Is Quantum Theory As Fundamental As It Seems? by Michael James Goodband

Hi Joy,

The comment by Jens with the references to Geoffrey Dixon and Cohl Furey, and my reply above, really crystallizes the point I made in my essay and that I was trying to make earlier to you about the colour group. Furey's algebraic space R*C*H*O reproduces the spins (H) and charge eigenvalues (O) for the particle symmetry group SU(3)*SU(2)*U(1) as would apply to a continuous field (R) with a cyclical (C) component, i.e. a wave (I mention this pattern in my book - it's in the Bodleian). This is what I would expect for QT being fundamental, but it isn't.

The equivalent physical manifold realisation S0*S1*S3*S7 arises in 11D GR with no added fields - Einstein's vision with the number of dimensions specified :) - and QFT is derived through dimensional compactification and a representational change. BUT the colour space is the S3 fibre of S7, which means that it cannot be SU(3) but locally SO(3). These 2 features seem to be linked: continuous QT matter fields and SU(3); or discrete particles and SO(3). Take out QT as being fundamental and the space-based approach seems destined to dispute the colour group - that would seem to include your framework.

Michael

Dear Anton,

The notion of a self-creating particle that you allude to could be one way of viewing the problematic dynamics that I identify in my essay for a "bare" particle being a topological defect in space. In my pure GR theory, such topological defects have the appearance of Planck scale black holes bearing charges, but their spinning creates an ergo-region that can contain virtual radiation. Despite virtual matter appearing in QT, it is actually a relativistic concept m^2

Dear Hector,

The difference between mathematical models and the way nature really works, is precisely the point I'm making. Maths models of certain forms have mathematical restrictions that nature doesn't - maths can be incomplete but reality isn't - and we our free to fix our models by changing their maths form.

The Gödel connection is not as adventurous as it may at first appear, but does depend on a *very* careful meta-science analysis (review paper) of what it means to construct a scientific theory such that it accurately models the physical world. This is related to Einstein's point that he makes in the EPR scenario, but involves being *far* more careful about the specification of the 1-to-1 correspondence between features of reality and a mathematical theory - this is captured in my usage of physically-real term.

Precisely because of the points you make, the domain of applicability of Gödel's original incompleteness proof to science theories is restricted to causal dynamic systems that implement arithmetic changes over countable numbers of objects of different types. As long as the different object types are physically identifiable as being different types, then they can be classified into different sets in a mathematical theory, where the cardinality of the sets gives the countable number of objects present in the physical world (an accurate 1-to-1 correspondence). Note that ZF set theory is not the appropriate set theory for science theories as in reality objects occur as different types, and ZF doesn't support urelements or types.

The modelling of causal changes in object type A->B necessarily gives a form of logical implication in a 1-to-1 model of reality, and by carefully tracking the mathematical modelling of causal changes to the numbers of objects in reality, the conditions for when Gödel's theorem applies *within* a science theory itself (the critical meta-science bit is to parallel Gödel's meta-mathematics exactly) can be itemised. These conditions for the original form of Gödel's proof are very restrictive but can exist for real physical systems - I show that this condition can be used to divide physics into Object Physics (where it doesn't apply) and Agent Physics (where it does).

The axiomisation required to apply Godel's incompleteness to a scientific theory is limited to the core features required to denote different object types in sets, such that arithmetic over the numbers of elements in the sets is supported - this just comes from the axiomisation of set theory and arithmetic. Application of the proof to real scientific theories with further mathematical features - that are not necessarily axiomised - is then explicitly dependent upon the corollary to Godel's theorem: as long as the additional axioms don't change the integer arithmetic captured in the core set of axioms about object numbers, then Godel's incompleteness theorem will still apply. Meta-science analysis of a physical system can identify whether the arithmetic axioms would be changed by the extra mathematical features of a scientific theory without necessarily having to axiomise the theory, as all is required is to constrain what they *must not* be like for the proof to still apply, ie. they must not effect the integer arithmetic over object numbers.

Consistency in a physically-real scientific theory with a 1-to-1 correspondence with the numbers of objects of different types is then, as you say, the key issue. For a 1-to-1 denotation of object A in reality, the logical truth value of A (true) in the maths means that object A exists in reality, and conversely not-A (false) means that object A does not exist in reality. In this context of a 1-to-1 physically-real scientific theory of arithmetic changes in object numbers, an inconsistency in the theory would necessarily imply that a statement of the form, A and not-A, could be derived. This statement has the meaning that object A *both* exists and doesn't exist at the same time.

Now our observation of reality has been very time limited so far, and so an object existing and not-existing at the same time might arise, but it hasn't been observed so far. As this sort of inconsistency would imply that real magic was physically possible, the induction from our time limited observations of reality to a general statement of truth about reality seems safe - all science implicitly makes this assumption, otherwise the pursuit of science would be somewhat pointless. In a 1-to-1 physically-real scientific theory this gives the required form of consistency for Godel's incompleteness theorem to apply and for the theory to be *known* to be incomplete over arithmetic changes in object numbers.

In the 11D pure geometric theory considered for physics unification (STUFT), particles arise as topological objects that either exist or not, are of 12 different types (corresponding to the fundamental particles), and are countable. So when the full set of conditions for arithmetic changes in object number required by Godel occur, the theory is provably mathematically incomplete. But by changing the maths terms used in the theory, and including the observation of an undecidable wave property for particles, this restriction can be bypassed to give a scientifically complete theory. The notable feature is that this change - integer valued terms to real valued terms - gives a (meta-science) *derivation* of quantum field theory. As applications of Godel's incompleteness theorem go, it is hard to imagine a more significant and dramatic example.

