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

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

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

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 include commutator structure between position elements [q_i, q_j] = αħ, for α a constant, and similar commutators for momentum, this can be generalized by the Gelfand theorem and Connes' noncommutative geometry.

For gravity this is clearly an important aspect of quantum gravity. Of course we have a lot of funny ideas about this. In your paper you have the "digital" as a world with quantum gravity. This appears evident by just looking at the Schwarzschild metric element 1 - r_0/r, for r_0 = 2GM/c^2, the Killing vector K_t = (1/sqrt{1 - r_0/r})∂_t. This forms a natural operator for a Schrodinger type of equation Hψ = -iħK_tψ. However, the momentum operator p_r = -iħ∂_r does not commute with K_t, and there is a noncommutative geometry. Another way of looking at this is the element r_0 exhibits fluctuations so that r_0 = + δr_0, where δr_0 = f*sqrt{Għ/c^3} = f*L_p for f \in [0, 1]. So a fluctuation of mass-energy in a region of space there is then a fluctuation in the proper time ds ~ (1 - δr_0/r)dt^2, which is a noncommutative situation in energy and time. So there are curiously two different ways of looking at this.

We have two snags with our ideas of quantum gravity. One in string theory, where the action is formulated as

L = ∫d^nx sqrt{-1}(R + α'R_{abcd}R^{abcd} + O(α'^2))

Requires there to be a classical background, or R_{ab} = λg_{ab}. This background dependence is a major criticism which has been lodged at string theory. However, the LQG folks who raise this complaint have problems of their own. In the assignment of a degree of freedom with each strut in a discrete spacetime there is a vacuum E = 3kT/2 element, which when summed up results in a huge entropy to spacetime. For this reason LQG is not able to reconstruct classical spacetime. String theory on the other hand employs holographic principle which vastly reduces the number of degrees of freedom to horizons and boundaries and these problems are avoided.

The string perturbation series is also problematic. It appears almost incomputable. We may then be able to work with some finite series, as in an effective theory. The near horizon for a black hole in an AdS_n is where the spacetime becomes AdS_2xS^{n-2}. The AdS_2 has an equivalency to the CFT_1 with isometries of the SL(2,C) group. This is the elementary group which constructs the quantum SLOCC quantum bit structures equivalent to BPS black holes. The CFT_1 is the Diff(S^1) which is the Virasoro algebra, and in this case with two copies bounded on a conformal map of S^1 to a strip. This defines the Hartle-Hawking quantum states.

The Hartle-Hawking state is constructed by a map from Calabi-Yau three-fold. This constructs the states according to a type of modular form which is related to the partition of integers. This modular form in a Dirchlet L-series has the Riemann zeta function, where its zeros determine the eigenvalues.. The 3-fold in the conjugacy classes of a maximal tori on the F_4 gives the cycle [0, e^{2π/3}, e^{4π/3}], which defines the Eisenstein series E(z), E(2z) E(3z) and the partition function for the quantum states of the AdS_2 ~ CFT_1 spacetime. This is quantum gravity to one loop. This is also equivalently determined by the G_2 group, which in the E_8 is the centralizer of the F_4 group. This extends the work which I present in the paper I wrote for FQXi Building up to AdS_7 will take us up to 6 loop calculations, and extended to 11-dimensional SUGRA to 7 loops.

What comes after that? Frankly, nothing for in effect we run out of algebra. However, there is something which is going on. The hyperbolic dynamics on H_2 ~ AdS_2 is S-dual to a quartic theory of fermions. This is not my work, but was demonstrated by Zamolodchikov (among the other amazing things that guys did), and physically it means the underlying physics of strings in the AdS_2, or equivalently on the boundary as CFT_1, is that of a fermion condensate.

The high temperature domain for the string is the Hagedorn temperature. The density of states for a string with respect to modes n is

η(n) ~ exp(4πn sqrt{α'})

that defines a partition function Z =~ ∫ η(n)exp(-n/T)dn. The Temperature is computed by 1/T = ∂Z/∂n and the path integral diverges for a temperature greater than

T_H = 4π sqrt{α'}

which is the Hagedorn temperature. This is proportional to the reciprocal of the string length. The entropy of the system is the logarithm of the density of states the S ~ 1/nT_H, which in the large n limit is zero. The modes number is given by n = 1/(sqrt{d}M_s), for d the number of degrees of freedom and M_s the string mass. String theory on the AdS_2 transitions into a theory of fermions at this high energy. Strings are then similar to the topological states, such as Skyrmion states.

