Given the prior discussion of Joy Christian's work on FQXi I thought it might help to clarify how my S10 unified field theory arrives at the same conclusion: it is all about the Hopf spheres S0, S1, S3 and S7. This might initially look like yet another case of putting mathematics before physics - the cart before the horse - that many of the essay entrants have pointed at as being a problem with physics. However, I arrive at this conclusion from the physics side, where the crux is identifying the form of Quantum Theory as being that due to a change in mathematical representation from natural-number terms denoting particle numbers, to real-number terms denoting particle numbers, in order to escape Gödel's incompleteness theorem where the wave property of a particle is the non-derivable feature in classical physics. This would mean that the matter fields of Quantum Theory are not fundamental and so cannot be just introduced into a unified theory of physics - matter must originate by some other means. This suggests re-considering extensions to GR, but not to forget the physics!

The metric of GR is conceptually a grid laid out over a surface in order to define distance measurements. An example which grounds the physics is to imagine an inflated party balloon and drawing lines of latitude and longitude on the balloon with a felt tip pen. This grid defines a metric field for the surface which expands and contracts as the balloon inflates and deflates. The Einstein tensor gives how this metric field changes with the volume of the balloon, where the form of the Einstein tensor is based upon the physical assumptions that the space is both homogeneous and isotropic. Add the physical conditions that the space is finite but without a boundary, and spheres are the simplest surfaces meeting these conditions. Now to say that there is no space-time in GR is analogous to imagining that the balloon blinks out of existence leaving the ink of the pen lines hanging in thin air - a mathematical map without its physical territory!

So we leave the territory where it is, keep the physical conditions specifying spheres, and then remember that Kaluza and Klein successfully unified gravity and electromagnetism by extending the number of dimensions with a closed S1 dimension associated with the U(1) symmetry group of electromagnetism. The condition of space being a closed sphere S^N and the S1 group space of electromagnetism gives the key to particles, as any theory where a symmetric space S^N is broken in some way to give a space containing S1 will give rise to topological monopoles. Such particle-like objects would be of the form of a hole in the space, like an air bubble in water.

In the same way the U(1) electromagnetic group space S1 corresponds to a compactified dimension in the orginal Kaluza-Klein theory, the S3 group space of the SU(2) isospin group would also correspond to compactified dimensions. There is the apparent problem that the colour group SU(3) doesn't have a simple correspondence to some space, so we will just denote it X for now. If the operative 'hole' of a wormhole is inserted into some sphere S^N it will change the topology to that of a higher dimensional torus S^3*S^M where the form of the closed spatial universe is the sphere S3. The problem is then to solve for S^M to get the particles as topological defects and for the colour space X to be something sensible - the solution is S^7 for which the colour space X = S^3 corresponds to colour group SO(3). As the space of monopoles and anti-monopoles is S^0 = {-1, 1} the Hopf spheres S0, S1, S3 and S7 are all physically realised. So physics arrives at the mathematical condition, it is just then a lot simpler to say it as the meta-principle: it is all about the Hopf spheres!

The S3 is the physical space of a closed universe and S7 consists of the compactified dimensions of a Kaluza-Klein theory, which I refer to as the particle space as it gives the properties of the topological monopoles as particles. The S^3 closed universe is locally flat R^3 and has S^7 particle dimensions at every point x. However, to get the conditions for particles as topological monopoles, there must exist a non-trivial global map from S7 to the spatial S3 universe. In local terms in R^3, this means that the orientation of S7 changes between two spatial points x1 and x2; in GR this change would be denoted by the metric, whereas in the dimensionally reduced theory it would be called the Higgs field. This physical S^7 space at every point x in the locally flat R^3 space apears to give the point at which to start considering comparisons with Joy's work.

Michael James Goodband

    Hi Michael,

    Nice summary. As Tom says: Where were you when we needed you? :-)

    The discussion of my work, both here at FQXi and elsewhere, is usually at a very superficial level. Most people seem to get stuck at the most basic EPR correlation, when I want to talk about local causality of any conceivable quantum correlations, no matter what the underling quantum state. This can *only* be done by recognizing the exceptional properties of the parallelized 7-sphere---especially its closed-ness under multiplication, as well as that of its fibres, S^3, S^1, and S^0.

