Dear Lawrence,

sorry for taking so long to respond. Unfortunately, my time is limited at the moment, and your posts always take some careful picking apart for me to understand.

That said, I'm either not quite there yet, or we're not talking about quite the same thing (although perhaps you're suggesting an alternative interpretation of my setting). The CHSH-polytope I reference is a classical entity, living in the probability space spanned by the value-assignments to the observables in the CHSH-experiment (i. e. the hidden variable vectors, and their convex combinations, which just give the probability distributions over experimental outcomes).

The Kirwan polytope lives in a space of (eigenvalues of) quantum states, and contains information about how a given system is entangled. Are you saying that there exists an entanglement polytope that's the same as the CHSH-polytope? If so, I'm afraid that's not quite clear to me. I mean, I can see that the four-qubit entanglement polytope must be contained in it, but that's trivially the case, because it's just the unit (hyper-)cube. Do you think there's more of a connection than that?

7 days later

It has been a while since I checked FQXi. I am a little disappointed in how the essay contest is developing.

The CHSH polytope is based on the relationship

I_{chsh} = A_1Г--B_1 + A_1Г--B_2 + A_2Г--B_1 - A_2Г--B_2,

for Alice and Bob experiments with two outcomes. This curiously is a type of metric that can be interpreted as pseudo-Euclidean. This is also a measure of entropy, for it may be expressed according to conditional probabilities. An arbitrary two-qubit state after Schmidt decomposition can always be written as

|П€_nвџ© = c_0|n_+, n_+вџ© + c_1|n_в€', n_в€'вџ©.

We choose the measurement settings in the following way

A_1 = m_1В·Пѓ, A_2 = m_2В·Пѓ,

B_1 = (1/в€љ2)(m_1В·Пѓ + m_2В·Пѓ), B_2 = (1/в€љ2)(m_1В·Пѓ в€' m_2В·Пѓ).

Here n, m_1 and m_2 are the unit vectors perpendicular to each other. Now find the expectation value of the CHSH operator in the state |П€_nвџ©. We get

вџЁП€_n|I_{chsh}|П€_nвџ© = 2в€љ2C.

The expectation of I then has this bound.

This CHSH polytope is I think related to the Kirwan polytope. The CSHS comes from the relationship with different basis measurements, while the Kirwan polytope is based on eigenstates. For x and z measurements we can think of there being two copies of the Kirwan polytope. The CHSH is then a discrete lattice for some form of covering space.

Cheers LC

Dear Jochen, Very interesting paper. To understand Assumption 1 I had to get rid of the idea that measurements are usually repeated under identical conditions; in this case the measurement may change with n and this is necessary to derive the paradox. It may be helpful to think in terms of settings (as in the EPR-Bell-Bohm situation), so that the choice of a measurement is a choice of the settings. This introduces a tacit Free Choice assumption into the argument. The contradiction is reminiscent of, perhaps even equivalent to, the so-called paradox of predictability, see e.g. the review by Rummens and Cuypers, Determinism and the Paradox of Predictability, Erkenntnis 72, 233-249 (2010), https://link.springer.com/article/10.1007/s10670-009-9199-1. I must admit that I find inferences or suggestions of the kind that an undecidable proposition can be modeled by a quantum superposition suspicious - the former are very general, the latter arise is a very specific mathematical context (Hilbert space) and one needs additional arguments to really make the inference. As far as I know, no one has managed to do this convincingly.

Having said this, I will continue follow your work with great interest. Best wishes, Klaas

    Dear Klaas,

    thank you for your comment! I'm glad you found something of interest in my essay.

    Regarding the importance of free choice, indeed you can make an argument that one might be constrained to make only measurements for which f(n,k) yields a definite value, thus never running into the sorts of phenomena following from its indefiniteness. That we do, apparently, run into them would then be evidence that there is no such constraint (which of course doesn't entail that we have free choice).

    However, one does not need to think about changing measurement conditions to derive the paradox, merely about the totality of all possible measurements on the system---whether they are ever performed or not. The index n is then essentially just an attempt to enumerate these measurements, with the argument then showing that no such enumeration can be complete.

