Lawrence, this is possibly your most readable essay yet in these contests, but you have still managed to maintain a high level of novel mathematical ideas. I liked the historical introduction that puts the relationship between Mathematics and physics in perspective.
The HoTT ideas are very interesting and they mesh well with my own ideas of higher category theory as a system of multiple quantisation so it is good to see this presented.
You should do well.
Thanks for the positive word. I wrote the bulk of this essay up in a single day. I spent another day correcting it. I tried to keep the mathematics somewhat "physical," in that the discussion on incompleteness and numbers is oriented towards what one might actually encounter in computing these things.
I read your essay early on, and as with most essays I have read I have yet to assign a score to it. So far you are running at the top. I will probably have to re-read yours. I have been a bit busy and unable to attend to this contest that much.
Cheers LC
Lawrence
Nice work. I see you mentioned entanglement of quantum black holes. In light of twistor-scattering amplitudes, I suspect there is an operad structure, to be defined for all such quantum black holes. In the complex case, the operad is already defined (by Loday), and can be viewed as a chain of punctured Riemann spheres. This has an interpretation in terms of associahedra and binary tree diagrams. In essence, once can tile a moduli space with associahedra. The vertices of the associahedra correspond to interactions that contribute to the relevant n-object scattering amplitudes.
The connection with the Bott periodicity of large N, or SU(N), entanglements with associahedra is with the some sort of projection of this thing. The Stasheff polytope K_5 has 14 vertices, and this might be seen as composed of two copies of 7 elements mapped to the Fano plane, with each of these "sevens" associated with a projective point "в€ћ", or as associated with eight elements by the Hopf fibration. So the associahedra might be some sort of "double cover" with the Bott periodicity on A&D type Lie groups.
I think to construct such operads one has to work more from the ground up. Spacetime as a monoid, groupoid or so called magma might lead to the sort of spacetime algebra that leads to the sort of category theory. The set of possible operations would then form these "trees" and are associated with certain homotopies and categories. My essay discusses this in an informal sense, and homotopy as a route to category theory with monad/monoid structure.
I can discuss this more tomorrow or later. I see that you have an essay as well, which I will try to get to in the coming days. These are coming in faster than I can read them.
Cheers LC
Dear Lawrence,
very good essay, I have to go more deeply into the details to post a substantial comment. Also thanks for the emails (I will answer soon).
In the meantime I also wrote an essay which appears now. Her is the link to
Best Torsten
The construction of spacetime, from say, category theory would be elegant. Through the lens of noncommutative geometry, one comes very close to the spirit of category theory, as classic geometrical points are replaced by pure states of a C*-algebra. If one takes the lessons learned from D-branes seriously, gauge symmetries arise from configurations of branes, and natural brane coordinates are noncommutative. In fact, the brane coordinates arise from a noncommutative C*-algebra. Let's take the case of N coincident branes, which have a U(N) gauge symmetry as long as the branes are coincident. The N D-brane positions in spacetime are packaged in a matrix, say X, in the adjoint representation of the unbroken U(N) group. Upon diagonalizing X, the N eigenvalues give the classical spacetime positions of the N D-branes, corresponding to the ground state of the system.
In category theory language, the noncommutative C*-algebra of NxN complex matrices Mat(N,C) can be interpreted as the algebra of noncommutative functions over a finite point space, with N objects. The elements of the noncommutative C*-algebra serve as morphisms over these N objects. Therefore, we can define a category C with N objects in Obj(C) and Hom(C) consisting of Mat(N,C) morphisms.
Michael,
I have been a bit tied up with a number of things. Your discussion 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
Hi LC,
Once again, you made an excellent work with a very profound Essay. Here are my comments:
1) This should have been an excellent Essay also in previous contests "It From Bit or Bit From It?" and "Is Reality Digital or Analog? ".
2) My recent work on black holes, that you know, goes in the same direction that "topology and the computation of topological charges and indices, quantum numbers, and connection to logical switching theory are likely to supplant concerns of geometry, metrics and infinitesimal structure of manifolds."
3) On the other hand, I am not sure that such a statement goes in one specific direction. I try to clarify: you claim that
i) "This means that the fundamental description of reality is not with space, spacetime or anything geometric. Geometry or metric space is something which is a measure of entanglement of quantum bits with black holes and the inability to follow the entanglement phase. Geometry is then not fundamental."
But you also claim that:
ii) "Spacetime is built up from entanglements".
