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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...

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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

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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...

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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

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Hi Joy and Tom,

I too am grateful to you for engaging in meaningful discussion. I was previously unaware of Joy's work (and Buridan's principle ) and your comments have advanced my thinking to the point where I am certain that the key issue really is a mathematics description problem in trying "to reconcile local discrete measures with globally continuous functions".

In the spirit...

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Hi Joy and Tom,

I too am grateful to you for engaging in meaningful discussion. I was previously unaware of Joy's work (and Buridan's principle ) and your comments have advanced my thinking to the point where I am certain that the key issue really is a mathematics description problem in trying "to reconcile local discrete measures with globally continuous functions".

In the spirit of the essay contest of questioning assumptions, I realised I have been making an assumption about Joy's work. This is partly because the initial point of comparison was the dependence on the 4 spheres S0, S1, S3, S7. In my case, these are physical spaces in a classical metric field theory where the Relativity meta-principle - make no preference - selects them all, and the unification principle gives only one possible unification which yields these spaces (STUFT). The fact that the dimensionally reduced version of this KKT derives the Standard Model Lagrangian with the correct electroweak vacuum (and Weinberg angle) and spectrum of 12 fermionic particle-like objects in classical physics is a nice feature (and the coupling constants, including the Higgs scalar coupling which predicted the classical Higgs boson mass to be 123GeV). Then comes the *real issue*, what is Quantum Theory all about?

I think this is the real point of comparison of my work (primarily in Agent Physics but also as presented in Science Theories) with Joy's, where the QT context for Bell's analysis has distracted me from Joy's functional analysis of hidden variable theories being more general than *just* QT. Bell's analysis started from the existence of QT and asked whether there exist a hidden variable theory that can account for the same correlations between observables as QT. However, a functional analysis whose only conditions are local causation and correlations between observables can surely be applied to any assumption of a hidden variable theory in science (ie. without the pre-condition that is replacing QT)?

The reason for considering this possibility is that my physics-based analysis of numerous physical systems identifies a recurring feature of a self-consistent (causally closed) dynamic state residing on the giant connected component of some physical network. Any physically-real theory of these systems can be proven to be incomplete because of the discrete character of the dynamics of the network - this specifically includes the classical physics of particles, as in my essay. It seems to only make sense for the possible undecidable proposition in the physically-real theory to describe a collective property of the dynamic state residing on the giant connected component. This can potentially give a description problem in physically-real terms, because the inputs to the network cause discrete changes to propagate through the giant network component, with its undecidable feature, to the outputs. Encountering a network state with undecidable properties would surely have some effect on the outputs, such as altering the correlations between the outputs observed?

Joy's functional analysis of correlations between observables in a non-relativistic context would seem to be wholly appropriate to this situation. The combination of my work and Joy's functional analysis leads me to the proposition: the presence of the undecidable property on the core network component causes correlations between the network outputs that cannot be accounted for in a discrete theory in physically-real terms. Assuming that the correlations can be accounted for if only we knew some extra missing terms constitutes an assumption of a hidden variable theory. The follow on from the above proposition is that the richer functional structure of continuous functions can account for the correlations in output, where such terms do not directly correspond to the inherently discrete physical components of the network system and so are non-physically-real terms (like the wave-function of QT).

Extending the functional analysis to this scenario could potentially provide a mathematical proof (or disproof) of my proposition that the presence of an undecidable feature on a discrete network system is the *cause* of the correlations that cannot be accounted for by a discrete hidden variable theory. I show that the required network conditions can occur in biology, psychology and economics ... with the prediction following on from this proposition that there will exist correlations between observables in these system which cannot be accounted for by a physically-real scientific theory. These disciplines implicitly make the assumption that there will exist a hidden variable theory that will account for all experimental observations. It seems to me that Joy's work provides the basis for the construction of experimental tests of these assumptions throughout science.

Best

Michael

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FQXi'ers - On the connection of my work to Joy Christian's (in parts because it's long)

Joy expressed Bell's analysis about whether there could exist a 'complete' 'local' theory that could replace Quantum Theory, as Bell considering functions of the form

A(n,l): R3 * L -> S0 (see eqn 1.1)

R3 is a co-ordinate based denotation of the flat Euclidean space E3 in...

