On a Final Theory of Mathematics and Physics by Michael Rios

I have the papers by Gunaydin, and the U Duality paper starts out with rather familiar topics. The 57 dimensions is an extension of the 56 dimensions by one. The 248 of E8 is with the rank or grade -2, -1, 0, 1, 2 given as 248 = 1тКХ56тКХ133тКХ1тКХ56тКХ1 according to the E7 decomposition with 133. The 57 of E_{7(7)} contains 28 gauge charges plus their magnetic duals. The additional dimension is some sort of parameterization. This is though a charge space, not as I can see a space in the same meaning of spacetime. These are in a sense 56 U(1) electromagnetisms for charges and magnetic monopoles that are parametrized by this additional dimension.

The decomposition is of course for e_{8(8)} = so(16) + 128, where the 128 consists of the 56 here, plus 70 scalars and 2 gravities with mass and NUT charges. We may then think of the additional dimension here as a gravitational.

The inclusions O(n) тКВ U(n) тКВ Sp(n) = USp(2n) and USp(n) тКВ U(n) тКВ O(2n) are define a cyclic sequence of 8 used in a geometric proof of the Bott periodicity theorem. The sympletic and unitary groups are related by Sp(n) тЙ... U(2n) тИй Sp(2n, C). This appears to be good news for my idea about Bott periodicity, which is motivated by physical ideas of a phase transition in entanglements that give rise to spacetime.

As you know Bott periodicity is a cornerstone of K-homology. I know about this, but I am not a working expert on K-theory. Largely I am in part a bit of a "meatball" mathematician, which is to say that I am not as steeped in the epsilons and deltas as mathematicians are. I try in some ways to straddle between physical concepts and idea and mathematical truth.

Cheers LC

I mention K-homology as the framework for studying the worldvolume geometry a la Connes. The natural coordinates for branes are noncommutative, and classical spacetime is spectral. One builds a manifold from noncommutative and nonassociative C*-algebras. This is in agreement with your comments on magmas, as observable algebras are refined magma structures.

I presume you refer in part to the paper by Chamseddine, Connes, and Mukhanov. This is an interesting development. It find in particular the fact that they have discovered an 8-fold cyclic structure in this also interesting. This suggests something related to Bott periodicity, maybe something involved with normed division algebras, maybe E8 ... . This idea of there being a quantum of geometry is curiously dual to the smoothness of geometry one gets with the moduli space developments of Uhlenbeck, Donaldson and others with gauge fields on 4-manifolds.

This may connect up with the experimental results by the Fermi and Integral spacecrafts by NASA and ESA respectively. The observation of photons at different wavelengths found they arrived from very distant Оі-ray burstars, from very short to longer optical wavelengths, at the same time. This is dual to the prediction of loop quantum gravity, which predicted a dispersion of photons due to the graininess of spacetime. The observation is of spacetime structure with a very long baseline. In effect the transverse momentum that scatters off spacetime, or that involved with detecting these photons, is tiny. This compares to a measurement of particle that attempts to localize it in a tiny region of spacetime. This results in a massive transverse momentum transfer and the particle scatters wildly. If one attempts to localize the particle to near the Planck scale then spacetime time will behave in a quantum or discrete manner.

I am going to write on the AdS/CFT holographic principle. There is an interesting way to think about this, which is not in the literature or at least I have not seen it. It is to see that the AdS_5 is a moduli space for the CFT. In this way a Yang-Mills gauge field is a boundary of a moduli space, and the set of gauge connections as themselves the holographic projection. I will try to leave off with how this connects to the Cartan matrix for the E8 group.

The AdS/CFT holographic principle is related to the results on gauge fields in four dimensions. The moduli space for a quaternion H or SU(2)^2 = SO(4) bundle is a 5-sphere, and a hyperbolic spacetime form of this is the AdS_5

We consider pairs of quaternions with R^8 ~ H^2. This has the quaternion inner product rule

= p_1q-bar_1 + p_2q-bar_2

that is a real valued product on R^8 The vectors with real norm = 1 form the seven sphere S^7, which is a 3-fibration over S^4 ~ HP^4. This is just a quaternion version of the Riemann sphere. The principle bundle ПЂ:S^7 --- > S^4 is formed from the imaginary part of one quaternion as an internal space over the other set of quaternions with unit norm. The bundle map then sends one set of quaterionions into an H^1 inner product (q_1, q_2) --- > and the remainder is the projectivization of H^2 - = (pq_1, pq_2). The fibration is the left adjoint action of the SU(2) group. The bundle contains vertical and horizontal portions for the group action and the base manifold respectively. The vertical tangent bundle is then the set of p in Im(H) that defines a (pq_1, pq_2) as a 4-space with the group action by p on every element. The horizontal component, defined by H with elements (q_1, q_2), have the one form

