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

Logically in theoretical analysis we have symmetries imply conservation laws. This is just how the theorem is provem. From an empirical perspective the observation of a conservation law supports the existence of a symmetry. In that case you do not have a proof, as is the case with all science.

I am not sure what you are trying to say about Bell inequalities.

Cheers LC

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

A symmetry in QFT or strings implies a root-weight structure which are eigenvalues for particle states. The particle states are constants of the motion for the Hamiltonian of the system. The Hamiltonian is then the Cartan center of the algebraic generator of the symmetry. The Cartan center in adjoint rep consists of matrices which commute with the rest of the algebra.

Cheers LC

Edwin,

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

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

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

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

http://arxiv.org/abs/1101.3559

which illustrates how n-partite entanglements are equivalent to black hole types. What I have done is to look at the equivalency with quantum entanglements and AdS spacetimes. There are discrete group realizations which enter into the problem.

I have worked further to characterize this discrete system. It is related to something called the Calabi-Yau manifold. A black hole has a set of quantal units of area on its event horizon. These are identified with quantum states, and are a partition of integers for counting the number of states on a black hole horizon. The area of a black hole is composed of little quanta of areas given by a sum of integers n_i >= 0,

A = 4 π a(n_1 + n_2 + ... n_m)

where this total number N = n_1 + n_2 + ... n_m can be written according to the integer partition. Another way of thinking about this is that the string modes can exist in a distribution which is an integer parition. This is the holographic principle in action, where the event horizon or stretched horizon is composed of a "gas" of strings.

The density of states for a string is tr(w^N) , which for N = \sum_nα_{-n}α_n the string number operator. Given there are 24 string operator the computation of this generating function is tr(w^N) = f(w)^{-24} for

F(w) = Π_{n=1}^∞(1 - w^n)

This is a form of the Dedekind η-function and the remaining calculation leads to a form of the Hardy-Ramanujan approximation for the integer partitions. Recent results by Ono, Brunier, Folsom, and Kent in the role of modular forms in number theory has result in an exact theory for integer partitions

The partition function for the bosonic string is a Dedekind η-function, which is usually approximated so it has a form of the Hardy-Ramanujan function. However, if one considers it under various the discrete group actions the resulting generator function is a product of Eisenstein functions. The Eisenstein function is has eigenvalue -2 with the euclideanized hyerpbolic Laplacian -y^2(∂^2/∂x^2 + ∂^2/∂y^2). This function, F(z) under the action of (1/2π)(-i∂/∂z + 1/y] is an anti-holomorphic function that is an integer partition.

So things are moving forwards, I hope. 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. These are in particular with the AdS ~ CFT correspondence, correspondence between n-partitite entangelements and black hole types, elliptic curve structure with quantum states, Calabi-Yau forms, integer partitions, which all point to the prospect that the quantum states of supergravity are the zeros of the Riemann ς-function.

My essay here is doing pretty well. It might be a bit overly technical, which is why it is not in the top 6, but has at least been in the top 6 through 12. I will confess that I do think some essays currently ahead of my paper do not particularly warrant the positions they hold, but that is outside my power to change. I would say the papers now ahead of mine I do regard as most reasonable are:

Is Reality Digital or Analog? by Jarmo Matti Mäkelä

Reality Is Ultimately Digital, and Its Program Is Still Undebugged by Tommaso Bolognesi

A Functional Virtual Reality by Efthimios Harokopos

The World is Either Algorithmic or Mostly Random by Hector Zenil

Continuous Spacetime From Discrete Holographic Models by Moshe Rozali

A Universe Programmed with Strings of Qubits by Philip Gibbs

What Mathematics Is Most Pertinent For Describing Nature? by Felix M Lev

Cheers LC

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

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