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FQXi'ers - On the connection of my work to Joy Christian's (in parts because it's long)

Joy expressed Bell's analysis about whether there could exist a 'complete' 'local' theory that could replace Quantum Theory, as Bell considering functions of the form

A(n,l): R3 * L -> S0 (see eqn 1.1)

R3 is a co-ordinate based denotation of the flat Euclidean space E3 in which the 3-vector orientation n resides, the space L is a space of 'hidden' variables which gives 'complete' states and S0 is the function co-domain for the observable A. Joy identified that the function co-domain S0 is rather trivially wrong, it should be S2 sub S3 (see eqn 2) as the possible orientations of the 3-vector n define a 2-sphere. Quantum correlations follow from the topology of the spaces.

My work effectively addresses what is meant by 'hidden' variable and 'complete' states in physics, neither of which were sufficiently well-defined by Bell. There are effectively 2 different underlying meanings for 'complete'

1) mathematical completeness - every theorem in a formal system can be derived

2) scientific completeness - every observation can be predicted

Bell fails to specify which he means. This shortcoming can be viewed as originating with the original EPR paper which gives the meaning of 'complete' as: "every element of the physical reality must have a counterpart in the physical theory". But this isn't physics! It fails to specify how you would verify that this was true - namely by experiment. This is why I use 'physically-real' to specify a term in a theory that *directly* corresponds to an observable feature in reality (I took this term from a QT textbook discussion of Bell's theorem). Any mathematical term which does not have this 1-to-1 correspondence is a 'non-physically-real' term, eg. the wave-function denotes strictly countable numbers of electrons by the real-numbers, and since 0.5 of an electron is never measured in an experiment, the wave-function is a non-physically-real term.

There are also 2 possible meanings for 'hidden' and Bell fails to specify which

1) hidden from the subset A, B of observables considered in a correlation experiment - in which case the hidden variable could be found in the future by a new experiment

2) hidden from all possible observables - in which case it is a non-physically-real term!

Option 2 is what is implicitly meant by Bell, but how such a conspiracy of Nature could arise is not considered. The missing element is dynamics, which is because the physical space of the EPR scenario isn't just Euclidean space E3, but a Euclidian sub-space of Minkowski space-time M4. There are an infinite number of ways of picking out E3 from M4, parameterised by the velocity v of the reference frame, i.e. E3(v) sub M4. Joy didn't make this correction either, but it doesn't change his correlation results because they depend upon integrating over the space of the 'hidden' variable to get expectation values. This may suggest that the parameterisation E3(v) could just be dismissed. However, this would effectively amount to setting v=0 for all the reference frames of the scenario, but without any relative motion there is no dynamics and so nothing happens - thus setting v=0 is unphysical!

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Dear Michael,

''The experimental observation of electrons passing through slits generating the wave characteristic of an interference pattern challenges this divide, although this is only really clear when the electron beam intensity is reduced to the point of a single electron passing through the slits at a time.''

If we were to assume that we live in a universe which creates...

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Dear Michael,

''The experimental observation of electrons passing through slits generating the wave characteristic of an interference pattern challenges this divide, although this is only really clear when the electron beam intensity is reduced to the point of a single electron passing through the slits at a time.''

If we were to assume that we live in a universe which creates itself out of nothing, without any outside interference so particles have to create themselves, each other, then it seems logical to assume that particles and particle properties must be as much the product as the source of their interactions. If so, if particles in such universe only exist to each other if, as far and as long as they interact, exchange energy, with all particles within their interaction horizon, then the electron, on nearing the slits, would 'see' its world split into two slightly different worlds, worlds which from both sides of both slits interfere with its path.

As to its wave character, the Uncertainty Principle (UP) is interpreted to say that virtual particles can appear by borrowing the energy to exist from the vacuum, for a time inversely proportional to their energy. This suggests that real particles can be thought of as virtual particles which by alternately borrowing and lending each other the energy to exist, force each other to reappear again and again after every disappearance, at about the same place. As in this view particles express and at the same time preserve each other's mass by continuously exchanging energy (the sign of which then alternates), the origin of mass is obvious, as is the equality of inertial and gravitational mass.

The hidden variables Einstein wanted to exist to avoid indeterminism can be identified as the energy exchange by means of which particles express and preserve each other's properties. However, the unpredictability Einstein wished to eradicate remains if particles indeed are as much the effect as the cause of their interactions. It is because their exchange is unobservable as long as the particles are at equilibrium, because it serves to preserve the status quo, that we have been able to remain ignorant of it: because we've always assumed that particles only are cause of forces. Only when their equilibrium is disturbed so the frequencies they exchange energy at changes and net energy is emitted or absorbed, the effects of this exchange become observable. In the study I propose a mass definition based on the UP: the less indefinite the position of a particle (or mass center of an object) is, the greater its mass is, a definition which might make it possible to derive the equations of relativity from the UP. For other interesting features of a self-creating universe, see my essay or the more extended study at www.quantumgravity.nl (which also contains some remarks on (CTRL+F) 'Gödel).

