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Questioning the Foundations Essay Contest (2012)
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"1 + 1 = 2" A Step in the Wrong Direction? by Jens Koeplinger and John Shuster
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Author Jens Koeplinger wrote on Aug. 29, 2012 @ 12:40 GMT
Essay AbstractFundamental questions in physics can be asked anytime, anywhere. Often they arise at the interface of physics, mathematics, and philosophy – where scrapping conversation turns into testable hypothesis. This essay explores the idea that the primitive act of counting "1, 2, 3 ..." makes an implicit assumption that ultimately causes some of the challenges faced in quantum mechanics today. A hypothesis for what could be done differently is developed during a humorous, yet serious, conversation among a physics student, a math student, an ex-philosophy student, and a city councilor. Beginning with a physics student's ill-fated attempt at bargaining for a lower price, the essay touches upon beauty in numbers and nature; repetition, inversion, and algebraic closure in mathematics; and observability in quantum mechanics. A surprising property of the complex numbers will be shown to indicate incompleteness or inadequacy in regard to resolving certain questions in quantum mechanics. A new kind of number and arithmetic may be needed, and a proposal for such is sketched using the E8 lattice.
Author BioJens Koeplinger received a "Diplom" (M. Sci.) in physics at the University of Heidelberg, Germany, in 1999. When not exploring possibilities in physical mathematics, he is working daytime hours as IT Systems Analyst for AT&T, and off-hours developing mobile apps at "Dirty Little Cyborg". John A Shuster earned an A.B. in math and economics (physics minor) from Kenyon College (OH) in 1971, then did graduate work in operations research at the University of Rochester (NY). He is a retired Systems Analyst who enjoys grandchildren, travel, writing, and exploring new math systems.
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Joe Fisher wrote on Aug. 29, 2012 @ 19:38 GMT
Dear Messrs Koeplinger and Shuster,
Despite the dismal fact that I do not know an awful lot about mathematics, I loved reading your exceptionally well written, humorous, yet perceptively cogent essay, and I do hope it garners one of the prizes. I hesitate to mention what my picayune quibble with the essay might be lest you might question my motive for bringing it up, but I will risk elaborating on it if you do not mind. In my essay Sequence Consequence, I concentrate on reality. I believe that one real Universe having one real appearance can be perpetually occurring in a real here for a real now in one real dimension once. Real stuff has always to be in one real dimension. I think that if there were three abstract spatial dimensions, it would be difficult if not impossible to determine how abstract stuff was distributed. Would heavy abstract stuff helpingly remain in dimension A, moderate abstract stuff stay in Dimension B, and light abstract stuff linger in dimension C. I prefer to think that only 1 of anything could only ever exist once. Unfortunately, the most confounding illogical code seems to be the numeric representation of numbers. For instance, a single line is used to depict each of the numbers from zero to nine. The number 0 could visibly equal the number1if only the number of lines used to construct both numbers was considered. Does the space inside of the 0 have a value? Is that spatial value greater, equal, or less than the space between the 0 and the 1? There has been a standardization of the measured speed of light. Why has there never been a set standard for the presentation of numbers?
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Author Jens Koeplinger replied on Aug. 29, 2012 @ 22:48 GMT
Dear Joe,
Thank you for your kind words! I'm glad you enjoyed the essay, we sure had a lot of fun writing it. Relating nature's observed geometric dimensionality and magnitudes to abstract defined algebraic dimensionality and numbers is one of the big riddles to be solved, we feel as well. Thanks for sharing your thoughts towards the numbers 0 and 1, and for referring to your essay.
Jens
Frank Makinson wrote on Aug. 29, 2012 @ 19:47 GMT
Jens,
Your essay reminds me of the many times over the past decade that I have been trying to communicate with "System International (SI) loyalists" that their base units are not suitable for "scientific units of measure." One of my communicants even stated "they are totally anthropocentric, arbitrary, and non-natural base units, from the POV of physical law."
My paper had not been...
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Jens,
Your essay reminds me of the many times over the past decade that I have been trying to communicate with "System International (SI) loyalists" that their base units are not suitable for "scientific units of measure." One of my communicants even stated "they are totally anthropocentric, arbitrary, and non-natural base units, from the POV of physical law."