    As for superposition, you're right to point out that the structure of linear operators on Hilbert space is a quite specific one; but in the end, the project of deriving a theory from underlying principles is one to derive the specific from the general---for comparison, the structure of Lorentz transformations on Minkowski space is also quite specific, while following from the very general principle of relativity, together with the constancy of the speed of light.

    Indeed, if one views Hilbert space as a concrete realization of an abstract propositional structure---the orthomodular lattice of its subspaces---then one can show that this essentially follows from the notion that there exists a maximum amount of information that can be extracted from any system (https://link.springer.com/article/10.1007/s10702-005-1129-0). This is both connected to undecidability (as in Chaitin's principle, you can't derive---under a suitable measure of complexity---a theorem more complex than the set of axioms), and superposition, with the failure of the distributive law in quantum logic.

    For simple (that is, not subject to Gödelian phenomena) axiom systems, this correspondence was demonstrated by Brukner (https://link.springer.com/article/10.1007/s11047-009-9118-z) and Paterek et al (https://iopscience.iop.org/article/10.1088/1367-2630/12/1/013019/meta), who show an explicit way to encode axioms in a quantum system and demonstrate that a given measurement will produce random outcomes whenever the corresponding proposition is not derivable from the axioms. In a sense, my work is simply an extension of this to cases where undecidability is not due to the limitations of the axiom system, but to the inherent limitations imposed by the limitative theorems of metamathematics (although in the treatment using Lawvere's theorem, one can pass over first establishing a correspondence with formal axiomatic systems).

    Having said that, I of course don't claim to have a complete reconstruction of quantum theory in hand. There are still different options possible---for instance, it's not easy to see why one should use Hilbert spaces over the complex field, and not over the reals or quaternions. In that sense, perhaps one should think of the connections I point out, as of yet, as 'family resemblances', rather than strict formal equivalences. I view them as enticing prospects that seem sufficiently promising for me to carry on exploring this point of view; but I would not be the first wanderer to be deceived by tantalizing lights in the dark forest.

    Cheers

    Jochen

    Dear Jochen,

    You wrote a really excellent Essay, have my sincere congrats. You chose a very hot topic by discussing it from an original point of view. Your approach of reconstructing quantum mechanics is in a certain sense similar to my attempt of reconstructing quantum gravity through its fundamental bricks, that are black holes. In addition, from the philosophical point of view my position is near local realism too. You deserves my highest score, I wish you good luck in the contest.

    Cheers, Ch.

      You are welcome, I agree fully , it is difficult to understand these infinities inside this physicality that we observe and try to understand. We search a kind of universal partition with these numbers and these foundamental mathematical and physical objects. I consider the 3d coded spheres and a gravitational coded aether sent from this central cosmological sphere, it is there that these finite series of spheres are coded by a kind a infinite eternal consciousness that we cannot define, we can just understand this physicality and its laws. The reals, irrationals, rationals, imaginaries, primes, p adics analyses , harmonics of fourier and this and that seem under a specific universal partition but we know so few still, I consider that these 3D quantum spheres of this aether play between the zero absolute and the planck temperature and they have codes permitting the geonmetries, topologies and properties of matters in this space time, I have considered the Ricci flow, the lie derivatives, the poincare conjecture, the topological and euclidian spaces, the lie groups, the heat equations and other mathematical Tools , and I have invented with a person the assymetric Ricci flow, that permits to create the unique things and all Shapes , I try to find the good mathematics for this formalisation and the good partition, but it is not easy.I have quantize with this logic the quantum gravitation, I have just considered different distances like if our actual standard model was just emergent due to codes farer .

      Best Regards

      Dear Jochen,

      I really enjoyed reading your essay. I particularly liked your clean-cut presentation of the principles of finiteness and extensibility. You might like to have a look at my essay wherein I outline finiteness as a program to (re)construct an alternative, indeterministic classical physics (a program that we are developing with Nicolas Gisin). It would be nice to find an analogous (but of course not completely identical) feature of extensibility in indeterministic classical physics. We can maybe discuss this.

      Meanwhile, congratulations again, top rate so far!

      Flavio

        Dear Christian,

        thanks for your kind words. I agree that out approaches seem to share the same spirit, of trying to find the 'lynchpin' issues from which everything else may (hopefully) unfold, and come to be explicable. Sometimes, a complicated problem may resist forceful attempts to crack it, while crumbling under a gentle tap at the right place. Let's hope we've found the right place to tap!

