A Historical Approach to Research in Fundamental Physics by Emile Grgin

Terry,

You write:

"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 agree that the interpretation of what the progression from one number system to another implies - physically - is important. This is even the 'raison d'etre' for any new number system.

But I have a different view of number systems in general - both mathematically and physically.

Mathematically, saying that a particular structure, S, is the number system of some theory is a powerful statement. It tells us to forget about any simpler underlying number system and accept as syntactically valid only those statements (postulates and theorems) that can be formulated exclusively in the number system S. This does not apply to the ephemeral statements within proofs of theorems, but theorems that can be proved within S alone are invariably more impressive - at least aesthetically. Consider, for example, a complex projective space. It is equivalent to two real projective spaces only in information capacity related to geometric objects. But is not equivalent structurally: Division by complex numbers is not automatically contained in a pair of real projective spaces.

Physically, what I said above still applies, but there is more. The Catesian and polar representations of complex numbers are mathematically equivalent (we use what suits us better), but the polar representation is distinguished in quantum mechanics: The unobservable phase factor gives rise to the gauge group U(1) while the norm is subject to Born's interpretation. The same is true for quantions, which also amit a polar representation: The unobservable phase factor gives rise to the gauge group U(1)xSU(2) while the norm is subject to Zovko's interpretation and gives rise to the equations of motion.

Dear Eckard,

What was your hint?

If it's that I read your essay, I did. I read it then and I reread it now.

If it's that I comment on your essay, I can't because I always keep quiet when I have nothing useful to say.

I can tell you, though, that I enjoyed very much the part where you talk about the things you know. I will even reread it more carefully. Thank you for sharing those insights.

Regards,

Emile.

Daryl,

I see only one connection between our essays: We both speak of the arrow of time.

I have the impression that this concept can be interpreted in more than one way.

The standard interpretation has to do with the time parameter that appears in the equations of motion. If the entropy is preserved, the motion is reversible. If the entropy increases, it is not -- which means that a direction of time is distinguished. We call it the future. This is clear. What is not clear to me, even after having read much on the subject, is why many physicists are uncomfortable with the problem 1) in your essay. If you have solved this problem, I would like to understand the solution, but for that I would have to go to your articles and study them carefully -- a project that will have to wait.

In my work, an arrow ot time appears at the microscopic level. I don't see how it could propagate to the macroscopic level. My impression is that the existence of this arrow is related to the failure of the P and T symmetries in the electroweak theory. Time will tell.

Regards, Emile.

Darryl,

We seem to be turning in circles.

You asked me two days ago to read your essay. I told you I read it, but that I would have to read your articles to understand it.

Today you come back and tell me again to read your essay. All I can do is tell you again that I've read it, but that I would have to read your articles to understand it.

The first three are not readily accessible to me, so I went to the fourth on ArXiv:0902.4667 but got the message that Windows cannot find it.

Then I reread your essay in the hope of doing better in the second pass. No luck! I don't understand it.

I do believe that you see a connection between your work and mine, but this is normal since our final objectives coincide. The differences between all posted essays are not in the final objectives but in the initial premises. As far as I could tell so far, they are all different.

Emile

Dear Eckard,

In my 16:36 GMT answer to your post of Oct 10,14:54 GMT I overlooked your question, namely:

"Isn't it better to correctly re-transform originally measurable quantities from their arbitrarily chosen mathematical representation back into the original realistic domain as do engineers like me?"

Sorry. I did not ignore this question intentionally. I 'got stuck' trying to figure out what your hint was, and then I forgot.

Concerning the beginning "Isn't it better to correctly re-transform ...?", I dont see why you would even ask this question. We are all completely free to make any transformations we find useful in our work. If the transformation is reversible, nothing is either lost or arbitrarily inserted -- so no one will ever object. And if the transformation is very important (but please, not in the author's opinion alone, but in that of the world), the author's name ends up attached to it -- as in "Fourier transform", "Laplace transform", etc.. That's the way things are.

Being thus puzzled by the motivation behind your question, I consider the two possible interpretations.