My paper had not been published when I had that one particular communications, he had a draft, but I provided the individual with the IEEE citation after the paper was published. I have his comments on file, but his constant exposure to SI units has completely immunized him to even thinking the SI base units are unsuitable for scientific purposes.
IEEE paper titled, "A methodology to define physical constants using mathematical constants"
IEEE Methdologyor postprint
Postprint MethodologyIEEE no longer allows authors to post the published version anywhere.
E8 based QM theory isn't complete, but it presents a structure that contemporary QM and string theory does not have.
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Author Jens Koeplinger replied on Aug. 29, 2012 @ 22:58 GMT
Hello Frank, thank you for pointing out your research. Aside from formal publication, do you have a reference to freely available material that would give the reader here an overview of your thoughts? You must admit that accusing the IEEE as having an "anthropocentric" bias is a bit odd, given that the customers of an Engineering society are humans after all. It's like accusing Barnes & Noble of selling books. Re "E8 based QM", I wish I knew what that is ... Best wishes, Jens
Frank Makinson replied on Aug. 30, 2012 @ 04:08 GMT
Jens, John,
I did not state the IEEE has an anthropocentric bias. I don't know if the person I quoted is an IEEE member, that used the term anthropocentric, but I do know he is an electrical engineer.
The IEEE publication I cited was the culmination of over ten years of trying to get the concept published. It was rejected by several publications before I submitted to an IEEE publication, and it was rejected. I rewrote the introduction and submitted the paper to another IEEE publication and it was accepted.
Presenting a physical law in the form of two right triangles is not taught in the text books.
The methodology in the IEEE paper disposes of anthropocentric bias in how base units of measure should be derived. Physicists are trying to derive physical laws that govern the characteristics of the universe, and the use of man-defined base units does not help. My topic, 1294, discusses the multi-century assumption that SI units are suitable for scientific units. Even the BIPM admits they are not based upon fundamental physical constants; they don't know how to correct it. My emails to various BIPM officials have never been answered.
The BIPM is a bureaucracy that exists for the purpose of preserving artifacts that represent purely anthropocentric base units of measure. SI units are fine for commerce. Bureaucracies do not take any action that will eliminate their existence.
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Jens Koeplinger replied on Sep. 10, 2012 @ 02:28 GMT
Ok - thank you for clarifying. It would still be good to have a public overview of your work somewhere. It could be as simple as a personal web page or so. I'd be glad to have a look. Thanks, Jens
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Roger wrote on Aug. 30, 2012 @ 03:48 GMT
Another essay on considering the context of numbers as opposed to just considering their value as a number is at
http://fqxi.org/community/forum/topic/1375
Thanks.
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Author Jens Koeplinger replied on Aug. 30, 2012 @ 11:11 GMT
To give the reader information of what you're advertising, your article is titled: "Thought Experiments in the Abstract Field of the Mathematics of Infinities Produce Experimental Artifacts Suggesting That Their Use in the Real-World Science of Physics Should Be Reexamined".
Anonymous wrote on Sep. 10, 2012 @ 07:07 GMT
Another dream where we wake up just before the answer is given.
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Author Jens Koeplinger replied on Sep. 10, 2012 @ 13:58 GMT
:) Well then I suppose we better get to work and find the answer while awake!
Seriously, going beyond discussing a physical assumptions that may be wrong, to actually provide a fully working answer, that would be truly amazing. We decided against writing about published ideas that are developed further. With that, the essay sides in favor of inspiration, but at the expense of presenting a working model. Those familiar with my line of research know of course where the work with John is heading: From the 4D Euclidean quantum gravity model that needed a geometry, to the octonionic background geometry that needed a quantum theory, to the nonassociative quantum theory in one dimension that needs math yet to be determined in order to go higher-dimensional. "Just" about a year ago did I learn about a technique that lets us do exponentiation and differentiation on one-dimensional lattices. In a very optimistic estimate, this essay is a half-way point for our work of making this a reality on the E8 lattice. Next to formal publications (
http://www.jenskoeplinger.com/P ) we're working in a glass house (
http://groups.yahoo.com/group/hypercomplex/ ). Open-source research, so to speak - contributors welcome, to the least we appreciate if you post to our group if you've done related work, or work inspired by us.
