Essay Abstract
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.
Author Bio
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
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 system. More remarkably after some studying of what my system was saying I concluded that it not only unified physics but also mathematics, although yet again I got to know about unifying mathematics much later upon reading Woit's blog.
The reason why I thought it unifies mathematics is because as I tried to put my system in an old fashion mathematical formalism and not by the simulation that I am using, I discovered that it connects to just about every major branch of mathematics. from category to probability to intersection theory to network, you name it.
The starting thread that connected to all these branches was that my system was really nothing but a generalized Buffon's needle. Now, Buffon's needle once generalized even more it connects to geometric probability ( which connect to endless branches of math) to TWISTORS(surprise!) to many others. However, my system shows that while you can go to the manifested complex number of twistors, but you do not need to, geometric probability is enough at least to get major results.
My last essay was titled
"Reality is nothing but a mathematical structure, literally"
I will have much more evidences in this contest.
References for connection between Buffon's needle and twister
http://en.wikipedia.org/wiki/Buffon's_needle
http://en.wikipedia.org/wiki/Geometric_probability
http://en.wikipedia.org/wiki/Integral_geometry
http://en.wikipedia.org/wiki/Radon_transform
http://en.wikipedia.org/wiki/Penrose_transform
http://en.wikipedia.org/wiki/Twistor_theory
Also, more conformation came from this concept
http://arxiv.org/pdf/quant-ph/0608251.pdf
Thank you for reading.
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
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.
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.
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.
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
Very good but advanced essay even for mathematicians.
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.
Hi Joe
I do admit quantum gravity, the Langland's program, etc. are indeed abstract. It is in this abstraction that it is surprising to see connections being made, between each tower of abstraction. For surely, one would expect, the probability is that each discipline of abstraction would have little to no connection with another. But as time has shown, some mathematics, no matter how abstract and obscure, eventually is rediscovered and interpreted in physics.
Ramanujan's mock theta functions are a fine example of this unity of abstractions. Mock theta functions were written in his lost notebook, with little explanation of their origin or purpose in mathematics, much less physics. However, almost 100 years later, in calculating the entropy of quantum black holes in string theory, one finds mock theta functions. This led last year to the proof of Umbral Moonshine, a variant of Monstrous Moonshine (see Finding Moonshine by Marcus du Sautoy). So there is something deep occurring here, and nobody really understands the bigger structure in its entirety.
Thanks Alex. I attempted to express cutting edge examples of the unreasonable effectiveness in mathematics. In time, I am hopeful, there will be a concise, elegant framework in which to view such mathematical coincidences.
Dear Dr. Rios,
"The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to:
• Encourage and support rigorous, innovative, and influential thinking connected with foundational questions;"
All of mathematics is abstract. There is no way abstractions can be unified to explain how the real Universe is occurring. My superb theorem of inert light unifies everyone's understanding of reality. Please stop ridiculously supporting any utility of abstract quantum codswallop.
Joe Fisher
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.
Efthimios
Thank you for your review. I tried my best to convey some of the mystery and excitement of the ongoing profound relationship between pure mathematics and theoretical physics. I will read your essay now.
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 example, that Math, like music, favours creativity, variety of concepts, forms, and their relations, while Physics is dominated by minimality, and by the urge to find the smallest single explanation for the dynamics of the physical universe.
On the other hand, if unification makes sense also in mathematics (in a way that you do not seem to fully address in your essay), and the two paths eventually converge to the same formal structure, then would you go as far as concluding, with Tegmark, that the physical world IS a mathematical structure?
A final point, inspired by this a quote from Murray Gell-Mann:
"Life can emerge from physics and chemistry plus a lot of accidents. The human mind can arise from neurobiology and a lot of accidents, the way the chemical bond arises from physics and certain accidents. Doesn't diminish the importance of these subjects to know they follow from more fundamental things plus accidents."
'Accidents' are primarily responsible for the way several important aspects of the universe (e.g. humans) look like.
Should accidents be also part of the ultimate theory of everything?
If so, the motives-based unifying theory that you envisage in your essay would still be quite preliminary. When concepts such as accidents, emergence, evolution, 'history' come on stage, I believe that notions of algorithmic, computational universe, with their rich array of emergent properties, become quite appropriate and effective.
Best regards
Tommaso
PS
I find that revealing the assigned scores is bad practice - you'll notice that this happens only when the score is very high. Taking at least a look at the essay from someone who commented your essay - that is good practice! Thank you and good luck!
Yes, a conclusion could be made that is in line with Tegmark's, albeit with a detailed description of the mathematics underlying such a system. It would be a simulation hypothesis, with details that are subject to testability.
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
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.
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)âŠ--SL(n)âŠ--...âŠ--SL(N).
The SLOCC is then in general given by the C^NâŠ--...âŠ--C^N/SL(n)âŠ--...âŠ--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)âŠ--...âŠ--SL(N), which is the gauge-(like) transformation of the states.
The 3 and 4 qubit systems are
G_3/H_3 = SL(n)âŠ-- SL(n)âŠ--SL(N)/U(1)âŠ--U(1)âŠ--U(1)
G_4/H_4 = SO(4,4)/SO(2,2)âŠ--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)}/âŠ--_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