• [deleted]

  • Post #23

To follow up with the historical aspect of your essay, I think that in the ancient world the concepts of the universe beyond the Earth mirrored the type of building activity of the time. Hellenic and Roman construction was static for the most part. People built large structures which were long lasting. Of course if one goes back further in time this is even more the case, such as the Egyptian pyramids --- which will last in some part for up to 100,000 years. Ancient cosmology, such as Ptolenmy's world was geometrically fixed for eternity. In later times the tendency was more process oriented. Before Kepler watch making, optics, and other machines were a growing industry. Which mirrors the emergent physics of the time which was more process modelling.

The later physics of the mid-19th century to the current time is motivated by the need to unify distinct categories. In many ways this was accompanied by Charles Darwin, who demonstrated how distinct species of life were related to each other. Maxwell with his displacement current, Einstein with his unification of time and space, and then gravity with spacetime ilustrate the trajectory of field theory. Quantum mechanics might be seen as a generalization of the least action principle, where the variation in a path is replaced by a wave mechanics and interference principles.

Cheers LC

  • [deleted]

  • Post #24

Emile,

your essay needed to be exhaustive since you chose the historic path. History often does not repeat and one only can learn the right lessons from the mistakes made. Innovation and excellence usually comes with unbiased attitude and freedom of approach. Physics has hardly been in the limelight as one finds Nobel awards going for lifetime achiements, etc. May i suggest that freshness of ideas is what is required. My guess is these may well come from Cosmological ideas, as several unexplained facts are hanging around there that have connection with fundamental aspects of Phyiscs. Particle Physics approach is escalating in costs while the early universe study contains all that knowledge that we now require. i am sorry if i have presented these ideas, in contrast to your emphasis on the historic approach. In fact that is a normal or usual approach and that is why we are producing so called new research that is merely like corollaries to what is already known, no path breaking ideas are emerging!

Dear Narendra Nath,

I wholeheartedly agree with three of your mutually related statements, so much so that that I see them as a theorem:

(1) History does not repeat itself.

(2) We can learn lessons from past mistakes.

(3) We need fresh ideas.

Proof:

Any good idea that has already generated a paradigm shift has been squeezed dry of fresh juice. On the other hand, a barren idea could not have been exploited to the hilt because there there was nothing in it to begin with. It remains youthful forever -- and keeps coming back. Only a fresh idea has a remote chance of being fruitful.

Quod erat demonstrandum.

This is why I look at history to make sure I would waste no time on either good ideas or bad ideas. It follows that I have no choice but to follow fresh ideas. Eventually, others will decide whether my work belongs to the good heap or the bad heap, but that's not my concern. The main thing is that the interim be mine: I thoroughly enjoy what I am doing.

Thanks for your post. Emile

PS. Freshness is a necessary condition, not a sufficient one.

Dear Ray,

In preparation for reading your essay, I was just going over the posts related to it when I noticed your note to Florin:

{

Emile's ideas are interesting. It will take time for me to process these ideas. Off the top of my head, I expect to need pairs of quantions to rewrite the Dirac Equation in terms of quantions instead of gamma matrices. Still, a pair of quantions is only four dimensional.

}

Since you don't currently have access to my book (Florin has that advantage), let me clarify the relation between Dirac's equation and quantions.

First: The approach.

It is not a question of re-writing Dirac's equation in terms of quantions (the first time around, I wouldn't have known how to begin), but of obtaining a quantionic equation of motion from some explicitly stated axiom, or principle. There is only one such principle that brings differentiation into the algebra of quantions. I call Zovko's interpretation (it is simply the extension of Born's interpretation to quantions). It yields a unique quantionic field equation that has an arbitrary mass parameter and four real Minkowskian vector potentials.

Second: The relation to Dirac's equation.

Having a quantionic equation of motion is nice, but where is the beef? Who says it has anything to do with physics? To find the answer, write both equations (quantionic and Dirac's) in matrix form and side by side. The relationship jumps out of the page: A quantion is equivalent to a Dirac spinor, but not trivially: The relationship involves complex conjugation.

Third: An insight.

The properties of quantions are such that it makes a lot of sense to view the algebra of quantions as a number system. Now, a field which has only one degree of freedom in the underlying number system is called a scalar field. Thus, Dirac's spinorial equation is equivalent to the unique quantionic scalar equation that implies the equation of continuity. Therefore, in quantionic physics, a fermion is the simplest object. It is a scalar particle.

So much for this.