Hope this helps describing where we come from in a bit more detail! Thanks, Jens
Rick Lockyer wrote on Sep. 10, 2012 @ 14:54 GMT
Hi Jens,
I liked your essay, but it did leave me wanting more explanation. I think due to the length limitations the story line took up too much space. Knowing you a bit, I am quite sure you could have provided more content. I do appreciate your intent using the method you did, and had mused myself about presenting
my essay within a story line with the theme Crazy Uncle O’s Magical Mystery Tour of Physical Reality. The wife talked me out of it. Just as well, without the prop I had to leave out quite a bit of content I wanted to put in to be able to shoe horn it into 9 pages.
I look forward to announcements on your blog http://groups.yahoo.com/group/hypercomplex/ about further developments. Keep up the good work.
Regards,
Rick
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Author Jens Koeplinger replied on Sep. 11, 2012 @ 13:41 GMT
Hello Rick - thank you for leaving your note. John and I noticed your essay as well. You understand that I am disappointed about seeing no mention, favorable or otherwise, of my analysis of your work (
arxiv:1103.4748 ). It is of course your choice on what to write about, and what to ignore. Jens
Rick Lockyer replied on Sep. 13, 2012 @ 16:07 GMT
Jens,
Trust me when I say I did not intentionally fail to mention your paper, there just was no space available. I chose to discus algebraic invariance in terms of what I called Iso(). I was remiss not to include your paper in the Reference section, and have posted such in my essay blog. This also was not an intentional act, I just tried to give references related to the essay content. It is easy to leave things out of the references, like your omission of
this in the very same arxiv paper. No worries, it is all good.
Rick
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Author Jens Koeplinger wrote on Sep. 14, 2012 @ 01:58 GMT
Ok, I appreciate your note. You wrote at some point about your vision: "Algebra, analysis, topology and groups are interlocking parts. The most fundamental is the algebra, for it sets the tone for the remainder." Your 'octonion variance sieve' can indeed be expressed elegantly using derivation algebras. Those exhibit properties similar to what one would expect from arithmetic. There are a couple of formal bugs in my paper on this part of your work, in its current version on the arXiv at least; but since it has attracted no feedback whatsoever I'm somewhat demotivated towards fixing them. Maybe that explains my negativity.... I do believe that your octonion variance sieve works, and that - for differential equations - there are solution spaces that don't simply collapse into the quaternion case. Best wishes, Jens
Author Jens Koeplinger replied on Sep. 14, 2012 @ 01:59 GMT
(this was meant as a reply to Rick's earlier post)
Joy Christian replied on Sep. 14, 2012 @ 03:02 GMT
Hello Jens,
Please do not be "demotivated" towards fixing the bugs in your arXiv paper. It has been more comprehensible to me than Rick's own writings and explanations (because of my own limitations as a physicist rather than a mathematician, and because of my associative, Clifford-algebraic perspective derived from the works of Hestenes and Lounesto). So, please, do revise your arXiv paper if necessary because it has been useful at least to me. In particular, I would be interested in understanding how the solution spaces for some differential equations do not collapse into the quaternion case. This is not what I would expect from my topological perspective of the octonionic 7-sphere.
Best,
Joy
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Author Jens Koeplinger replied on Sep. 14, 2012 @ 21:23 GMT
Hello Joy - thank you so much for posting!
Regarding the bugs, it's good to see your interest. I'll try to get to them soon. Essentially the problem is that I'm treating polynomial functions and algebras as if they were the same thing. E.g., on the right-hand side of (5.7), a set of functions { f[N], u, v } is of course not contained in the quaternions. Oops - that doesn't make sense. What I meant to write was that the multiplication rules used in the polynomials f are quaternion, therefore making the f[N] quaternionic polynomial functions.
Then regarding where the approach collapses into the quaternions, I admit that my work is incomplete in that I only state in (5.9) that
der( Df ) contained in H
does not necessarily require that the polynomial f is quaternionic. In order to be complete it needs to be shown exactly where f may be octonionic and whether there exist any interesting differential operators D such that (5.9) still holds *and* Df is not already quaternion. Rick is proposing such a construct for his recovery of the Maxwell equations; and I've checked his multiplication rules by hand and found no error. But that doesn't make it formal proof, of course ...
Thank you again for your interest!
