A Historical Approach to Research in Fundamental Physics by Emile Grgin

[deleted]

Emile,

Thanks for clarifying my position, I meant it like this: "Why then in this case two real numbers cannot achieve what a complex number can".

I struggled also a bit in the beginning with the very point Ray and Cristi raised, and with another issue I want to clarify: are quantions new physics? Superficially it looks that they are not, but this is deceiving. Let me explain.

Quantions are not just like useless rearranging the chairs on the deck of Titanic. Quantions do not have their motivation in clever reinterpretation of Pauli matrices, but in a systematic approach for discovering non-unitary realizations of QM. This led to quantions which turned out to be intimately linked with spinnors. It would have been very unfortunate if there was no link with Pauli matrices and Dirac equation. (If Dirac equation were not discovered by now, quantions would have led uniquely to it.) But do they lead to additional physics? The hope is that they will. The standard nonrelativistic QM is unsatisfactory from several points of view and let's present 2 of them: (1) space and time are outside concepts, (2) the measurement problem or the emergence of classical physics from QM.

The first issue is naturally solved by quantions, and there is hope for the second one. Relativistic QM should be the correct framework of nature. Instead of the standard Hilbert space, one deals with the non-commutative geometry framework, and this holds big promises as quantions make intuitive the non-commutative geometry approach.

Florin

Cristinel Stoica

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 Grgin

Florian, your clarifications are very good. Thank you for helping out. Concerning your point number (2), I doubt very much that quantions will help with the measurement problem, but then, we are both guessing.

I like your meraphor about the rearrangement of chairs because I've heard too many times that quantions are just that.

Emile

[deleted]

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

Emile Grgin

Dear Cristi,

In your post of Oct. 4, 13:14 GMT, you are not asking a question but making a statement. You say that it it is owing to familiarity that I believe in quantions more than others do, not because they are more powerful.

OK, I'll give you the opportunity to prove that quantions do not add anything to our present understanding of physics.

Let's hit the reset button and start from scratch.

My first move consists of the following two statements:

Statement # 1.

A new number system, which is an extension of the field of complex numbers and is located midway between the complex numbers and the quaternions exists. I call it the algebra of quantions. Mathematician did not discover it because it is not a division algebra and does not touch anything in the mainstream of mathematical research. Physicists did not discover it even though the purely algebraic part of this number system is weel-known. It is the algebra of complex two by two matrices.

Statement #2.

If we consider a quantionic field on some initially unspecified manifold, then: (1) The algebra of quantions tells us that the local metric of the manifold must be Minkowskian. (2) This algebra gives rise to a vector field of future-oriented vectors. By Zovko's interpretation, this vector field is to be viewed as a current -- which means that it must satisfy the equation of continuty (= vanishing divergence). Since this equation is not identically satisfied, it yields a differential condition (a field equation) that the field must satisfy. Given that the complex numbers are a substructure of the quaternions, we may restrict the field equation so obtained to the infinitesimal neighborhood of complex numbers. Doing so yields the Schroedinger equation with an arbitrary potential and an arbitrary mass parameter. In the general case, a non-singular transformation of field variables transforms the quantionic field equation into Dirac's equation with an arbitrary mass parameter and four different vector potentials. One of these is readily identified as the electromagnetic potential.

THAT'S ALL!

(a) Anyone who says that this ain't so has to find an error in my math. Well, good luck!

(b) 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! No borrowing anything from phenomenology: not the concept of time, not space, not a wave function, not electomagnetism, nothing, period. I didn't borrow anything.

(c) Anyone who says that this is not factually new because I did not get any results that are not already known is factually correct. But that person is speaking from a point of view which is not that of research in fundamental physics. The aim of such research is to actually discover new paradigms and show that they hold water -- not merely to vent opinions based on intuition alone.

Speaking of opinions, mine is that your objections belong to category (c). Please correct me if I misunderstood.

Best regards,

Emile.

Emile Grgin

Florin,

If you have a hunch, that something else. I hope it works out.

I don't understand what's happening: I think I am doing things right, and then my posts end up under "anonymous", like the one above to Cristi. Let's see what will happen to this one.

Emile

Cristinel Stoica

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

P.S. The sign you are not logged in, when you right the message, is that there is above a box labeled "Your name:". When you are logged in, this box disappears.

Emile Grgin

Dear Cristi,

I think I have to admit defeat and accept the unpleasant fact that I am unable to make myself understood by some (probably very many) physicists, including you. After having tried hard to explain that my work in foundations ended up in extending the number system of quantum mechanics (just as QM extended the number system of classical mechanics), you tell me that you like quantions but (I quote) "I just wanted to be fair about other structures and results too." Since fairness is an argument I've never seen before (I've seen it in labor arbitrarions, but not yet in physics or mathematics), I don't understand some of your reasoning. This also applies to the first sentence of your essay: "I propose a gentle (requiring minimal changes to both theories) reconciliation of Quantum Theory and General Relativity." Since the theories in questions are the most rigid in all of physics, I suspect I will never understand how they could be minimally modified. Reading the essay did not help.

My suggestion: Let's broadmindedly accept the not very terrible fact that the probability of our understanding each other's objectives in physics in the very near future is not high enough to justify the effort. This does not prevent me from wishing you success in your approach to the problem we are both working on.

Best regards,

Emile.

[deleted]

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.

[deleted]

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Terry Padden

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.

Emile Grgin

Terry,

I will definitely read your essay, but I can comment on your post immediately.