    This is perhaps *the* fundamental conceptual difference between our respective uses of these spheres. While I too arrived at them through physical considerations (by analysing the conceptual arguments of Einstein, EPR, and Bell), what I ended up with are the *parallelized* spheres, which are---so to speak---as flat as a sheet of paper. More precisely, their curvature tensors vanish identically, while torsions within them remaining non-zero. Thus the theory of gravity more appropriate in the context of my work is the teleparallel gravity, not the usual general relativity. It turns out that without parallelization local causality cannot be maintained for all conceivable quantum correlations, or even for the basic EPR correlation. Parallelization is the *only* way to meet Bell's challenge. Unfortunately this fact is not yet widely appreciated, even by some supporters of my work.

    Best,

    Joy

    As the potential skunk at the picnic, and the possibly alluded to deluded 'supporter' and one who is minimally familiar with both of your work, it is not clear to me that a shared appreciation of S0, S1, S3, and S7 doth a marriage make. I too believe that these normed division algebras are important and, with Rick Lockyer's view of Octonions, see them as applying to my own work. Just sayin'.

    Hello Professor Mickael from UK,

    Ok let's play, they need help :)

    Your essay shows us a very good knowledge of several theories, existings.But I see several irrationalities. Why BH particules ? the derivations cannot give us a quantum BH, a BH is a sphere , with a volume, central to galaxies, with rotations. So indeed Godel is right, but his reasoning is subtle, indeed a lot of people confound the theorems of uncompleteness of Godel with the physical axiomatizations. the axiom of truth becomes an essential. Is it important to insert not coherent derivations or superimposings for a kind of confusions.

    My perception is that a lot of persons utilize this uncompleteness of Godel to imply an, ocean of confusions. In fact, the coherences must be formalized with a kind of universal wisdom !!! Is it necessary to imply the confusions when the truth is so evident and simple? it is the question after all.

    The Uncompleteness is simple in its pure meaning.

    until soon and spherically yours of course.

    Hi Edwin,

    My comments were by no means an attack on you (there are supporters of my work outside FQXi and the cyberspace, including within the main-stream Bell community). But, yes, as I have said before, you are among those who have not yet understood my argument.

    I have nothing against your own ideas as long as you acknowledge that your model is manifestly non-local and it can never be local. But you are unlikely to acknowledge this because you are unable to see the blatant non-locality of your model. I thought we had agreed to disagree about this.

    In any case, both Michael's work and mine stand on its own. They neither need to be married, nor stand in conflict. At this stage I am as curious as Michael to witness some of the same broad conclusions emerging from two very different explorations.

    Best,

    Joy

    Hi Joy,

    That was tongue-in-cheek and not to be taken seriously, but I don't think that Michael's approach to normed division algebras bears much relation to yours. I believe that he and I see SO(3) as more appropriate to our theories than SU(3) but his theory and mine are very far apart in other ways. But you are correct, that it is interesting that normed division algebras are becoming significant in this fashion. And I do not think your theories are either married or in conflict. I really don't see much overlap except for the shared appreciation of this topology. My remark was spurred by Tom's "where were you?" with the implication that his use of Sn spheres would have helped your case. Perhaps, but I doubt it. Although there was a period about 18 months ago when you and others were arguing about the definitions of particular topologies when Michael would probably have been on your side. And he does agree with you that I haven't solved the non-locality problem.

    I'll bow out of this discussion with the best wishes for your model and for Michael's theory. Both are very impressive. The problem with both is their complexity, requiring so much effort to comprehend. They are beautiful accomplishments. Congratulations to both of you. I truly admire you both for the obvious intellectual effort required to produce these works.

    Best,

    Edwin Eugene Klingman

    Hi Joy (Part 1),

    I have been contemplating your work in the links you gave. My slip in saying the Hopf spheres I think was my subconscious trying to get my attention: with the particle/anti-particle space being S^0={-1,1} and the space of cyclic waves being S^1, the existence of wave-particle duality seems to be saying the fibre-bundle of the first Hopf sphere. This implies that the first Hopf sphere provides an underlying context for an analysis of Quantum Theory, such as yours.

