The Topological Unified Field Theory on the Complex Hopf Fibration

Jarek Duda

@Jenny Lorraine Nielsen , hello, congratulations for the article!
I also pursue models of particles as topological excitations - since 2009, in 2012 had FQxI essay, since 2021 finally quantitatively combining Landau-de Gennes superfluid liquid crystal-like field with Skyrme-like kinetic term to get e.g. electromagnetism for quantized topological charges (updated https://arxiv.org/pdf/2108.07896 , fresh conference talk ).
String hadronization seems the clearest way - they assume in LHC collision there is formed quark string, and it decays into particles ... this quark string nonperturbatively is modeled as topological vortex, so we just need to find the correspondence - between what topological vortex can decay into, and what they observe in LHC collisions.

The big question is distinguishing fundamental from effective e.g. symmetry groups - living in 4D, the basic symmetry is SO(1,3) Lorentz group of rotations and boosts - field rather has to transform with these fundamental symmetries, and most used can be seen this way (below), but you are using much higher symmetries - do you see them as fundamental or effective?

Lorraine Ford

Jenny Lorraine Nielsen
No excuses. I encountered this individual a few years ago, on FQxI. So, I know that what he says IS drivel.

Steve Dufourny

Jenny Lorraine Nielsen Congrats on your paper , it is a beautiful and general work. Even though I approach things with a different logic through the spherical topological geometric algebras I invented for my theory of spherisation and quantum spheres, I appreciate your contribution.

I believe we cannot truly use the name TOE (Theory of Everything), because there are deep limitations in philosophy, physics, and mathematics. To be frank, I think these limitations will remain even if humanity survives and researches in physics and mathematics for another 10,000 years.

Spacetime is of course important, but maybe ,and I insist maybe , we are in a kind of philosophical prison, due to the way General Relativity (GR) has been considered like a primary essence, especially with the Einstein Field Equations (EFE) and correlations with Quantum Field Theory (QFT). GR and QFT indeed work very well with our measurements and observations, but the real issue for me lies in their philosophical origin and if we have deeper logics hidden and parameters.

When Einstein developed his wonderful theories of relativity, many scientists exclaimed, “Wow, he has understood the universe ,even, in some way, God by considering light as the primary essence.” After that, scientists worked with geometric algebras such as Hopf and Lie......, or developed geometrodynamics with points (like Wheeler), or strings and branes from Witten. And so today, most people view the Planck scale as a framework of points or 1D strings, connected by a 1D cosmic field of GR.

But maybe this is a mistake , focusing only on this single path of fractalisation and extrapolation of dimensions. The same applies to information: whether one calls it God or a mathematical accident, most people interpret it through this “light-based” perspective.

For me, the problem is philosophical , it concerns the notion of primary essence, the fundamental objects, and the true origin of the universe.

That being said, I really liked how you attempt to unify gravity with the other forces without relying on a metric, but rather by reducing GR in 4D. Congrats again , it’s a beautiful idea, Regards

Steve Dufourny

Hi, I discussed with David Chester on facebook about your works, he told me this, « I suggested to Jenny to move from S⁷ to S⁹ based on information from my published paper on E8. I also told her a while back that I don't think S⁹ is sufficient to obtain the standard model and gravity. Some creative ideas, but I think there's a lot of incorrect statements, pieces that don't make sense together, claims of proof with unjustified axioms, and derivations of constants with circular logic.
I thought Jenny was talking about torsion more than nonmetricity. She mentions torsion 228 times here, but only nonmetricity once. She has a vague comment on nonmetricity in relation to gravitational redshift, but there's not really any details of what she means. »
He has good ideas, I think ,and so do you. I am trying to develop my theory with a tool I invented: spherical topological geometric algebras, with spheres as fundamental objects for tensors, vectors, and scalars. Using associativity, commutativity, and non-commutativity, I aim to harmonize a general framework for the motions, oscillations .....of these spheres. I try to remain consistent with QFT, EFE, GR, and the SM. It’s not easy to resolve the divergences of quantum gravity, which is why I strongly believe the problem is philosophical and that we need to introduce deeper, hidden parameters. This idea of consciousness also intrigues me , I have read and studied many approaches and papers , but the philosophy of the origin of the universe seems to remain a problem because of deep limitations.