I have the impression you understood my essay pretty much the way I like it to be understood. Thus, there being no 'corrections' to be made, let me answer your question

"Are you asserting that Qantions are an ultimate mathematical structure ?"

No, I am not. In general, I am never asserting anything beyond what I can prove. What's more, I am not even interested in unprovable assertions. For example, You say:

"For me, science including maths and physics will always be open ended."

To comment on this statement, I have to break it up into three:

For science I agree with you because emergent phenomena give rise to new science, and I can't see how the superstructure of such phenomena could ever hit a ceiling.

For mathematics I also agree with you because Goedel proved it.

For fundamental physics I neither agree nor disagree with you. I plead total ignorance. And I have no need to fill this ignorance with beliefs.

Next quote:

"Progress will always be possible - but only if the foundations are sound. I am sure in mathematics they are not. They are incomplete."

I fully agree with this statement taken in isolation, but to the extent that it refers the preceeding one about physics, I have no opinion. I take it that the incompleteness you mean is Goedel's. Now, Is Goedel's theorem applicable in fundamental physics? Maybe it is, but I don't see by what mechanism. A necessary condition for that is the presence of a set of relevant objects whose cardinality is aleph zero. Yet, there are only 10^180 Planck cells in the entire visible universe. This is humongous on the human scale, but not even worth mentioning in comparison with infinity. Even if every cell could communicate directly with every other cell, the whole thing would have the complexity of tic-tac-toe (from the standpoint of aleph zero, of course). So, the question is: Does the mathematical set of natural numbers have a concrete realization in ultimate physics? My final answer is I DON'T KNOW, but if I were forced by some nasty deity to bet my life on it, I would probably hit the NO button (and that might well be the end of me, but I would never know it).

You might have gotten the impression that I view quantions as the ultimate mathematical structure in physics because I said something to the effect that quantions seem to be the last number system (relevant to physics). They indeed seem to be that because they leave no wiggle room for deformation, and offer no opening into which one could add some structure --- which is not the case with complex numbers (otherwise they would not generalize to quantions). Now, "ultimate number system" is not the same as "ultimate structure". And for the latter, I don't have anything to say.

Regards,

Emile.

[deleted]

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

Emile Grgin

Steve,

You ask:

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

The answer is absolutely not!

At least I don't think so.

If I am not mistaken.

I hope this will help, more or less.

Sincerely, Emile.

[deleted]

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

[deleted]

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

Emile Grgin

Dear Steven,

You are reading much too much into my essay. I am definitely not suggesting that "the Ultimate Possible in Physics is pure mathematics."

I am not suggesting anything about Ultimate Anything because I don't know the future. If I did, I would not be suggesting it but telling it for a fee.

While I don't know the future, I know enough of the past to extract trends from the history of physics and mathematics. Since these trends happen to admit extrapolation into the near future, they suggest an approach that is likely to lead us to the next paradigm in fundamental physics. And this is plenty for me. The ultimate paradigm is none of my business.

Another point: Your linking Florin's objectives and mine is unfair to both him and me. I noticed that Florin has a very wide range of interests in physics and the knowledge to go with it. One of these is my work, which pleases me very much. My interests, on the other hand, are limited to a single question: What is the NEXT paradigm. Florin will speak for himself if you ask him.

I am sorry to disappoint you by not having any reflections concerning your question. But, more to the point, why should that question be addressed to me? It has absolutely nothing to do with anything I ever wrote (or even thought).

Good luck with the contest!

Emile.

[deleted]

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.

Terry Padden

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.

Emile Grgin

Hi Peter,

(Note: I was forced by the server to split this post in two. Please view them as one.)

Your first sentence -- to the effect that the mathematical presentation of quantions in my essay is difficult to follow -- is an understatement. I think it is impossible to follow for anyone not already familiar with it (Florian Moldoveanu being probably the only one who is on this forum). It is much too condensed to serve as a primer. Since you may be wondering why I even put it there, the following explanation is meant to anticipate your question.

It is by seeking trends in the sequence of major paradigms in the histories of physics and mathematics (though I wrote only about physics) that I arrived at the approach to research which led to quantions. Having outined this trend analysis, I had to emphasize that my essay is not meant to tell others what to do while not doing it myself (there are more than enough of those essays), but to show that the approach in question does yield physically meaninful results. A book being needed to do this cogently, a few pages cannot do it justice. Yet, without these few pages, the essay would not even belong to the contest.

You say that you don't see clear relationships to conventional mathematics being made in conventional ways.

Very true. This is because I did not work along conventional lines (I did not take it for granted that the structures mathematicians have investigated so far are the only ones relevant to physics at its foundations). Thousands of physicists and mathematicians actively engaged in research have been very familiar with the conventional math you mention for as long as quantum mechanics exists. So was I, in my graduate student days (and you may add twistors to that). When I realized that the conventional approach was over-crowded, I decided not to delude myself into believing that I could make a dent where others didn't in the only subject of interest to me: A true unification of quantum mechanics and relativity. I thus had to strike out on my own (though an initial part of the journey was made in collaboration with Aage Petersen, whose memory I cherish). Not surprisingly, the outcome (i.e., the number system that brings about the unification in question) is NOT a conventional mathematical structure. It looks like four different structures when viewed from four different 'angles', and each one has a different physical interpretation -- and this is what supporst the unification of two very different theories. I wish I could write a short tutorial on the subject, but it is impossible. Maybe later when some more dust settles. There is just too much deductive work that relates the conceptual view of mechanics developed in first abstract algebraic paper on the subject (2001) and the second book, whose last chapter is about the complementary roles played by Dirac's spinors and quantions (2007).

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