Essay Abstract

Research that aims at identifying new fundamental ideas in physics can greatly profit from a historical approach. The present essay develops this idea by conceptually analyzing the major physical theories created since antiquity and by distilling from them the research trends that have been unmistakably successful. The author's approach to research is based on extrapolating these trends into the future. It is a method that led to a unification of quantum mechanics and relativity based on a new number system structurally located between the complex numbers and the quaternions. Following a brief description of the concrete results obtained so far, the question of what's ultimately possible in physics is addressed by speculatively generalizing the results in question.

Author Bio

Emile Grgin studied mathematics and theoretical physics at the University of Zagreb and in Peter Bergmann's General Relativity Group at Syracuse University. He worked mostly as independent consultant in applications of physics. He is now an independent researcher in the field of structural unification of relativity and quantum mechanics.

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One can exquisitely model one cycle of a sine curve with an odd-power polynomial to achieve an algebraic value arbitrarily close to the trigonometric value. Said model is worse than useless beyond that single cycle, for it will perfectly go in the wrong direction. One need not resort to abstractions. Economics is an arbitrarily perfect interpolation that disgorges Scientific Socialism, the Boys from Chicago in Pinochet's Chile, and the United States' $100 trillion debt set to explode. Bad luck goes "BOOM.".

Discovery is excluded from spread sheets, PERT charts, budget allocations, and business plans. All discovery is insubordination. Allowing management to dictate research from casebooks is not a formula for success, it the disaster of Bell Labs become Lucent Technologies become an empty building. Pharma has fewer drugs to introduce each year not more, despite every amplification of research oversight to render it more focused, documented, and efficient.

Mankind is purely incapable of predicting its future. Intelligences of a committee add like resistors in parallel. The Battle of the Somme was Napoleonic infantry marched against German machine guns. Xerox PARC was a local disaster, HP rejected Wozniac's motherboard, the world needed more than seven IBM computers,. Your formula is indeed a limit of physics. Its implementation will terminate physics as assuredly as the One True Church knew Jupiter's four Galilean moons orbited the Earth. It was in the Pope's casebook.

"Distill research trends that have been unmistakably successful" then do the opposite. That is where a blue rose awaits plucking.

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Dear Dr. Grgin,

I am very happy you decided to enter this contest and make the case for the marvelous unification of quantum mechanics and relativity.

As I have said in my essay, your profound ideas have inspired me a great deal and it is my true belief you are trully deserving the Nobel prize for them.

Good luck!!!

Dear Florin

Except for the over-overexaggeration at the end (you seem to be shoving under the rug the fact that that my work is "work in progress", not a finished product) I am very pleased with your having understood my approach and objectives. I am also delighted to see that it has helped you formulate your own approach. You have my best wishes.

Emile.

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Dear Dr. Grgin,

Florin has a great deal of respect for you, and I respect Florin. I skimmed over your paper, but will read it more thoroughly next week.

You define a quantion as a quadruple of complex numbers. If I understand this properly, your quantions would be a two-dimensional isomorphism with the Pauli sigma spin matrices or the two-dimensional real components of a four-dimensional Quaternion.

I understand that you are trying to avoid the Clifford divisor algebras, but these are the best understood algebras. Quaternions are very important to spacetime, Maxwell's equations, and the Dirac gamma matrices. It seems reasonable that if we can build twistors (the 4-D equivalent of Dirac 4-spinors) out of pairs of Pauli sigma spinors, then we should be able to represent Spacetime with pairs of quantions.

A personal bias of mine is that I think that 8-dimensional Octonions must be relevant because Einstein's Field Equations of General Relativity are ten coupled tensor equations. These ten tensors seem to reflect Octonion symmetries. Now we need quadruples of quantions to represent an Octonion. However, in their "Gravitation" book, Misner, Thorne and Wheeler tried to reduce the number of needed tensor equations down to six (reducing the number of necessary equations because we have 4 spacetime dimensions). We could represent six tensor equations with a Quaternion, and may thus be able to reduce down to a pair of quantions.

Now I need to read your paper more thoroughly and see if I understand your ideas!

Good luck in the contest!