The primary informations also seem essential and the fact to understand where they are and from what, I consider a kind of infinite eternal consciousness, a kind of mind of God like Einstein said, a kind of God of Spinoza , but I respect the thinkers thinking in a mathematical accident. We don t know in fact simply the truth and we cannot affirm our assumptions. Regards

Jenny Lorraine Nielsen

Steve Dufourny
Hey I mention Chester in the acknowledgments. I was working with the infinite diffeological complex Hopf fibration and believe the entire infinite fibration is relevant as I outline in the theory; however S⁹ is sufficient to hold neutrino interactions.

S⁷ holds all of the standard model gauge groups but S⁹ holds SO(10) which makes particle interactions easier. Chester continues to make arguments about how So(10) is not sufficient, claiming that it cannot hold both spacetime and U(1) due to overlapping operators, which is incorrect and a solid counter argument is provided in my appendix.
Chester did not originate any aspect of this theory but his conversations were helpful. He has his own TOE.

Steve Dufourny

with the tool I invented, the spherical topological geometric algebras, I try to unify universally but not easy due to deep philosophical limitations and probably hidden parameters, A GUT or TOE seems so a difficult problem even in extrapolating with the SO groups , spheres, SU and the isometries or others and geometrical algebras. I consider Kaluza Klein works and gauge fields and vectors, scalars, tensors for the spaces and dimensions with extrapolations and reductions with partitions, I consider also the works of Bob Coecke and professor Miranda and their ideas about quantum music and the bloch sphere. Nor easy for the matrices and their fluctuations. The chirality is important also and the ranking of families and properties.
There are a lot of challenges. The Harmonic analysis is relevant too on spheres and couplings. A big puzzle,regards

Steve Dufourny

David Chester what you tell is important if we want to be logically consistent.

The groups, subgroup must be concrete with the generators if I can say. It is mainly about the consistences of isomorphisms. the spacetime and generaltors considering Lorentz need to be consistent with the combinations for the spacetime and internal systems.
Nontriviality and subgroups........
I ask me what is the analysis if we consider the Coleman mandula theorem for the internal and spacetime symmetries for the matrices , I d say that Poincare invariance , nontriviality ,mass gap and others need to be consistent for the internal generators and spacetime generators considering Lie algebras and lorentz spacetime for example.The scales, quant and cosmol must absolutelly converge. The isomorphisms, automorphisms or topologies must converge. It is like if we told that we must demonstrate that the generators conserve the laws, charges......

We cannot simply assert an overlap of spacetime and internal Lie generators without either violating Coleman–Mandula or showing why its assumptions do not hold. Or we mnust prove these violations simply.
The transformations , rotations must respect the conservations of charges for all states probably. So can we operate and change or remorve the generators, I don t believe personally. Cartan, Jacobi, ,,,,d agree it seems

Jenny Lorraine Nielsen

Steve Dufourny
I don’t assume CM violation. I show that the assumptions of CM are necessarily violated by any Lie algebra that generates Aharonov Bohm and relativistic EM

Jenny Lorraine Nielsen

Steve Dufourny I do not assert anything as a heuristic in the paper. I write detailed mathematical proofs of any conjecture. These proofs may be argued with, but I don’t “merely assert” anything

marcovici alexandru

R vs. R2.
Yes, the real line R and the plane R2 both have the same cardinality (they’re both uncountably infinite). That’s why you can “zipper” the decimals of two real numbers into one — Cantor showed this over 100 years ago. But this is a set-theoretic fact about cardinalities. Geometry/topology adds more structure: in R2 you have two independent coordinates, continuity, smoothness, and so on. That’s what distinguishes “line-like” from “surface-like.”

