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**Lawrence Crowell**: *on* 6/10/15 at 21:56pm UTC, wrote The paper by Yang-Hui He & John McKay centers initially upon the linear...

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FQXi FORUM

October 15, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: On a Final Theory of Mathematics and Physics by Michael Rios [refresh]

TOPIC: On a Final Theory of Mathematics and Physics by Michael Rios [refresh]

Since ancient times, mathematics has proven unreasonably effective in its description of physical phenomena. As humankind enters a period of advancement where the completion of the much coveted theory of quantum gravity is at hand, there is mounting evidence this ultimate theory of physics will also be a unified theory of mathematics.

A Mathematician with research interests in homogeneous supergravity, M-theory and exceptional mathematical structures.

Dear Michael Rios,

In my opinion, not one-dimensional strings or loops, nor two-dimensional membrens may be sufficient for a physical theory. You may need 3-dimensional continuum, spontaneously vibrating, as was discussed in my last year's essay: Titled:

On the Emergence of Physical-World from the Ultimate-Reality by Hasmukh K. Tank.

fqxi.org/community/forum/topic/2001

This continuum, and the vibrations in it may not be mechanical. They can be like the electromagnetic waves, or some new kind of waves. I hope, your reading of this essay may trigger some useful idea for your theory.

And my views on current string theory are discussed in this year's essay, titled:

"On the connection between physics and mathematics"

Do you agree with my view, or you would like to correct them?

Yours sincerely,

Hasmukh k. Tank

report post as inappropriate

In my opinion, not one-dimensional strings or loops, nor two-dimensional membrens may be sufficient for a physical theory. You may need 3-dimensional continuum, spontaneously vibrating, as was discussed in my last year's essay: Titled:

On the Emergence of Physical-World from the Ultimate-Reality by Hasmukh K. Tank.

fqxi.org/community/forum/topic/2001

This continuum, and the vibrations in it may not be mechanical. They can be like the electromagnetic waves, or some new kind of waves. I hope, your reading of this essay may trigger some useful idea for your theory.

And my views on current string theory are discussed in this year's essay, titled:

"On the connection between physics and mathematics"

Do you agree with my view, or you would like to correct them?

Yours sincerely,

Hasmukh k. Tank

report post as inappropriate

Hasmukh

Indeed, even in eleven dimensions (D=11) there are 5-dimensional branes solutions. Here's how to see it. 11-dimensional supergravity contains a C-field (M-theory 3-form), which gives rise to a field strength that is a 4-form,**F**. A (p+2)-form field strength couples to sources that are p-dimensional. So the 4-form field strength couples to **2-brane** sources. The 4-form field strength has a magnetic dual that is a (D-p-2)-form, which in D=11 gives a 7-form field strength **F***. This couples to **5-brane** sources.

However, bosonic string theory lives in 26-dimensions, so its connection to D=11 M-theory is still murky. It was conjectured that there might exist a D=27*bosonic M-theory* that contains the bosonic string theory in D=26 as a compactification limit. As argued by Horowitz and Susskind, the theory contains a C-field (bosonic M-theory 3-form), which gives a 4-form that once again couples to **2-branes**. These 2-branes have magnetic duals that are **21-dimensional branes**.

Further mathematical studies will likely shed light on the validity of such a bosonic M-theory. So far, from lattice and sporadic group arguments, there does seem to be hints of a 27-dimensional structure.

Indeed, even in eleven dimensions (D=11) there are 5-dimensional branes solutions. Here's how to see it. 11-dimensional supergravity contains a C-field (M-theory 3-form), which gives rise to a field strength that is a 4-form,

However, bosonic string theory lives in 26-dimensions, so its connection to D=11 M-theory is still murky. It was conjectured that there might exist a D=27

Further mathematical studies will likely shed light on the validity of such a bosonic M-theory. So far, from lattice and sporadic group arguments, there does seem to be hints of a 27-dimensional structure.

Dear Michael,

Thank you for this excellent and informative essay. It really captures the essence of mathematics and physics.

The reason why I like your approach is because it is very much what I find in my approach, which recreates physics from a simple mathematical structure. And that confirms Tegmark's conjecture, although I got to his only after I came up with my...

view entire post

Thank you for this excellent and informative essay. It really captures the essence of mathematics and physics.

The reason why I like your approach is because it is very much what I find in my approach, which recreates physics from a simple mathematical structure. And that confirms Tegmark's conjecture, although I got to his only after I came up with my...

view entire post

report post as inappropriate

adel

Thanks for the comments. Twistors, in light of the new research into Yang-Mills scattering amplitudes, is a hot area of study. Take a look at the twistor references in my paper, such a Witten's on twistor strings and Arkani-Hamed and Trnka's amplituhedron paper. There are more mathematical mysteries still to be unveiled.

Thanks for the comments. Twistors, in light of the new research into Yang-Mills scattering amplitudes, is a hot area of study. Take a look at the twistor references in my paper, such a Witten's on twistor strings and Arkani-Hamed and Trnka's amplituhedron paper. There are more mathematical mysteries still to be unveiled.

You have more unified fields of study but there still goes one unification under the radius, since the point is one.

With regards,

Miss. Sujatha Jagannathan

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With regards,

Miss. Sujatha Jagannathan

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Miss Sujatha

Yes, the unification I'm interested in should reveal a deep connection between number theory and higher dimensional geometry. The concept of a point, in this framework, is closer to that as described in noncommutative geometry, where a pure state in a C*-algebra is the natural generalization of a point in classical geometry.

Yes, the unification I'm interested in should reveal a deep connection between number theory and higher dimensional geometry. The concept of a point, in this framework, is closer to that as described in noncommutative geometry, where a pure state in a C*-algebra is the natural generalization of a point in classical geometry.

I have pondered what the amplituhedron has to do with associahedron. The amplituhedron is a result from the application of YM theory or Yangians in the Grassmannian G_{4,2}. This object is the basis for my work with Bott periodicity in large N and is an object of twistor theory in the double fibration M^{3,1} < --- C^3 --- > G_{4,2}. It would be of interest if the 8-fold cyclicity were continued into the domain of exceptional and sporadic groups. You indicated on my essay page that the associahedron is involved with a binary tree that is categorically the same or a monad for punctured Riemann spheres. The Grassmannian is an equivalency of planes rather than lines, and much the same structure should then exist for twistor theory.

