On a Final Theory of Mathematics and Physics by Michael Rios

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

DEar Micharl,

I just gave you a good mark. I hope you will find the time to read my own essay.

Best,

Michel

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 link does not seem to work.

The J3(O) matrix is diagonalized by F4 into real eigenvalues. The F4 is the automorphism group of the Jordan 3x3. The sequence

F4: 0 --- > A_4 --- > F_{52/16} --- > OP^2 -- > 0

Means the eigenvalues found with F4 can be found in part with the subgroup A_4 = SO(9) or SO(8,1).

The E6 contains SO(10) or SO(9,1) . As with spacetime and the Lorentz group SO(3,1) the rotations SO(3) on a spatial surface that in turn foliate spacetime. These foliations define a thickened small region of spacetime with some lapse function, and the two spatial boundaries define two SO(3)s. In this way the SO(9,1) embeds SO(9) in a pairing. This is a physical reason why the E6 ÔèéSO(9,1) is over the CxO. Similarly the E7 is over HxO, the quarter-octonions.

For E6 the eigenvalues are not necessarily real. These complex numbers may appear perplexing, and further violate unitarity. However, in this C* algebraic form of the Weyl-Heisenberg algebra (a, a^ÔÇá, N, c):

[a, a^ÔÇá] = 2c

[N,a] = Ôê'a

[N,a^ÔÇá] =aÔÇá

[c,Ôï...] = 0,

where the standard theory has c = ›. We can see this does not result in a larger phase space volume or demolish the Hessian transformation principles of conservative physics. The Hopf algebra coproduct is introduced:

╬"a = aÔè--I + IÔè--a,

which is a coproduct "duplication" of the operator into left and right forms of the operator. Now the coproduct can be deformed forming a quantum group:

╬"aq = aqÔè--qc + qÔê'cÔè--aq

where q is the deformation parameter related to a geometric-like series:

[x]q = q^x Ôê' q^{-x}/(qÔê'q^{Ôê'1})

The elements of the q-deformation can be functions of any form. In standard QM these would be complex variables or functions, but they can also be real valued.

The deformation ╬"a(q) = a(q)Ôè--q^c + q^{Ôê'c}Ôè--a(q) is related to Bogoliubov operators. We may write

q^c = Ie^{2¤Çi¤å} = I(cos(2¤Ç¤å) + i sin(2¤Ç¤å)),

for ¤å a parameter related to acceleration. Of course in addition

(q^c)^{-1} = Ie^{-2¤Çi¤å} = I(cos(2¤Ç¤å) - i sin(2¤Ç¤å)).

This gives

╬"a(q) = a(q)Ôè--I(cos(2¤Ç¤å) + i sin(2¤Ç¤å)) + (cos(2¤Ç¤å) - i sin(2¤Ç¤å))IÔè--a(q)

= (a(q)Ôè--I + IÔè--a(q))cos(2¤Ç¤å) + i(a(q)Ôè--I - IÔè--a(q))sin(2¤Ç¤å)

We may of course form the Hermitian conjugate

╬"a^ÔÇá(q) = a^ÔÇá(q)Ôè--q^{-c} + qÔè--a^ÔÇá(q)

= (a^ÔÇá(q)Ôè--I + IÔè--a^ÔÇá(q))cos(2¤Ç¤å) + i(a^ÔÇá(q)Ôè--I - IÔè--a^ÔÇá(q))sin(2¤Ç¤å).

The commutator [╬"a(q), ╬"a^ÔÇá(q)] is

[╬"a(q), ╬"a^ÔÇá(q)] = ([a(q), a^ÔÇá(q)] Ôè--I + IÔè--[a(q), a^ÔÇá(q)])(cos^2(2¤Ç¤å) + sin^2(2¤Ç¤å)),

which gives a unit product.

The hyperbolic trigonometric functions obtained from ¤å --- > i¤å are squeeze parameters and we have a C* form of the Bogoliubov operators

b = au + a^ÔÇáv, b^ÔÇá = a^ÔÇáu + av

for u = cosh(2¤Ç¤å) and v = sinh(2¤Ç¤å). The commutator of b and b^ÔÇá is

[b, b^ÔÇá] = u^2 - v^2 = 1.

The q-deformed algebra has the same u^2 - v^2 = 1. This then means we have the operators employed in computing Hawking and Unruh radiation.

Cheers LC

I got the ppt file to work. For extremal BHs this will work with the 28 charges and 28 magnetic monopoles in the 57.

LC

I think this has to do with gauge theoretic aspects of the quaternions. I attached some time back a post on how quaternions naturally give rise to YM gauge fields. For the momentum p_{aa'} = О»_aО»_a' and the polarization Оµ^{aa'} we have the transversality condition p_{aa'}Оµ^{aa'} = 0. In addition there is the gauge transformation Оµ_{aa'} --- > Оµ_{aa'} + k p_{aa'}, k = constant so that p_{aa'}Оµ^{aa'} = 0 is invariant for a massless field. We can then find the gauge field according to these as

F^{ОјОЅ} = Пѓ^Ој_{aa'}Пѓ^ОЅ_{bb'}( p_{aa'}Оµ^{bb'} - p_{bb'}Оµ^{aa'})

The H^2 space constructs the gauge field and its dual to give the intersection form ∫F/\F = 8πk. Working in C^4 does not bring that structure in.

The 8-dim spacetime enters into this picture as the E8 Cartan matrix. The intersection form is an invariant with respect to that. This is wrapped up in the theory of gauge field in four dimensions. There is underlying this a four dimensions of C^4 or H^2 in complex variables, but underlying this are 8 dimensions in pairs of real variables.

LC

The symmetry is really SO(4,2) ~ SU(2,2). The group on the left is the isometry group of AdS_5 = SO(4,2)/SO(4,1). The group on the right is the group for twistor space with PT^ = SU(2,2)/(SU(2,1)xU(1)). We may think of the quaternion form of this as a transition from C^4 to H^2 with SL(4,C) --- > SL(2,H). This has four complex dimensions, eight real dimensions. We might of course be so bold as to double down, with SU(2,2,C) so that SL(4,C) --- > SL(8,C), but with the above identification of pairs we have that this is SL(4,H) ~ SL(2,O).

LC