    I think that you short-changed yourself with the meta-principle you gave earlier. Although in mathematical terms the S^7 case (eqn 1.32 of your 9_Origins.pdf attachment) is more general than the S^3 case (eqn 1.28) as S^7 contains S^3 subspace, the assertion of S^7 ONLY precludes the possibility in physics that the two spaces have different origins such that the S^3 is not a physical subspace of S^7. A real sphere example is where the space of the particle symmetries is S^7 - as in my S10 unified field theory (STUFT for short) - and the space of the rotation group is S^3. The rotation group is not a subgroup of the particle symmetries and so BOTH S^3 and S^7 occur as they have a different origin. So the most general statement of your work is not solely in terms of S^7, but S^3 (1.28) AND S^7 (1.32). With the first Hopf sphere providing an underlying context for the wave-particle duality of Quantum Theory, your work would then seem to independently contain S0, S1, S3, S7 and not just as subspaces of S7 (as parallized spheres).

    In the context of the spheres being real physical surfaces, the presence of BOTH S^7 and S^3 is critical as the homotopy group for the map S^7 -> S^3 shows that it just involves the S^4 base-space PI_7(S^3) = PI_4(S^3) = Z_2 and gives a chiral non-trivial vacuum looking for all the world like the electroweak vacuum and gives the correct value of the Weinberg angle just in geometric terms. This breaks the symmetry of the S^7 and gives a 3 by 4 table of topological monopoles looking like the particles.

    In metric field terms, your eqn 1.53 together with the closure condition of eqn 1.55 specify the spheres S0, S1, S3, S7 as a collection of closed spaces. The principles of GR seem to be captured by the meta-principle: make no preference. This means no preferred speed, ie. the speed of light is always the same, no preferred location (homogeneity) and no preferred direction (isotropy) - these also say no boundary to the space. Applying this no preference condition to the 4 spheres, says all of them. With space being S^3 and the 'particle space' being S^7 the above map S^7 -> S^3 gives a non-trivial vacuum winding and topological monopoles and anti-monopoles with space S^0. The pre-condition of the S^7 -> S^3 map and the lack of an independent S^1 are both addressed by the unification principle: the S^3 of space and the S^7 'particle space' are unified in a sphere S^10 which then has a hole inserted to give S^3*S^7 with the above mapping. In GR, such a scenario would be cyclical between the unified S^10 phase and the 'broken' S^3 * S^7 phase, thus giving the independent occurrence of S^1 in a 10+1 dimensional extension to GR.

    This gives the physically-real side I address in extended GR where spheres are spheres, particles are particles and waves are waves.

    Michael

      Joy (Part 2),

      There is a critical dividing line between your work and mine which doesn't seem to have been fully appreciated yet: essentially my extended GR is on the physically-real side and your work is on the non-physically-real side - as in QT. Although the physically-real side of extended GR is consistent I prove that it is incomplete because the calculation of the mass reduction of the topological monopoles as Planck scale rotating black holes is subject to Gödel's incompleteness theorem. This both means that the reduced masses of the particle-like objects cannot be calculated from classical physics, and that the self-consistent dynamic state of the particle-like objects can possess a non-derivable feature.

      Assuming that this feature is a wave property, Gödel's proof can be circumvented by changing from physically-real terms denoting the countable number of particles at specific points in space, to non-physically-real terms denoting them as real-number valued fields spread throughout space with a wave property, ie. a wave function. The mathematical conditions of Gödel's proof can then be used to prove no hidden variable theory and the mathematical conversion from wave function to particle number cannot be derived in classical physics - hence has 'weird' descriptions like 'collapse of the wave-function' which make little physical sense.

      However, in the switch in integrals from physically-real terms to the non-physically-real term of the wave function there is a critical flatness condition. The problematic expansion is about a black hole, and the event horizon and ergo-region *cannot* be denoted by a continuous field term in space, so the replacement which derives QFT *only* holds in flat space away from the black hole. This firstly means that QFT *cannot* be unified with GR. But there is a further problem with this approximation of QFT, in that it excludes the ergo-region with its sign reversal of the metric term gtt that allows apparently non-local causation to make sense in physical causation terms. In an approximation that excludes this ergo-region, you *are* going to have a causation issue. The purpose of the representational change to the non-physically-real wave function and QT is to get a consistent and complete theory, but the represenational replacement doesn't even consider this causation issue.