Thus, in my opinion, statements can be inverted. One could claim that "Entanglement of quantum bits is something which is a measure of geometry or metric space". In other words, this could be a sort of duality and/or complementarity of the fundamental description of reality. Geometry and entanglement could be two different aspects of the fundamental description of reality, but we could be unable to decide which one is the most fundamental.
In any case, I stress again that you wrote an intriguing Essay deserving the top score that I am going to give you.
I wish you best luck in the Contest.
Cheers,
Ch.
Dear Christian,
Event horizons are a way that a quantum state or EPR pair of qubits can be entangled with a black hole. The entanglement is coarse grained, for one does not know which of the qubits on the horizon one's EPR state in the pair you are entangled with. Any attempt to find out runs into limitations of the Heisenberg microscope argument. So horizons are an ensemble or Bayesian set of priors for quantum states. This is sort of the meaning of how geometry is built up from entanglements. The converse has some element to it, in that entanglements are coset structures of geometric elements.
I will look at your new entry and score in the next few days. I have been on travel lately and time is a bit tight.
Cheers LC
Hello Lawrence,
As a layperson I am not going to pretend to understand these new approaches and the maths being utilised but in your conclusions you state interestingly,
"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent whole-
ness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos [21].
This chaos is a set of statements that are purely self-referential and have truth or falsehood by no logical
reason.
Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where
some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of
universe. These logical coherent forms are then accidents similar to Chaitan0 s philosophy of mathematics.
It is very dicult to understand how this could be scienti cally demonstratedｬ yet maybe regularities inｼｯpｾｼpｾphysics described by mathematics exist for no reason at allｮ｢ｼｯpｾｼpｾﾂasicallyｬ the mystery of the quantum vacuum is really expressed as the ｧhierarchy problemｧｮ ﾉ think if you stare at the so called hierarchal gap then that is where the state of physics is at todayｮ ﾉ like the idea of entanglement defining geometry somehow and this may be inherent in the blank space ｨvoidｩ where the particle desert existsｮﾔhe ﾂig ﾇap ｨor whatever you may call itｩ maybe the potential pool from which the multiverses are generaed or at least some indication that the multiverse do indeed existsｮ ﾍaybe ﾇﾕﾔ convergences are different for different ﾕniversesｮ ﾁlsoｬ maybe it is a matter of physics becoming more aligned with mathematics toward the quantum gravity realmｮ ﾉn that we live in a ｴﾄ world our physics ｨﾆrom ﾇreek physikaｩ is generally found to work ｨempiricallyｩ from the platform of the sensory ｨsensationｩ because of the physical nature of things at our low energyｮ ﾓensory as apart from intelligenceｮ ﾁt some point the physics may blur into the mathematics and the senseｭdata ｨempiricalｩ will be left behindｮ ﾔhis implies that intelligence will discover things by ｧexplanatory powerｧ and that the empirical apparatus ｨmachinesｩ no longer yield anything useful in upward and onward explorationsｮ ﾍaybe ｴﾄ is unique in that it implies a physical world in which physics is a workable contextｮ ﾉf so then there might be an ｢ end of the machineｨsｩｧ such as the ﾌﾈﾃ where it just cannot yield the pure mathematical result as a physical ｨsensationｭobservationｩｮ ﾔhen we are at the ｢end of physics｢ in a different sense and a brave new world of mathematics ｨexplaining thinsｩ is awaitingｮ ｨﾍaybe the meaning of life is not ｴｲ but ｴﾄ ｻｭｩ ｩ ﾉf you look at the hierarchal gap as a ｧzen koanｧ then there is more there than meets the eyeｮｼｯpｾｼpｾmark
Mark,
There are some indications that this might be happening. With high energy physics the LHC and tests of the standard model, supersymmetry and maybe some hints of exotic physics are probably the last of the sort of direct tests physics has enjoyed or wanted. The future may see theories tested in increasingly oblique or indirect ways. I hope that progress can be had this way through the 21st century.
I see the prospect that physics becomes purely a mental game a bit disturbing. If we end up in the future where we are unable to make any test, no matter how indirect, of our physical theories we will be in a problem. In some ways it will be the end of physics as a science. Science does depend upon experimentation.
Theoretically the heirarchy problem is solved with supersymmetry. It is just an open question of whether the LHC with 13TeV beam can find evidence of supersymmetry.
Cheers LC
Dear Lawrence,
Now I had time to read yiour essay. I agree with a comment above: it is one of your best. Here are my own comments:
1. You spoke about Bott periodicity but SU(N) has a 2-periodicity. It is the SO(N) group which admits an eight-fold periodicity (with integer coefficients, it is 4-periodic like the symplectic group for rational coefficients).