Jens
Author Jens Koeplinger replied on Sep. 14, 2012 @ 21:27 GMT
corr: I incorrectly quoted; here's the correct quote:
(5.9) der( Df ) contained in der( H )
Joy Christian replied on Sep. 16, 2012 @ 20:10 GMT
Hi Jens,
Thanks for your explanation. I look forward to the revised version of your paper.
Concerning the topic of your essay, I have a somewhat different take on the connection between quantum mechanics and the division algebras. A summary of my view can be found in the
first chapter of
my book, if you are interested.
All the best,
Joy
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Author Jens Koeplinger replied on Sep. 16, 2012 @ 22:24 GMT
Thanks for pointing out your book - I must admit that I'm familiar with some aspect of your work (though by far not all of it). When you describe correlations on the 7-sphere, in octonion space, I will look in your work for how you propose to recover spacetime. The avenue where your work is intriguing to me goes as follows: Correlations on the 7-sphere describe the complete quantum mechanical...
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Thanks for pointing out your book - I must admit that I'm familiar with some aspect of your work (though by far not all of it). When you describe correlations on the 7-sphere, in octonion space, I will look in your work for how you propose to recover spacetime. The avenue where your work is intriguing to me goes as follows: Correlations on the 7-sphere describe the complete quantum mechanical state of a system. Electromagnetism uses a certain four dimensional subspace that just so happened to be parametrizable by four independent variables. When performing distance measurements between two such sets of four-vectors (four-parameters), these parameters exhibit Minkowskian metric. In my hypothesis (which still is aligned with yours, I believe) human bias assumes such a four-parameter space as foundational and describes the remaining non-electromagnetic forces as deviations thereof. In that thought, certain properties of canonical description of physical law become artifacts (here I am starting to speculate): The Higgs mechanism in the Weak Force or Yang-Mills instantons in the Strong Force, both of which assume existence of more than one vacuum metric. While my understanding of these mechanisms is very poor, I am nevertheless not satisfied an assumption that physical reality could be built on 4-spaces with both Minkowskian and Euclidean metric. I do know that it is not valid to attempt a microscopic understanding of a field theory this way; but I am not satisfied with this, either.
It could be that nature is that way, and I am simply an unsatisfied person. But if my hypothesis were to be true, then - starting with quantum systems on the 7-sphere - I would assume that by keeping some parameters unchanged but rotating others, such rotation would be transitioning through the entirety of physical forces that may exist in reality. Maintaining the anthropocentric bias and assuming electromagnetism as foundational, these forces would then appear as gravity, weak, and strong force we know today.
In 2006/2007 I gave it a first attempt using complex octonions and pairwise multiplication of a differential operator and a wave function. I don't think anymore that it can be made to work into a reasonable quantum theory, and it uses complexified octonions ("conic sedenions") and not on pure octonions. Let me point it out nevertheless and describe in the next paragraph how it could become quantum systems on the 7-sphere, in pure octonion space: http://www.jenskoeplinger.com/P/Paper-Koepl-2006-7v1.pdf . There, a one-parameter angle alpha transitions the Minkowskian Dirac equation into a 4D Euclidean counterpart and back. The nice part of this work is that the resulting force, next to electromagnetism, indeed describes gravity (in the weak-field limit it becomes linearized gravity, as it should). The bad part, of course, is that I've given up trying to make it into a working quantum theory.
However, your work becomes relevant to me if an octonionic exponentiation a^b would exist, where both a and b are general octonions. In that case, the same reasoning from my 2006/2007 papers could be applied to such generalized exponentiation, in which case I would also have a quantum theory that I'm happy with (chapter 3 here: http://arxiv.org/abs/0910.3347 ). Pairwise multiplication c*d and exponentiation c^d have the curious property in the quaternions and octonions, in that these morphisms intersect when using only basis elements: Take e.g. c = i_1 and d = i_2, both imaginary basis units of the quaternions (and octonions). The product c*d is defined as i_3. The exponential c^d is (i_1)^(i_2) which you can define as exp(i_2 * ln(i_1)) which also becomes i_3. All the reasoning from my old 2006/2007 papers would then become valid, only this time on the octonions and the 7-sphere alone from my 2011 paper, with no need to complexify or other ugliness. I would be comfortable in defending such a quantum theory --- Problem is: No such arithmetic for a^b exists today where both a and b may be generalized octonions.