П‰ = Im(q_1dq-bar_1 + q_2dq-bar_2)

and curvature form

О© = dq_1/\dq-bar_1 + dq_2/\dq-bar_2

where we consider a "slice" where dq_2 = 0. The horizontal bundle subspace is spanned then by ∂/∂q_1 and this curvature form is the pullback HP^1 of a self-dual form. The horizontal action is then two copies of the sp(2), sp(2)+sp(2) ~ sp(4) action on S^7 projected onto S^4. These group actions form the quotient of the H^2 group SL(2,H) ~ SO(5,1) with SL(2,H)/(sp(4) ~ SO(5)). This is the moduli space, which has the dimension predicted by the Atiyah-Singer theorem.

This is the Euclideanized form of the theory, for we really want a Lorentzian version of these spaces. The z_0 component of the quaternions is modified so the conjugate of z_0 is -z_0. This gives us the norm zz-bar = -|z_0|^2 + z_1^2 + z_2^2 + z_3^2, and the unit condition is replaced with the zero or null condition on a light cone. Our moduli space is then

SL(2,H)/Sp(4) ~ SO(5,1)/SO(5) = B^5 = the five dimensional ball,

and in the Lorentzian form it is

SL(2,H)/Sp(3,1) ~ SO(5,1)/SO(4,1) = AdS_5.

Of course AdS_5 is the anti-de Sitter spacetime in 5 dimensions. One perspective on this is to say that one of the Sp(2) symplectic groups with Sp(2) ~ SU(2) is replaced with sp(1,1) ~ SU(1,1) for boosts instead of rotations. Another perspective is to say that two copies of SU(2), or SO(4) is contained in sp(3,1), but where the Lorentzian change in metric is not on the group actions.

This theory is connected with the Dirac operator, and Connes' work is centered around this as well. I think these two developments are related to each other. Maybe these two are related to each other according to S-duality or П„ = Оё/2ПЂ - 4ПЂi/g^2 for a general coupling constant for the two theories. Connes' work with the Dirac matrices may connect up with Atyah's work on the Dirac operator and the elliptic bundle in some sort of dual theory.

Cheers LC

I am a bit short on time. SL(2,H) is the group of the quaternions in my above discussion on AdS_5. These group actions form the quotient of the H^2 group SL(2,H) ~ SO(5,1) with SL(2,H)/(sp(4) ~ SO(5)). SL(2,H) is the isometry group of the 5-dim moduli space, and for a Lorentzian form with SO(4,2) this is the isometry group of AdS_5.

The SL(2,H) is really a double twistor theory, It also connects with the 57 dimensions on E_{6(6)}. The FTS dim[Der(H)тК--J3(O) + aut(H)] = 5тИЩ26 + 5 = 133 is E_{7(7)} and this embeds the bf 57 for the gravitons plus 28 "electromagnetisms" and their magneto-duals or 28 dimensions for a gauge theory (an SO(8) gauge theory as well?).

Cheers LC

Thanks for this bracing high level survey of the latest work connecting maths and physics. If Wigner was impressed with the mysterious connections between group theory and quantum physics he would be fully amazed at where it has now led. I hope the end point will bring some unification that restores the possibility that a single human mind can get the whole picture.

I have been reading Witten's paper Perturbative Gauge Theory As A String Theory In Twistor Space (http://arxiv.org/pdf/hep-th/0312171v2.pdf ). This is a long paper to read.

My sense of what you describe as twistor theory based on H^2 is that H^2 ~ C^4, and of course we have a different meaning to projective spaces. The standard theory has T = C^4 with projective twistor space CP^3 = P(C^4) < --- F_12(T) --- > G_{2,4}(C). The G_{2,4}(C) = U(4)/U(2)^2, or in signatured spacetime SU(2,2)/SU(2)xSU(1,1). The quotient between SU(2,2) = SO(4,2) and SU(2)xSU(1,1) ~ SO(3,1) is the Grassmannian space of this space of 2-planes in 4-space. The quotient SO(4,2)/SO(4,1) is the AdS_5 or moduli space for the quaternion bundle. This has a relationship to G_{2,4}(C) as its quotient is with SO(3,1) вЉ‚ SO(4,1). The CP^3 is projective twistor space PT and the manifold F_{12}(T) is a five dimensional space that through this double fibration maps information in this space to CP^3.