In reading your essay, I noticed a point of contact with a problem I still have in the quantionic approach to the classification of particles. I will talk about it on your page when I finish reading your essay later today.

  • [deleted]

  • Post #28

Emile response to my comments appear to surprise me as to what he means by freshness in a historic approach. May be he has some hidden freshness that i have missed in his detailed presentation. i am definitly getting old for freshness and so Emile may well be correct as i miss the freshness of ideas he has presented!

Narendra,

Quoting you:

" May i suggest that freshness of ideas is what is required."

Yes, you may suggest it, and you did suggest it, and I said I wholeheartedly agreed.

Moreover, I thought we both meant freshness of a physical or mathematical idea that yields theoretical results not obtained by less fresh ideas already investigared. If you meant something else and I did not correctly guess what it is, then (1) I apologize for wasting your time, and (2) bow out of the discussion because, whatever it is, it would be outside my very narrow area of expertise.

Best regards, Emile.

  • [deleted]

  • Post #30

Dear Emile,

I wrote the post below to Florin Moldoveanu. I have only been introduced to this concept at this point. There seem to be deep connections with Jordan algebras and I think this might be some aspect of how to work with E_8 physics. I don't have your book, and at this time I am just introduced to these ideas So I don't know if you might agree or disagree with what I write below, but I figure I might as well try to communicate a possible insight. Cheers LC

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

  • [deleted]

  • Post #31

Dear Emile, i appreciate your attempt, no doubt about it. In your comment, may i say that ideas are mainly conceptual and not mathemetical or phsical. The tools we use in Physics may be mathematical or observational.You have no need at all to bow out of discussions. It is the way we all learn irrespective of our age. Stopping from participation shows some avoidance on your part to accept an alternate view or suggestion. Let us keep our options always open and enjoy the life with its usual ups and downs. best wishes and all possible encouragement from my side. May be you try going through my essay on this website and i will appreciate your critical comments/analysis of the same very much.

Dear Narendra Nath,

I apologize for not having stated why I cannot get involved in discussions outside my narrow field of interest. The reason is very prosaic: I am currently pressed for time.

I read your essay, as well as many other ones, but taking a little time for reading is not the same as taking a lot of time for actively participating in discussions on different subjects -- which I would do with pleasure in some less hectic time of my life. What you say about keeping our options open is certainly true. This is not where my problem is.

With my best regards, Emile.

Dear Emile

Wonderful Essay, ..and essential conditions of structural unification regarding Symmetry, Completeness, and Irreducibility.

I'm also impressed by our support of learning lessons from past mistakes, and of the need for new ideas.

One area I wonder if you could consider is that of reviewing abandoned theories. A postulate may be rejected on evidence which later proves mistaken. In historical research on a fundamental model I've been developing I came across the matter of stellar aberration, cited by Lorentz and others as disproving an important postulate originating from Fresnel (author of the original STR equation), but later proved wrongly so. By then physics had moved on and the case for the prosecution not reopened.

My rather too light hearted essay495 touches on this, but, if you are interested in it's genuine potential for unification in line with and support of your own philosophy, please go to; http://vixra.org/abs/0909.0047

I'd be most honoured of your view on compliance with your postulates.

Peter Jackson

Dear Lawrence,

This is response to your post of Sept. 29, 03:25 GTM.

Let me begin with a piece of reverse advertisment concering my book. If you are now working on something else and intend to get to quantions later, I would suggest you wait and then contact me before looking for the book. I might have an update by then. The book is not 'wrong', it's just 'old fashioned'. For an animal as young as quantions, two years is not not a short time.

In my essay, I emphasized the importance of a phase I referred to as "polishing", or "finalization" (I don't remember which word I used). Briefly, a new theory is usually formulated in the mathematical formalism the author has at hand, but before going ahead with it, it makes sense to develop a formalism that does most of the work for us. After a couple of passes, I now have a 'sufficiently optimal' one for quantions. For example, the derivation of the quantionic version of Dirac's equation using matrix algebra is spread out over 40 pages in the book, but comes out of a few lines of linear algebra in the new formalism. This is not just about saving paper: It makes the deductive distance between the principle at the source of the quantionic Dirac equation (I call it "Zovko's intepretation") and the equation herself so short that it comfortably fits into our brain's working memory. We then have a much better understanding of what's going on and the correct intuitions for going ahead with new research. An since I am always afraid of having made an algebraic error, or of having overlooked something, it's quite a relief to see, as in the above example, that the new elegant derivation and the original messy one agree in their results.