Sincerely,

Ray Munroe - author of a Geometrical Approach Towards A TOE

Dr. Grgin:

I am mathematician by formation but been doing computer science for living for way too long (25 years) so I am out of shape on the formula manipulation and algebra side. Never the less I find your essay very clear. Although my essay to this contest propouses there is no such a thing as a final physical theory, I think your aproach does lead to a great step forward towards better physics.

Like Florin says: you might well deserve the Novel prize. Now speaking for my self, if for an unexplainable reazon I get a better prize on this contest than you I would gladly give it to you.

Best...

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Dear Emile,

I noticed that in your essay you mention 'structure' or 'structural' about thirty times. I agree with your view on the continuously increasing role of 'structure' in mathematics (and hence in physics):

Author Emile Grgin wrote on Sep. 25, 2009 @ 11:42 GMT

"The 'structural' view of mathematics--according to which mathematical objects are not defined ontologically but only by their mutual relations--was introduced for two special cases in the 1830's (Galois and Hamilton) and had become standard by the 1940's (Bourbaki). This is 'the mathematics of mathematicians'. It answers the question 'What?'--as in 'What is possible?' or 'What exists?'. The key word is 'structure'. In this conception of mathematics, equations not anchored in mathematical structures are meaningless.

Just as mathematics was becoming more and more structural, its role in physics was becoming more structural as well."

However, I want to draw your attention to the fact that, *so far*, all structures in mathematics (including those introduced by Galois, Hamilton, and you) are of completely numeric origin. And this is not surprising at all, since all of the present day mathematics stands on the foundation of numbers. Obviously, this constrains us in many ways: in the case of a finite-dimensional vector space, for example, there exists essentially only one metric (Euclidean) consistent with the underlying algebraic structure (see, for example, section 3 in our paper.

As implied by my post of Sept. 20 (under my essay), practically all concepts of 'structure' within the current mathematics have emerged in the *continuous setting*, even algebraic ones, since the final/formal separation of algebraic from topological concepts where accomplished by Bourbaki relatively late. In other words, within the conventional mathematical setting, we don't really have *sufficiently general* concepts of structure--especially as it relates to the representation of processes/objects--completely independent of the 'continuous' setting. This situation, in part, motivated the development of the ETS formalism (outlined in my essay), which introduces a quite general concept of 'discrete', or 'relational', representational structure.

Dear Ray,

Thank you very much for your comments.

I suspect that other readers will have views similar to those you expressed in the first two paragraphs. Since mine have a diferent slant (for being more complete), I hope you won't mind it if I take this opportunity to clarify my views and objectives by discussing several points taken from those two paragraphs. After reading them, other people might not have to bring up the same issues.

By the way, this forum is great. It offers us an excellent opportunity to interactively extend the character limit necessarily imposed on our essays.

(1) "You define a quantion ..."

This is a good beginning from the purely mathematical point of view. In mathematical texts, a definition is a definition is a definition. The motivation for introducing a particular definition is extra-mathematical. Some authors do us a favor by not hiding their heuristics, but, following Gauss's extremely dry writing style, many mathematicians consider it bad taste to enliven their dry logic with the human intuitions that led then to the novel structures they are presenting. In physics, this attitude is not acceptable. If I give you only the definition of quantions, why should you be interested? The first question you would naturally ask is "Where do they come from?" Did I pull them out of a hat and then notice, to my own surprise, that they have physical interpretations? Or were they revealed to me in my sleep by the goddes Kundalini? None of this. The algebra of quantions is the unique generalization of the field of complex numbers that satisfies the meta-physical principle at the foundation of all my work (not to be confused with "metaphysical"). It is the "composition principle", that I don't have to explain here because Florian did it in his essay. It took me several years to make the journey from the principle in question to the mathematics of quantions. This derivation is spread over four atricles published in 2001 (available online) and a book published in 2005. It was a messy business. Not having the slightest idea of what to expect, I was struggling with several new concepts at the same time. Maybe I should take some time off to write a self-contained article streamlining that proof.

(2) "... your quantions would be an isomorphism with the Pauli matrices..." Yes, you are right, but this is is half of the story. The algebraic half.