        Cheers

        Jochen

        Dear Flavio,

        thanks for your kind words! I'm aware of your work with Nicolas Gisin, although I haven't yet had the time to study it thoroughly. I'll take your essay as an opportunity to rectify this; if there is a connection between our approaches, maybe we can narrow in on what, precisely, it is that separates the classical and the quantum. I'll get back to you after I've had a look at your essay.

        Cheers

        Jochen

        You can see easily that I consider coded particles like causes of our reality. I don t consider strings and 1D main Cosmic filds creating these geonetries, topologies, matters and properties, nor a geometrodynamics. I beleive that the strings are a fashion philosophically speaking like if all was fields, Waves, oscillations, I prefer to consider particles coded in a gravitational aether where the space dispappears and is coded if I can say. I doubt that this universe is an enormous heat and that we have only photons like primordial essence and after vibrations, oscillations creating this physicality. I beleive that the strings and thsi GR have created a kind of prison for the thinkers, but it is just my opinion of course, we have for me a deeper logic to all this puzzle.

        Hi, All this is very interesting about the polytopes, and the plays of maths. We search after all what are the foundamentals of this universe. The polytops can converge, but for this we must be sure about their properties and if they are foundamental, we know that we have many different polytopes , like the Lie Groups also and this E8 for example,or the infinite polytopes, the abstracts ones or the complex polytopes also. And dualities appear also. Now the real question is , must we consider these polytops really considering the QFT ? is it just a tool to rank and study better the fields in our standard model ? the real question is there, and we can extrapolate philosophically deeper, are we sure that all is made of Waves and fields ? like in the strings theory , or in the geometrodynamics, because if we have coded particles instead of fields creating our physicality , so we must consider particles and not fields implyinmg these geometries, topologies,properties of matters and so the effects possible in extrapolating the maths. The maths are Always interesting but they must be utilised with the biggest wisdom considering the interpretations and assumptions, we cannot extrapolate and conclude all what we want.The problem foundamental for me is that we consider still these geometrisations due to fields , like if we had a 1D main field from this Cosmic scale and permitting with the oscillations to create the reality with these 1D strings at this planck scale, all is false if the particles are coded and in 3D, don t forget that we can create all SHAPEs, geonetries, topologies with coded 3D particles, 3D spheres for example, now imagine this, imagine that the codes of geometrisations and properties are inside these particles , imagine a Ricci flow, the Hamilton Ricci flow, a kind of assymetric Ricci flow to create the unique things, imagine too this poincare conjecture and the heat equation and imagine the plays of maths with the topological and euclidian spaces, and the lie derivatives and lie groups, we can create all geometries and topologies also, so we arrive at big philosophical questions about these foundamental objects and the main cause of these objects and their properties. You can tell all what you want with polytopes, we cannot affirm that it is foundamental simply. The same for my reasoning considering these 3D spheres coded at this planck scale considering a gravitational coded aether sent from the central cosmological sphere. I beleive that we must prove what we extrapolate simply and at this moment we are limited simply. The aim is not to create mathematical partitions but to find the real universal partition, it is totally different at my humble opinion. The convex polytopes and the linear transformations must be sure after all, and the vectors and scalars also, the problem is that we cannot affirm in fact, so the same for the extrapolations and assumptions. The secret maybe if I can is to superimpose a deeper logic to this universe , the fields, strings, geometrodynamics and the fact to consider only photons like main essence imply a prison for the majority of thinkers, that is why we cannot explain our unknowns mainly for me.Think beyond the box and maybe consider coded particles, the Waves particles duality is respected because they are inm motions and in contact in a superfluid these particles....Regards

        Jochen. I enjoyed your paper. You may find my essay. "Clarification of Physics--" interesting. I introduce a self creating system, a new basic level to the current epistemically horizon and show how it fits into the creation of a multiverse that includes "our" physical universe. I would appreciate your comments on my essay. John D Crowell

          here is a general post about the entropical spherical informations

          Entropical spherical informations and general universal communications , the sortings, superimposings, synchronisations and the link with quantum 3D spheres and the general spherisation of the universe .Why and how ? sources, signals and encodings .....