********************

Yes, twistors and quantions are related --- the way two branches of a tree are related: They come out of the same trunk, but then have their own lives. This is worked out in detail in my first book ("The algebra of quantions" 2005). Here is a synopsis:

The geometric track:

Slightly rearranging history for the sake of conceptual coherence, we may say that Roger Penrose started with the conformal compactification of Minkowski space, whose invariance group is SO(2,4). Since the Lie algebras so(2,4) and su(2,2) are isomorphic, Penrose started investigating the pseudo-unitary space of quadruples of complex numbers {u,v,w.z} whose norm is defined as uu*+vv*-ww*-zz*. These are the original twistors. They are to the conformal group what Pauli spinors are to the rotation group.

The algebraic track:

The composition principle applied to the abstract structure extracted from classical mechanics yields, as a unique abstract solution, an algebra Petersen and myself called "quantal algebra". See:

"Inherently Relativistic Quantum Theory" Part I. The Algebra of observables. Fizika B (Zagreb) 10 113-138 (2001). On line at http://fizika.hfd.hr)

Given an abstract algebra, the next step is classification, which consists in finding all its concrete realizations. This is done in

"Inherently Relativistic Quantum Theory" Part II. Classification of solutions. Fizika B (Zagreb) 10 139-160 (2001). On line at http://fizika.hfd.hr).

The Lie algebras of the solutions are su(n), so(6), and so(2,4). There is nothing else (unless I made a mistake -- but probably not, because the proof, which is a modification of Cartan's proof, is not very complicated). The solution so(6) seems spurious to me, but it would be interesting if someone else saw in it something I missed. The solution so(2,4) brings us into the world of relativity. This is encouraging because the starting point was the abstract structure of observables. Not only that relativity was not postulated, geometry was not either. Yet, there is no mystery. An abtract simple Lie algebra makes no reference to geometry, yet Cartan's classification brings all spaces with a Pythagorean (or pseudo-pythagorean) metric to the light of day.

Given su(2,4), one can continue along the twistorial branch. But there is also another branch, which takes us from the conformal group SO(2,4) to the Lorentz group SO(1,3). This can trivially be done geometrically (by freezing two dimensions), but the quantal algebra offers a non-trivial option. It is a non-standard complexification possible only because the metric (+,+,-,-,-,-) admits an imaginary unit sqrt(-I) which is different from sqrt(-1). The latter leads to complex numbers, the former to quantions.

Coming back to your question concerning the quantion/twistor relationship, I just described their common algebraic source. They are brothers. Will they meet again in the future? I don't have the silghtest idea. If they do, I will be very pleased. It seems to me that they parted company like the brothers Brutus and Cassius in Shakespeare's Julius Caesar with the words:

And whether we shall meet again I know not.

Therefore our everlasting farewell take:

For ever, and forever, farewell, Cassius.

For if we do meet again, why, we shall smile;

If not, why then, this parting was well made.

********************

Quoting you:

" I think the composability requirement maybe satisfied for the exceptional algebra and its anti-Hermitian pair under G_2 holonomy."

If my proof is correct, no. But even at the cost of being embarrassed for having overlooked something, I would prefer it if you were right. It could be a source of some interesting physics. Beyond this, I have nothing useful to say.

Best regards, and thank you for bringing up this discussion.

Emile.

  • [deleted]

  • Post #35

Dear Peter,

I am very happy to hear (I mean read) that you like the three conditions of structural unification.

I put your paper on my reading list for tomorrow, bu I'll probably get to it tonight. I just hope I will have something useful to say about it. I know I will learn something, at least from the reference you mention.

As for reviving old ideas discarded for the wrong reason, this is exactly what quantions do. It was not my idea to extend the field of complex numbers at the foundation of quantum mechanics to a structurally riched number system. I saw in Adler's monograph on the subject that the idea of substituting quaternions for complex numbers is almost as old as quantum mechanics itself. It just happens (for a good structural reason) that quaternions do not support a unification with relativity. By itself, the idea of generalizing the number system of quantum mechanics does not tell us how to do it. One could only try the systems already known in mathematics -- and the field of quaternions was the only one that did not complain. While quaternions did not work, I was sufficiently familiar with the attempt to recognize, very early in the game, that quantions ought to be viewed as a number system. This worked out -- so far at least, but there is still much to be done before we can be sure that the approach will not hit a very thick brick wall.

Regards, Emile.

  • [deleted]

  • Post #36

Dear Emile, dear Lev,

Perhaps I am the only lonely one who tries to revive a mathematics beyond numbers in my essay. Be not misled by the matter of inner ear.