The 8 fold cyclicity I think means there are only particle states corresponding to the E8, E8xE8 or J3(O) group. In effect there is only one electron in the universe. Feynman said that a particle in a path integral description winds all over the universe. This is one point of my description of the double slit experiment with a particle winding around the slit. However, on the cosmological particle horizon this one electron is frozen or trapped so it has a vast number of multiple appearances in the O-region. The same holds for quarks, photons, and so forth. There is only one electron, but it appears in a multiple set of paths we interpret as an ensemble of electrons.

Cheers LC

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The 8 fold cyclicity I think means there are only particle states corresponding to the E8, E8xE8 or J3(O) group. In effect there is only one electron in the universe. Feynman said that a particle in a path integral description winds all over the universe. This is one point of my description of the double slit experiment with a particle winding around the slit. However, on the cosmological particle horizon this one electron is frozen or trapped so it has a vast number of multiple appearances in the O-region. The same holds for quarks, photons, and so forth. There is only one electron, but it appears in a multiple set of paths we interpret as an ensemble of electrons.

Cheers LC

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Before comparing associahedra to the amplituhedron, it is better to return to Witten's original twistor string paper and study his instanton diagrams. Specifically, look at fig. 3, where in the complex case, lines are replaced by Riemann spheres with an internal "twistor" field tube connecting them. Projective twistor space, CP^3, contains copies of CP^2, which further contains copies of CP^1. The CP^1's are the Riemann spheres that are interpreted as instantons in Witten's diagrams. So one can draw MHV amplitudes in terms of points and lines, as long as one remembers that line=CP^1 and plane=CP^2. Knowing this, one can write out combinatorial diagrams for MHV amplitudes, either in terms of Riemann spheres with tubes or points and lines. It's a bit easier to write out the point/line diagrams, which can further be translated to chorded polygon diagrams, which have a binary tree equivalent. This is how one proceeds to build (signed) associahedra for MHV amplitudes. Counting the diagrams, one recovers the same numbers found by the usual formulae given by Hodges et. al.

If one prefers, one can alternatively use quaternion two space, H^2 instead of the usual twistor space C^4. This gives a projective twistor space, HP^1, a 4-sphere. In this representation, there is only a single line for points to localize on, HP^1, and configurations of these lines lie in a higher space, HP^2. This lends itself to generalization into the octonions, where we consider a twistor space O^2, with OP^1 (8-sphere) projective twistor space. Such 8-sphere lines configure in an OP^2, the Cayley plane. Here, collinear configurations of points are transformed by E6(-26). OP^3 doesn't exist by topological restrictions, hence octonionic amplitudes would maximally be configurations of points on 8-sphere instantons. Such amplitudes could be realized in a bosonic M-theory in D=27.

If one prefers, one can alternatively use quaternion two space, H^2 instead of the usual twistor space C^4. This gives a projective twistor space, HP^1, a 4-sphere. In this representation, there is only a single line for points to localize on, HP^1, and configurations of these lines lie in a higher space, HP^2. This lends itself to generalization into the octonions, where we consider a twistor space O^2, with OP^1 (8-sphere) projective twistor space. Such 8-sphere lines configure in an OP^2, the Cayley plane. Here, collinear configurations of points are transformed by E6(-26). OP^3 doesn't exist by topological restrictions, hence octonionic amplitudes would maximally be configurations of points on 8-sphere instantons. Such amplitudes could be realized in a bosonic M-theory in D=27.

Michael,

I have been a bit tied up with a number of things. I think that somehow the associahedron and structures with 3-way products or higher (trees etc) comes from a type of sheaf or Gerbe on the system. The Gerbe gives a WZW type of action that I always thought had connections to associative or nonassociative systems. I will take a look at Witten's paper, which I read again this past summer. I agree that this all connects with Witten's "Twistor revolution" in string theory. I will write more later about replacing C^4 with H^2. This replaces projective complex spaces with projective quaternions.

Your discussion on my essay blog about D-branes and the NxN matrix of their symmetry in U(N) (SU(N)) or SO(N) is close to what I have been working on. The Bott periodicity of these matrix systems gives an 8-fold structure. This 8-fold system has a connection of E8. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

Cheers LC

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I have been a bit tied up with a number of things. I think that somehow the associahedron and structures with 3-way products or higher (trees etc) comes from a type of sheaf or Gerbe on the system. The Gerbe gives a WZW type of action that I always thought had connections to associative or nonassociative systems. I will take a look at Witten's paper, which I read again this past summer. I agree that this all connects with Witten's "Twistor revolution" in string theory. I will write more later about replacing C^4 with H^2. This replaces projective complex spaces with projective quaternions.

Your discussion on my essay blog about D-branes and the NxN matrix of their symmetry in U(N) (SU(N)) or SO(N) is close to what I have been working on. The Bott periodicity of these matrix systems gives an 8-fold structure. This 8-fold system has a connection of E8. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

Cheers LC

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The qubit entanglement interpretations of the exceptional structures is indeed suggestive. For the 3-qubit entanglement interpretation, one must keep in mind this interpretation is a special case of the Freudenthal triple system (FTS) with E_7(C) symmetry. More specifically, the 3-qubit system is seen when the FTS has *diagonalized* 3x3 Jordan C*-algebra components. Given a general element of the FTS, the interpretation is more general and not yet given in the literature. This carries over to the 57-dimensional (non-linear) representation of E_8, that builds on the FTS.

What is known is that F_4 and E_6 give LOCC and SLOCC transformations for octonionic qutrits. E7 and E_8, from what I can see algebraically, operate on generalized octonionic dyons.

What is known is that F_4 and E_6 give LOCC and SLOCC transformations for octonionic qutrits. E7 and E_8, from what I can see algebraically, operate on generalized octonionic dyons.

I think you wrote one of the best, if not the best essay I have read so far. I rated it at 9 and I did not give a 10 because I have not seen all the others.

If you get a chance, please take a look at my essay and rate it. I would be much interested in your expert comments. I also think, like you do, that Newton was the first to discover/use math in physics with his fluxions.

I hope that your good work that is right on the subject will be rewarded.

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If you get a chance, please take a look at my essay and rate it. I would be much interested in your expert comments. I also think, like you do, that Newton was the first to discover/use math in physics with his fluxions.

I hope that your good work that is right on the subject will be rewarded.

report post as inappropriate

Dear Michael,

your essay is particularly effective in suggesting the idea that the deeper we go in investigating fundamental physics on one hand, and in creating unifying concepts in mathematics on the other, the more we find that the latter ‘cover’ the former.

I wonder, however, whether unification is as desirable in Math as it is in Physics.