2. Your double slit experiment is very interesting. You view it from the topological point of view. Maybe one should remark that this approach wa already done by Berry and others using geometric phases.
3. you discussed it that HOTT will overcome the continuum approach. But homotopy needs a continuous family of maps (the deformation). It is central point in the approach and many results using implicitely the continuum (like Cerf theory, Whiteheads theorem etc.)
4. I also don't understand why you want to change from continuum to discrete. I showed in a previous essay that a smooth manifold contains only finitely many information (from topology). Furthermore, the dynamics in quantm mechanics (or field theory) is smooth (and continuous). Only the spectrum of the operators is discrete.
5. HOTT is a good approach but this proposal don't change the logic. Therefore Gödel works. Fro your approach, you need model theory (including forcing) to go over it. I remembered on a approach of Landsman to quantum mechanics using this approach. But my friend (and co-worker) Jerzy is the real expert.
I like your part explaining the Turing machine (and the relation to the Entscheidungsproblem)
Very good work
Torsten
In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have any reference to infinitesimals or infinities. The calculus is based on the limit where the difference between two points becomes infinitesimally small. Physical experiments have not direct bearing on this.
It is the case that homotopy does involve curves that are smoothly deformed into each other, but this is used to get the value of the homotopy group that is usually Z_2 or Z, where Z could be interpreted as just unbounded and infinity is avoided. The homotopies are then more directly related to the actual measured aspects of physics.
Spacetime is a bit odd with regards to this. The Planck scale does indicate that one can't isolate a qubit in a region smaller than sqrt{G徴/c^3}. The Heisenberg microscope argument indicates that if one tries to isolate the Planck unit of area a quantum state is contained that it will scatter violently. This illustrates that using a large value of momentum to isolate particle demonstrates that spacetime has a discrete structure. This has an interpretation in the generalized uncertainty in string theory. On the other hand the FERMI and Integral spacecraft measurements of distant burstars found no dispersion of photons predicted by loop quantum gravity. This is a discrete form of quantum gravity, and it appears to be in trouble. In this experiment a very large ruler (measurements out to a billion light years) found that spacetime appears very continuous. This suggests a more general form of the uncertainty principle, where at one limit spacetime is continuous, and on the other limit discrete.
The problem is that physics is not completely discrete or continuous. One of these FQXI essay contests went into this. The main thrust of my essay though is that the physical observables we measure, and physics is an experimental science, are discrete. Mathematics has what I might call a "body" and a "soul." The body is what is computed, and can be computed on a computer. The soul is all of the continuum stuff, calculus, infinitesimals etc, which have a weaker connection to experiments. I am not committed to any metaphysics about whether the soul exists or not. That is to say I have no belief or lack thereof with respect to what some might call Platonism.
LC
Lawrence,
Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.
I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:
If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.
I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.
I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.
In principle I agree with you very much. In particular I like your body-and-soul picture
Best
Torsten
In the subject of gauge theory a central aspect is the interaction form. These types of continuous homotopy or homotopy-like constructions involve curves that can be adjusted in certain ways so that an index is invariant or constant.
I am back home, but of course I have a lot of things to attend to here. I will try to expand on things in the future. The whole subject involves orbit spaces, or quotents of groups or spaces. The subject of four-manifolds is centered around the moduli, a 5-dim space that in a hyperbolic setting can be the AdS_5. Of course the hyperbolic setting is not Hausdorff and there are other problems. However, this is a form of orbit space that is mapped to the quantum SLOCC types of theory.
Cheers LC
Dear Lawrence,
You start with Goedel "no mathematical system can ever prove all possible atements as theorems about itself" and you propose HOTT (homotopy and type theory together) which of course fits the great categorization process at work in mathematics. I found a very recent preprint of Yuri Manin pointing the same direction http://xxx.lanl.gov/pdf/1501.00897.pdf
"Information is physical" and you seem to suggest that "mathematics is physical", and both are quantum (in your conclusion). I like your approach and thank you for a very original and readable text with non-trivial concepts.
This year, I am exploring the most discrete and anomalous/sporadic object ever found. I hope you can comment on it.
Best.
Michel
This is in line with motives, categories and fundamental quantities as discrete elements from homotopy or varieties. This is as you say in line with category theory. In fact I think that ultimately the fundamental observables in the universe are topological categories, similar to Etale or Grothendieke theory.