the guess is that one can change between some axioms continuously
how do you add remove properties might not be so stepwise definite
the end result might be nice and operable , (the writher might have cancer and be terminally ill )
the idea / ontoloogy ('m not sure if use this word right ) is to try not separate to much the paper from the writer of the paper

a yes, i remembered something

so lets say you write the law of universal attraction

F= ma
what would ... when you write this thing down does this equation contain your contribution to the law *in a verry small amount

what / how would be your existance contained in the equation , you cannot separate / substract yourself from the universe and write something that does not take also yourself in to account .

the movement of the hand is in itself an acceleration and change of gravitational fields,
you are basically changing the spacetime while trying to argue about it.

Jenny Lorraine Nielsen

marcovici alexandru can you please ask a clear question? F=ma is derivable via the canonical equations of my theory although I focused on deriving QFT, QM and GR from the theory in the paper.

marcovici alexandru

marcovici alexandru
i read in the last years an article about cancer being a vestigial / atavistic ( somewhat caused by genes) disease , also the evolution of physical laws [...]

sorry i meant by that of
newton's law of universal attraction

F=aM
F_{x}=(\frac{GM_{}} {r^2})M

F_{xy}=G\frac{M_xM_y}{r^2}

if that little a (normally acceleration) is a term containing more details of how the movement is started or actuated than it might be right

so no questions being adresed -
just an observation
i'm a generalisation enthusiast (and educational system tester)
with hearing problems
i have not yet started my work in research studies officially at university
so i have not yet gotten chance to look in detail at your 123 pages online instant downloadable paper

more comments.

imagine going to a library asking the person for the latest research in [ you name your field] topological quantum gravity field theories .

the librarian is going saying to you wait a little to look it up ( old style files system archive or on his mainframe computer)
than after founding some titles he or she is saying -

yes i have such and such articles , witch one do you like to read?, after your answer the follow up question- in the library or at home ?

than taking the article from the library office counter (because that's how you have answered )
opening the backpack and aranging room for the article and than placing the article and than closing the backpack , than going home and reading it carefully maybe with some friends help for about a month than returning it to the library with your notes and comments . to be sent o to the (or reviewable by the) authors of the article

the same with climate ~~diabetes~~ change

ok here is a question :

F_{xyz}?=G_Δ\frac{M_xM_yM_x}{ \frac {r_{xy}^2r_{yz}^2r_{zx}^2}{1+1+1} }

the mean of distances with some other constant that can be deduced from (related with ) the normal constant

or some generalization of the two body expression function with individual pairwise forces
F{xyz} ?=Ψ_{x,y,z}(F_{xy},F_{yz},F_{zx})

which one is a better expression that could be used for an actual calculation of motion of large scale objects like stars and galaxies

dark matter and dark energy are very frequent recalled problems that are being tried to be solved.
modify this thing to obtain that thing

the things that can be said are of a different order of magnitude compared with the things that are actual and right
filter of reality / experiment and evidence based science is constructing little by little from the body of knowledge present , mathematics goes by the hope- of let's make structure maybe latter will connect with reality

a new line of thought might be possible lets make mathematical structures with reality ,
are there some resonances that reflects what is reality ?

Steve Dufourny

Jenny Lorraine Nielsen Thanks for explaining. I really enjoyed your paper. I think both you and David Chester are very interesting general and innovative, broad-minded thinkers who are exploring innovative creative ways to solve unknowns. Congratulations.

I don’t know if you’re familiar with it, but I read Bob Coecke’s book Quantum in Pictures, along with some of his papers. The “quantum guitar” he invented also intrigues me,I play guitar and piano, I found it especially fascinating. He is very relevant, and I think you would enjoy both his book and the creativity of his ideas. Maybe you’ve already read it or come across his work.