One could argue, for...

view entire post

your essay is particularly effective in suggesting the idea that the deeper we go in investigating fundamental physics on one hand, and in creating unifying concepts in mathematics on the other, the more we find that the latter ‘cover’ the former.

I wonder, however, whether unification is as desirable in Math as it is in Physics.

One could argue, for...

view entire post

report post as inappropriate

The Freudenthal system gives a form of 3-entanglement with the hyperdeterminant. The hyperdeterminant in for the matrix M that is N^m is invariant under the action of SL(N)⊗SL(n)⊗...⊗SL(N).

The SLOCC is then in general given by the C^N⊗...⊗C^N/SL(n)⊗...⊗SL(N). The standard example for qubits is with N = 2. The Freudenthal triple system occurs for m = 3. The LOCC is given by the quotient of the global group by the local qubit group of transformations, SL(n)⊗...⊗SL(N), which is the gauge-(like) transformation of the states.

The 3 and 4 qubit systems are

G_3/H_3 = SL(n)⊗ SL(n)⊗SL(N)/U(1)⊗U(1)⊗U(1)

G_4/H_4 = SO(4,4)/SO(2,2)⊗SO(2,2).

The further decomposition of the 4-qubit system on the algebra level is so(4,4) = ⊕_4 sl(2,R)⊕(2,2,2,2) in a Cartan decomposition. The Kostant-Sekeguchi theorem works for standard Lie groups. For exceptional Lie groups te 3 and 4-quibit systems are quotients E_{7(7)}/SU(8) and E_{8(8)}/SO(16). These seem to naturally work in much the same way as the above. The 4-qubit system decomposes further into where the algebra is

e_{8(8)} = so(16)⊕128

with E_{8(8)}/⊗_8 SL(2,R). However this does not permit the commutation of independent 128 elements, such as (2,2,1,2,1,1,1,2) and (1,1,2,1,2,2,2,1). There is somehow “more structure” here. It we could get a version of the K-S theorem to work here we could have a more complete LOCC theory for exceptional groups.

Appealiing to the Freudenthal system again, I had thought some years ago (around 2011) that by looking at the theory is H^2 instead of C^2, again thinking with twistors, that this might be a way around this problem. I ended up finding a 57 dimensional representation of E8, or some subgroup therein, but I was not able to accomplish what I wanted. I can’t remember exactly how this worked, and I actually abandoned this.

Cheers LC

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The SLOCC is then in general given by the C^N⊗...⊗C^N/SL(n)⊗...⊗SL(N). The standard example for qubits is with N = 2. The Freudenthal triple system occurs for m = 3. The LOCC is given by the quotient of the global group by the local qubit group of transformations, SL(n)⊗...⊗SL(N), which is the gauge-(like) transformation of the states.

The 3 and 4 qubit systems are

G_3/H_3 = SL(n)⊗ SL(n)⊗SL(N)/U(1)⊗U(1)⊗U(1)

G_4/H_4 = SO(4,4)/SO(2,2)⊗SO(2,2).

The further decomposition of the 4-qubit system on the algebra level is so(4,4) = ⊕_4 sl(2,R)⊕(2,2,2,2) in a Cartan decomposition. The Kostant-Sekeguchi theorem works for standard Lie groups. For exceptional Lie groups te 3 and 4-quibit systems are quotients E_{7(7)}/SU(8) and E_{8(8)}/SO(16). These seem to naturally work in much the same way as the above. The 4-qubit system decomposes further into where the algebra is

e_{8(8)} = so(16)⊕128

with E_{8(8)}/⊗_8 SL(2,R). However this does not permit the commutation of independent 128 elements, such as (2,2,1,2,1,1,1,2) and (1,1,2,1,2,2,2,1). There is somehow “more structure” here. It we could get a version of the K-S theorem to work here we could have a more complete LOCC theory for exceptional groups.

Appealiing to the Freudenthal system again, I had thought some years ago (around 2011) that by looking at the theory is H^2 instead of C^2, again thinking with twistors, that this might be a way around this problem. I ended up finding a 57 dimensional representation of E8, or some subgroup therein, but I was not able to accomplish what I wanted. I can’t remember exactly how this worked, and I actually abandoned this.

Cheers LC

report post as inappropriate

Yes, the entangled qubit interpretation is an approximation to a deeper structure, as the full Freudenthal triple system has more degrees of freedom. For the interpretation to work, even at the 3-qubit level, one is no longer working over the real forms of the FTS, but a complexification thereof. For M-theory compactification on the 7-torus, D=4, N=8 SUGRA has E_7(7) symmetry, while for homogeneous D=4, N=2 SUGRA one recovers E_7(-25). Upon complexification of the two relevant FTS for these cases, one recovers a single FTS with E_7(C) symmetry. The single (complex) variable extension of this FTS is a 57-dimensional structure with nonlinear E_8(C) symmetry.

As of yet, there is no interpretation for the fully complexified FTS with E_7(C) symmetry. It would describe some generalization of the SUGRA systems where the charges are complex valued. Moreover, when the FTS has diagonalized electric and magnetic components, the FTS resembles a 3-qubit system. The complexified FTS, as a bonus, also gives the hermitian symmetric domain that generalizes the non-compact E_7 Riemannian symmetric spaces used in the D=4 SUGRAs.

As of yet, there is no interpretation for the fully complexified FTS with E_7(C) symmetry. It would describe some generalization of the SUGRA systems where the charges are complex valued. Moreover, when the FTS has diagonalized electric and magnetic components, the FTS resembles a 3-qubit system. The complexified FTS, as a bonus, also gives the hermitian symmetric domain that generalizes the non-compact E_7 Riemannian symmetric spaces used in the D=4 SUGRAs.

What are the references for E_{7(7)} and E_{7(-25)} with N = 8 and 4 SUGRA? Are there other compactificiation schemes besides the 7-torus? For instance are there schemes with S^2xK3^2 or some other scheme with CY manifolds?

For K = R, C, H, O we can decompose h(K) into

h^n(K) = R⊕h^{n-1}(K)⊕K^{n-1}

following Baez in his “Octonions” so we form a type of matrix with diagonals in h_{n-1} and off diagonals in K^{n-1} that comprises the spin factor R⊕K^{n-1}. In this isomorphism we may assign K^{n} into a partition K^{n,n+1}. In this way we have one field ψ \in h^{n-1} and another φ \in K^{n-1} forming the diagonals and off diagonals respectively. In this way the projective Fano plane OP^2 and its “Poincare-like dual” as a line in the heavenly sphere OP^1 form a CP^3 with the J^3(O) = h^3(O) construction.

