Heuristic rule for constructing physics axiomatization by Florin Moldoveanu

Florin Moldoveanu

Lawrence,

I have tried to make the quantionic case as clear as I could in my paper, and if there is something unclear I can gladly explain it. The basic idea was of the 2 products and how the structure is self-preserving when 2 systems interact. The original insight was however the 1 to 1 mapping between observables and generators (energy vs. Hamiltonian for example), but by tinkering with this Grgin and Petersen arrived at the 2 product and composability idea. This has nothing to do directly with quantions, and quantions were discovered much later as arriving from a different kind of complexification procedure to construct the state space from the algebra of observables. In this complexification procedure, one of the elements of the algebra plays the role of sqrt(-1). This so called internal complexification was only possible in non-unitary representations of the quantal algebra, and simple closure arguments in the 2 products restricted the dimensionality to either 3 or 6. The 3 case led back to standard QM and it was actually used by Joy Christian in his papers. The 6 case can be only SO(2,4). Once the sqrt(-1) element is chosen in the SO(2,4) space, the centrelizer leads to quantions. The norm of the quantions can be understood as a probability current density and if its divergence is zero, it leads directly to Dirac and Schrodinger equation. (I think there is another solution leading to Klein Gordon, but I need to complete the computation). The divergent case is properly understood only in the non-commutative geometry framework and it is much more complicated.

In passing, quaternions do not have the composability property and in quanternionic QM there is no 1-to-1 correspondence between observables and generators because the derivative operator is not uniquely definable.

[deleted]

The initial set up by Grgin and Peterson is an interesting map between the octonions and thse objects which structurally lie in between complex numbers and quaternions. What you display of quantion algebra in eqns 14-17 what I referred to seems to more or less apply with product rules.

I have frankly only read half way through your paper at this point. My main question here is whether this structure provides a way of illustrating how octonionic fields are some underlying topological aspect of quantum field theory. It is interesting to me that this leads to SO(2,4) as the centralizer, which is the AdS_5 group, with connections to conformal field theory. I also question whether this complexification can be illustrated as a way of finding a renormalization group flow similar to the Zamalodchikov c*-theorem.

If I am asking the wrong question or have some misconception of things you might disabuse me of such. Quantions are one of the number of structures I have in the last couple of months had on my stack of things to address, which Grgin's paper posted last week piqued my interest.

Cheers LC

Florin Moldoveanu

Quantions have very little to do with octonions. Only real, complex numbers and quantions have the composability property. This may be related to the Cayley-Dickson construction for complex numbers, and of a similar construction for quantions.

SO(2.4) in quantions does not seem to be (yet) related with the renormalization group, but they are related with the conformal compactification

of the Minkowski space.

Robert Oldershaw

Hi Florian,

I have finally discovered your paper and downloaded a hard copy.

The issues you discuss look very interesting at first reading and I look forward to studying your approach more carefully in the next two days.

Hopefully I will have some substantive comments to offer in the near future.

RLO

[deleted]

I read your article: arXiv:0901.0332v2 [quant-ph] 16 Jan 2009 on quantions again. This is very much related to twistor theory in a way. The distinction between the A(z) and M(z) appears to be similar to a type of geometric quantization. The set of PT^{+/-} and PN is capture in the definition of the inverse, where the lack of division algebra for determinant = M(q) = 0 along null directions defined in some way by PN. So this is related to twistor theory in some way. I am not prepared to comment on whether there is some "map" or isomorphism between the two systems

I can't help but think there is a lot more going on with physics. Witten has proposed twistor string theory, where twisters are a form of D-brane. There is also some interest in extending twistor theory into the domain of exceptional algebras. This construction leading to quantions has references to Lie-Jordan algebras and Jordan products. So I have been reading this with an eye to the prospect this might have implications which are beyond what is currently formulated. The quantion 2x2 matrix could well enough be extended to the J^2(O) matrix

|z_1 O|

|O* z_2|

and the octonion elements considered as generalizations of the elements z and z* in your equation 24. The extension to J^3(O) is by the standard BFSS matrix model with

J^3(O) = R⊕J^2(O)⊕O^2.