Regards, Anton

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Dear Michael,

I just read your very interesting essay. A few thoughts come to mind.

1. Concerning your invocation of Godel’s theorem in regard to physical measurement on page one, I have a question which might be rather clumsy to state. I am thinking that perhaps two different ideas of “proof” or “derivation” are involved. Suppose you have a true statement about...

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Dear Michael,

I just read your very interesting essay. A few thoughts come to mind.

1. Concerning your invocation of Godel’s theorem in regard to physical measurement on page one, I have a question which might be rather clumsy to state. I am thinking that perhaps two different ideas of “proof” or “derivation” are involved. Suppose you have a true statement about natural numbers, which is undecidable. It seems that you could always test the statement for a particular choice of natural numbers: just substitute in the chosen numbers and see if you get an identity. The undecidability comes from the fact that you can’t prove the statement is true in general; i.e. for an arbitrary, unspecified choice of numbers.

In your terms, you couldn’t derive it from the axioms of the theory. Now the “scientific,” or “physical,” proof of the statement is just the inductive reasoning (in the sense of Newton) based on the experiment. So I suppose what you are saying is that no matter what the axioms of your physical theory, there will be statements which are “scientifically provable” in the sense of measurement that don’t follow from the theory?

2. Does this imply that it is impossible to have a physical theory with a finite number of physical laws to which other, independent, true laws cannot be added?

3. You mention of using “successor, predecessor, zero and projection functions” to build up number theoretic functions and a “growing network of object reactions that include processes which implement arithmetic increases in the numbers of some objects.” You also mention the “path integrals of quantum field theory” in a similar context. This reminds me of “quantum causal theories” of quantum gravity. Examples are causal set theory and causal dynamical triangulations. This is interesting here because such theories often involve countable “classical universes,” and this appears to make a difference in light of your discussion on pages 3-4. For more perspective on this viewpoint, see my essay here.

4. Your discussion of black holes and mass reduction is very interesting. Janko Kokosar speculates about a “black-hole” theory of fundamental particles in his essay, but he believes that the Higgs mechanism ruins this. I don’t know if your ideas change this consideration.

5. The essays of Torsten Asselmeyer-Maluga and Jerzy Krol might be interesting in the context of your discussion on pages 7-8.

Your essay gives me many more ideas to think about. Thanks for the great read! Take care,

Ben Dribus

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Space-Time: real or emergent?

This may well depend upon what you think these terms mean. Nothing prevents clear understanding quite like speaking the same language, but using the same words with different meanings. So instead of using these ambiguous terms and hoping everyone chooses the same meaning, let's call the manifold of GR an "absolute" space in the sense that the "absolute" scale...

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Space-Time: real or emergent?

This may well depend upon what you think these terms mean. Nothing prevents clear understanding quite like speaking the same language, but using the same words with different meanings. So instead of using these ambiguous terms and hoping everyone chooses the same meaning, let's call the manifold of GR an "absolute" space in the sense that the "absolute" scale of physical distances in GR is not in fact defined - this is seen in the scale factor of the universe in GR not being specified.

In any model with an "absolute" space that includes compactified dimensions, e.g. M4*X (the model referred to in the essay considers local manifold M4*S7 but the point here is more general) this problem of not having an "absolute" scale in GR is resolved by measuring ALL physical distances in M4 using the compactification scale of X - virtually always assumed to be the Planck scale (the model of the essay arrives at a Planck unit system through consistency). Of course, this explicitly depends upon grasping the meaning of the word "relative" from which the name Relativity is derived, surely not a difficult conceptual point?

As I said under Edwin's thread of 30 Sept, in any system of measurement units the intrinsic error of measurement is ½ the unit used, and when this is expressed in invariant form (i.e. 2nd Casimir invariant spin) for the Planck unit system this gives the Heisenberg uncertainty relations. It should hopefully be obvious that this implies that the relations are fundamentally NOT about quantum theory, but about the structure of space - namely that there are compactified dimensions. This also has the effect of converting an assumed smooth "absolute" manifold of M4*X (in GR) into a 4D space-time manifold MX4 expressed in terms of physical units of measurement that is NOT smooth, and has Planck's constant "built-in": precisely the features of an "emergent" space-time (i.e. MX4 "emergent" from M4*X), with an exotic differential structure such as those of the 4-manifolds discussed in Torsten Asselmeyer-Maluga's essay. The Cellular Automata view of this "absolute" M4*X manifold is one where the neighbouring spatial cells overlap each other by ½ a cell (I mentioned this in the context of Vladimir Tamari's CA model) to give an "emergent" view of a MX4 space-time manifold that is NOT smooth.