The double fibration has the equivariant action of SL(4,C). For H^2 this action is likely replaced by SL(2,H). Both of these are dim = 15 and ~ SU(4) or SU(2,2). The structure above has to be modified in some ways. I would propose that the product of elements in the two H's is given by the Jordon product or the general product

U*V = Uв--¦V - ВЅ(U tr(V) + V tr(U)) + ВЅ(trU tr(V) - trV tr(U))I.

The Freudenthal determinant of a matrix U is det(U)I = tr[(U*U)в--¦U], which gives the eigenvalued problem

det(U - О»I) = О»^3 - О»^2Tr U - О»tr(U*U) - det(U)I = 0

This cubic involves the eigenvalues О»1, О»2, О»3 with eigen-vectors x, y, z. The determinant det(U - О»I) = (U - О»I)в--¦[(U - О»I)*(U - О»I)], where have (U - О»I)*(U - О»I) = 0, so that if U - О»I = P_О» a projector on the Fano plane then this is an orthogonality condtion. We then have for instance two projectors P_{О»1} = xx^† and P_{О»2} = yy^† so that P_{О»1}в--¦P_{О»2} = 0 as an orthogonality condition.

Given K the Jordan algebra h_n(K) = RвЉ•h_{n-1}(K)вЉ•K^n. Given T = C^4 this is h_5(C) = RвЉ•h_4(C)вЉ•C^4. By SL(4,C) ~ SL(2,H) we have a representation

h_3(H) = RвЉ•h_2(H)вЉ•H^2.

The quaternion x = (p, q) is then such that

xx^† =

|p_1p_1^†, p_1q_2^†|

|q_2p_1^†, p_2q_2^†|

as a 2x2 matrix of 2-spinors or quaternions. For this exterior product between x with positive helicity and x' with negative helicity this is a null quaternion momentum for a massless particle. Inner products may also be easily defined for + and - helicity fields. Further analysis along these lines will lead to maximal helicity violating amplitudes and the BCFW theory.

The h_3(H) is is a matrix of the form

|П†, П€|

|П€, r|

for П† \in h_2(H), П€ \in H^2 and r \in R. The two quaternions with four components defines four dimensions complex or 8 real dimensions, we have 2 + 1 dimensions from the remaining parts of the matrix, which gives the HP^{2|3} =~ CP^{4|3} coordinates (8 - 3 = 5).

The matrix U \in J3(O) is diagonalizable by the F4 group. The partition of the matrix as

U = sum_iP_i = sum_i(λ_i x_ix_i^†).

The matrix above, is a 5x5 matrix that transforms under SO(10) or SO(9,1). The SO(9) вЉ‚ SO(9), which is also a subgroup of F4. This serves to conserve the eigenvalues, and further it is a subgroup of the E6 exotic group.

Cheers LC

The short exact sequence

F4: B_4 --- > F_{52/16} ---- > OP^2

connects the B_4 = SO(9) (SO(8,1)) with the Moufang plane. An irrep of SO(9) is 8вЉ•8 for a "line" ~ OP \in OP^2. The old geometry of Euclid tells us that two points give a line, and the duality between OP and OP^2 involves a triality of 8вЉ•8вЉ•8. This is a manifestation of how F4 diagonalizes the 3x3 J3(O). What is interesting is that h_2(O) embeds in a larger matrix (the 3x3) and the SO(9) embeds in SO(9,1). SO(9,1) вЉ‚ E6, so E6 seems to be a decent candidate for the automorphism of h_3(HвЉ•O) or

All of that is similar to Susskind's idea of the boosted frame in SO(10) stringy SUGRA, which is a holographic result.

I planned on going deeper into these issues, for I did a lot of calculations. However, I thought I would bring a bit of physics into this picture. The concern is with the physical interpretation of nonassociative operators. In my essay I discuss the question of hypercomputation, which is interesting if you are into quantum information. These Malament-Hogarth spacetimes. Since the inner horizon r_- of a Kerr-Newman black hole is continuous with I^в€ћ in the exterior region an arbitrarily large or infinite set of computation states can be transmitted to an observer as they cross r_-. Of course this pertains to eternal black holes, but I will stick with the idealization for now. Thus a computer running a recursively enumerable or nonhalting program can be "emulated" by a computer that crosses r_-. In effect this is a sort of Universal Turing Machine.

