A Historical Approach to Research in Fundamental Physics by Emile Grgin

Dear Florin, dear Dr. Grgin,

I agree with the affirmation that it is important the structure, not the underlying vector space. For example, in Hestenes' version of Dirac's equation, the usual Dirac spinors (elements of a complex vector space) are replaced by even elements of the Dirac algebra (identifiable with elements of the Pauli algebra).

Dirac spinors, as vectors, have no richer algebraic structure (the product of two spinors is not a spinor) while spinors regarded as even elements of the Dirac algebra not only form an algebra, they are very similar to the complex numbers they generalize.

Of course I agree that a new structure may provide at least fresh insight and, in some cases, more compact notations and calculations. After reading some of your papers, wanting to understand more, I took the liberty to ask you some questions, to see if the quantions' potentiality corresponds with my own exigencies. I like to do my homework. I hope I did not ask too many or inappropriate questions, please take it as a proof of interest. You both collaborated admirably in trying to answer mine and Ray's questions.

Best regards,

Cristi

Emile,

I have a hunch of how to approach the measurement problem in relativistic (quantionic) QM. From 10,000 feet, the non-unitary evolution during measurement should occur naturally in a non-unitary representation of QM, but the actual mechanism I expect to be subtler.

Decoherence is only half the solution. My hunch is that Gleason's theorem breaks down in general in relativistic QM, and its failure will generate superselection rules which will solve the remaining mystery of the measurement problem. But this means we have to look past the Zovko interpretation in the divergent case.

Florin

Dear Dr. Grgin,

I did not say "stop using quantions, because spinors and Clifford algebras can do the same job, factually". But it seemed to me that you misrepresented the state of art in physics, when compared it with quantions. I tried to show this in some examples (please correct me if I am wrong):

You compare quantions with complex numbers and quaternions, and say that only quantions tell the Minkowski metric. Why not compare quantions with the Dirac algebra, which also tells the Minkowski metric on the underlying vector space, or with the Infeld-van der Waerden spinors, or with complex quaternions and Pauli algebra.

It is true that quantions yeld a special timelike future oriented vector, and from its conservation you derived the Dirac equation. But is equally true that a special future-pointing timelike vector occurs also when we highlight a special Pauli subalgebra of the Dirac algebra. It also occurs if we choose a hermitian (1/2,1/2) spinor (=vector). Infeld and van der Waerden derived Dirac's equation from the conservation of this vector in 1933. Compare your derivation of Dirac's equation with this one, and not with Dirac's, which you say that it is not "conceptually justified".

You say that quantions unify quantum mechanics with relativity (the special one?). I mentioned that Wigner and Bargmann obtained the spinor fields (used to describe particles of various spins in quantum mechanics), as representations of the proper orthochronous Poincare group. These spinor fields obey an equation, which for 1/2 spin is Dirac's, and for 0 spin is Klein-Gordon's. It sounds to me like a unification of special relativity with quantum mechanics. Why compare quantions with nonrelativistic QM? Compare them with the relativistic one, which unifies QM and relativity.

You say: "Anyone who says that this is not conceptually new is welcome prove it by deriving the same results from the field of complex numbers and Born's approximation (Born's approximation is the spacial case of Zovko's). No cheating!"

Again, you compare quantions with complex numbers, avoiding other structures I mentioned, which, as I said, lead to Dirac's equation in an elegant manner.

It was the comparisons you used that concerned me all the time. If you have other reasons to believe in quantions, I am glad, because they are a nice structure and I think they should be researched. I have nothing against them, I just wanted to be fair about other structures and results too. In my opinion, the comparisons you made with the state of art are very selective. That's all.

Best regards,

Cristi

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Dear Emile and Cristi,

I think Cristi does have a valid point. It may not have anything to do with fairness, but great claims deserve great scrutiny and this should include positioning the claim into the state of the art knowledge. Positioning the work in the proper context does not reduce the value of the conceptual breakthrough. Special relativity for example is not diminished by the fact Lorenz already discovered his transformation earlier.

Quantions are (for now) a conceptual breakthrough. Any such endeavor takes time to fully appreciate because: (1) it is always an uphill battle to push a new paradigm, and (2) you have to invest some time to study it and understand it. However, ultimately it will have to produce new results. I am a firm believer quantions would explain new things in the near future. This is not blind fate, it is based on the intuition I am building about them.

There is more to quantions than as an even dimensional subalgebra. If this is all that was to it, it is no wonder why Hestenes did not made additional progress since he discovered the U(1)XSU(2) symmetry in 1966. [This is not meant to diminish Hestenes' results after the electroweak symmetry discovery because he certainly achieved his research goals resulting in the intuitive geometrical algebra approach leading to the recent gauge theory of gravity. I am only pointing out that the link between his "space-time algebra" and Connes' non-commutative geometry was not investigated and this is where I think the true future value of quantions resides.]

The value of quantions does not reside at all in the ability to derive Dirac's equation. Deriving Dirac from quantions is only a sanity/reality check along their development.

Their main strength is in their uniqueness proof as the only mathematical structure which is compatible with quantum mechanics and relativity. To fully understand that you have to first understand what is quantum mechanics. I am not talking about the textbook Hilbert space formulation. QM is a vast domain: Jordan algebra approach, operator algebras, geometric quantization, lattice approach, Hilbert space, GNS construction, Tomitza-Takesaki, Gleason - K-S - Bell theorems, non-commutative geometry. Composability leads to an intuitive coherent axiomatization of QM. On one hand you have complex numbers for the non-relativistic case; on the other you have quantions at the center of the relativistic case.

