Discrete Time and Kleinian Structures in Duality Between Spacetime and Particle Physics by Lawrence B Crowell

Dear Lawrence,

Hope my rating of your essay helps a lot.

Constantinos

Thanks dear Lawrence,Hi Christi,

It's fascinating these distributions of numbers.

Laurent and Taylor shall agree.....the series of exp......2pi i....and the sphere still and always.....the series are rational in logic of proportionalities......now if we take the theory of rests.....and the singularities....see the main central sphere and its fractal and finite serie......the function is purelly analytic....if the serie is finite...if the serie is infinite(correlation with increases of mass and thus entropy with a number universal )....now the sense of rotations .....and a little of Smirnov ....-3/16 i.......3/8 pi.....2pi i(-3/16 i)......the real axis are relevant with a finite number of spheres .......

Best Regards

Steve

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.....

Regards

Steve

Dear Crowell,

very interesting essay. I see some analogy with my idea [link:www.fqxi.org/community/forum/topic/901]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,

Donatello

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.

Constantinos

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,

Tom

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. The black hole horizon is then a boundary in space which contains all the field theoretic information in that space, but all on a region of one dimension less. As such the horizon turns out to be partitioned into units of area A = Għ/c^3 (ħ and c are "one" in naturalized units) which hold some integer number of microstates, and the total number of states is given by an integer partition of microstates. Dimension for space is then 3 by necessity, and as this is embedded in a Lorentz manifold the dimension of spacetime is 4.

The Anti de Sitter Spacetime (AdS) AdS_4 reduces to AdS_2xS^2 near a black hole contained in the AdS_4. This is a BTZ type of black hole, and the AdS_2xS^2 has a correspondence with the condition for an extremal black hole. So the quantum states on the AdS_4, or AdS_2 in this reduction, are in entanglements which reflect the BPS type of the black hole, 1/2 or 1/4 supersymmetric etc. In 11 dimensions there are two dual AdS/CFT correspondences AdS_4xS^7 and AdS_7xS^4. On the AdS^7 there is a hyperbolic form of the G_2 holonomy which is matched by the elliptical G_2 holonomy on the S^7. The G_2 group, the smallest exceptional group, has 7 roots and defines the solution of a cubic form.

With the S^7 this can be reduced to S^6 with the infinite momentum or lightcone frame, so the gauge fixes field in 10-dimensions. The G_2 holonomy becomes SU(3)xS^6 (which I have in my essay) and for a general six dimension, eg S^6 replaced by K_3xK_3 etc, the quantization condition on the AdS_2 is given by the dual description of the Calabi-Yau (CY) manifold in 6 dimensions.

At this stage things become rather mathematical, and in the last couple of months there have been some papers posted which are alarmingly close to what I am working on. The CY form is a cubic form given by the G_2 action. This then leads to some Eisenstein series calculations for the modular system that computes the partition of those integer areas on the horizon. The 3-fold action of the CY form then gives an Eisenstein series which gives a modular definition of these integers of quantum states on horizons that should be identical to the integer partition theorem proof, recently found by Ono et al.

The 10^{500} vacuum states comes about from estimates on the number of possible CY manifolds and orbifold configurations for strings and Dp-branes. The modes of a string, such as its vibration state etc, are T-dual to the winding number of that string on a compactified region. This is what lead Polchinski back in 2000 to say there were about 10^{100} such configurations, and the estimate is up to about 10^{1000} these days.

The relationship between this "multiverse" and the many worlds aspect of measurement is something which Tegmark has suggested. He suggests that ultimately the splitting of the universe into these multiverses or an ensemble of spacetime cosmologies is a matter of state reduction. There is potentially some merit to the idea, for the observable universe has this classical aspect to it, which indicates some sort of cosmic state reduction. Tegmark of course takes this far further to argue there is something like a platonia of all possible structures that exist. I am not sure about these speculations to be honest.

Cheers LC

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,

Alan

Lawrence,

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.

Constantinos

Ah! Lawrence,

Just when I thought we had this issue resolved you come back with more Delphic pronouncements. What am I "...computing something as variable ..."? What "partition function" I am using when I am not using any? Can there be any other way than the "usual way of computing things..."? And what is wrong with "... what you are doing puts this upside down a bit", if there are no mathematical flaws in the arguments and the 'physics of the physical makes sense'? Isn't this in fact how we progress?

I can understand the need to "see the epicycles" in the "ellipses that I am showing". It may be a necessary and comforting exercise for those that know how to use epicycles to calculate orbits. I confess I don't. Perhaps you can show us where they lie! Were you to do that, nothing in what I have done will get any clearer. But what has been done may get clearer.

Constantinos

Dear Lawrence,

From what you write, it is clear that you have found a connection between the derivation of Planck's Law in my essay and statistical thermodynamics. That is wonderful! I do hope this does raise some eyebrows with insightful delight. I just find it amazing that Planck's Law can be derived more simply using continuous methods. And isn't this what we badly need in Physics? Perhaps this connection that you found may provide us with greater insight that leads to a 'physical view that makes sense'.

Thanks for that!

Constantinos

Hi Lawrence,

I agree with you in general, although I use different language and a different method of getting to the same conclusions. I always have to read your stuff over and over to make sure the connection is there, not because it's opaque but because you pack a lot of meaning into fairly spare symbols.

To this point of whether string theory solutions are real, or mathematical artifacts (quantum amplitudes), I think it just doesn't matter. That is, to make an analogy, in E = mc^2, c^2 is also a mathematical artifact; i.e., it doesn't affect the meaning of the statement E = m. Similarly, when we speak of the fundamental domain of string theory as a hyberbolic 2-space, the range of solutions can be infinite without violating the meaning of string theory as a unifying principle that retrodicts physical results. (By the same comparison, the theory of common ancestry known as evolution is a retrodictive theory, because we cannot observe random mutation and natural selection in real time, though we can model it and compare the model to past events.)

So a 10 dimension boundary condition should be sufficient to limit the domain solutions if the real solutions can be shown finite given the boundary -- which is precisely within the spirit of Einstein's "finite but unbounded" relativistic universe. A finite set of solutions obtained by construction rather than heuristics should give us the means to at least perform thought experiments and create accurate computer models.

So ... going back to my research, which I am in the process of expanding and trying to compact (wish I had your talent for that!) into more comprehensible and standard terms:

My energies have been directed toward a tight proof of the 10 dimension limit. A key idea is the construction of the modulus 12 subgroup of Sophie Germain primes (detailed in my time barrier paper) such that the 2 dimension submanifold self similar to the mod 12 subgroup possesses properties of linear independence over Q and infinite orientability on C*. This latter property allows n-dimension continuation over non-compact manifolds.

The role of the mod 12 structure is then clear, as an eternally recurring zero of an ordered set of kissing spheres admitting non-lattice numbers. That is, every sphere, S > 1, contains at least one 12-vertex lattice, S^2 --> S^oo.

To elaborate, supposes -- instead of analyzing by differentiation and integration -- we allow discrete elements of a contnuous function to self organize in the n dimension Euclidean space. The least 3 dimension sphere, S^2, with 2^n degrees of freedom -- 3(2^n) -- is the zeroth member of S^n, for being the least of the elliptic (zero) manifolds. Lesser Euclidean dimensions have hyperbolic (negative) characteristics and parabolic (positive) manifolds are hyperspatial.

Expressing as a table:

Geometry Order Dimension Topology

point - 3 0 none

line - 2 1 S^0

plane - 1 2 S^1

sphere 0 3 S^2

hypersphere 1 4 S^3

(I hope the table doesn't break up. I can't preview on this computer.)

Best,

Tom