We can then in this way proceed with K’⊗K, where for K’ = H and K = O we have a manner by which one can describe J^3(O)⊗G, of octonion basis with a gauge group action. In this way we can have E_8(C) and E_8(H), where the latter is a quaternionic extension of the FTS. I attach a little note here that illustrates in an elementary manner how quaternions naturally give rise to gauge theory. I have another one of these notes that includes the Dirac field, and how quantum mechanics has in some ways a more natural expression as quaternions. Stephen Adler went into this some.

I was going to write more, but this morning has me rather tightly scheduled.

LC

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For K = R, C, H, O we can decompose h(K) into

h^n(K) = R⊕h^{n-1}(K)⊕K^{n-1}

following Baez in his “Octonions” so we form a type of matrix with diagonals in h_{n-1} and off diagonals in K^{n-1} that comprises the spin factor R⊕K^{n-1}. In this isomorphism we may assign K^{n} into a partition K^{n,n+1}. In this way we have one field ψ \in h^{n-1} and another φ \in K^{n-1} forming the diagonals and off diagonals respectively. In this way the projective Fano plane OP^2 and its “Poincare-like dual” as a line in the heavenly sphere OP^1 form a CP^3 with the J^3(O) = h^3(O) construction.

We can then in this way proceed with K’⊗K, where for K’ = H and K = O we have a manner by which one can describe J^3(O)⊗G, of octonion basis with a gauge group action. In this way we can have E_8(C) and E_8(H), where the latter is a quaternionic extension of the FTS. I attach a little note here that illustrates in an elementary manner how quaternions naturally give rise to gauge theory. I have another one of these notes that includes the Dirac field, and how quantum mechanics has in some ways a more natural expression as quaternions. Stephen Adler went into this some.

I was going to write more, but this morning has me rather tightly scheduled.

LC

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For SUGRA references on non-compact real forms of E_7 see references [29]-[32] in my essay. The E_7(7) case is a 7-torus compactification while I'm not certain about the other compactification manifold for E_7(-25).

The 56-representations correspond to the FTS Lie algebra gradings of E_7. There are other gradings, however, that have a spin factor-like structure, such as:

e_7(-25) = 1 + 32 + so(2,10) + R + 32* + 1

This grading suggests a (2,10) spacetime signature with 32 spinor. An E_8 grading that generalizes this is:

e_8(-24) = 14 + 64 + so(3,11) + R + 64* + 14

Here one sees a (3,11) spacetime signature with 64 spinor. This grading is manifest in Lisi's E_8 model. Morever, it has been shown by Sezgin there exists a super Yang-Mills theory with (3,11) signature.

The 56-representations correspond to the FTS Lie algebra gradings of E_7. There are other gradings, however, that have a spin factor-like structure, such as:

e_7(-25) = 1 + 32 + so(2,10) + R + 32* + 1

This grading suggests a (2,10) spacetime signature with 32 spinor. An E_8 grading that generalizes this is:

e_8(-24) = 14 + 64 + so(3,11) + R + 64* + 14

Here one sees a (3,11) spacetime signature with 64 spinor. This grading is manifest in Lisi's E_8 model. Morever, it has been shown by Sezgin there exists a super Yang-Mills theory with (3,11) signature.

I forgot to attach my notes on how gauge fields emerge from quaterions.

LC

attachments: 1_quaternion_notes.pdf

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LC

attachments: 1_quaternion_notes.pdf

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It appears that the E_{7(-25)} and E_{8(24)} could serve as the global groups G_4 and G_3 fior 4, 3 qubits with SO(2,10) and SO(3,11) as the local gauge-like transformations with

G_4/H_4 = E_{7(-25)}/(E_{6(-78)}xU(1))

G_3/H_3 = E_{8(-24)}/(E_{7(-25)}xSL(2,C))

with E__{6(-78)} the maximal compact subgroup or as E_{7(-25)}/SO(2,10) with a further decomposition. The \bf 56 of the E_{7(-25)} in the 4-qubit correspond to 28 = 1 graviton plus 27 gauge potentials and 28 = 1-NUT parameter gravi-like boson plus 27 magnetic dual gauge potentials. The intertwining of the E_{7(-25)} as the global and local groups for the 4 and 3 qubit case seems interesting.

An additional feature that might be of interest is that the decompositions you cite have even orthogonal Lie groups, and this would fit within my idea of there being a Bott-like periodicity with the exceptional cases. The open question is still whether the nonvanishing commutation of the spinors can be accounted for in some general K-S theorem. It might be that the nonvansishing of these commutators is a blessing in disguise, for it could mean there is a massive reduction in the number of fundamental degrees of freedom.. This restriction might recover something like the K-S theorem. I am on travel this week and into next, so I am a bit constrained with time right now.

In doing all of this, I tend to like to also focus on the physical aspects of these things. I am not the greatest master of group theory, especially large Lie groups and sporadic groups. I think there is some sort of phase change that occurs at the horizon of black holes for accelerated frames. There is a phase change in analogue with symmetry protected states and the phase change from conducting phase to a Mott insulator phase.

Cheers LC

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G_4/H_4 = E_{7(-25)}/(E_{6(-78)}xU(1))

G_3/H_3 = E_{8(-24)}/(E_{7(-25)}xSL(2,C))

with E__{6(-78)} the maximal compact subgroup or as E_{7(-25)}/SO(2,10) with a further decomposition. The \bf 56 of the E_{7(-25)} in the 4-qubit correspond to 28 = 1 graviton plus 27 gauge potentials and 28 = 1-NUT parameter gravi-like boson plus 27 magnetic dual gauge potentials. The intertwining of the E_{7(-25)} as the global and local groups for the 4 and 3 qubit case seems interesting.

An additional feature that might be of interest is that the decompositions you cite have even orthogonal Lie groups, and this would fit within my idea of there being a Bott-like periodicity with the exceptional cases. The open question is still whether the nonvanishing commutation of the spinors can be accounted for in some general K-S theorem. It might be that the nonvansishing of these commutators is a blessing in disguise, for it could mean there is a massive reduction in the number of fundamental degrees of freedom.. This restriction might recover something like the K-S theorem. I am on travel this week and into next, so I am a bit constrained with time right now.

In doing all of this, I tend to like to also focus on the physical aspects of these things. I am not the greatest master of group theory, especially large Lie groups and sporadic groups. I think there is some sort of phase change that occurs at the horizon of black holes for accelerated frames. There is a phase change in analogue with symmetry protected states and the phase change from conducting phase to a Mott insulator phase.