Best regards,

Jenny Lorraine Nielsen

Zachory I derive all constants from my theory. I compare to experiment only to check it I match the empirical values.

For a while, I had a derivation of the VEV (which is foundational) derived separately from other constants in the paper, which led to some confusion. This has always been derived from the theory itself.

Jenny Lorraine Nielsen

David Chester In the gauge section and the appendix, I successfully show that any Lie algebra which successfully generates relativistic EM must share a generator with the spacetime generators. I do not violate CM, I just make distinct assumptions from CM. CM assumes that there can be no overlap and explains the result of that assumptioon. The assumptions of CM do not match nature and as axioms were unjustified by CM itself. I have well derived theorems based on known facts about nature which show a lack of a shared operator is unacceptable.

It’s possible to write a working theory without an overlap but such a theory is necessarily not unified by a single algebra and thus is forever a set of patchwork pieces without a unifying equation or structure

Steve Dufourny

Jenny Lorraine Nielsen Hello, Like I am curious, what are your thoughts about the singularities. Personally, I do not believe in singularities where the laws of physics supposedly break down in regions of our spacetime. In my view, these singularities arise from the mathematics used, which produces undefined results or infinities. It is, in my humble opinion, a consequence of the mathematical framework employed in Einstein’s equations.

Many people are fascinated by these mathematical constructions because the breakdown of the laws of physics seems to provide limits that invite speculation and extrapolation , a philosophical choice, in a sense. If spacetime curves in response to mass and energy, then when extreme conditions and mathematical extrapolations are applied, the equations predict singularities ,but all of this remains hypothetical.
Thus, we encounter the so-called Big Bang singularity and gravitational black-hole singularities. However, this depends on how we use geometric algebra, how we define volumes and scalars, and how we interpret event horizons. I believe these singularities result not from the true breakdown of physical laws, but from the limitations of the models and mathematics we employ.

In my opinion, the laws of physics themselves do not break down, rather, we need to introduce additional parameters to complete our understanding within the Standard Model (SM), Quantum Field Theory (QFT), and General Relativity (GR). The gravitational collapse, to me, is not a path toward a singularity but a bridge to new parameters. I believe there must be a finite structure beyond that bridge, which is why I consider dark energy (DE) and dark matter (DM) important in my theoretical view.

In this sense, I do not believe that singularities, as they are commonly interpreted, truly exist , not because the laws of physics are broken, but because they point toward the unknown, toward a bridge leading us to deeper physical understanding. You see it is how we intepret the philosophical choices and the generality of our laws.

Best Regards

Steve

Jenny Lorraine Nielsen

Really compelling formulation — I appreciate how you’ve structured the internal symmetry algebra cleanly and respected the conventional assumption of separability between gauge and spacetime sectors.

That said, one breadcrumb I’ve been following suggests something intriguing: when attempting to generate relativistic electromagnetism from a Lie algebra alone, it seems that a shared generator between the gauge and spacetime symmetries becomes necessary. In other words, overlap might not just be permissible — it might be essential for true unification.

CM tends to assume non-overlap as a default axiom, but that assumption may itself be structurally limiting. In my own work, I found that excluding shared operators prevents the emergence of a unified algebraic structure — any such theory ends up modular, not unified.

If you’re open to exploring this, it might be worth asking:
• What physical or mathematical cost is incurred by not allowing shared generators?
• Could a minimal overlap yield deeper unification without violating known constraints?
• What happens to your system’s algebra if a shared generator is introduced deliberately?

Curious to hear your take if you’ve thought about this from a bundle or commutator perspective. Could be an interesting point of convergence!

Ulla Mattfolk

I have began to wonder if spacetime is higher dimensional, maybe even coming from dark energy? <<<is SO(10) enough then?
see Kerr BH with an 'empty' zone around as ex.
you write: Cosmology is explained topologically: dark energy emerges from torsion in the S 1 fiber (Theorem 13), and dark matter arises from global holonomy of S 9 cycles (Theorem 14).

what gives the torsion? describe the S1 fiber more?

