrote on Mar. 16, 2011 @ 00:39 GMT
Dear Crowell,

very interesting essay. I see some analogy with my idea link[\link]", originated on the attempt to interpret the fact that, using Witten's word, in AdS/CFT "quantum phenomena are encoded in classical geometry". I find that the kinematic and the quantum behaviour of a field is encoded in a four dimensional boundary, in agreement with the Holographic principle.

Good luck,


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Author Lawrence B Crowell replied on Mar. 16, 2011 @ 03:00 GMT

I have read the first two section of your paper and the ending, though rather hurriedly I must confess. This does look rather interesting. My only pause with the gravitational part is how degrees of freedom are counted, but this seems to be a comparatively minor issue at this time.

Cheers LC

Constantinos Ragazas wrote on Mar. 17, 2011 @ 01:46 GMT
Congratulations Lawrence! I am happy that you placed comfortably well in the final group. And good luck going forward.

I know this is not the time to be asking, but I am very anxious to have your reflections on the two very short posts I linked in my Mar. 12 post to you. One especially seems to suggest that the CSL hypothesis contradicts the Photon Hypothesis.


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Author Lawrence B Crowell wrote on Mar. 17, 2011 @ 13:41 GMT

The analysis you do involves once again the use of a scale factor as a dynamical variable. In particular this occurs with the Δt, which is a spread in a Euclidean time you related to temperature. The partition function Z has the β = kT in the generator, but the calculation of temperature only happens with the appropriate calculation 1/T = k∂Z/∂E. I can guarantee that this will be a sticking point with reviewers.

Cheers LC

Constantinos Ragazas replied on Mar. 19, 2011 @ 00:25 GMT

You write,

“The analysis you do involves once again the use of a scale factor as a dynamical variable. … I can guarantee that this will be a sticking point with reviewers.”

Are you referring to my use of time as a continuous variable? I am not doing statistical thermodynamics. No use for partition functions or discrete lumps of time or energy in any of the results.

I view time as a human concept. It is what we wish it to be, as long as we have no contradictions in our reasoning. We can have our theories with discrete time, and we can have our theories with continuous time. I prefer the simplicity of my results using continuous time.

But I also have serious philosophical dilemmas with 'discrete physical time'. What is between discrete moments of time? Do we pass from one discrete moment to the next instantly? Is there then 'no time' between discrete moments of time? And if there is 'no time' between 'moments of time', wont that then be 'continuous time'?



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Author Lawrence B Crowell replied on Mar. 19, 2011 @ 12:31 GMT
The fact that you equate a spread in time Δt with a temperature by Δt ~ ħ/kT has by implications an association of the time spread with a statistical quantity. This is what might prove to be a difficulty with reviewers of your papers.

Cheers LC

Constantinos Ragazas replied on Mar. 19, 2011 @ 17:26 GMT

Thanks for your post. I think we are close to resolving this issue.

You write, “The fact that you equate a spread in time Δt with a temperature by Δt ~ ħ/kT “

If I were to blot out any reference to Δt ~ ħ/kT in all of my papers, would that satisfy your objection? The truth of the matter is that non of my results depends on this association. I could have never ever made any mention of this. I only did because I found it to be an interesting curiosity that comes out of my derivations as a consequence.


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Alan Lowey wrote on Mar. 19, 2011 @ 11:13 GMT
Dear Lawrence,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,


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Author Lawrence B Crowell replied on Mar. 19, 2011 @ 12:24 GMT
The closest thing I can think of with respect to the Archimedes screw is the fibration on a manifold. A spacetime manifold has at each point an internal space of N dimensions. In the case of electromagnetism N = 1, for weak force N = 2, and for the strong nuclear force N = 3. This can be thought of as a line or fiber above every point on the manifold or space. Then a closed orbit on the space results in a phase which when lifted onto the fibration results in a sort of spiral.

In the case of gravity things are a bit strange, for the fibration is given by the symmetries of the space itself, and further the algebraic structure is not compact. So things become a little thornier. This is one reason why gravitation is tough to quantize.

Cheers LC

Alan Lowey replied on Mar. 23, 2011 @ 13:10 GMT
Dear Lawrence,

thanks for the reply but I'm a particles in empty space man. The Archimedes screw idea does away with a spacetime continuum. I have a mechanism to explain the orbit of Mercury quandry as well, it's the 'inclination hypothesis' i.e. that gravity is stronger towards the plane of rotation of a celestial body.

Best wishes,


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Author Lawrence B Crowell wrote on Mar. 22, 2011 @ 02:01 GMT

The use of sphere packing is a sort of physical argument with respect to qubits. The Planck scale is a limit to the scale where one can identify a qubit. It is not possible to identify quantum information on a scale smaller than the Planck unit. Sphere packing is a configuration where each sphere with a qubit defines a quantum error correction code. The polytope is then a root...