Cheers LC

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The qubit interpretation is only valid in the complex case (e.g., E_7(C)). Otherwise, one has entangled real qubits i.e., rebits. The complex case has much more degrees of freedom than the entangled qubit picture, however. In other words, there is a deeper quantum information interpretation here, more similar to dyonic topological quantum computation. These systems are far more general than any existing quantum computational framework. One must define new systems to make proper sense of them.

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

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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

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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

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Cheers LC

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Dear Michael,

Grothendieck's motives " may serve as an unreasonably effective tool leading to a unified theory of physics and mathematics". This is a very ambitious but credible hypothesis that can at least lead to new conceptual bridges between maths and physics. You have the great merit to introduce the topic in a short non technical essay with the relevant references. I would love to understand quantum field theory in terms of the associahedron and related structures. "Locality and unitary are emergent features": does it mean that non-locality which is specific to quantum theory is not part of the picture?

A few topics you introduced are also in my essay of this year that you may find useful to read. It makes a big use of Grothendieck's dessins d'enfants. Trying to see the relationship to motives, I found "Gauge Theories and Dessins d'Enfants: Beyond the Torus" by S. Bose et al. May be there are other references you may mention to me.

I wish you the best for your enjoyable essay.

Michel

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Grothendieck's motives " may serve as an unreasonably effective tool leading to a unified theory of physics and mathematics". This is a very ambitious but credible hypothesis that can at least lead to new conceptual bridges between maths and physics. You have the great merit to introduce the topic in a short non technical essay with the relevant references. I would love to understand quantum field theory in terms of the associahedron and related structures. "Locality and unitary are emergent features": does it mean that non-locality which is specific to quantum theory is not part of the picture?

A few topics you introduced are also in my essay of this year that you may find useful to read. It makes a big use of Grothendieck's dessins d'enfants. Trying to see the relationship to motives, I found "Gauge Theories and Dessins d'Enfants: Beyond the Torus" by S. Bose et al. May be there are other references you may mention to me.

I wish you the best for your enjoyable essay.

Michel

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Hello Michel

I'm glad you find these topics of interest.

Please read:

Positive Amplitudes In The Amplituhedron

Nima Arkani-Hamed, Andrew Hodges, Jaroslav Trnka

arXiv:1412.8478 [hep-th]

Emergent spacetime from modular motives

Rolf Schimmrigk

arXiv:0812.4450 [hep-th]

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

Robert de Mello Koch, Sanjaye Ramgoolam

arXiv:1002.1634 [hep-th]

I'm glad you find these topics of interest.

Please read:

Positive Amplitudes In The Amplituhedron

Nima Arkani-Hamed, Andrew Hodges, Jaroslav Trnka

arXiv:1412.8478 [hep-th]

Emergent spacetime from modular motives

Rolf Schimmrigk

arXiv:0812.4450 [hep-th]

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

Robert de Mello Koch, Sanjaye Ramgoolam

arXiv:1002.1634 [hep-th]

DEar Micharl,

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

Best,

Michel

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

Best,

Michel

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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.

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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.

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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

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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

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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.

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Philip

Thank you for the encouraging words. There does seem to be a deep underlying structure slowly revealing itself. At this level of the game, when disparate areas of mathematics are elegantly united, one can move further into the unknown with a sense of assurance the arduous journey will be fruitful.

Thank you for the encouraging words. There does seem to be a deep underlying structure slowly revealing itself. At this level of the game, when disparate areas of mathematics are elegantly united, one can move further into the unknown with a sense of assurance the arduous journey will be fruitful.

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.

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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.

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Christian

Thank your for your review. Indeed quantum black holes are very important. It's especially amusing that one can construct quantum black holes with E6, E7 and E8 symmetry. The charge space for the E8 quantum black hole, for example, is 57-dimensional. To regard such objects as building blocks for quantum gravity would imply one take higher dimensional geometry seriously.

Thank your for your review. Indeed quantum black holes are very important. It's especially amusing that one can construct quantum black holes with E6, E7 and E8 symmetry. The charge space for the E8 quantum black hole, for example, is 57-dimensional. To regard such objects as building blocks for quantum gravity would imply one take higher dimensional geometry seriously.

Dear Michael,

Thanks for replying.

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

Cheers, Ch.

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Thanks for replying.

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

Cheers, Ch.

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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

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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

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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 LC

attachments: morse_indices_and_entanglement.pdf

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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 LC

attachments: morse_indices_and_entanglement.pdf

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Lawrence

Nice work. It's a nice exercise to prove eigenvalues of the exceptional Jordan eigenvalue problem are real. You have enough content in your paper to show this. Essentially, one can use the trace to define an inner product on J(3,O) and use positive definiteness to get your result. However, once you study the eigenvalue problem in the split-octonion case, you can no longer argue by such means. In general, the eigenvalue problem one must study is over the full exceptional Jordan C*-algebra.

See Sparling's paper A Primordial Theory for insight on "fat points" of the split Cayley plane.

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Nice work. It's a nice exercise to prove eigenvalues of the exceptional Jordan eigenvalue problem are real. You have enough content in your paper to show this. Essentially, one can use the trace to define an inner product on J(3,O) and use positive definiteness to get your result. However, once you study the eigenvalue problem in the split-octonion case, you can no longer argue by such means. In general, the eigenvalue problem one must study is over the full exceptional Jordan C*-algebra.

See Sparling's paper A Primordial Theory for insight on "fat points" of the split Cayley plane.

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

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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

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Lawrence

Your Morse index tunneling picture looks promising. You might want to take a look at the attractor mechanism in supergravity for further insight.

Your Morse index tunneling picture looks promising. You might want to take a look at the attractor mechanism in supergravity for further insight.

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...

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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...

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Indeed E6(C) contains the SO(9,1) and SO(5,5) determinant preserving groups of the octonion and split-octonion 2x2 Jordan algebras, respectively. From a twistor space perspective, the SO(9,1) transforms points on curves in OP^2, while not preserving distance, but merely preserving collinearity. This is sufficient to attempt to extend Witten's twistor amplitudes to the Cayley plane. (Picture 8-sphere D-instantons here.)