I see no reason why this construction can't be extended into higher level systems. I think the composability requirement maybe satisfied for the exceptional algebra and its anti-Hermitian pair under G_2 holonomy. Further the analogue of the M(q), would be related to the characteristic polynomial det(Ω − λI) = λ^3 − (trΩ)λ^2 + tr(Ω*Ω)λ − (detΩ)I = 0, which defines the Lagrangian for the J^3 system. In an extended sense this is the same as M(q), which defines a proper interval, equivalently the Lagrangian, for a particle moving in spacetime. I am going to see if I can make this work.

The problem with restricted systems of quantum gravity is they do not contain enough information, and they run into difficulties. This happened with twistor theory and symptoms of this are evident with loop variables. I am not going to go into the reasons and history of this, but it is fairly clear this is the case. This does not mean these theories are wrong, but they are limited pieces of the puzzle. The problem with restricting a theory of quantum gravity to SO(3,1) or even SO(4,2) ~ SU(2,2), which is where both twistor theory and quantions live, is there is insufficient information. Conversely, there is this massive thing called string/M-theory which has too much. This is less of a theory than it is a sort of framework. It is my sense that ideas from theories such as twisters, LQG and maybe here with quantions provide the necessary constraints on string theory to maybe make it calculate some workable physics.

Cheers LC

Florin Moldoveanu

Hi Robert,

Looking forward to your comments.

PS: Thank you for 't Hooft's paper.

Florin Moldoveanu

Lawrence,

I am currently trying to understand the link between quations and geometric quantization and I even have a conjecture I hope to prove in this area.

Because A(z) and M(z) are different, quantions can also be cast as a Hopf algebra and this promises again additional insights.

Lawrence Crowell

Connections with Hopf algebras would be interesting. This would bring things in contact with K-theory.

Cheers LC

[deleted]

My comments were not precise enough to elicit your response. Briefly, i seek your priorities among the three tools broadly needed to work out a problem in Physics: 1. Conceptual aspect leading to some precepts;2. Mathematical formulation: 3. Experimental data and its critical analysis. How you prefer to proceed helps you to come with a final product. Speciality approach is self-confining and may mislead you to ignore factors that may be significant for your overall consideration.

Florin Moldoveanu

Dear Narendra,

I did answer you on Sep. 24, 2009 @ 22:22 GM, but maybe I misunderstood your question.

Let me try to explain how my approach scores against your 3 tools.

1. Conceptual aspect: The working conceptual assumption in my approach is that nature is relational at core and nature is just mathematics reorganized in a different way. All mathematical structures are unique: there is only one Pythagoras theorem for example and only one 2D curvature-free geometry. However, only some mathematical structures are distinguished and play a special role in Nature: Minkowski space, Hilbert space, etc. Why is that? Asking for the difference between mathematics and nature singles out precisely those and only those distinguished mathematical structures. All other mathematical structures do get to play a role in nature, but not a global one.

2. Mathematical formulation. From the heuristic rule one derives physics principles which in turn have mathematical consequences. Various physical theories (like special relativity, or quantum mechanics) have many alternative axiomatizations. What I have found possible is that one can select a distinguished axiomatization using the 3 principles. The best example is quantum mechanics which becomes intuitive in the new axiomatization (work in preparation).

3. Experimental data and its critical analysis. My approach is not philosophy, but it is old fashion physics rooted in experiments. I am asking for all mathematical properties of the physical world which are universally valid in nature and not universally valid in math. Being universally valid in nature means they have to pass all past, present, and future experimental tests. I am not asking for new experiments and because of that my approach will not answer all physics questions. It is therefore definitely not a theory of everything, but only physics axiomatization. If you consider set theory as the basis of mathematics, set theory would not help you solve for example Riemann's conjecture. I am only looking for the mathematical foundation of nature, just as set theory is the mathematical foundation of modern math. I can only speculate that maybe this would lead to TOE, I really do not know either way, but I can hope for it.