Regards,

Michael

Hi Jonathan,

On comparison with Dixon and Furey, they are attempting to match up the symmetry groups SU(3), SU(2), U(1) with the algebras O, H, C, which they do but fail to find the 3 families of particles. I contend that this is because the colour group isn't really SU(3) but Spin(3). I followed field theory practise of focusing on the local structure and gave SO(3), which in retrospect wasn't helpful because simple representation arguments rule out SO(3) as the colour group BUT they do not apply to its double cover Spin(3) [the group space of SO(3) is S3 with opposite points identified, which won't give the colour and anti-colour in a meson, but the group space of Spin(3) is the full S3, and so includes the required opposite points]. With colour group Spin(3), the group spaces are S3, S3, S1 and fit into S7. In conceptual terms, remove the unbroken U(1) of electromagnetism from S7 to give S6 and map to a spatial sphere S2 - the homotopy group is PI6(S2) = Z12 = Z3*Z4 and so gives a 3 by 4 table of topological monopoles. The non-associativity of the octonions is conceptually the reason why there are 3 families of 4 particles each: trying to match a colour group of SU(3) with the octonions won't give the correct spectrum of 12 particles in this way. Geometry triumphs over Algebra?

I have also found the point of view for comparison with Joy's work: view the hidden domain as being of finite size with an enclosing S2 surface. The map of the rotation group space S3 to this S2 has 2 orientations for homotopy group PI3(S2)=Z2 [note this is stamped on by the fibre bundle mapping with homotopy group PI3(S2)=Z but the Z2 one fits a general SN pattern and is still there]. The map of a compactified S7 space associated with particle symmetries to this S2 also has 2 orientations, as homotopy group is PI7(S2)=Z2. When the hidden domain surface S2 only encloses empty space, the symmetry operations of rotation (S3) or particle symmetries (S7) are free to act everywhere to rotate +1 orientation into -1 orientation as they are reachable through S3 or S7. BUT when the hidden domain encloses holes in space (as in STUFT) or singularities where the symmetry operators don't apply, this is not possible and the hidden domain S2 surface will have an orientation.

As STUFT is about a physical space with a metric, Joy's Clifford algebra approach where the hidden variable IS the orientation, effectively constitutes the correct framework for analysing singlet states of topological monopoles in STUFT. Specifically, the topological orientation feature that seems to have raised objections IS precisely the feature generated by the topological monopoles of STUFT. The hidden feature is whether the particles are AB or BA, but AB = -BA as both the quaternions and octonions are non-commutative and this is effectively manifest on the S2 surface of the hidden domain as an orientation - the hidden variable.

I am in the process of preparing a short article outlining the topological features of STUFT and this comparison with Joy's work - it is taking longer than expected.

Michael

Hi Joy,

I have yet to accomplish the final critical step in the comparison as I am currently distracted by what looks as though it will prove to be a fruitless essay contest, despite the essay presenting the correct false assumption that is preventing progress towards physics unification (also outlined in the essay).

A hidden domain containing either holes or singularities and being bounded by S2, to which the S3 spin space or S7 particle symmetry space is mapped is the basis for comparison as it is the basis of the topological monopoles. The global mapping of S7 to the spatial S3 breaks the particle symmetry space S7 into its subspaces S3, S3, S1, but also breaks the isospin symmetry with the second S3 space - so that can be crossed out. This leaves intact the particle physics symmetries of spin (S3), colour (also S3 for colour group Spin(3)) and electromagnetism (S1) which are the subspaces of S7. Interpreting your results in this context would mean that for any number of correlated particles within the S2 of a hidden domain, these 3 spaces fit together into S7. For the simple topological monopole/anti-monopole (S0) case with charges arising from S7 and spin (S3), this doesn't look likely ...

BUT this doesn't take into account the entire point of my essay: the discrete monopole theory is mathematically incomplete, and for them being particles (S0) the wave property (S1) is the undecidable feature and gives wave-particle duality - the Hopf fibre bundle S1. Particles would then simultaneously be in representations of: S0 - particle; S1 - wave; S3 - spin; S7 - charge, which in itself gives a significant uniqueness condition (as stated in the book). The combining of spaces S3 (spin), S3 (colour), S1 (electromagnetism) in S7 for any number of correlated wave-particles (fibre bundle S1) within a S2 hidden domain then has this extra significant factor.

The non-associativity of the octonions would seem significant in resolving this general embedding issue, which appears to be the crux of establishing equivalence, but is currently the open problem ...

Michael

Rick,

The statement AB = -BA was made in the context of a singlet state of 2 particles (for which it is true in both H and O), and not for general A and B. The A and B were a reference to the observers Alice and Bob when they measured the spins or particle charge vectors - a mixture of terminology, sorry for the confusion.

The topological mapping and geometry arguments outlined above give the limit of what can be reached in such conceptual terms. The issue of whether the wave(S1)-particle(S0) fibre-bundle for the particles inside the S2 hidden domain provides the necessary element that can combine the unbroken symmetry spaces spin S3, colour S3 and electromagnetic S1 into S7 requires algebraic proof. Having reached what feels like a delineation between geometry and algebra, apparently there is one.

The spin space S3 has a spatial origin, whereas colour space S3 and electromagnetism S1 have an origin in the particle symmetry space S7 in STUFT - so the spaces are in a physical sense "alien" to each other. It seems to be the very features you describe which make it even conceivable for these "alien" spaces (spin and the particle symmetry spaces) to cohabit the same S7 and so meet the general correlation condition of Joy's work.

Tom

The particle physics is inside the hidden domain, but for those particles being topological monopoles, that topology is projected onto the S2 surface enclosing the hidden domain. Sort of doing particle physics, but without the particles. Although the process seems to also require turning the wave(S1)-particle(S0) duality similarly into a topological condition projected onto the S2 domain surface. Any ideas?

Michael