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’} = λ_aλ_a’ and the polarization ε^{aa’} we have the transversality condition p_{aa’}ε^{aa’} = 0. In addition there is the gauge transformation ε_{aa’} --- > ε_{aa’} + k p_{aa’}, k = constant so that p_{aa’}ε^{aa’} = 0 is invariant for a massless field. We can then find the gauge field according to these as

F^{μν} = σ^μ_{aa’}σ^ν_{bb’}( p_{aa’}ε^{bb’} - p_{bb’}ε^{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

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F^{μν} = σ^μ_{aa’}σ^ν_{bb’}( p_{aa’}ε^{bb’} - p_{bb’}ε^{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

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From a projective space description the degree one, genus zero curves in OP^2 are 8-spheres. If Witten's formalism carries over to the Cayley plane, MHV amplitudes should localize on these curves. But first, I agree, it is better to develop the H^2 case with SO(5,1) symmetry.

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

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LC

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Using J(2,H), gives determinant preserving group SL(2,H)=SO(5,1) in Coll(HP^2)=Str_0(J(3,H))=SL(3,H)=SU*(6). Like the SO(3,1) case, where one represents the momentum vectors as bi-spinors in J(2,C), a given vector is lightlike if and only if its determinant is zero. Hence, in the quaternion case, SL(2,H)=SO(5,1) preserves the lightlike property of each quaternionic bi-spinor.

If one...

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If one...

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The inclusions

R: SO(4,3)->SO(3,2)->SO(2,1)

C: SO(5,3)->SO(4,2)->SO(3,1)

H: SO(7,3)->SO(6,2)->SO(5,1)

Have as their middle group the isometry group for AdS_4, AdS_5 and AdS_7. These all share a certain relationship with each other. The AdS_5 gives as 10 dimensional theory for AdS_5xS^5, while the other two give 11-dimensional theory for AdS_4xS^7 and AdS_7xS^4. These last two are dual theories and these have a relationship with the 10 dimension theory in sense that SO(9) is a subset of SO(9,1).

The unification of these implies they are physically an aspect of the octonion. The groups in these inclusions SO(2,1), SO(3,1), SO(5,1) and SO(9,1), dimensions = 3, 4, 6, 10 correspond to physics in these respective dimensions, with 4 for gravity, 6 the CY manifold of compactification. This seems to hinge on the collination of the Moufang OP^2 by E6(-26) which contains SL(2,H) ~ SO(9,1).

There seem to be interesting connections here, and I have this suspicion that an aspect of this is related to the 5-dim moduli space for SU(2) or SO(4) bundles on 4-manifolds. The Lorentzian case gives the hyperbolic SO(4,2)/SO(4,1) ~ AdS_5.

More later after I try to bend some more metal on this.

Cheers LC

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R: SO(4,3)->SO(3,2)->SO(2,1)

C: SO(5,3)->SO(4,2)->SO(3,1)

H: SO(7,3)->SO(6,2)->SO(5,1)

Have as their middle group the isometry group for AdS_4, AdS_5 and AdS_7. These all share a certain relationship with each other. The AdS_5 gives as 10 dimensional theory for AdS_5xS^5, while the other two give 11-dimensional theory for AdS_4xS^7 and AdS_7xS^4. These last two are dual theories and these have a relationship with the 10 dimension theory in sense that SO(9) is a subset of SO(9,1).

The unification of these implies they are physically an aspect of the octonion. The groups in these inclusions SO(2,1), SO(3,1), SO(5,1) and SO(9,1), dimensions = 3, 4, 6, 10 correspond to physics in these respective dimensions, with 4 for gravity, 6 the CY manifold of compactification. This seems to hinge on the collination of the Moufang OP^2 by E6(-26) which contains SL(2,H) ~ SO(9,1).

There seem to be interesting connections here, and I have this suspicion that an aspect of this is related to the 5-dim moduli space for SU(2) or SO(4) bundles on 4-manifolds. The Lorentzian case gives the hyperbolic SO(4,2)/SO(4,1) ~ AdS_5.

More later after I try to bend some more metal on this.

Cheers LC

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What I have is a couple of questions. We have that a group G decomposes into H and K or G = H⊗K⊗M with the algebraic g = h⊕k⊕m. This Cartan decomposition for the exceptional groups E6(C) and E7(C) are H = E6 and K = SO(10)xSO(2) and H = E7 and K = E6xSO(2). We also have for G = E8(C) that H = E8 and K = E7xSU(2). These are symmetric spaces and are moduli of the form E6/SO(10)xU(1),...

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The symmetric spaces you mention are for the compact (complex) cases. One recovers these groups using the complexified octonions. The SLOCC groups you listed are produced by the octonions. You can use the split-octonions for E6(6), E7(7) and E8(8). You'll notice E7(7) corresponds to the symmetry for D=4, N=8 SUGRA.

For the eigenvalue problem, recall that E6(-26) corresponds to the reduced structure group of the exceptional Jordan algebra. Apply its determinant preserving property to the Jordan eigenvalue problem to see how it acts on eigenvalues and eigenmatrices. This will show you how to proceed with your elliptic curve analogy.

For the eigenvalue problem, recall that E6(-26) corresponds to the reduced structure group of the exceptional Jordan algebra. Apply its determinant preserving property to the Jordan eigenvalue problem to see how it acts on eigenvalues and eigenmatrices. This will show you how to proceed with your elliptic curve analogy.

So I am not completely bat shit crazy here. What you say about E6(-26) appears workable since it has F4 as the mcs, which in turn diagonalizes the J3(O) in the reals. As Einstein once said, "I need a little think." This stuff is not easy for me, particularly since I have been primarily employed in applied work.

Cheers LC

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Cheers LC

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The E6(-26) scales the eigenmatrices, while the F4 fixes their lengths, as measured by the Frobenius norm (trace norm) over J(3,O). There is another geometrical property preserved, as you can check for yourself. Hint: imagine each eigenmatrix assigned to a vertex.

I will probably have to communicate by email, as I think this contest is ending soon.

The f4 algebra is the set of isometries of C\otimes OP^2, which is in a sense the "complexification" of the OP^2 that has F4 as its isometry group. The 78 dimensions of E6 is 52 + 26 so that E6 = F4x26, where the 26 are vectors corresponding to the Killing form = -26. The collineations e6 correspond to isometries of OP^2 by F4 as SO(9,1) and SO(9). The points that are collinear in OP^2 correspond in C^3 to SL(4,C). These are points collinear in C^3, This complex conformal group defines the set of lines connected to a loop. This is the standard system of two vertices + and + and two vertices _ and -.

This will then define collineations of points on algebraic curves in CP^3. I suspect in some way this may lead to the elliptic curve.