If the question was rhetorical, but actually meant to enjoin physicists to think like engineers, I have no comment. Not even my parents could tell me how to think.

If the question was genuine, and you actually want to know, I will answer it in a broader setting:

A child asks why the grass is green. The botanist answers in terms of plant physiology.

The child goes on to study physiology, and then asks about its underlying mechanisms. The biologist answers in terms of ... etc... etc.... until, after five or six steps, he reaches atomic physics.

The student asks why the constituents of atoms behave as they do. The fundamental physicist answers in terms of quarks, gluons, vector bosons, etc., (which, by the way, are not "arbitrarily chosen mathematical representation"; they are the only surviving representations after countless other ones that did not numerically relate to measurable quantities had to be mercilessly discarded).

The graduate student asks Why quarks? Why intermediate vector bosons? Why ...? But there are no definitive answers yet. For the time being, the buck stops here.

You want to know why fundamental physicists do not take over engineers' way of thinking. The reason is that they've already been there and thoroughly exploited that approach in Archimedes's time, some 2000 years ago. At that time, the buck stopped at engineering. It no longer does. Sorry, but that's the way it is! Returning in our days to engineering-level thinking would close the circle, and we all know that circular arguments are for the birds -- but even that is questionable, considering they can't even tell us what came first, the chicken or the egg (or is it the other way around?).

I hope this answers your question. And, of course, good luck with your essay.

Regards, Emile.

Hi Chris,

I am very pleased about your having enjoyed my essay, and thanks for your brownie points and good wishes.

I had already read Peter Jackson's essay, and I must admit that the "dressing" you mention was very much an obstacle to understanding. But then, prodded by your suggestion, I just read his longer article, "Doppler Assisted Quantum Unification allowing Relativistic Invariance". It is much better, but not in my style of work -- which implies that I have no intelligent comments about it. Peter's views are very broad (which is a valid style in my opinion) while I focus on details (valid or not, I made it my style because I enjoy it most).

Sorry you did not submit an essay, for I would wish you good luck too.

Regards, Emile.

Dear Jayakar,

Thank you very much for your thoughts and wishes.

Maybe I did not understand exactly what you meant in the first paragraph, but there is no problem with deriving the wave function and the Schroedinger equation from quantions. The two norms are not in the way because only one is relevant in this derivation. It is the algebraic norm. Of course, you might have meant something else. But it is very true that the two norms throw a new light on the gauge groups you mention, in particular on the root systems. This has not been published yet because these results must agree with everything else, and it takes a lot of time to double check everything.

For the other ideas you have, you are probably right, but I don't know because I look at things from a somewhat different angle -- at least for the time being.

Best regards, Emile.

Terry,

If by "progressive approach" you mean the march Naturals --> Rationals --> Reals --> etc., then I may or may not agree that it is mathematically open ended. That would depend on what we minimally expect of number systems.

Physically, however, I don't expect the sequence to be open ended. I expect it to end at quantions. This may sound self-serving, but it is not meant to be. The argument has to do with the idea of rigidity. Complex numbers still contain some soft spots. This is what makes it possible to generalize them to quantions, which are much more structured. But quations appear to be absolutely rigid. Without soft spots, the only possible generalizations are structure-reducing, but that's not interesting. My objective is to have as much physics as possible encoded in the number system.

I am curious about your view that we need a change at the fundamental level without disturbing the Naturals.

(1) Why would we need such a change? Not that I have any objection, but not having an objection is not equivalent to agreeing. To agree, I would have to see a reasonably convincing heuristic argument, or an example, or something.

(2) Something more fundamental (= structurally more primitive) than the Naturals might be Wolfram's cellular automata. Lev woud say it's his linguistic constructions, but, unlike you, he wants to get rid of numerics. This is OK with me, but I happen to be interested in an intellectual game in which outcomes are numerical. In other words, Lev and I work in different professions. If he thinks otherwise, it is his responsibility to derive interesting physical consequences that have experimental interpretations -- and I mean numerical one, as opposed to hand-waving contributions.