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T H Ray replied on Mar. 22, 2011 @ 17:17 GMT

You hit the problem of string theory experimental falsification right on the head. String theory does not live in the domain where Planck's constant resides.

Where quantum information is not an irreducible given but rather the result of hyperbolic geometric projection -- which is my strategy -- we can speak of normalization on S^3, which is the first member of my kissing order calculation. It's long been known that low dimension topology has some rather unique properties, which is part of the reason it took so long to prove the Poincare Conjecture, and why we have Calabi-Yau manifolds.

By setting the threshold of observation at 3 dimensions (instead of 2) we get "3 for 2" in a manner of speaking, and a projection of length 1 to 4 dimensions. IOW, 3 + 1 dimensions in a kinetic theory recovers Minkowski space by a strict algebraic method. This method has the added advantage of projective determinacy to the Lebesgue measure domain (where we can deal with PLanck's constant), because by construction, all even dimension models (up to our limit of 9 + 1) reduce to a 0 + 1 spacetime. But look (table in previous post) -- the first even dimension domain is - 3 + 1, leaving the - 2 hyperbolic fundamental domain of string theory. The mathematics is sound, because when we allow the point a - 3 geometry, and knowing that complex analysis interprets the point as a line, we get the string domain naturally coupled with a time metric, without having to impose boundary conditions. Then with analytical continuation over n-dimension manifolds, we get irreversible dynamics while nevertheless preserving by Poincare Recurrence the classical time symmetry of 3 + 1, as a result of recursion native to the real number line. (PR is a better tool than straight commutative symmetry, because no experiment could in principle reverse the time metric along the same path, since the experimenter and apparatus are never at rest relative to the metric.)

The next major task, as alluded to previously, is to fix a finite range of vacuum solutions in the Lebesgue measure domain and put together a program to test them (even if only in thought experiment). It would be serendipitous if the range turned out to to be [0, 1] :-)

I'll look up the Hardy-Ramanujan reference to see if I can connect to it.



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Author Lawrence B Crowell replied on Mar. 23, 2011 @ 03:04 GMT
This sounds a bit like the paper of yours I read over a year ago. It seems that this PR idea is meant to conserve information. Is the kissing number meant as a way of error correction code for this?

Cheers LC

T H Ray replied on Mar. 23, 2011 @ 11:47 GMT

That is correct. Remember, though, that in my construction, information and time are identical. The redundancy of information recurrence in a dynamical system provides sufficient randomization in a nearly closed ensemble, to contribute useful feedback for error correction.

So time symmetry/information conservation in the 3 + 1 domain characterizes classical physics -- while gravity is the result of n-dimension dissipation over manifolds n > 4. My specifically physical definition of time: "n-dimension infinitely orientable metric on self avoiding random walk" implies that dissipation over Riemannian manifolds in Euclidean space (all Riemannian manifolds are orientiable) increases entropy toward the center of mass which in sphere kissing terms is the center sphere. All the information we can recover, though, is on the manifold of S^3, which leads to holography.

Do you remember back in the article introducing Verlinde's results to FQXi, that Lubos Motl objected to the entropic gravity model on the grounds that gravity is symmetric? I allowed that he is right -- and right he is -- it's a problem I recognized years ago, and for which I supplied an algebraic model (see my "Time Counts" FQXi essay and my "time barrier" preprint) that shows a very slight asymmetry using the escape velocity scalar. Briefly, every point of spacetime has a unique and changing value of escape velocity; every instantaneous gravitational relation between any 2 bodies, however, is projectively asymmetric by a tiny but nonzero amount. I mean, that instead of taking c^2 as a spacetime area, I take the value c as a metric and calculate the spatial area projected on each point -- represented by the difference between antipodal escape velocities. This method has the virtue not only of explaining Einstein's approach to "the relativistic theory of the asymmetric field," it explains noncommutative geometry in physical terms. That is, backward-forward projection is not arithmetically commutative. As a consequence, we need the closed algebra of C* to fully capture time and information conservation in both classical and quantum mechanics.

As noted previously, I've identified a 10 dimension (9 + 1) limit to the 4 dimension (3 + 1) horizon outlined above. This limit is an extradimensional analogue to 3 + 1 Minkowski space.

Yes, I get your Hardy-Ramanujan connection now. Note in my ICCS 2006 paper (5.5.2), the linear transformation map A --> B[(0)mod 2] to the even part of B and the odd part of A, and A --> B[(1)mod2] to the odd part of B and the even part of A. As a result, the unit diameter of S^3 has no zero dimension center point. That would leave the embedded hyperbolic space where your string theoretic model can live, projecting holographic information to the S^3 boundary.