          The complexity appears with the quantities of informations and can be ranked between the minimal and maximal of informations . For this let s consider a main universal emission from the central cosmological sphere, it is there that this infinite energy codes and transform thsi energy in matters, 3D finite series of spheres for me in a gravitational coded aether where this space disappears playing between the cold and heat generally.The source is from there and the aether is the source but it encodes also and recepts in function of evolutive codes and properties disered to create the diversity and communications of evolution in logic.

          The works of Shannon can converge and the uncertainty can be better understood at my humble opinion seeing the complexity and number of these finite series having probably the same number than our cosmological finite series of spheres, there is like an universal link between this finite number,

          the redondance and the equiprobability can be better understood if we know the real universal meaning of this general thought

          The thermodynamics can converge considering two main constants for this gravitational aether, like codes playing between this zero absolute and this planck temperature, it is an assumption but when we consider all the properties of these series, we can understand better the synchronisations, the sortings, the superimposings with all the motions, rotations , oscillations of these 3D spheres.

          The second principle in thermodynamics become relevant , Q/T correlated with this entropy and we can converge with the entropy of Shannon and the topological entropy in considering several mathematical Tools of ranking, like the lie derivatives, the topological and euclidian spaces, the Ricci flow and an assymetric Ricci flow, the poincare conjecture , the lie groups and others mathematical Tools. See that the motions, rotations , oscillations, volumes, densities, mass, angles, senses of rotations, moments, and other physical properties can help for the rankings and for a better understanding of communications ,uncertainties and probabilities.

          The potential of these series so become the key and the distribution also of informations in function of codes of evolution and properties of matters. It is a question of internal energy and distribution of this energy in function of internal codes and informations. The relevance becomes the infinity of combinations.

          Regards

          6 days later

          Dear John,

          thank you for your comment. I'm glad you enjoyed my thoughts. The notion of self-creating systems is a very interesting one, and close to my thoughts in some regards, so I'll definitely have a look at your essay.

          Good luck in the contest!

          Cheers

          Jochen

          Dear Jochen.

          It is only now that I am daring to make comments on your excellent essay because your approach is quite another one as mine. But I think one is learning most with an open mind, so I made the following notes while reading your essay:

          "the Newtonian, classical framework, can no longer be upheld". Why not, it is describing our daily macro reality quite good and can be used in many ways. In my opinion, it is the mixing up of the micro and macro, and then especially the quantum "world" that is leading to such remarks. The mainstream of 14 interpretations are created by agents inside an emergent phenomenon (reality) while they are themselves part of it and so are trying to UNIFY the two, while when we are splitting up our emergent reality and its source (in my perception: Total Simultaneity) we can treat them apart. A TOE needs not to be a unified theory of ALL but can be a theory that is dealing with different entities.

          Quote

          1. Finiteness: There is a finite maximum of information that can be obtained about any given system. 2. Extensibility: It is always possible to acquire new information about any system

          Unquote.

          I would add the following Finiteness: there is a finite maximum of information that can be obtained by an agent about a given system at a certain moment. Extensibility: no changes it already covers my addition in 1. (you also indicate this with "old" information.

          The "Horizons of understanding" are in my perception limited by the borders of our emerging reality, the Planck units and the velocity of light. Heisenberg's uncertainty relation is just describing that at a certain moment there is the only position of a particle, the velocity is a relationship with a new moment in time and space. So velocity can only be measured inside the emergent phenomenon of reality. Superposition: more quantum states can be added together, every quantum state can be represented as a sum of two or more other distinct states. We are reaching out to the borderlines of our reality and nearing Total Simultaneity (unreachable) where each point can be represented as an INFINITY of distinct states. (phase space) Measurements are "events" that are time and space-restricted, so results are always from the past, and only dealing about ONE distinct state.

          I think that the first part of your argument " But this means that there exist some state sg and measurement mg such that the value of mg(sg) cannot be predicted by f." should be extended to Any state sg and its measurement mg cannot lead to any "prediction". Because sg and mg are both events from the deterministic past, the future is containing still ALL probabilities, so is always indeterministic.