      It would seem that my flat space condition on the representational replacement was just the beginning of the 'flattening' required to get QT fully consistent and complete. There is going to have to be some condition in QT to resolve this causation issue, but I currently don't understand the parallelization of S0, S1,S3 and S7 well enough to understand how it does it.

      Best,

      Michael

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        • Post #30

        Joy wrote, " ... what I ended up with are the *parallelized* spheres, which are---so to speak---as flat as a sheet of paper. More precisely, their curvature tensors vanish identically, while torsions within them remaining non-zero."

        This seminal point really isn't easy to grasp, but grasped it must be. I think what Michael says below, "The pre-condition of the S^7 -> S^3 map and the lack of an independent S^1 are both addressed by the unification principle" -- i.e., the "no preference" principle necessitated by general relativity (nice, Michael) -- nails down why Riemann curvature vanishes everywhere while torsion is preserved in the 2-dimensions of S^1 as a twist, which demands such a nonorientable surface embedded in the orientable measure space. Classical orientation entanglement is preserved, as in spinor theory.

        I realize that Michael continues with S^3 X S^7 to S^10, but we don't need it for the present discussion, because the limit of parellizability is S^7. (I would comment, though, that an earlier topology paper of mine concluded that the 10-dimension limit is identical to the 4-dimension horizon but that is also, no pun intended, outside the scope of this discussion.)

        Yeah, I'm shooting from the hip. It looks like something important is going to start percolating here very quickly, though.

        Tom

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        • Post #31

        Michael,

        Quick reply, while I am still digesting ...

        Truncate your theory to the S^7 limit, and I think you will find that Joy's framework satisifies both the completeness criterion for a physical theory (as described by EPR) and Godel completeness.

        Tom

        Hello Dr. Goodband

        You plainly seem to have taken the bull by the horns, but with my limited technical knowledge I could follow some (but not all) of the skilful action and wish you good luck in finishing off the Quantum Bull and do a great favor to physics.

        In my fqxi essay Fix Physics! and my earlier Beautiful Universe Theory (BU). I have tried to trace the steps by which modern physics went 'wrong' with suggestions on how it may be revamped. One major cause is the false particle-wave duality concept: It is not individual electrons or photons that show up as dots in a double-slit image-field - just sensor atoms reaching saturation point. Eric Reiter independently reached (and experimentally proved) the same idea. His fqxi essay shows how the point photon concept is simply wrong. With the fall of the point photon the probability interpretation as a physical 'fact' and much puzzlement in QM falls by the side.

        You speak of a "finite physical network that is potentially infinite in theory, this situation could only arise in the context of an infinite network expansion about some object." This is very much like the node lattice of my (BU) theory. Indeed the 5th dimension of Kaluza-Klein has been somewhere interpreted as nodes of such an ether lattice.

        Despite the mostly qualitative nature of my work, I would be honoured if you can read and comment on it.

        Best wishes,

        Vladimir Tamari

        • [deleted]

        • Post #33

        Hello,

        I am surprised that several "said" responsible scientists are so irrational in their superimposings.probably it is due to a lack of generality, I don't know, or the bad strategies simply.In all case, I am surprised by this comportment. But Like I love Jesus Christ and Buddah , I pardon you all. I am understanding your strategy after all.It is simply logic your comportment.I know this Good team band. And I accept. I continue, I persevere like a real searcher. Like a real generalist, a real universalit. I just show you what are my sciences, in fact I am happy to give courses to these strategists.In fact it exists the true and the false. I pray for them in fact.I pray in a pure spherical universality. I am going dear friends to go at new York, I will put an ocean of flowers and plants in this town.It is the country of the freedom. I will go !

        Soon furthermore.and REVOLUTION SPHERIZATION WITH SCIENCES AND UNIVERSAL CONSCIOUSNESS.

        Spherically yours.

        ps to the Institute of Advanced Studies....be rational and respect the real generalists please.Don't make films in your heads but simply respect me.Change of strategy.

        Regards

        • [deleted]

        • Post #34

        or probably that they confound what is the good equations and the general equations.Or perhaps that they show simply an ocean of words for the confusions and their own knowledges showed in live.Or it is just for this monney also.Or this or that.