Incidentally, Florian mentioned that quantions were first discovered by Benjamin Peirce in1882. I am sorry I pointed this out in my book because several colleagues, including Florian, made too much of it. Peirce classified all complex associative algebras up to a certain order. Since quantions are associative and are within the range he considered, they are necessarily one of the 120-or-so algebras he identified and systematized. But so what? Saying that a particular associative algebra belongs to the set of associative algebras is not very informative. Peirces's classification does not suggest that the algebra in question might be relevant to physics, not that it is a distinguished generalization of the field of complex numbers. Again, sorry I did not ignore this red herring. What happened is that, working in isolation, I was less than self-confident -- so I was looking for support wherever I could find in other people's work. Looking back, this was rather foolish, but, as our friend Steve would say, "c'est la vie".

The other half of the story is differential (as opposed to algebraic). In the field of real numbers, we have the differential calculus, which is very intuitive and finds applications everywhere, even in the "dismal science" of economics. In the field of complex numbers, which is more stuctured, derivation is less intuitive, has fewer applications, and gives rise to analytic functions. This type of derivation does not generalize to other division algebras, like the quaternions. Non-commutativity prevents it. But then, it does exist in the algebra of quantions, where it beautifully outwits non-commutativity. In the quantionic domain, derivation is even less intuitive than in the complex numbers, and does not enjoy all its nice properties. Yet, it has exactly the properties needed to give rise in a very simple way to the Klein-Gordon equation and to a very general quantionic field equation which manifests itself as the Schroedinger or Dirac equation. It has no conceivable application anywhere else! This is what I like best about it. It seems final -- as in final theory.

(3) "...you are trying to avoid the Clifford algebras, but they are the best understood ..."

First: I am not 'trying' to avoid anything in particular. I am 'actually' avoiding everything that's not in the deductive path of arguments that follow exclusively from the meta-physical principle of composability. This requires no "avoiding effort" whatsoever. If some piece of mathematics is not on the path in question, it might as well not exist as far as I am concerned. But since you mention Clifford algebras, let me just say that I would have no chance of extracting from them the paradigm I am interested in: the merging of QM and relativity. If so many physicists and mathematicians much abler than myself have not found it over the decades these algebras have been investigated, I can conclude only one thing: The paradigm in question is not hidden in the Clifford algebras. Ergo, these algeras are not relevant to my current work.

Second: Being "best understood" is not an argument. I'll bet you that when God created relativistic quantum mechanics, He (or She, or It, or Whatever) did not consider what Ray Munroe will one day understand best, and then use it as the fundamental structure of the universe. I think I made this point implicitly in my essay: The Greeks understood the circle best, so they used it to construct a celestial mechanics. It was an "engineered solution". Insisting on using a particular structure for knowing it best, or for being deeply in love with it, is taking a very very very long bet. This is, at least, what I conclude from the history of physics.

(4) "Quaternions are very important to spacetime,..." I fail to see how. Real quaternions go with the rotation group, not with the Lorentz group. And if you take complex quaternions and then slice them at an angle that replaces the 3 imaginary units by the 3 Pauli matrices, you land in the algebraic part of quantions. So, why bother even mentioning quaternions. I won't hurt their feeling by not being deferential to them.

(5) "... then we should be able to represent spacetime with pairs of quantions". If by spacetime you mean the affine Minkowski space, then, fortunately, NO. I say "fortunately" because otherwise the door to general relativity would be closed. If you mean the linear Minkowski space, it's already done.

Re the third paragraph, I can only wish you good luck. You obviously know that area better than I.

Best regards, Emile.

Dear Juan,

Thank you very much for your good wishes. Of course, I am particularly pleased to hear that my essay was clear. I am always concerned about being misunderstood -- and not without reason.

Incidentally, you might like to read my comments on Ray's post. They expand my essay.

Now, your second paragraph is a bit too strong. Please keep in mind that Florin is more enthusiastic than myself about quantions. Not that I have any particular reason to doubt them, but I am aware of too many examples in the history of physics when some idea remained promising for a long time, but then had to be dropped for not delivering the last chapter. I am not even near the last chapter on quantions.

Bast regards, Emile.

Dear Lev,

I am eager to read your essay very carefully. I just put it on my reading list for Monday evening.

Yes, I know that I sound like a broken record stuck in a groove with "structure" in it. This is due in part to my having had some rather frustrating discussions about my work with colleagues whose research falls in the category of Kuhn's "normal science". From that point of view, my goal seems ridiculous. I could never convincingly explain that I am working on a problem whose solution is neither a number nor a mathematical expression, but a structure that has no standard name because it has not yet been introduced by mathematicians.

