On a Final Theory of Mathematics and Physics by Michael Rios

Is quantum information meaningful in the quaternionic case, such as E_7(H), or OxH? I am by default thinking of the exceptional groups as complex or quaterionic.

The "mod" on the K-S theorem might have something to do with Hermitean domains. As I look into this it appears that this might be a more general way of looking at the Cartan theory of decompositions of Lie groups. If this could be made to work in the exceptional domain that would be interesting.

LC

I suppose I would like to have some reference for that. I have certain ideas along these lines. The one limitation I have is that I am not that highly knowledgeable on these large groups, though I know some of this.

Cheers LC

It is possible to compute (anti)hermitean eigenvalues with the FTS.

I presume by *-Jordan QM you mean a C* version of the FTS as some generalization of a quantum group. The quantum group properties are then generalized into this form.

LC

The bioctonion plane has E_6 symmetry, which is is a decent GUT type of theory. E_6 embeds SO(10). For gravitation one might have to go to quateroctonions HâŠ--O with E_7 symmetry or ocyo-octonions with E_8 symmetry and dimension = 128. The 128 dimensions, the last in the secquence of O, RâŠ--O, CâŠ--O HâŠ--O with dim(P^2(KâŠ--O) ) = 16, 32, 64, 128. In this way one can get the G_3/H_3 and G_4/H_4 coset constructions.

The pairing up of division algebras K' ,K permits the definition of a Lie algebra with derivations (derivatives) such der(K')âŠ--J3(K)⊕(K'_tf)âŠ--J3(Y), where tf means tracefree. Is it then possible with this sort of gadget to construct E_{7(-25} and E_{8(24)}? If so then maybe the qubit SLOCC coset construction can go through.

I think that Hermitian symmetry forms might be a way of appealing to the Cartan decompositions in the Kostant-Sekiguchi theorem. I am not sure how this Tits construction of the Freudenthal triple system can facilitate that.

LC

It is possible. In noncommutative and nonassociative geometry, one uses derivations of the algebra to define vector fields and differential forms. Your decomposition for division algebra pairs, follows from the definition of a derivation for the observable algebra over that pair.

For example, over the quater-octonions, we have an observable agebra derivation of dimension 5 x dim tf(J(3,O))+dim(Aut(H))=5x26+3=133. This gives the Lie algebra for E_7(-25).

For the octo-octonions, we have 9 x dim tf(J(3,O))+dim(Aut(O))=9x26+14=248. This gives the Lie algebra for E_8(-24).

I discovered these derivations long ago, as generalizations of the E_6(-26) derivations given by P. Ramond and S. Cato. I presume the E_6(-26) derivation was introduced to them by F. Gürsey. Such derivations make use of the associator over J(3,O). This is why one can rightfully refer to the geometry as nonassociative, as it is built from a nonassociative structure algebra.

Hi again,

I had an earlier post, now my essay has been posted. I know it is probably not to your liking. But I think it will give you more confidence in your own system.

Essay

Thanks and good luck.

This is application of the Freudenthal-Tits der(K)вЉ--J3(O)вЉ•(K_tf)вЉ--J3(O)_tf for K = C would then I presume be dim(Der(C)вЉ--J3(O) + aut(C)) = 3в€™26 + 0 = 78 corresponding to E_{6(-26)}.

If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan's decomposition with [h,h] вЉ‚ h, [h,k] вЉ‚ k, [k,k] вЉ‚ h

N∩G/g = N∩K/k.

For map Ој:P(H) --- > k on P(H) the projective Hilbert space. The differential dОј = = П‰(V, V') is a symplectic form. The variation of ||Ој||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(Ој).

In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,1)/SO(3,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.

In computing the topologies of four manifolds the important quantity evaluated is the intersection form. This may be realized with cobordism of 3-spaces in the 4-dim space. This computes across, admittedly I am being a bit glib here, across the Morse indices.

Physically the idea is that entanglement is itself uncertain in quantum gravity. The fluctuations of spacetime with certain topologies correspond to different entanglement geometries. Consequently the entanglement of states with a quantum black hole are themselves uncertain. This might be one route to looking at this problem called the firewall.

Cheers LC

Dear Michael,

Wonderful essay! You gave a great historical discussion for modern mathematics and M-theory and new connections between mathematics and physics. I also totally agree with your final message about the importance of collaborations between mathematicians and physicists. I would be honored to get your opinion on my essay.

Best regards,

Mohammed

Quasi conformal functions, as I recall, just means the function is not defined on a circular disk, but on an elliptical one. As I recall this is a generalization of the analytic condition on C.

Do you have references for this? The idea that I am working or playing with is that the geometry of entanglement changes or evolves. Physically this is due to topological changes in an embedding 3-space. The AdS_5 ~ CFT^4 indicates that the geometry of the CFT, or QFT, on the boundary is given by the interior AdS_4. I think the entanglement geometry of the QFT or CFT_4 is equivalent to the Morse indices of a spacetime. I am not sure how to formulate this, but I thnk that the E8 Cartan matrix at the heart of the intersection form for 4-manifolds connects this.

LC

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