If this is so then gravity is an effective theory with a classical background. The middle or semi-digital aspects of the world are a form of effective theory. If gravitation or quantum gravity is an emergent theory, we might also ask the same about quantum mechanics? This is based on some aspects of my paper, which I did not illuminate much. Lightcone structure is a projective structure in the completion of the AdS_n spacetimes by quotient geometry. 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 = P x| H. 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

\left(\matrix{

1 & a & b\cr

0 & 1 & c\cr

0 & 0 &1}\right)

which may be extended to n-dimensional systems to form the 2n+1 dimensional Heisenberg group H_n of n + 2 entries

\left(\matrix{

1 & a & b\cr

0 & I_n & c\cr

0 & 0 & 1}\right)

where for O(n,2) the Heisenberg group is H_{2n+3}. The elements a and c are then n+2 dimensional row and column vectors of O(n,2). These are Borel groups, which emerge from the quotient space AdS_n/Γ, where the discrete group Γ is a manifestation of the Calabi-Yau 3-cycle, and which as it turns out gives an integer partition for the set of quantum states in the AdS spacetime. So both spacetime and quantum structure as we know them are emergent.

If we return to our more ordinary world, where gravity is classical and for that matter flat and ignored, quantum mechanics does bring to us a series of difficulties. I tend to agree with you that interpretations of quantum mechanics do not appear effective, for they have no empirical means of falsification. The quantum world may be seen equivalently as a many worlds splitting off continually or as Bohmian be-able particles guided on some path by a pilot wave. The simple fact is that quantum physics assumes two things: The first is that a measurement apparatus is infinite, or has an infinite number of atoms or degrees of freedom, and further that an infinite number of measurements can be conducted. These two assumptions are clearly idealizations.

The difference between a superposition and entanglement is the following. We consider a two slit experiment where a photon wave function interacts with a screen. The wave vector is of the form

|ψ> = e^{ikx}|1> + e^{ik'x}|2>

as a superposition of states for the slits labeled 1 and 2. The normalization is assumed. The state vector is normalized as

= 1 = + + e^{i(k' + k)x} + e^{-i(k' + k)x}

The overlaps and are multiplied by the oscillatory terms which are the interference probabilities one measures on the photoplate. We now consider the classic situation where one tries to measure which slit the photon traverses. We have a device with detects the photon at one of the slit openings. We consider another superposed quantum state. This is a spin space that is

|φ> = (1/sqrt{2})(|+> + |->).

This photon quantum state becomes entangled with this spin state. So we have

|ψ,φ> = e^{ikx}|1>|+> + e^{ik'x}|2>|->

which means if the photon passes through slit number 1 the spin is + and if it passes through slit 2 the spin is in the - state. Now consider the norm of this state vector

= + + e^{i(k' + k)x} + e^{-i(k'+k)x}.

The spin states |+> and |-> are orthogonal and thus and are zero. This means the overlap or interference terms are removed. In effect the superposition has been replaced by an entanglement.

So we may think of the these two entangled systems as that for an electron and the other for a C-60 buckyball in two different states of some sort. One of these particles is pretty clearly in the quantum domain, while the other pushes the envelope of what is quantum. However, people have performed two slit experiments with buckballs, where they have to be supercold. We do not have to cool down electrons. So we might imagine the two slit experiment with electron where one slit contains a buckyball that has some phonon state entangled with the electron being present or not. We may then think of there being an atomic force microscope which then measures the buckball and ... up the scale to the Schrodinger cat. There is a process of entanglement which proceeds up the chain. The scale in length or time diminishes, or the complement in momentum and energy diminishes, as the ratio of mass or action between the system and apparatus approaches zero.

So the curious thing is that we really are operating in the quantum world all along. However, we only see one of the outcomes; we do not see the measurement apparatus in two states or the alive/dead cat. This then leads us to the emergence of the next level in the world, the classical world. While everything is ultimately quantum mechanical, "all the way down," there is the emergence of this classical world which we observe through our senses. It is also the world which we first came to understand with the progression from Galileo and Kepler and culminating in Newton. Of course the Bohmist might object to the idea of the classical world as an illusion, for they say the quantum world is ultimately classical-like or objective in some sense of nonlocal hidden variables. In that language, the classical world is a domain where the Bohm quantum potential is zero. From a many worlds perspective the observer is eigen-branched along only one entanglement path.

So this is how I would interpret this layering of continuous and discrete structures. At the emergence of gravity this seems to connect with the semi-digital. The extremely high energy world consists of quantum states given I think by the zeros of the Riemann zeta function. However, the fields are continuous, so there is I think at this level a complementarity between the continuous and discrete. Once gravity is classical then you have a 1/2 continuum and 1/2 discrete perspective. This then leads to the classical world which appears continuous.

Cheers LC