So, you see, I am very much tuned in to your work that describes quantum systems on the 7-sphere. If my speculation is right, then your work proves existence and completeness of such quantum theory. Can we find an arithmetic that works the way I need it to evaluate my hypothesis? There are some indications that such arithmetic would need to be built on the E8 lattice.
Now simply strip out all the parts that we're not sure about, or don't yet, then you essentially arrive at our essay. :)
Fun times? You bet! Jens
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Joy Christian replied on Sep. 17, 2012 @ 04:46 GMT
Jens,
I didn't realize how attuned you are with my approach to quantum theory.
Thanks for summarizing your own views. As Rick noted earlier, your essay does not do justice to all the things you have thought about. I also appreciate the rigorous mathematical standards you uphold before accepting your own ideas. This is in sharp contrast with the cavalier attitude towards mathematics and logic I see in some of the other essays in this contest.
The connection of my approach to quantum correlations with what we actually observe in spacetime can be found in
this paper of mine---see, especially, the construction in equations (111) through (117). In my view, quantum correlations are the *evidence* of the fact that we "live in" an octonionic world. There would be no stronger-than-classical correlations otherwise.
I don't quite understand your demand for the exponentiation of octonions from a physical point of view. But it seems like an interesting mathematical problem nevertheless. Can you please summarize briefly why such an exponentiation is important from a physical point of view?
By the way, you may be interested in checking out Michael Goodband's essay in this contest. His approach is also based on the division algebras---i.e., on the parallelizable spheres S0, S1, S3, and S7.
Joy
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Author Jens Koeplinger replied on Sep. 17, 2012 @ 21:17 GMT
Interesting - thank you so much for the direct reference to your paper. There is one particular interpretation of your work that I'm focused on (and again, please excuse my tunnel vision). I see how you embed the four spin three-vectors from an EPR ansatz into equatorial 6-spheres of your general 7-sphere configuration space. You don't yet handle time in the paper/equations referenced, please...
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Interesting - thank you so much for the direct reference to your paper. There is one particular interpretation of your work that I'm focused on (and again, please excuse my tunnel vision). I see how you embed the four spin three-vectors from an EPR ansatz into equatorial 6-spheres of your general 7-sphere configuration space. You don't yet handle time in the paper/equations referenced, please correct me if I'm wrong. Glancing at your equations (113) through (116) it seems that you have at least one independent degree of freedom left; which is enough for an observer time dimension. You mention GHZ states as candidates for the entanglement scenario you're describing. A more simple proposal could be bound quark states: Ignoring quark/anti-quark states, the smallest bound quark states observed in nature have three constituents. The quark/anti-quark symmetry (if so) could then be driven by an additional time parameter. In your model, it seems straightforward to measure distances between the various n_1, n_2, n_3, n_4: The distances are simply from a 3-dimensional Euclidean metric. I'm not sure how time would enter such a system, and how to do (special) relativity in your model. ... That would be something worth tinkering around for me, to take one of your spins (say, n_1), and try to introduce a time parameter somehow to describe it in relativistic form. Have you done such a thing yet?
Regarding your question why I'm looking at exponentiation: It is purely a proposal at this point. With a modified Born rule ( http://arxiv.org/abs/0910.3347 sections 3.2 and 3.3 ) that requires invariance of the conventional eigenvalue relation (del Psi = m Psi) under changes of octonion algebras, I'm happy with crafting a quantum theory that I would be able to defend as "similar, but simpler" as compared to canonical quantum theory. Theoretical reductionism is key for me, but in order to achieve this I would need a generalized octonion exponential. Here, "octonion exponential" is a placeholder term for a new kind of morphism, to be found, between two octonions. This new morphism must reduce, in the special case of a complex number subspace, to complex number exponentiation. Therefore the terminology choice "exponentiation". If nature were one dimensional, and could be described in such a complex number subspace, and if you were ask for the influence of 1/r potentials on a test particle, then such quantum theory proposal results in what would be the Dirac equation in one dimension (seciton 3.3, equation 3.1). So my ansatz isn't necessarily wrong. Whether anything of this works at all for the description of nature, though, hinges on the existence of such octonion exponential.