Mark,

Last question first, to get it out of the way:

No, octonions are not a subalgebra of quantions, nor are the two in any way related. It is true that both quantions and octonions are defined by eight real numbers, but this is a coincidence without deeper meaning. On the other hand, Dirac spinors, which are also defined by eight real nubers, are physically equivalent to quantions. Mathematically, however, they are very different (for example, the product of two 4-spinors is not a 4-spinor while the product of two quantions is a quantion).

Your other questions are not really applicable to my work because I am interested in a very specific research topic: Does a structural unification of quantum mechanics and relativity exist? Or, to put it in another way, Does a mathematical structure which encompasses both quantum mechanics and relativity exist?

If it does, it cannot be based on quaternions because these objects are not relativistic. You say that they have a well-established connection with physics. That's true, but that connection is at best with quantm mechanics (if one insists on doing QM the hard way) and not at all with relativity. But even without this argument, it would be very foolish to waste time in seeking the key to unification in the quaternions. It would be like prospecting for gold on Times Square in New York City. It's too late for both.

As for the octonions' connection to string theory, it's as irrelevant to structural unification as it is to chemistry. The domain of structural unification is that of standard experimental physics while string theory lives in the vicinity of Planck's length. Apples and oranges.

This answers your specific questions. Let's now discuss why I go from the complex numbers straight to a non-division algebra, skipping over the other normed division algebras.

First observation:

The skipping in question is true only from the point of view of mathematical tradition, which insists on having positive definite algebraic norms. But Nature does not care very much about our traditions. It was also a mathematical tradition, until 1908, to consider only positive definite geometric norms (Pythagora's theorem). In 1908, Minkowski's reinterpretation of Einstein's work taught us that Nature thinks differently. Now, if we drop the tradition, quantions are structurally closer to the complex numbers than the quaternions. Therefore, there is no skipping from the point of view of physics. Going from complex numbers to quaternions would be skipping. And, indeed, quaternions are already much too far from the complex numbers to support structural unification. Only a small modification of the complex numbers is needed, and that is provided by quantions.

Second observation:

I never arbitrarily decided to drop the complex numbers. My initial objective was to figure out whether quantum mechanics and relativity could be structurally unified. This was a project on which I had to work alone -- probably because nobody believed it could be done. It is well known that quantm mechanics is rigid, which means that it has no soft spots that could be modified to admit relativity. Intellectually, I was as aware of this as any other physicist. Intuitively, I did not believe that this was the whole story. A careful analysis of quantum mechanics done in a manner unconventional in physics but very conventional in mathematics led me to the answer: The "soft spot" is not in the axioms of quantum mechanics; it is in the underlying numer system (of complex numbers). What's more, this analysis led uniquely to the new number system. I called it the algebra of quantions.

Emile

Mark,

In answering your questions last night, I was already more asleep than awake. Afraid of having said something I wouldn't if it had been the other way around, I just reread your post and mine. Fortunately, there are no very great sins of comission and only two of omission, both easily fixed:

(1) I forgot to thank you for your questions. They are specific and, as it seems to me, representative of what others might also ask. Well, thank you, and don't hesitate to come back with questions in the same style.

(2) My saying that Dirac spinors and quantions are physically equivalent but mathematically very different may be more puzzling than useful. A well-known example ought to help: The Heisenberg and Schroedinger pictures are physically equivalent, but the Hilbert space and the Jordan algebra of self-adjoint operators are mathematically very different.

Nevertheless, equivalence, or equality, are rarely perfect -- an imperfection humoristically conveyed by the phrase "more equal". When referring to perfect equality, we use the word "identity". Coming back to quantions, let's make two observations related to equivalence: (a) Quantions and Dirac spinors are identical from the point of view of information contents (4 arbitrary complex quantionic components = 4 arbitrary complex spinor components) and the one-to-one correspondence is well defined. (b) In the big picture, quantions are "more equal" than spinors because we only have to add an interpretation (Zovko's, which generalizes Born's) to obtain the Schroedinger and Dirac equations with all possible potentials. And if the vector potentials are interpreted as differential connections, the corresponding gauge group is U(1) x SU(2).

Referring to your post, you see that my not postulating U(1) and SU(2) does not mean that these objects do not enter the picture.

Emile