All best,


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Author Lawrence B Crowell wrote on Mar. 25, 2011 @ 02:07 GMT
Tom, In reference to your last post:

The Verlinde result involves the screen, or in effect the event horizon. The Birckhoff theorem tells us that if you place a black screen around any spherically symmetric gravity field, that the actual configuration of the source of that field does not matter, and all are equivalent to a black hole. So classically we have that a black hole is the...

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Anonymous wrote on Apr. 5, 2011 @ 00:56 GMT
Philip Gibbs posted 24-Cell and the Kochen Specker theorem, which is a continuation of correspondence between n-partite qubit entanglements and BPS black holes. This concerns an interesting paper which illustrates how the Kochen-Specker theorem, similar to the Bell theorem, is a consequence of the symmetries of the 24-Cell.

There is in my opinion a “stringy” interpretation of this. ...

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Author Yuri Danoyan+ replied on Apr. 11, 2011 @ 16:11 GMT
Dear Lawrence

To my mind 24 Cell theory-redundant theory.

I can imagine 12 Cell theory because in Standard Model well known 2 facts:

1.As is well known all matter is made of 12 particles: 6 quarks and 6 leptons.

2.Standard Model describes their electromagnetic, weak and strong interactions, 12 leptons (including antiparticles), 12 quarks (including antiparticles) and 12 bosons (8 gluons Three gauge boson and a photon).


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Author Lawrence B Crowell replied on Apr. 12, 2011 @ 12:13 GMT
Tha naonymous above is me, the author. I sometimes find these web pages do not always automatically reference the author and put anon instead.

The number of gluons in the SU(3) QCD gauge theory is due to the dimension of the group dim(S((n)) = n^2 – 1, which is 8. This emerges from the 8(+)1 irreducible representation of the group, here (+) means oplus. The other representation is 3(x)3, where (x) means otimes, which describes doublets for quarks and leptons.

F_4 is a much larger group, and has dimension dim(F_4) = 52. 48 of this 52 are the roots of the F_4 as a complex or self dual system. This is considerably larger than the SU(3). The F_4 lattice is a four dimensional body-centered cubic lattice. This lattice defines a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the 24-cell. The Weyl group of F_4 is the symmetry group of the 24-cell of F_4 which is generated by reflections through the hyperplanes orthogonal to the F_4 roots. This is a solvable group of order 1152. This can be seen in that each of the 48 roots has 24 reflections. These are the Hurwitz integral quaterions.

Cheers LC

Author Yuri Danoyan+ replied on Apr. 13, 2011 @ 03:10 GMT
Dear LC

If Feynman, and earlier Stueckelberg, proposed an interpretation of the positron as an electron moving backward in time, all anti-particle will be redundant and N=12 would be right solution.


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Author Lawrence B Crowell wrote on Apr. 13, 2011 @ 12:35 GMT
@ Yuri,

There are three types of symmetries in the world. The first type is the external symmetry of spacetime. These external symmetries are the SO(3,1) ~ SL(2,C)xZ_2 of relativity. The other type of symmetry is the internal symmetry, or symmetries, of gauge theory. These are similar in form to the external symmetries of gravity, but there is a change in signature and the meaning of...

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Author Lawrence B Crowell replied on Apr. 13, 2011 @ 12:43 GMT
I ended this before concluding with the nature of F_4 and the 24 cell. The 24-cell is the kissing sphere condition in four dimensions. This is the minimal packing of spheres in a 4 dimensional polytope. In three dimensions it is the octahedron, which was the Kepler conjecture proven last decade. Musin proved around 2002 the case in four dimensions, and the minimal sphere packing in four dimensions is the 24-cell. The group structure of the 24 is the F_4 group of the Hurwitz quaternions. This also sits in a larger structure, related to the Freudenthal determinant which is the magic square of the Cayley sequence of exceptional Lie groups.

Cheers LC

Author Yuri Danoyan+ replied on Apr. 13, 2011 @ 18:53 GMT
My opinion about supersymmetry see here



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Author Lawrence B Crowell wrote on Apr. 24, 2011 @ 01:25 GMT
The argument involves quantum uncertainty between different coordinates. Principally this is between time and position. Suppose there is a theory that time does not exist. All that exist with some geometric content is space. Suppose there is another theory where only time exists, but not space. It might then be that these two theories are complementary sets. They are complementary in the...

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Author Yuri Danoyan+ replied on Apr. 24, 2011 @ 19:46 GMT
It seems to me very interesting


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