I see there being in a sense what I call the "body" of mathematics, which are those aspects of mathematics that can be, at least in principle, solved on a computer, and the "soul," which is the continuum mathematics of infinitesimals and infinities. My essay concentrates on the body, and not so much on the soul. I think for physical science the body is more directly associated with what is observed in the universe.
The "body-soul" duality I tend to advocate is something one can "wear" as needed. I might by virtue of some argument want to invoke a mathematical objectivity of sets, continuous spaces and even to the point of Platonism. At other times I may put this entirely aside. In my essay I largely put this aside.
Garrison Keillor has a feature on his show "Prairie Home Companion" called Guy Noir with the opening line, "On a dark night in a city that knows how to keep its secrets, one man seeks answers to life's persistent questions; Guy Noir, Private Eye." That is about my sense of the question about the relationship between physics and mathematics. We may never know for sure. Further, the universe may have a kernel of structure, symmetry and order to it that appears in a fractal-like form at different scales, but where nature also has this inherently chaotic or disordered nature to it as well, which I think is distilled down to the stochastic nature of quantum measurement.
I will try to get to your essay in the near future. I just got back from some travelling.
Cheers LC
Dear Lawrence,
Every scientist has his own way and velocity in going through the wonderful secrets of nature. At FQXi you already wrote many excellent essays like "Discrete time and Kleinian structures in Duality Between Spacetime and Particle Physics". I wonder if you already looked seriously at the concept of an orbifold? I see that it plays a role in the VOA associated to some sporadic groups. I also found http://arxiv.org/abs/math/0505431 for your topic of this year.
I appreciate much the impetus you gave to my essay. After my first participation I learned how it works and don't take care to much of the lazzy inappropriate votes. You received from me the best endorsement. The goal is a continuing friendly discussion about the topics of mutual interest.
Best.
Michel
Dear Michel,
Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup is then over certain primes, such as either Heegner primes or maybe primes in the sequence for the monster group. This is of course related to the Kleinian groups and the compactification of the AdS_5.
The AdS_5 compactification issue is something I started to return to. I gave up on this after the FQXi contest over this because it did not seem to gather much traction. The AdS_5 = SO(4,2)/SO(4,1) is a moduli space. The Euclidean form of this S^5 =~ SO(6)/SO(5) is the moduli space for the complex SU(2) or quaterion valued bundle in four dimensions. The AdS_5 is then a moduli space, and the conformal completion of this spacetime is dual to the structure of conformal fields on the boundary Einstein spacetime. This moduli is an orbit space, and this is the geometry of quantum entanglements.
If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan's decomposition with [h,h] вЉ‚ h, [h,k] вЉ‚ k, [k,k] вЉ‚ h
N∩G/g = N∩K/k.
For map Ој:P(H) --- > k on P(H) the projective Hilbert space. The differential dОј = = П‰(V, V') is a symplectic form. The variation of ||Ој||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(Ој).
In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,2)/SO(4,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.
With your presentation of the П€-problem and the connection between the Bell theorem and Grothendieck's construction, you push this into moonshine group О"^+_0(2). This leads to the conclusion or conjecture, I am not entirely clear which, that the moonshine for the baby monster group is coincident with the the Bell theorem. The connection to the modular discriminant is interesting. This then gets extended to О"^+_0(5). Your statement on page 7 that g(q) = П†(q)^24 is much the same with what I wrote above. There is a bit here that I do not entirely follow, but the ideas are intriguing. I would be interested in knowing if the hyperbolic tilings of О"^+_0(5) have a bearing on the discrete group structure of AdS_5.
You may be familiar with Arkani-Hamed and Trka's amplitudhedron. The permutations arguments that you make give me some suspicion that this is related to that subject as well. This would be particularly the case is the О"^+_0(5) is related to the tiling and permutation of links on AdS_5 given that the isometry group of AdS_5 is SO(4,2) ~ SU(2,2) which can be called the twistor group. This is connected with Witten's so called "Twistor-string revolution."
Thanks for the paper reference. That looks pretty challenging to read. I am not quite at the level of a serious mathematician, though I am fairly good at math and well versed in a number of areas.
Cheers LC
Dear Lawrence,
I am really impressed by your knowkedge of so many things related to string theory. I propose that we start a collaboration because we have so many things to share and we are also quite complementary. I was very enthousiastic in writing the essay because new relations between several parts of maths and physics was taking place as by magic and also thanks to the computer. This is unreasonable in some sense!
My best wishes,
Michel