          Entanglement and Bell's Theorem are also explained in my "Total Simultaneity Interpretation", and I hope that after reading these remarks you may be interested to know more about it. Of course, my essay is not so clearly written as yours, but it is quite a new approach to the essence of reality I think so I would be very obliged to hear your opinion.

          You can find it HERE .

          Best reagards

          Wilhelmus de Wilde

            Wow Jochen, this was great!

            You managed to tie together undecidability and epistemic horizons in a way I have never seen before, and which intuitively rang true to me. For what it's worth, I believe that the way quantum mechanics will ultimately escape from the problem of supporting so many radically different ontologies is precisely through the kind of reconstruction you propose, and the fact that you managed to tie this to the kinds of epistemic horizons discovered by Godel and Turing blew me away.

            I have responded to your kind comment on my piece over there.

            Best of luck in the contest!

            Rick

              Dear Wilhelmus,

              thank you for your considered comments. When I say that the Newtonian framework can't be upheld anymore, I mean in an absolute sense---you're right to point out that for almost every practical matter, a Newtonian calculation will yield an adequate answer. But Newtonian mechanics can't be universally valid---necessarily, if my arguments work out. It has to break down at some point, and needs to be amended---or perhaps completed: in the same way as special relativity is a consistent completion of Newtonian mechanics in the realm of velocities approaching that of light, quantum mechanics can be viewed as a consistent completion of Newtonian mechanics in the realm where we're close to extracting the maximum information from a system.

              I agree that a theory of everything may apply different concepts to different domains, but these concepts must be consistent, so as to not 'crack at the seems', so to speak. All domains of physical reality ultimately interact, even if perhaps in a mediated way, and thus, our descriptions of each must match up at these interaction points. Hence, Newtonian mechanics needs modification, even if these modifications are practically unnoticeable in everyday life.

              You make a good point regarding the embeddedness of the observer within the phenomena they observe. That's in fact another way to think about such phenomena, worked out by Thomas Breuer, Maria Luisa Dalla Chiara, and others.

              Regarding measurements, we must surely admit the fact that in a very large number of cases, we can exactly predict what outcome a measurement will yield, and that outcome will indeed be observed. So, to that end, there are some future possibilities that are excluded by the present state of affairs.

              I'll have a look at your essay.

              Thanks again for your comments, and good luck in the contest!

              Cheers

              Jochen

              Dear Rick,

              thanks for this enthusiastic reply! I'm happy you found something that rang true for you in my essay. The sort of approach to quantum mechanics I pursue has been on the margins for a long time, and still does attract some skepticism (well deserved, in many cases), but I hope that we've gotten to the point that nobody gets thrown out of any offices for making the suggestion (as Wheeler was by Gödel). If my contribution helps with that just a little, I'll be satisfied.

              Cheers

              Jochen

              Hi Jochen,

              great essay that demonstrates that empirical data restrict the freedom to extend quantum theory by some deterministic hidden variables already for one and the same kind of QM-experiment. I think you made a very good job to decrease chances for getting thrown out of an office for making a certain suggestion.

              If you like i would be happy if you could comment on my essay where i also try to link undecidability to quantum events (although not as elegant as you have done).

              Hope you are well an healthy.

                5 days later

                "... since Cantor, we know that there isn't just one infinity, but ranks of them-- ..."

                Sorry, I don't consider a fabrication a scientific finding.

                BTW Scangolies is evidently wrong when he attributes the idea of infinities of different size to Georg Cantor: Already Bernard Bolzano (1781-1848) wrote this, cf. Paradoxien des Unendlichen, Reclam, Leipzig (1851).

                In order to get rid of myths and confusion, we should be careful: It was Leibniz (1646-1716) who introduced what he called the lowest level of infinity: something that is larger than anything, in other words the relative infinity.

                I am suggesting let's learn from Leibniz' sucess story and calculate as if the unbounded plurality of thinkable references was identical with not just Salviati's notion of being infinite, the logical property of simply being endless. But be careful and understand what you are doing. Don't derive nonsense.

                Try and prove McEathern and Kadin wrong. I claim having revealed that Fourier was partially wrong. Maybe, some consequences might be devastating for castles in the air?

                Eckard Blumschein