A Historical Approach to Research in Fundamental Physics by Emile Grgin

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.

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 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...

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 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

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

Dear Emile,

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

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

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