        In all case, I know this team and why they make that. Simply the hate probably.

        I pray for their souls. They need help in fact. Furthermore you imagine their credibility if I am recognized.I understand their strategy and their fear.Probably that they are going to discuss between a kind of team band for a kind of pseudo politeness.Probably also that they think that they understand the maths of de sitter and riemann or ....but do they understand the maths of Dufourny.I doubt.

        spherically yours

        Regards

        Hi Michael,

        You have raised a number of interesting issues. I will number my responses to them for clarity.

        (1) You wrote: "...the assertion of S^7 ONLY precludes the possibility in physics that the two spaces have different origins such that the S^3 is not a physical subspace of S^7."

        The "S^7 only" assertion is not strictly necessary for my analysis to go through. However, from the point of view of quantum correlations, separation of S^3 from S^7 in the manner you have suggested seems to be unjustified. In my picture quantum correlations are correlations among events occurring within spacetime---or equivalently among the clicks of a network of detectors. As far as EPR type correlations are concerned these events can be viewed as occurring within S^3. But S^3 is definitely not enough to reproduce quantum correlations beyond those exhibited by the 2-level systems. For example, the correlations exhibited by the GHZ state can only be reproduced as events occurring within a parallelized 7-sphere. To be sure, the clicks we observe appear to us as occurring within R^3. So the "extra" dimensions of S^7 are certainly hidden from us in that sense, but these dimensions are not necessarily compactified as in your work. In fact I tend to view the correlations exhibited by states like GHZ as the *evidence* that the rotation group of the physical space is S^7, not S^3, with the latter being only a special case of S^7. Still, this does not seem to necessitate the "S^7 only" assertion.

        (2) Like Tom, I very much like your meta-principle: "make no preference. This means no preferred speed, ie. the speed of light is always the same, no preferred location (homogeneity) and no preferred direction (isotropy) - these also say no boundary to the space." Absolutely marvellous!

        (3) But the following separation is potentially in conflict with my analysis: "With space being S^3 and the 'particle space' being S^7..."

        For the reasons explained above, in my analysis the separation of S^3 as "physical space" from S^7 as "particle space" is not justified. All measurement events are occurring within S^7, but we only see them as occurring within R^3. This, however, does not seem to be in conflict with your earlier statement that "physical S^7 space at every point x in the locally flat R^3 space appears to give the point at which to start considering comparisons with Joy's work."

        (2) Much of what you say in your Part 2 below has to do with the 'flattening' you require to get QT fully consistent and complete. The flattening required for my analysis to go through has to do with "absolute parallelism", as in teleparallel gravity. Since both S^3 and S^7 are simply-connected manifolds, absolute parallelism is equivalent to their curvature tensors vanishing identically, with torsions within them remaining non-zero in general. This is automatically the case if we view S^3 and S^7 as sets of unit quaternions and octonions, respectively. The very algebra of quaternions and octonions then provides means to define orthonormal frames at each point of these manifolds. This however induces torsional twists within them, and it is these twists in the manifolds that are responsible for what we observe as strong quantum correlations. The latter have nothing to do with non-locality or entanglement per se, because the distant events within S^3 and S^7 are now causally linked by distant parallelism in a non-mysterious way. In other words, in my picture the correlations between distant events are no more mysterious than the innocent correlation between Dr. Bertlmann's socks discussed by Bell.

        Best,

        Joy

        A few number of persons understand really the uncompleteness of Godel.If now people thinks that they have understood just because they make politeness between them.So indeed there is a probelm.

        And the rule of the Institute of Advanced Studies is to be rational and deterministic. This Institute cannot be corrupted. and cannot be irrational.

        Furthermore, the responsability of an institute like this one cannot imply confusions.