Steve Dufourny

Ulla Mattfolk I am intrigued with the paper of Jenny and it is a very good one, her non metricity is relevant , I ask me if the gravitational field is encoded in non-metricity rather than curvature/torsion, so there is probably a way to solve the divergences .The aspects of gravity must give a gravitational dynamics and converge for the scales, cosmol and quant, ......maybe the UV behaviors also can be analysed with partitions and mathematical tools to have a kind of UV finite theory. Higher derivatives for the unitarity and ghosts are the key maybe, I try but not easy with the spherical topological geometric algebras I invented. the aim is to have a kind of finiteness of UV in the 4D quantum gravity. The gauge symmetries , divergences, unitarity,anomalies analysis,this finiteness of UV, causality and locality........, regards

Steve Dufourny

Ulla Mattfolk In a sense we need a proof of renormalizability in 4D gravity with concrete higher derivative terms and it is there that the ideas of David Chester and Jenny Lorraine Nielsen are excellent because the finite sets can be found. I tell me that the einstein gravity propagator is the key if we change it maybe to renormalise correctly ,
The massless spin 2 graviton is so the key if we consider these gravitons quantas of gravitational waves and I have difficulties to reach the solution due to the ghost problem in 4D. That is I replace the pointparticles with a structure , the spheres but there are series and problems due to fact that we have not a finite size simply.

Steve Dufourny

Ulla Mattfolk I believe that if we consider the spacetime with a structure and different than this GR , so in considering the DE it can be relevant because we add structures , and series of spheres at the quantum scale. The informations also seem relevant philosophcially in considering the antigravitation and maybe a kind of fifth force, I am not sure of course and so we relativate the primary essence of our relatity that we observe and measure relativitically speaking.We don t change the logic of our GR and QFT, but we add parameters to complete them......

Steve Dufourny

Ulla Mattfolk I have taken time to read the paper of Jenny Lorraine Nielsen.It is a very good general work trying to unify all unknowns. The Theorem 4: Compact topology discretizes the spectrum of field modes, ensuring regularization of ultraviolet behavior. As a corollary, this compactness implies UV finiteness. In her theory Topological Unified Field Theory (TUFT) based on the principal fiber bundle S1 →S9 →CP4,
The graviton is a topologically stable excitation in the reasoning with a spin and tensor structure spin 2 boson massless obeying the Bose-Einstein statistics and possess two helicity states, ±2. I like the idea on the topological and diffeological bundle S1 → S9 → CP4, and if this graviton couples universally to energy momentum tensor, so it is relevant about the interference modes on the S1 fiber , that gives a natural topological mechanism
for gravitational interaction like she explains. And that is why I am intrigued by the logic of non metricity and UV finiteness

Steve Dufourny

Steve Dufourny we arrive also at the hard problem of consciousness and its philosophical origin because if the infinite eternal consciousness is a reality and that we are in the mind of a kind of god, so the key is to find how the main primary informations act and how and the fields of this consciousness because it is maybe the primary essence of all, like if all was conscious at its level and after we have specific mechanisms in dimensions. I doubt that the spacetime of the GR is the primary essence , so we need to add hidden parameters for the antigravitation , the quantum gravitation and consciousness. It is there that this spacetime completed with DE is relevant with deeper structures, and after it is a question to find the correct higher derivatives, groups, subgroups to solve these unknowns.

Ulla Mattfolk

Steve Dufourny Lawrence Crowell once thought around the interior of a BH could form DE, also the redshift of spacetime and the gravitational waves can be understood as expansion of spacetime alone, and this has no impact or very small impact on matter due to their low energy?

In teleparallell gravity (L. Heisenberg) we can also have a situation in flat space and still torsion. Is this torsion then influence from darkmatter as cosmological constant? You say torsion for DE, but also DE without DE, and this cannot be the case here?

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