More later,

LC

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The f4 algebra is the set of isometries of C\otimes OP^2, which is in a sense the "complexification" of the OP^2 that has F4 as its isometry group. The 78 dimensions of E6 is 52 + 26 so that E6 = F4x26, where the 26 are vectors corresponding to the Killing form = -26. The collineations e6 correspond to isometries of OP^2 by F4 as SO(9,1) and SO(9). The points that are collinear in OP^2 correspond in C^3 to SL(4,C). These are points collinear in C^3, This complex conformal group defines the set of lines connected to a loop. This is the standard system of two vertices + and + and two vertices _ and -.

This will then define collineations of points on algebraic curves in CP^3. I suspect in some way this may lead to the elliptic curve.

More later,

LC

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If we consider the Jordan-Wigner QM as diagonalized by F4, this seems to fit with the F4 scheme for the Kochen-Specker contextuality result. If we shift to E6 are we then generalizing the nature of quantum mechanics? Would the occurrence of real and imaginary eigenvalues correspond to the Bogoliubov coefficients in Hawking radiation.

I am curious about the physical implications of this.

Cheers LC

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I am curious about the physical implications of this.

Cheers LC

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Indeed F4 is the group of unitary transformations for a Jordan formulation of three-dimensional octonion quantum mechanics. E6(-26) is the group of collineations for OP^2, which includes F4 as a subgroup. In other words, one can consider F4 as a rotation group and E6(-26) as a Lorentz group, as is typically done in the extremal black hole literature.

As I look at E6, this seems to be the case. F4 ~ SU(3,O) and E6 ~ SL(3,O), which implies there are now an additional set of operations. These are analogous to the case with SL(2,C) ~ SU(2)xU(1,1) or SU(2)xSL(2,R). The E6 is then SL(3,O) ~ SU(3,O)xSL(3,H). These SL(3,H) transformations are rapidities similar to boosts in relativity.

Kochen-Specker proved that the measurement of can't be...

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Kochen-Specker proved that the measurement of can't be...

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Michael,

I might use your email from now on. i think it is on your paper. Anyway, is what I wrote 6 days ago at all right? Don't worry telling me it is all wrong. I'd rather know that I am wrong that not to know.

Cheers LC

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I might use your email from now on. i think it is on your paper. Anyway, is what I wrote 6 days ago at all right? Don't worry telling me it is all wrong. I'd rather know that I am wrong that not to know.

Cheers LC

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Michael,

As I read and research deeper into this the more questions tend to occur. The paper by Ferrara, Gimon and Kallosh has been most interesting to study. I like the Hamiltonian dynamics approach they take. I also see this might connect to the Hessian methods I outlined earlier It appears that the entropy functions I(z) (z = (p, q) in pseudocomplex notation) is such that

I(z +...

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As I read and research deeper into this the more questions tend to occur. The paper by Ferrara, Gimon and Kallosh has been most interesting to study. I like the Hamiltonian dynamics approach they take. I also see this might connect to the Hessian methods I outlined earlier It appears that the entropy functions I(z) (z = (p, q) in pseudocomplex notation) is such that

I(z +...

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The connection between SL(4,C) and CP^3 is seen with

h_5(C) ~= h_4(C)⊕C^4⊕R

That is a 2x2 matrix of 4-component objects, spinors ψ and vectors φ in C^4 and h_4(C). The Jordan algebra h_4(C) ~ CP^3 and in conjuction with the C^4 tells us that we have a CP^{3|4} superspace for twistor theory.

It would appear best at first to consider

h_3(H) ~=...

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h_5(C) ~= h_4(C)⊕C^4⊕R

That is a 2x2 matrix of 4-component objects, spinors ψ and vectors φ in C^4 and h_4(C). The Jordan algebra h_4(C) ~ CP^3 and in conjuction with the C^4 tells us that we have a CP^{3|4} superspace for twistor theory.

It would appear best at first to consider

h_3(H) ~=...

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I can now see the connection to D-branes. The triplet system most fundamentally involves the cubic equation with eigenvalues. The d = 5 cosets of states G_5/K_5 involves G_5 = SL(3,K), K = R, C, H or O with quotients with SO(3), SU(3), USp(6), and F_4 for E6(-26) and USp(8) for E6(6). In four dimensions, with an additional vector, a similar quotient of states exists. These are G_4/K_4 with G_4...

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I have another question on all of this, which might in some ways be an insight. The invariant of E6(6) is this cubic invariant. The matrix

J3 = q_{ab}ω^{bc}q_{cd}ω^{de}q_{ef}ω^{fa}

is written with the 54 = 27⊕27, for 27 = 1⊕26, where these are sp(8). The action is then S = π sqrt{|J3|}. The determinant of this matrix is a 3-volume that has zero volume and if its bounding area ∂V = ∂J3/∂q = 0 ½ of supersymmetries are preserved. If the volume is zero and the area nonzero then ¼ SUSYs preserved and if both are nonzero then 1/8.

I have been thinking that this has a connection to the Weyl curvature and its action. Since this has gotten into conformal relativity this is a time where I can use this forum as a sound board to present some physics involving the Weyl curvature. Consider the Lagrangian

L = (1/2)sqrt{-g}C_{abcd}C^{abcd}

and the variation with respect to the metric g_{ij} is then

sqrt{-g}^{-1}δL/δg_{ij} = B^{ij} = 2C_{aijb}^{;ab} + C_{aijb}R^{ab} = 0.

This is comparatively simple for a conformally flat spacetime where R^{ab} = 0. An Einstein spacetime with R^{ab} = λg^{ab} is also a solution to the equation. This is the Bach equation.

Suppose these two actions are the same. This then equates a quadratic function with a cubic function. The above functional differential with the metric is essentially the same as a derivative with respect to the 26 charges of the sp(8). The metric is then given by the 26 electromagnetic charges. This then gives action as an elliptic curve. This seems to be a physical way of looking at ideas of using Shimura and symmetric varieties in physics.

LC

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J3 = q_{ab}ω^{bc}q_{cd}ω^{de}q_{ef}ω^{fa}

is written with the 54 = 27⊕27, for 27 = 1⊕26, where these are sp(8). The action is then S = π sqrt{|J3|}. The determinant of this matrix is a 3-volume that has zero volume and if its bounding area ∂V = ∂J3/∂q = 0 ½ of supersymmetries are preserved. If the volume is zero and the area nonzero then ¼ SUSYs preserved and if both are nonzero then 1/8.