        The axiom of dimensionality is not accepted at my knowledge. So why they insist ??? For the sell of books or what ?

        the incompleteness of Godel shows us how we can axiomatize our foundamentals. It is like the hidden variables in fact. We cannot confound the young evolution of our universal sphere, and so our unknowns.With bizare irrational superimposings where our universal laws loose their meaning. I don't understand why people interpret the incompleteness like that. It is sad in fact. We are just young at the universal scale.And so it is logic to have unknowns, but they are rational these hidden variables,not need of extradimensions. The axiom of dimensions is not a reality, only the 3D is rational and this time constant of evolution. The geometrical algebras are not there to imply confusions, but are there to improve our equations with determinism. The beauty of sciences is to discover the truths, not to imply confusions by pseudo parallelizations. The walls separating this infinite light without rotations above our physicality, and the light inside a sphere in evolution. The central spheres are the secret of codes of singularities. The informations can only be transmitted by this infinity inside the main central spheres of systems of uniqueness. Why so the people wants to invent false hidden varibales. It is the road towards our main codes, these central spheres which are important. The "infinite" light creates the "finite" light !!! all is connected by this light indeed , but the real interEst is to understand the physical dynamic in taking it like a project of optimization. Why hidden variables or bizare decoherences ???? The universe is a dterministic spacetime. Godel and Cantor are in a bar, do you think that they think that the glass of beer is an infinite system or a pure number entangled spheres.In the same time, they can add, derivate or integrate, or multiplicate these numbers.....so is it infinite, or is it relativistic in the meaning of this infinity and the infinities and the finite groups ??? all possesse a specific number of spheres, finite and precise !!!! the volumes of this entanglement in the pure serie of uniqueness so imply several interesting road considering my equations and the velocities of rotations and their sense of rot. differenciating the bosons and the fermions. IF THE NUMBER DOES NOT CHANGE FOR THE SERIES OF UNIQUENESS so we can see the quantization more the evolution by polarity between hv and m.

        Regards

        Michael, I am impressed with your work to the extent that I tried to order your book from Amazon U.K. through you directly (Amazon is out of stock) and the order was rejected with a message that it can't be sent to my U.S. address. What gives? -- are you only allowed to sell your own book in the U.K.? :-) Please send an ordering link to my Email, thomasray1209@comcast.net and I guarantee you a sale, if the shipping cost isn't prohibitive. Otherwise, any chance of Amazon U.S. making it available?

        Meantime, on the question of scientific realism, I too have looked at Joy Christian's framework with that question in mind. Rather than applying the Godel incompleteness theorem to the broad set of scientific theories which incorporate physically real terms (which would naively include the theoretical components of Joy's framework, i.e., the prediction of physically real quantum correlations) -- I find that mathematical completeness, as Joy describes, which meets the EPR criterion (every element of the mathematical theory corresponds to every element of the physical measure) also satisfies Godel completeness. I am willing to engage on this issue.

        I think it is important to understand that Joy's framework is noncontextual, and not merely an interpretation of observed quantum mechanical phenomena. His logical judgment on the state of quantum correlations is completely closed, exactly as the mathematically complete judgments of relativity in the classical domain. Christian's research, by taking a global (topological) approach to local realism, breaks down the distinction between local and global and prescribes an exact limit to the range of observables, just as relativity does ("all physics is local"), though in an extended universal domain unrestricted by classical mechanics.

        As a result, I find that Joy Christian meets Karl Popper's criteria for metaphysical realism (*Realism and the Aim of Science,* Routledge 1983). In turn, I think that your own variety of realism is satisfied, and that Joy Christian's result lies outside the set of constructs that would be subject to Godel incompleteness.

        All best,

        Tom

          Hi Joy,

          I will frame my point about separate occurrences of S3 and S7 in terms of the classic EPR scenario of correlated spin states between 2 particles, which without special characters I will denote as e^|e_ for electron spin up e^ and electron spin down e_ .

          My point is that this is just quantum mechanics, think quantum field theory. Just as the emission of a photon converts e^ to e_ the emission of a W-boson converts an electron into an neutrino, an up quark into a down quark etc. and there are also inter-family conversion reactions. Such interactions mean that the most general EPR 2 particle scenario in QFT is *not* of the form A^|A_ but A^|B_ where particles A and B can be of any type; A=B is just a special case in QFT.