Thanks for pointing to Michael Goodband's essay, I will read it soon. Sorry if some of this may take a little while; I'm currently transitioning to a new job, which is consuming a lot of my attention right now as I am leaving my current employer and prepare for what's next.
Best wishes, Jens
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Rick Lockyer replied on Sep. 19, 2012 @ 06:04 GMT
Jens,
This thread is drifting off your essay topic already, sorry for this continuance. You made a comment in this thread that my work recovered Maxwell’s equations. It is quite a bit more than that simple task. I have shown as a subset of the presentation EM current (M.E.), as well as all EM forces, work, energy, energy flux, conservation of energy and momentum. I have shown all forms of...
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Jens,
This thread is drifting off your essay topic already, sorry for this continuance. You made a comment in this thread that my work recovered Maxwell’s equations. It is quite a bit more than that simple task. I have shown as a subset of the presentation EM current (M.E.), as well as all EM forces, work, energy, energy flux, conservation of energy and momentum. I have shown all forms of the Octonion equivalent of the divergence of the 4D stress-energy-momentum tensor are made from the full compliment of algebraic invariants on its general form, no less nor more than expected. Beyond this, I demonstrated how my ensemble derivative directly indicates the EM Lorentz transformation. It is all in
The Algebra of Everything.
The 4D EM anti-symmetric second rank field tensor has 6 independent positions, necessary since the E and B fields are not of the same nature. This remains the case for any attempt to flatten things out. You mentioned needing only four parameters, which may be true if all you mean is 4 potential function components. But this does not imply 4D space (properly access the rank increase), nor does it require a Minkowski metric, clearly from what I have shown, where E lives on top of three O basis elements and B on top of three other basis elements. O covers the 4D tensor approach with leftovers, but the tensor approach can’t yield everything the algebra can, it is not general enough.
I am a bit puzzled by both you and Michael Goodband talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath. The metric for O and its subalgebras is the norm, which is positive definite all + signature so O has no isotropic algebraic elements. You do not get S7 with split Octonions that are not even a division algebra. Perhaps you could explain this sentiment to me. I can see where you are coming from since it is good politics, just do not see how you are going to get to where you seem to want to go.
Electrodynamics is fundamental, which is why I used it as a road map for my Octonion development. Its native algebra is O. The “non-electromagnetic forces” are explicitly given in native rectilinear coordinates within the endnotes of my essay.
Rick
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Author Jens Koeplinger replied on Sep. 19, 2012 @ 14:19 GMT
Rick - you wrote: "I am a bit puzzled by both you and Michael Goodband talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath." - I've double-checked the little note I left on Michael Goodband's discussion. All I am doing is refer him to the work of others that I believe is related ... so I don't know what of my writings you're referring...
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Rick - you wrote: "I am a bit puzzled by both you and Michael Goodband talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath." - I've double-checked the little note I left on Michael Goodband's discussion. All I am doing is refer him to the work of others that I believe is related ... so I don't know what of my writings you're referring to.
Then you wrote: "You do not get S7 with split Octonions that are not even a division algebra." - Sure you can "get S7 with split Octonions", e.g. when supplying Lorentz boosts as in Gogberashvili http://arxiv.org/abs/0808.2496 . So when Goodman proposes that he's working with some decomposition of S7 into S3xS4 or S3x(S3xS1), then that's not a problem to me a priori, on this high level at least. Whether he can actually make it to work is a whole different story, and I'm not competent to comment on that.
You conclude that same paragraph with a statement about me: "I can see where you are coming from since it is good politics [...]" - I'm lost, what are you driving at?
Overall, I am confused how you are - on one side - advertising your approach to recover Maxwell electromagnetism (amongst other claims, of course) from octonions, but - on the other side - criticize others for "talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath". In my analysis of your octonion variance sieve I remained on the mathematical side, without going into physics. The physics part is yours to defend, though I did have a peek. I do understand that you are defining an octonionic action
W = invariant F * j
(i.e., from a generalized octonionic force and flux) where W is built from an octonion differential operator
Del := { d/dx0, ..., d/dx7 }
and an octonion potential
V := { V_0, ..., V_7 }
such that (if I got it right) you define W as:
W = Del ( Del V + V Del ) ( Del V + V Del )
Here, the notation V Del means applying Del on V but using the commuted octonion multiplication rule. It is a product of three terms, but since the 2nd and 3rd term are the same we don't have to worry about nonassociativity. In turn, this almost instantly proves that it is an algebraic invariant under your octonion variance sieve.