Hypercomputation appears to be involved with black hole. An elementary argument can be seen in a canonical problem as the Zeno paradox situation where a switch is changed with each ВЅ, Вј, 1/8, 1/16 ... th of a second. The question is what state is the switch after one second. If that were to accelerate its flipping it would require an enormous amount of energy. If this could be done without it flying apart then at some point the energy imparted to it is sufficient that it becomes a black hole. Hyperturing machines then have some curious physical implications. The output of these types of machines appears to be inaccessible to cowards like us who remain outside the blackhole.

I mention in my essay that from a quantum computation perspective these quantum computer states may be shadow states. This is an old idea I came across some years ago. They are states that violate the Born rule in a sense, in that they do not correspond to observables. However, they are probability channels that influence scattering physics. It might be that quantum information corresponding to hypercomputation is of this nature.

I also have pondered whether nonassociativity plays a role here as well. Are quantum states with nonassociative algebra shadow states? Do they play a role of processing the interior of black hole quantum states to prevent violations such as firewalls? The connection to the M-matrix theory of Susskind, along with Banks et al, might be see in the what I wrote above. The nonassociativity might change the eigenstructure of this theory. The commutation of the matrices of SO(9), all which vanish in this theory, is changed. Nonassociativity liberates these constraints. The only variables in the SO(9), are not 9, but given by h_2(O) and this means only 2 fundamental degrees of freedom. However, this may only be present in extreme places such as the interior of a black hole.

There might be some unexplored territory with the quantum information aspects of this.

LC

Dear Michael,

Based on a recommendation of my friend Lawrence Crowell, I have read your nice Essay. Here are some comments:

1) Your idea that the ultimate theory of physics will also be a unified theory of mathematics is intriguing and in agreement with my Essay which uses general relativity as the most elegant example that physics is maths.

2) In a certain sense, I recently developed an independent approach to quantum gravity. In fact, a key point on the route for quantum gravity is to realize an ultimate model of quantum black hole (BH) as BHs are generally considered theoretical laboratories for ideas in quantum gravity. It is indeed an intuitive but general conviction that, in some respects, BHs are the fundamental bricks of quantum gravity in the same way that atoms are the fundamental bricks of quantum mechanics. My recent results have shown that the such an intuitive picture is more than a picture. I have indeed constructed a model of quantum BH somewhat similar to the historical semi-classical model of the structure of a hydrogen atom introduced by Bohr in 1913. In my model the "electrons" are the horizon's oscillations "triggered" by the emissions of Hawking quanta and by the absorptions of neighboring particles. Also, in my Bohr-like BH model, BH entropy is function of the BH principal quantum number. A recent review of my model, which will appear in a Special Issue of Advances in High Energy Physics, can be found here. I suspect that a final theory of quantum gravity should reproduce my results within a semi-classical approximation.

In any case, the reading of your pretty Essay has been very interesting and enjoyable for me. It surely deserves the highest score that I am going to give you.

I wish you best luck in the Contest.

Cheers, Ch.

Dear Michael,

Thanks for replying.

I hope you will have a chance to read my Essay.

Cheers, Ch.

Dear Michael,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

Michael,

I did some calculations a 2-3 weeks ago. I finally got around to writing them. This is a way of looking at these various models, where in some sense all of them are relevant. I would be curious to know what your assessment of this is. It is a very rough draft at best.

One of the intriguing things I think is the characteristic determinant equation

-det(A - λI) = λ^3 - Tr(A)λ^2 + σ(A)λ - det(A) = x

for x the root of an equation involving the commutator of the elements z \in O and the associator. This is suggestive of an elliptic curve if x = y^2. This may be a route to look at modular forms, Shimura forms in this sort of physics.

Cheers LCAttachment #1: morse_indices_and_entanglement.pdf

Michael,

The full C*-algebra problem means split Cayley plane is in effect a form of twistor space. The question I have is whether or not I can still have my idea about Morse theory, or maybe in this case Floer cohomology. I think that these large numbers of SLOCC entanglements with 1/2, 1/4, 1/8 supersymmetry are stable points in this general manifold. There may be quantum transitions between these, and the black hole horizon induces a type of uncertainty in the nature of entanglement. This might be a way around quantum monogamy that results in the firewall.

I'll take a look at the paper you referenced.

Cheers LC