Regards, Eckard

  • [deleted]

  • Post #37

Dear Emile,

On page 7 of your paper, you say "The algebra of quantions admits a derivation operator. Such an operator, D, is needed in equations of motion. Its two main properties are linearity and the Leibnitz identity D (FG) = (DF) G F (DG); where F and G are quantionic fields. Formally, D must be a quantion, but since the product of quantions is not commutative, the Leibnitz identity cannot be satisfied if F and D belong to the same algebra. This problem does not arise, however, because the algebra L is pared with a dual algebra, R; which commutes with it. Fields belong to L while D belongs to R: This is a theorem, not a matter of choice. It is analogous to the geometric duality of contravariant vectors and the covariant derivation operator." In my own work, the initial state fermion seems to be an 8-dimensional subset of a 12-dimensional framework, the interaction boson can have dimensionality up to 11, and therefore the final state fermion must be a different 8-dimensional subset of the 12-dimensional framework. Furthermore, bosons exist in a reciprocal space (compare the "Charges" in my Table 4 vs. Table 7) to fermions. Perhaps your geometry, my geometry, and the contravariant/ covariant geometries are related (although different dimensional sizes).

On page 8, you say "It follows that the Minkowski space generated by the algebra of quantions is more structured than the Minkowski space of relativity: It has an intrinsically distinguished arrow of time." It is good to have an arrow of time. Does the arrow of time look different for initial state fermions versus final state fermions?

I enjoyed your paper, but would have enjoyed more quantions. Now I need to look up these other references, such as arXiv:0901.0332v2 and your book.

Have Fun!

Ray Munroe

Dear Ray,

Of course, the first thing I noticed was a typo in the section you quoted. It sould be "...the algebra L is paired with a dual algebra, R;..", not "pared". I hope this will be obvious to everyone.

Well, it is certainly possible that the three geometries you mention are related. Now, if your geometry is related to Dirac's spinors, then it is related to quantions because 4-spinors and quantions are mutually related.

Concerning the arrow of time, the answer is easy: There is only one arrow of time. It come from the algebra of quantions, not from fermionic fields.

The quantionic approach is not sufficiently advanced to encompas interactions. Therefore, we cannoyt yet speak of initiaal and final states.

Concerning my book, what I said to Lawrence in my last post is true for all: Don't hurry to order it if you are not going to read it soon.

Regards, Emile.

  • [deleted]

  • Post #39

Thanks for the references. I am interested in the connection with twistors since both quantions and twistors involve SO(4,2) ~ SU(2,2). I have found that the E_6 group is involved with twistor-like equations formed from the Hermitian and anit-Hermiain Jordan J^3(O), which are intertwined with the G_2 group. This appears to be some sort of generalizaiton of the rule

= (2ħ)^{-1}(G(ψ, φ.) iΩ(ψ, φ)

for G(ψ, φ.) iΩ(ψ, Jφ), J^2 = -1, J = G^{-1}Ω. The two terms correspond on the algebra level the hermitian and antiHermitian J^3(O), With the G_2 holonomy this defines teh E_6 realizatio of the Jordan algebra J^3(HxO), and the space of physical states are projective twistor spaces. I will try to write this up in more detail in the forth coming days.

Cheers LC

  • [deleted]

  • Post #40

I keep forgetting this editor has fits and starts over carrot signs so this is meant to read

(ψ, φ)= (2ħ)^{-1}(G(ψ, φ) iΩ(ψ, φ)

LC

Hi Lawrence,

I don't know if it's a slip on your part or if I've been misunderstood. In case it's the latter, I'd like to fix it here. You say:

" since both quantions and twistors involve SO(4,2) ~ SU(2,2)."

Correction:

both quantions and twistors involve SO(4,2).

but only twistors involve the isomorphism SO(4,2) ~ SU(2,2.

Bye, Emile.

  • [deleted]

  • Post #42

The covering rule I included as a matter of fact. While it might not be explicitely used in quantions, it might still play some sort of role.

Cheers LC

  • [deleted]

  • Post #43

Dear Emile,

I understand about your book - this is continuing research, which is why I chose to post my book with a free partial preview. In my mind, my book's research is never finished, and the wording is never perfect. Nonetheless, your book is reasonably priced and might help my research.

Did my analogies help you to understand SU(3)? My references 3 (posted on my site) and 11 might be more helpful than my book. Ironically, I had trouble with "Hyperflavor" SO(2,4) (Section 7.2 of my book is wrong on the particular Lie algebra).

Good luck in the contest!

Ray Munroe