Your notation of this is different, with lots of indices; but I think my notation is faithful (or at least essentially right). Your achievement then follows from recovering Maxwell electromagnetism (and more; again, which is for you to defend) from that. This alone is a notable achievement! What confuses me is your criticism of others attempting to recover Maxwell electromagnetism, algebraic split-signatures, or Minkowski metric through other means. Just like yourself, some of these other approaches also start with octonions, and just as you are proposing an ad-hoc action W as I sketched above, these other approaches propose ad-hoc constructions as well.
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Author Jens Koeplinger replied on Sep. 19, 2012 @ 14:39 GMT
I meant Michael Goodband, sorry for the typo.
Rick Lockyer replied on Sep. 19, 2012 @ 16:03 GMT
Jens,
Any point on the unit 7-sphere may be multiplied by any other point on the unit 7-sphere to yield a product that is also on the unit 7-sphere. This is an outcome of the composition rule for Octonion Algebra N(A*B) = N(A) N(B) and the definition of the 7-sphere being all W such that N(W) = 1. The split Octonions to not abide by this composition rule. For normal Octonion Algebra, if N(A) = 0, then each coefficient of A is zero. For split Octonions, you may have N(A) = 0 with non zero components of A, and possible N(A*B) non zero.
On the form of the divergence of the stress-energy-momentum, it is not a double product of Octonion algebraic elements, it *must* be the group of basis element products, indexes as you state, with invariance rules applied. The stress-energy-momentum algebraic element post sieve has terms that would disappear if you simply squared the field elements due to anti-commutation of O. Specifically, A*A has no A5 A6 component, yet from the 4D EM roadmap we expect to have dyadic products like BxBy present.
My definition of Octonion Algebraic Invariance is an outcome of the multiplication rules for normal Octonion Algebra. It has no connection whatsoever to split Octonions. I think I have demonstrated Electrodynamics in O adequately, without anything to do with a Minkowski metric space. I do not believe anything close to this has been done with split Octonions, nor do I believe it ever will.
Hope this clarifies things.
Rick
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Joy Christian replied on Sep. 19, 2012 @ 19:46 GMT
Hi Jens,
Sorry it took me a while to get to your question (too many distractions!).
The question you have asked is actually related to what Rick is saying in his reply to you. It is not kosher to split octonions into 6+1=7 form by singling out a time dimension. You are correct to note that I do not handle time in my papers. This is because time is irrelevant as far as Bell's theorem is concerned. We are only concerned about correlations in space which take place at a given time. Relativistically this would mean correlations among points of a space-like hyper-surface, which is either a 3-sphere or a 7-sphere depending on the dimensions of the quantum system. In the special physical system considered in my equations (113) through (116) we do have one independent degree of freedom left over, and it could indeed be thought of as time dimension. But, as Rick says in his reply to you, I would be careful about breaking up the octonionic structure that way. I have not tried to do relativity because all my energy so far has gone into fighting my critics (I have been doing that for over five years now, with mostly abuse and derision as rewards). Relativization of my framework is certainly something worth looking at. I would be curious to know what you come up with.
Best,
Joy
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Author Jens Koeplinger replied on Sep. 19, 2012 @ 19:56 GMT
Ok, I see where I made a mistake in my attempt at a symbolic form of your generalized action W. Sorry for that, I was writing from memory and should better not do that.
Regarding other work on split-octonions and Maxwell's equations, I'm referencing several from my paper on your octonion variance sieve. There are approaches from direct coordinate products of split-octonions, as well as, spinors on Dirac-like equations on split-octonions. The work from S de Leo should also be quoted (I'll add the de Leo references in the forthcoming revision from the above exchange with Joy). Let me know if you're interested in more details about the different approaches.
Author Jens Koeplinger replied on Sep. 19, 2012 @ 20:18 GMT
Hi Joy - looks like our messages crossed over.
Yes I agree with everything you wrote, and thanks for your quick answer! I won't be that quick with any results, you bet ...
Regarding your critics, sometimes I wonder whether you really need to be fighting all the battles you're in; but then again, I'm cherry-picking from your work so there's not much I'm qualified to say elsewhere.