I'll come back after having studied your essay.

Thanks for your observations on structures. I will better understand them in a couple of days.

Emile.

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Dear Emile,

Thank you for the explanation. I understand that this contest had restrictive length limitations and it was difficult to include everything in these papers that we wanted to. I am not entirely sold on the Clifford divisor algebras. As you pointed out, someone would have already unified QM and GR if the Clifford algebras had allowed it. I'm simply trying to understand quantions and their possible applications relative to what I understand (and to your point, Nature or the Creator did not choose an algebraic system based on what I understand). I will read your paper thoroughly next week.

Good luck in the contest!

Ray Munroe

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Hi Dear Dr. Grgin,

It's avery beautiful essay .Full of pragamatism and rationality about the structures and the rules of math and physics where the unification appears .

Good luck .

Best Regards

Steve

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Dear Emile,

Please note that I was not implying that you overemphasized the concept of structure in mathematics.

Dear Lev,

I was under the impression you were, and that puzzled me a little. Thank you for correcting my misinterpretation.

Thank you, Steve, for your good wishes.

I am glad you enjoyed the essay.

Emile.

Dear Ray,

No, none of us should be easily sold on anyone else's ideas. But we usually speak of our own views with such conviction (which is natural) that it may sound like campaigning (which is unfortunate).

Concerning my own long-standing fear of being biased by what I know better, I put the antidote, under the heading "Symmetry", in the preface of my book referenced in the essay. Here is the first page of that preface:

Preface

In the title of this book, "structural unification" refers to a merging of two physical theories into a single mathematical structure. We impose the following conditions on such a merging:

Symmetry: The component theories must enter unification on the same footing. This is to eliminate approaches to unification that treat as more fundamental the theory which happens to be better known.

Completeness: Within its domain of applicability, the unification must yield no unphysical theorems. This is to eliminate unifications that require ad hoc adjustments when faced with the real world.

Irreducibility: The unification must yield more physics than can be derived from the given theories taken independently. This is to eliminate from consideration physically empty unifications.

Maxwell's electromagnetic theory is an example of structural unification: In covariant form, this theory is symmetric because the electromagnetic tensor subsumes the electric and magnetic fields with equal weights; it is complete because none of its theorems has to be rejected as unphysical for clashing with observations; it is irreducible because it yields at least electromagnetic radiation as new physics.

In contrast, quantum field theory is not a structural unification of quantum mechanics and relativity for not being a "single mathematical structure". It is not symmetric either: Canonical quantization grafts quantum properties on relativistic fields, thus treating these properties as after-thoughts. This cannot possibly be the way of nature.

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Dear Emile ,

You are welcome ,

Sincerely

Steve

I read your paper with considerable interest. I am always looking for connections between what I am thinking with others, or between what others are thinking. I posted this on Ray's blog site. Mine essay is at

http://www.fqxi.org/community/forum/topic/494

where I indicate aspects of how exception algebras may work with respect to the cosmological constant.

I have been giving the matter of quantions some study. I am not entirely decided about their status as yet. My sense is they are an interlinking between two complex number or quaternions in a way which defines norms differently. This might have something to do with S-matrix. So I will outline some aspects of S-matrix theory and black hole complementarity, and then try to make possible links to quantions.

The holographic principle and black hole complementarity are generalizations of the S-matrix. Susskind's treatment of strings falling onto a black hole according to a distant observer treats the S-matrix on a domain which is causally defined on an infinite domain of support according to the tortoise version of the radial Schwarzschild coordinate

r* = r - 2m ln|r - 2m|

The S-matrix requires an infinitely extended domain by which fields are causally related, which is "manufactured" by this coordinate. In these coordinates the string exhibits a range of strange behavior, which I am not going to review again in great detail. Yet the string ends up covering the black hole horizon and is frozen their according to this distant observer. To an infalling observer on a commoving frame with the string none of this is the case, but rather the string enters the black holes with no apparent change and then exhibits tidal forces of an extreme nature near the interior of singularity. The string is a form of S-matrix theory, and the two cases reflect the existence of two S-matrices, each according to state space elements which are incommensurate with each other, or according to noncommutative operators. This is one way of looking at the so called black hole complementarity principle. There is then a superposition of the string in these two bases of states, and for this reason the distant observer may see the string frozen above the event horizon and also "burned up" by Hawking radiation made of quanta scattered from the string according to the infalling observer's frame.