I have been thinking that this has a connection to the Weyl curvature and its action. Since this has gotten into conformal relativity this is a time where I can use this forum as a sound board to present some physics involving the Weyl curvature. Consider the Lagrangian

L = (1/2)sqrt{-g}C_{abcd}C^{abcd}

and the variation with respect to the metric g_{ij} is then

sqrt{-g}^{-1}δL/δg_{ij} = B^{ij} = 2C_{aijb}^{;ab} + C_{aijb}R^{ab} = 0.

This is comparatively simple for a conformally flat spacetime where R^{ab} = 0. An Einstein spacetime with R^{ab} = λg^{ab} is also a solution to the equation. This is the Bach equation.

Suppose these two actions are the same. This then equates a quadratic function with a cubic function. The above functional differential with the metric is essentially the same as a derivative with respect to the 26 charges of the sp(8). The metric is then given by the 26 electromagnetic charges. This then gives action as an elliptic curve. This seems to be a physical way of looking at ideas of using Shimura and symmetric varieties in physics.

LC

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The paper http://arxiv.org/pdf/1407.5977v2.pdf by Bonezzi, Corradini and Waldron is pretty interesting. Much of the start seems to hinge around

1/sqrt{det(H)} = exp(-½ Tr log(H)).

We have then that

I* = ½ Tr(H) = ½ ∫_0^∞ ds/s exp(sH).

This is an elliptic integral that diverges at s = 0. We then set the lower integration s = 1, which corresponds to the area of the string world sheet with s = t/x, for t the time and x the spatial coordinate on the sheet. The s = t/x is a form of the S-duality of the string, or a form of t and s channels in Mandelstam variables are equivalent.

The same seems plausible with the Grassmannian coordinates c. The equation 13

∫Dχexp(-Tr∫d^3cχF_A) = sum_A’δ(A – A’)det[δF_A/δA]

is integrated over the space of Grassmannian variables. This could be represented with a string target map. The Grassmannian coordinates c^a = δθ^a are anticommuting variables and the target map is given by a determinant of the c^a’s, similar to a Slater determinant of fermion states, that defines the string world sheet.

LC

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1/sqrt{det(H)} = exp(-½ Tr log(H)).

We have then that

I* = ½ Tr(H) = ½ ∫_0^∞ ds/s exp(sH).

This is an elliptic integral that diverges at s = 0. We then set the lower integration s = 1, which corresponds to the area of the string world sheet with s = t/x, for t the time and x the spatial coordinate on the sheet. The s = t/x is a form of the S-duality of the string, or a form of t and s channels in Mandelstam variables are equivalent.

The same seems plausible with the Grassmannian coordinates c. The equation 13

∫Dχexp(-Tr∫d^3cχF_A) = sum_A’δ(A – A’)det[δF_A/δA]

is integrated over the space of Grassmannian variables. This could be represented with a string target map. The Grassmannian coordinates c^a = δθ^a are anticommuting variables and the target map is given by a determinant of the c^a’s, similar to a Slater determinant of fermion states, that defines the string world sheet.

LC

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To introduce more physical ideas into this subject, in particular magic SUGRA I have been musing over this for a while. I think it is important to keep in contact with physical ideas. I do at the end here consider the prospect that this connects with J3 and E6. This has to do with the general uncertainty principle and the tension on the brane of our universe. If the tension is zero then the...

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Nice observations. Just keep in mind how branes are constructed in matrix theory. The geometry is not Riemannian; it is much more profound. Hence, motives.

I am looking into Motives. I have found the following http://www.ams.org/notices/200410/what-is.pdf to be of some help. The Wikipedia article assumes too much understanding of the preliminaries.

I do see that this has some connection to groupoids and magma constructions. I also think this has something to do with nature of the Leech lattice in J3(O). The 24-cell, indeed the very number 24, keeps cropping up. The F4 group has as its roots space the 24-cell, and I have been working on something involving F4 and its B4 reduction with respect to projective varieties of rays. These as putative hidden variables fail to define a Ω [φ, ψ] function as unit for a ψ-epistemic or 0 for a ψ-ontic theory. The F4 also diagonalizes J3(O).

There seems to be some sort of relationship of the sort

Leech lattice ---------------------------- modular forms

Golay Code ---------------------------- Elliptic curve (exotic varieties, Goppa codes)

Sporadic groups ------------------------ Special functions ( Jacobi θ-functions, Ramanujan mock Θ-function)

The space of quaternions H = {pm 1. pm i, pm j, pm k} ---> SL(2, 3) is 24 dimensional. In the sequence

F4: B4 --- > F_{52/16} ---> OP^2

the F4 when modded by the quaternion space, is then the space of J3(O) on the RHS minus the pairs of roots which is an elliptic curve.

My main interest is to connect the M_{24} in this system, but this is just one rung on the whole sporadic group ladder leading up to the FG or monster group.

LC

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I do see that this has some connection to groupoids and magma constructions. I also think this has something to do with nature of the Leech lattice in J3(O). The 24-cell, indeed the very number 24, keeps cropping up. The F4 group has as its roots space the 24-cell, and I have been working on something involving F4 and its B4 reduction with respect to projective varieties of rays. These as putative hidden variables fail to define a Ω [φ, ψ] function as unit for a ψ-epistemic or 0 for a ψ-ontic theory. The F4 also diagonalizes J3(O).

There seems to be some sort of relationship of the sort

Leech lattice ---------------------------- modular forms

Golay Code ---------------------------- Elliptic curve (exotic varieties, Goppa codes)

Sporadic groups ------------------------ Special functions ( Jacobi θ-functions, Ramanujan mock Θ-function)

The space of quaternions H = {pm 1. pm i, pm j, pm k} ---> SL(2, 3) is 24 dimensional. In the sequence

F4: B4 --- > F_{52/16} ---> OP^2

the F4 when modded by the quaternion space, is then the space of J3(O) on the RHS minus the pairs of roots which is an elliptic curve.

My main interest is to connect the M_{24} in this system, but this is just one rung on the whole sporadic group ladder leading up to the FG or monster group.

LC

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Yes, you have solid intuition so far. Take a look at:

arXiv:1505.06742

Recall, that the automorphism group of the Leech lattice sits nicely inside F4 over the rationals. The E8 and E8xE8 lattices have also been discussed in this context at N-Category Cafe.

arXiv:1505.06742

Recall, that the automorphism group of the Leech lattice sits nicely inside F4 over the rationals. The E8 and E8xE8 lattices have also been discussed in this context at N-Category Cafe.

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