Speciality approach is a tightrope walk approach. Ignore it and you end up doing only philosophy with no predictions. Follow it too narrowly and you set yourself for dead-ends and unimportant results. There is no prescribed way of doing it, for if there were, we would invent every day brilliant new theories. You can only judge the merit of an approach only after the fact by seeing what the fruits of that approach were. I do not really know if my proposed approach will bear fruit in the end. (But I certainly do hope it will.) Only time will tell that. But what I do know is that I do ask new questions from a novel angle. Even if my approach will turn out wrong in the end, new questions will open up new ways of looking at old physics puzzles, and maybe it will play some role in solving them.

[deleted]

Hi Florin ,

You are welcome ,it's sincerely.

I didin't know these Hilbert Problems ,the Riemann Hypothesis is interesting ,I always though what the zero don't exist .

It's a human invention I think .

About the fith H.Problem ,it's very interesting this axiomatization ,the physicality is essential I think dear Florin.The real numbers and sequences must be inserted for a real mathematical extrapolations .

Like the Riemann hypothesis ,with infinity and imaginaies more zeros ,it's not possible ,we need limits ,the sphere and its laws will help without any doubt .

Just to help you a little .

A triviality and an axiom are evidents when the physicality is encircle with its specifics numbers ,the naturals and prime numbers thus appears differently in a closed system and its oscillations ,periodocities ,rationalities .....

Best Regards

Steve

Robert Oldershaw

Greetings Florian,

When you are able to use your abstract axiomatization method to say something definite about the actual objects and processes of nature, please let me know of the results.

RLO

www.amherst.edu/~rloldershaw

Florin Moldoveanu

Hello Steve,

Thank you for your good wishes,

Best Regards,

Florin

Florin Moldoveanu

Robert,

I do have many things in progress. I am about done with my paper on axiomatizing QM, and with another one regarding global hyperbolicity. Then I am working on 3 other topics stemming from my approach. I hope to derive the Klein-Gordon equation from quantions, to investigate maximum entangle systems and their link with geometric quantization and composability, and to explore the link between relativistic QM with Hopf algebras.

If you are interested, you can check on my progress by searching for my papers on the archive from time to time.

[deleted]

Dear Florin,

In re-reading the posts above, I noticed the following statement of yours in your discussions with Lawrence (Sept 28, 01:27 GMT):

"I think there is another solution leading to Klein Gordon, but I need to complete the computation"

Let me bring you up to date: You are right!

This is not evident in my book of 2007, where Dirac's equation is derived only in the second pass (the first pass being the 2005 book). If you remember, I point out on page 5 of my essay that the last stage in the development of a theory is "finalization", which is not merely a matter of cosmetics. This is part of what I've been doing recently to quantions. In the new formalism, which seems final (third pass), the quantionic derivation of both Klein-Gordon's and Dirac's equation takes only two lines of algebra. Structural unification is also much clearer (even trivial). Seneral other things become essentially trivial as well.

Regards, Emile.

Florin Moldoveanu

Dear Dr. Grgin,

Congratulations!!! Can you share your proof with me? This is bitter sweet; you beat me to the punch line. Let me ask you a question. Is this related to SU(2,2) (and the second linking vector) as I suspect it is?

Regards,

Florin

[deleted]

Dear florin,

i thank you for the exhaustive response you have given to my three queries. I am partially convinced that you do have an approach to Physics through the tool of mathematics and do reject Maths that is not relevant to the physical conditions. It is fine that you choose your variable and boundary conditions properly. But ca one ever bs sure that nothing has been lefy out.You are right when you that time will tell if your approach is right when the predictions get verified. That is true for all the scientists worth the name. I certainly wish you all success.

My primary hesitency howver remains, Maths can not be the starting point in Physics. One must first work out a conceptual network based on existing proven knowledge, including data and theories. Ony then one should examine the relevance of appropriate maths to the problem at hand. We explain in Physics what Nature has already provided us to view, feel and experience. The human mind is at the centre of our knowledge. With any change in the way the mind is set up, we may arrive at a innovative aspect, not considered in the past. i hold such an approach the best for Physics. Lately, Physics has had hardly any breakthroughs, as partlyb reflected by the award of Nobel prizes for past works, life time studies, etc. AS serious questions/mysteries still remain to be tackled, specially in Cosmology, nanotechnology and other sub-disciplines, the freshness of approach to me is mor eimportant and it should not start with pure mathematics.

i have not checked my post and may be some errors/mistakes are left untackled.