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Emile

Good essay. I have a number of points I want to discuss here. The first one is this:

Your first part shows by historical (vertical) analysis why for you it is mathematics holding up progress in physics. My essay arrives at the same conclusion by a horizontal analysis emphasisng current issues facing mathematical science. In your second part you propose a resolution to the problem through the new Quantion structures you have developed. Your approach is what I call "progressive" in my essay. You accept the maths we have today as being correct as far as it goes - it just needs extending. Which you have done.

It seems to me the progressive approach is open-ended. New mathematical structures can always be extended - ad infinitum over time as the need arises. So the progressive approach only addresses current problems facing physics; in your case the current need to unify QM and GR - just as historically Maxwell united Electricity and Magnetism. Such unifications - although magnificent achievements in themselves - do not address ultimate questions, which is what we are asked to do. I include issues like "emergence" as being ones we should be able ultimately to address.

Are you asserting that Qantions are an ultimate mathematical structure ?

For me, science including maths and physics will always be open ended. Progress will always be possible - but only if the foundations are sound. I am sure in mathematics they are not. They are incomplete.

Hi all ,

It's super your discussions to all ,with Mr Christi Stoica ,the dream team increases thus the extrapolations dance with the ideas of all ,a beautiful synchronization arrives ....Florin ,Ray ,Mr Grgin,Lawrence ,Mr Stoica ,Mr Crane ,some I forget...the sing of primes ,naturals ,reals ,complexs imaginaries...arrives and goes towards a beautiful physical synchronisation ,the complemenatrity ,focus on fundamenatls, increases many things .

Dear Mr Grgin,

Does the mathematical set of natural numbers have a concrete realization in ultimate physics?

Here is a suggestion .

The multiplication of prime numbers gives naturals ,and complexification.

The physical serie is limited with prime numbers .The infinity is in the imaginaries thus a limit is necessary inside the closed system with 3D and a constant of building .Inside the complexification with naturals can be inserted but the imaginaries aren't considered due to the non physicality of course .

There the Goedel Theorem must be adapted in its axiomatisation of course with rationality and pragmatism .

In one word the unknew is the unknew and our rule ,our pure rule is to understand these 3D more a constant .

Sincerely

Steve

Dear Emile ,

Yes I understand better your works and researchs.I reread your essay too .

that helps .Indeed I see more clear .

Best Regards

Steve

Dear Emile,

Like your fellow contributor Florin you seem to suggest that the Ultimate Possible in Physics is pure mathematics. Does'nt that sound odd? Where are your connections to reality then?

We know for instance that the physical zero dimensional point is a non-existent one, since the smallest point we currently could observe would be the size of a photon. That would mean that all physical limits (as in calculus) should resolve into an whole multiple of photon constants. Since this is currently only partially reflected in physics, I cannot imagine how a more complex math is going to help.

Your reflections in this would be appreciated.

Good luck with the contest!

Steven Oostdijk

Hi Emil,

I find your mathematical presentation of quantions difficult to follow. I've read through the comments here as well, but couldn't find anything that clarified the distinctions between quantions, Clifford algebras, even subalgebras of the Clifford algebra (e.g. as Hestenes), ideals of Clifford algebras (spinors, but I see you explicitly say that quantions are algebraic objects, in contrast), or the various bundles over space-time that can be constructed starting from such algebraic objects. Are there any (anti-)commutation relations that are satisfied by the quantion algebra, or, if not, I don't see some other kind of presentation of the algebra anywhere? Is the quantion algebra freely generated, for example?

I've not read your paper thoroughly, and I've not followed any links, but it's not even clear to me whether a quantion algebra is defined at a point or requires a bundle structure (or something like it) for its definition. You seem to have a fair degree of mathematical sophistication, but I don't see clear relationships to conventional mathematics being made in conventional ways. Then I would feel more able to see how your algebra is unconventional. I worry, for example, that the Clifford algebras are universal for the algebraic anticommutation relation u.v+v.u=2(u,v), which leads naturally to spinors as ideals of Clifford algebras (and slightly less naturally to Hestenes approach in terms of constructing an action of the Clifford algebra on its even-graded subalgebra).

The other aspect of assessing a theory that I find difficult in your case is the extent to which your constructions make contact with quantum field theory, say. It's not clear to me that you satisfy a modern correspondence principle, that your theory must make contact with at least quantum mechanics in a clear way, at least in the sense of approximation.

I'm sorry this is not more helpful, and that it's somewhat late to the party. I couldn't bear to look at the rush of late entries to the competition until now. If I could understand what is different about quantions, it seems as if your paper might be interesting.

Emile

Thanks for a prompt, detailed, and perceptive response. You write as your "Ultimate Postulate"

"It thus appears that what is ultimately possible in observable physics ( ...) coincides with how deep one can go in mathematics and still find a fundamental structure which is both very rich (to support all of physics) and very specific (to guarantee finality)."

Agree

Then you write "In the quantionic approach the fundamental structure is a number system. I cannot think of something deeper that would not be too general (like set theory)."

I don't think the different number systems are important in themselves. What is important is the interpretation of what the progression from one number system to another implies - physically. These number systems are not fundamental. Abstractly they are all multiples of the Reals which derive from the Naturals. I am looking closely at the physics of the Naturals.