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

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

Sreenath,

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

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.

LC

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.

The partition of integers is important in 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. Makela's paper above is an interesting read on this.

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) = prod_{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. Recently a proof of the integer partition has been found by Ono, and his post docs Brunier, Folsom, and Kent of an exact formula for the integer partition function.

The black hole in the holographic setting has a stretched horizon which is a gas of strings. If we consider the string to be the bosonic string in d = 26 then 24 correspond to the SO(24) group for the graviton plus dilaton and a gauge field. So the Newtonian insight here seems to be pointing in this direction.

The SLOCC states are compositions of SL(2,R)'s, where a central group of importance is SO(8). The SO(8), or the split form SO(4,4) is important in the equivalency between entanglements and BPS black holes. The SO(8) is also in a triality situation within F_4, the exceptional group of the 24-cell. Now in my paper you might find that there is a 3-way structure with G_2, or the SU(3) holonomy which gives a discrete Klein structure for the entanglements with AdS_n. In the E_8 group the F_4 is the centralizer of G_2, or equivalently their elements are involutory. This means the triality structure is from the Calabi-Yau 3-form, and this periodicity constructs the Eisenstein E(z), E(2z), E(3z), then thus give a stringy approach to proving this theorem. This is then an exact quantum statistics for the stringy structure on the boundary and interior of the AdS_n.

For the AdS_2, which is the condition near a black hole horizon, say a BTZ black hole in AdS_3, the AdS_2 ~ CFT_1. This is the diffeomorphism on the circle, related to the Virasoro group. This is also the Hartle-Hawking state, which is given by this series. The eigenvalues of this are determined by the vanishing of a Riemann zeta function. The quantum states of the universe may then be determined by the Riemann zeta function.

I read your paper about a month ago, and I gave it a high score. I did not comment at the time because I was beginning to feel pretty badly. I will have to return to your sight to remember your paper, for I do recall being favorably impressed.

Cheers LC