[deleted]

Dear Florin,

Absolutely not! I mean it's not related to SU(2,2). And I don't see how it could be. Therefore, if you discover a connection I will be both amazed and pleased. Pleased because it would open up unexpected relationships between objects I think are unrelated.

SU(2,2) is the twistor branch on the SO(2,4) trunk. I informally call it the geometric branch. Since the Lie algebras so(2,4) and su(2,2) are isomorphic, twistors are related to the conformal group, which does not preserve the Minkowski metric.

Quantions are the algebraic branch. They are derived from so(2,4) by way of internal complexification. Their metric norm is Minkowskian.

You see now why I don't believe quantions and twistors will ever meet again, even though they are siblings -- their mother being so(2,4). Have a peek at Figure 26.1 on page 550 of my 2005 book.

As for the proof, it is not based on some "clever" new idea that could be easily conveyed. The idea of the proof is exactly the same as in the book. The difference is that I longer work with matrices. The formalism is different. Many things just 'fall into place' once that formalism is developed -- which I am doing these days, making sure that there is no mistake. As an analogy, consider 19th century differential geometry, where we (humans) had to do all the work, and compare it with tensor analysis, where the formalism does 99% of the work for us.

The secret to being good at mathematics is not smarts (there is plenty of that around) but laziness. The lazier you are, the harder you'll work at developing the tools that will do the work for you. The funny thing is that those tools end up BEING the next mathematics.

Incidentally, as part of the finalization effort, I think I also found a much simpler way of re-doing Part I of the four-part paper in Fizika, where composability is put to work. After I've gone through the details, making sure that I am not deluding myself, I'll be happy to send you the first draft.

Regards, Emile

PS: Why not drop the stuffy "Dr. Grgin"? On this forum we are all on first name basis, which is much nicer.

[deleted]

Dear Florin ,

Do you think it exists a maximum prime number ?

Personally yes .Possible it's the number of spheres perhaps.

This oscillation ,periodic is fascinating ,let's imagine a distribution in a spherical logic in a physicality with thermodynamics laws .

The periodicity of twins primes numbers is relevant about the oscillation .

Inside a sphere ,if we begin from the center ,the angle of distribution must be considered too ,it's relevant I think .Finite borders are necessary in my opinion .

Of course if the distribution of prime numbers is physically speaking in synchronization thus .....The thermodynamics probably and its coherences could help.The volume is relevant about too the specificities of the spheres like its volume .If a distribution is linked with this fact ,the volume can be more understood .Like our cosmological spheres which are a pure realities with a mass and a rotation .That's why I think what the series of spheres and the oscillations are main part of the puzzle .

The numbers ,physical ,spheres are specifics and their specifities too are correlated in its variables .Like an Universal distribution in a finite Universe .

The aim is not to find the impossibility but to understand this large numbers of spheres .

The mass ,the volume evidently are correlated in this distribution ,like forces at the quantum scale too where probably this number is the same, with of course some specificities for a specific rule, polarised, since more of 14 billions years if we consider the Big Bang distribution where time is a constant .

The numbers naturals are in the complexity where the multiplication isn't finish where the zero and the infinities are in consideration ....the link with the strong and weak interactions ,gravity and electromagnetism is the complexification thus evolution with the nat numbers ,the prime number are like in the strong interactions .Like the distribution of cosmological spheres where the volumes are variables with a division ,a fractal of spheres .The prime numbers probably were the fisrt serie at the Big Bang Hyp.In the quantum link it's the same with the main central coded sphere and its division or fractal of spheres ,the series is with prime numbers and after complexificate with the electromagnetism and gravity and natural numbers like some sequence of multiplication and periodicity with prime numbers,probably the senses of rotations is the key too with the evolution point of vue and the complexification .

What do you think dear Florin ,?

Steve

[deleted]

Correction to my post of 09:17 GMT.

As written in the fifth Paragraph:

" The difference is that I longer work with matrices."

As you probably figured out it should be:

"The difference is that I no longer work with matrices."

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