By grasping the real meaning of the word "relative" it can be seen that the "absolute" manifold M4*X of GR will give an "emergent" space-time MX4 with critical features of QT built-in, e.g. Planck's constant, and a non-trivial differential structure. It has been said many times before by many people that most of the features of QT are due to the geometry of space itself, and NOT specifically about the field formulation of QT. It is also well-established that to get the physics that we already know, the compactified manifold X must have 7 dimensions - for a closed manifold that means S7, as in the model of the essay. The given model recovers the Electroweak model EXACTLY - ALL couplings, ALL particles, ALL bosons masses, i.e. including the Higgs boson, Weinberg angle, Electroweak Vacuum and the quadratic and quartic terms of the Higgs potential.

Quantum Theory itself is JUST about the conversion of a natural-number valued term for the number of particles, to a real-number valued term that includes the undecidable wave-property, and the conversion back again - THAT'S IT! All other results produced by QT are due to geometry - this is why the same experimental results CAN be obtained without ever using this QT representation fiddle. If the derivation of results that agree with experiment no longer matters, then theoretical physics would amount to nothing more than a pointless self-serving exercise - at tax-payers expense!

The 40 years of failure to take physics beyond the Standard Model is due to attempting to make QT fundamental when it isn't. The false assumption that QT is fundamental IS the answer to the FQXi question posed (I'm not the only one saying it here!) - the structure of reality is not a popularity contest, the answer is the answer because it's right, not because it's popular.

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Hi Joy,

After my mentioning the rotation group map S3 to S2 for my spin defects, it occurred to me that there is a second class of experiment that might be worth considering: those involving two coupled gyroscopes, either mounted side-by-side or concentrically.

Unlike in the exploding ball case, the permanent mechanical connections allow for extra mechanical trickery. Your new paper...

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Hi Joy,

After my mentioning the rotation group map S3 to S2 for my spin defects, it occurred to me that there is a second class of experiment that might be worth considering: those involving two coupled gyroscopes, either mounted side-by-side or concentrically.

Unlike in the exploding ball case, the permanent mechanical connections allow for extra mechanical trickery. Your new paper considers the metric of the configuration space - not including the physical space about the configuration so our differing views on this are perhaps secondary - where the issue is whether opposite points on the S3 manifold of the configuration space can be distinguished - SU(2) - or not - SO(3). This difference seems to me, to ultimately feed through to the value of the hidden variable: L=±1 for the distinguishable case of SU(2); but L=+1 is identified with L=-1 for the indistinguishable case of SO(3). If the physical ability to distinguish between 2PI and 4PI rotations is just the issue, then can't that just be given by flipping a toggle switch? Fist 2PI rotation the switch is flipped into its set position, second 2PI rotation the switch is reset - so 2PI rotations are physically distinguishable from 4PI rotations. This sort of thing can be physically implemented for mounted gyros, but not the exploding balls.

The concentric gyro case seems to me to be the closest mechanical realisation of the S3 to S2 mapping of my spin defects - although I still can't see how to implement the required mechanical trickery to give the configuration the desired SU(2). Each gyro wouldn't be the normal disk, but the inner ring of a ring-bearing where the outer ring was mounted on gimbals to give the required S2 surface for the gyro rotation vector. With a hollow shell inside the first gyro, a second gyro could then be mounted inside it. Inside the inner gyro is a hollow shell to which the rotation group does not act because there are no mechanical objects there - hence a mapping from S3 rotation group space to physical S2. The difficulty here is that the homotopy group PI_3(S2) sees a clash between the Hopf fibre bundle with PI_3(S2)=Z and the general homotopy group relation PI_N(S^(N-1))=Z_2 for all N>2. This is the same issue between SO(3) and SU(2), where the Hopf fibre bundle configuration has 2PI invariance, whereas the configuration for PI_3(S2)=Z_2 has the desired 4PI invariance. The simple mechanical version has 2PI invariance, but the desired relative 4PI invariance could possibly be produced through suitable mechanical trickery.

If purely mechanical trickery isn't possible, then all the concentric rotating metal rings gives the possibility of using magnetic induction with suitably mounted magnets. How to do this currently eludes me, but it might be possible. Does this gyro version seem to be a potential option to you?

Best,

Michael

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