The ordered S-matrix defines each vertex, or particle, and its neighbor. In a linear chain a general state is an S-matrix channel of the form

|φ> = |p_1, . . . , p_i, . . . , p_j , . . . , p_n>

This state or S-matrix channel is related to but distinction from the channel

|φ'> = |p_1, . . . , p_j, . . . , p_i, . . . , p_n>

The particles or vertices p_i and p_j have exchanged their neighbors, which means some "relationship" structure to the amplitude has been fundamentally changed. The S-matrix is written according to S = 1 - 2πiT, so two states or channels |p_1, . . ., p_n> and |q_1, . . . , q_n> are related to each other by the S-matrix as

(p_1, . . . , p_n|Sjq_1, . . . , q_n> = (p1, . . . , p_n|(1 - 2πiT)|q_1, . . . , q_n>

= (p_1, . . . , p_n|q_1, . . . , q_n> - 2πi(p_1, . . . , p_n|T|q_1, . . . , q_n>:

For the < | the in channel and | > as the out channel p_n and q_1 are neighbors, and neighbors through the T-matrix. This eliminates an open vertex in the chain. The vertices or particles p_1 and q_n are the open elements in the chain and defines an "anchor" for the chain, and are thus defined as neighbors in this manner.

A four point function and the transition matrix defined by vertex operators T = V(p_1)ΔV(p_3) will contruct the Euler-beta function for coherent states of the S-matrix. This is the connection of course between string theory and the old bootstrap or S-matrix theory. Now for two S-matrices, which pertain to the different domains of causality on a black hole this theory is made more difficult. The S-matrix is a braiding operation of sorts between elements of a quantum group G. So we might model this as a commutator structure (braiding) between two elements a and b \in G. So we might denote this as ab --- ba. Now let us assume the states we observe are super-positions of incommensurate states involving two quantum groups G and G'. We will then have a structure of the sorts (ab)c --- a(bc), that exist in an associahedron I_2(5) with a homotopy structure. This homotopy then connects to a K-theoretic field theory, which I discuss in my paper

http://www.fqxi.org/community/forum/topic/494.

I will not belabor this part of the things, until later or somebody takes an active interest in what I am suggesting here.

The black hole complementarity principle. The complementarity is an odd structure, for Hawking radiation is due to a Bogoliubov transformation between basis elements. In this setting the theory of spacetime is classical and the fields scatter off the black hole or spacetime with an event horizon. The response of the black hole or spacetime is a metric back reaction, which is a classical response to a quantum scattering. Yet black hole complementarity has demonstrated that quantum information is preserved for the case of a BZT black hole in an anti-de Sitter spacetime. So a connection between the quantum principles of unitarity (or maybe more generally modularity) and a classical field theory which exhibits thermal physics (black hole entropy and Bekenstein bounds etc) exists within this AdS/CFT setting. Yet we do not as yet understand how quantum information is preserved. We just know that it is.

So the quantumal rules of Grgin seem to segue into the picture here. The permitted multiplication rules

(fαg)αh (gαh)αf (hαf)αg = 0

gα(fσh) = (gαf)σh fσ(gαh)

(fσg)σh − fσ(gσh) = agα(hαf)

Connects the Jordan exceptional algebra to a quantum algebraic system. The associator is then by the homotopy equivalence mapped to a quantum group as a system of permutations (related permutahedra) with one set of norms determined by the underlying permutative rules or associahedra and the other by standard rules of complex conjugation in quantum mechanics. So the associator is [f, g, h]σ = agα(hαf) which induces the map between the octonions and the quantion group. This seems like an interesting problem to develop.

Cheers LC

Dear Lawrence,

These few words are only to thank you for your comments. Unfortunately, it's probably not until tomorrow night that I will be able to read your essay carefully, and then your post. You will then hear again from me. I just hope I will have something useful to say.

Regards, Emile.

I should have mentioned that the homotopy principles here are codified by Stasheff polytopes. I have had the thought that operators in quantum gravity may have an underlying geometry or topology which differs from standard QM. The distinction between complex norms and distance norms wiht quantions might be an element of how this happens.

Cheers LC