          The observables to consider in the correlation analysis are both the spin eigenvalues of the rotation group SU(2) - group space S3 - and the particle types which are eigenvalues of some 'particle space'. I use this term in place of particle symmetry group, because grand unified theories assumed that it was going to be a group - a hidden assumption I could have raised in my essay - whereas my work says that it is the quotient group SU(4)/SU(3) isomorphic to S7. So there are 2 sets of observables with quantum correlations {^,_} and {A,B,...} where the values of the first set are the eigenvalues of the rotation group with space S3. In my case the second set contains eigenvalues of SU(4)/SU(3) ~ S7 (after the symmetry has been broken) and the S3 is clearly distinct from this S7.

          Your analysis should also apply to the quantum correlations between the observables in each of the 2 sets {^,_} and {A,B,...} for the most general EPR 2 particle scenario A^|B_ in the Standard Model QFT. Ultimately my question is whether there is a way to use your analysis in reverse to place a constraint on the origin of these observables?

          I.e. some argument of the form

          Parallelised S3 => group space S3 for the observables {^, _}

          Parallelised S7 => 'group space' S7 for the observables {A,B,...}

          A straightforward argument doesn't seem to work, which is why I am asking :-)

          Michael

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          • Post #39

          Hi Michael,

          You have framed your question very clearly. It reminds me of some passionate discussions I had last year on these pages with Ray B. Munroe, who is sadly no longer with us. He was a supporter of my use of 7-sphere, but he also saw things from the particle physics perspective and I had to explain my foundational perspective to him from scratch. Please allow me to do the same here, if not for you, at least for other readers who might to be interested.

          The issue at heart is local causality. This concept has been crystallized by various people over the years, starting with Einstein in his special relativity, and culminating in Bell's analysis of the EPR scenario. Bell used some earlier ideas of von Neumann to frame the concept for any realistic theory, and made it independent of any specific theory of physics, including quantum theory or quantum field theory, and independent even of the specifics of special and general relativities. He thus provided a very general, very reasonable classical, local-realistic framework, which does not depend on the specifics of a given set of observables. It depends only on the yes/no questions the experimentalists may ask and answer. Thus, for example, for the classic EPR-Bohm scenario involving a joint observable AB for observing spin up and spin down at two remote ends of the experiment, he formulated local causality in terms of the following factorizability condition:

          AB(a, b, L) = A(a, L) x B(b, L),

          where A(a, L) is independent of the remote context b as well as the remote result B, and likewise B(b, L) is independent of the remote context a as well as the remote result A. That is it. As you can see, his formulation of local causality only involves the measurement results A = yes/no and B = yes/no, apart from the measurement contexts a and b (such as the directions of the local polarizers), and the common cause L, which is the "hidden" variable or a complete EPR state.

          It should now be clear why the kind of details you have spelt out for more general scenarios involving particle productions etc are irrelevant for the central concerns of local causality. All that matters is how the yes/no answers to relevant questions are correlated, because any experiment in physics can always be reduced to a series of questions that can be answered in a "yes" or "no."

          Nevertheless, let us look at things from your perspective. Let us consider a scenario where an EPR 2-particle state is not of the form P^|P_ (in a variant of your notation) but of the form P^|Q _, where Q =/= P. For you, then, there are two sets of observables with quantum correlations, {^,_} and {P,Q,...}, where the first set contains eigenvalues of the rotation group S3, and the second set contains eigenvalues of SU(4)/SU(3) ~ S7. The question then is: Is Bell's local-realistic analysis applicable to this situation? Yes, absolutely. Is my topological correction to Bell's analysis applicable to this situation? Again, yes, absolutely.

          But here is a difficulty for you: Your set {^, _} is restricted to S3. It is, however, not possible in general to reproduce quantum correlations using my framework within S3 if the corresponding quantum systems have the spectrum of eigenvalues (or measurement results) more general than that of a 2-level system. So, ironically, there is no problem for the exotic set {P,Q,...}, for which the "group space" within your framework is S7, which is the most general available within my framework. It is the set {^, _} that will cause a locality problem for you, because, for a general quantum field, the spectrum of eigenvalues within {^, _} would be highly nontrivial. Within my framework, on the other hand, both {P,Q,...} and {^, _} fall under the same "group space" S7, and so there is no problem.

          So my framework does put the following constraint on the observables: If one restricts to the group space S3, then the only quantum systems for which local causality can be maintained are the 2-level systems. For more general systems S7 is inevitable.

          Best,

          Joy