Octonions are the simple and beautiful structure, fully agreed. We'll do our best not to break it. Here I'm referring back to the work with John, that is subject of the essay :), to eventually find a true octonionic exponential a^b where both a and b are octonion. Such an exponential would have a huge space of automorphisms, with all kinds of subspaces. Hopefully that will lend itself to a well-motivated introduction of observer time.
Thanks again, Jens
Author Jens Koeplinger replied on Sep. 19, 2012 @ 20:24 GMT
PS: I just wrote "find a true octonionic exponential a^b where both a and b are octonion". That was a bit sloppy in reference to the essay. The essay only envisions an arithmetic on the E8 lattice, which is formed by the integral octonions (except for a scaling factor).
Michael James Goodband replied on Sep. 19, 2012 @ 20:41 GMT
Hi Jens,
Thanks for your comment on my essay thread, and your pointers to the work of Geoffrey Dixon and Cohl Furey. Yes, it seems my scenario is related. Furey considers algebra R*C*H*O whereas I consider S0*S1*S3*S7 which obviously has the same underlying algebraic structure. The critical difference centres of my discrete S0 compared to continuous R, which is directly related to the issue of Quantum Theory not being fundamental. see reply Sept 19 for details.
The octonions look as though they are going to figure somehow, the question is just how. Obviously I think S0*S1*S3*S7 because they occur in purely geometric 11D GR without added fields. I claim that this is the form of unification of physics envisaged by Einstein - reasoned dispute welcome.
Michael
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Author Jens Koeplinger replied on Sep. 19, 2012 @ 21:05 GMT
Interesting. Thanks for leaving your note as well. In RxCxHxO the "R" is more or less fashion since you could simply rescale the universe to fit any such R :) - but using it as a discrete qualifier gives you a reflection symmetry, I understand.
If SU(3)xSU(2)xU(1) would really be the symmetry group of the Standard Model, in one shape or another, then I would think your S0*S1*S3*S7 or Dixon's/Furey's algebraic RxCxHxO are the way to go. So there's a good chance that you're on the right track. Personally I'm not convinced that this is the right ansatz; which is of course entirely my problem. I acknowledge SU(2)xU(1) from the electroweak force, of course, and SU(3) in itself from the Strong Force. But can they be really unified using conventional Field Theory? In my feeble understanding of the Higgs mechanism in the weak force, and Yang-Mills instantons in the Strong Force, this appears as if there is another "outer" symmetry (which could be as simple as a U(1) rotation or even a reflection) that isn't just a gauge group symmetry, but instead something that acts on the very base manifold that the Field Theory appears built over: Transitioning between Minkowskian and Euclidean spacetime geometry. With this concern, I literally don't want to start learning Field Theory properly, which leaves me poorly educated and essentially without formal help or substantial argument in physics.
So ... I wish you good luck! You've got a chance, no doubt. Best wishes, Jens
Thomas Howard Ray replied on Sep. 21, 2012 @ 11:57 GMT
Hi Jens,
Reading through this thread, I had flagged your comment to Joy, "Glancing at your equations (113) through (116) it seems that you have at least one independent degree of freedom left; which is enough for an observer time dimension."
Then reading on, I see that Joy replied to it. Indeed, I think this is a critical feature of Joy's framework; I know he has refrained from discussing the relativistic aspects of his research -- for me personally, however, it was necessary to verify early on that the framework is fully relativistic before agreeing that it is foundational. I do find it fully relativistic. Just as Perelman proved the Poincare Conjecture for S^3 without ever mentioning the conjecture (or having to), Joy has managed to incorporate relativity without ever mentioning spacetime.
On the matter of the extra degree of freedom imparted by the topological model, I hope you will be interested in an attachment I posted yesterday on my essay site, which explains the case in simple arithmetic. I am anxious to see if there might be gaps or hidden assumptions.
All best,
Tom
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Author Jens Koeplinger replied on Sep. 21, 2012 @ 16:20 GMT
Hello Tom - sorry I haven't gotten around to leaving a note on your essay! I'll also have a look at the attachment that you point out.
As Joy also cautioned, my thought of looking for time in his model may not be a valid one. Yes, he does write about spin orientations { n_1, ..., n_4 } at that section of his paper, and he constructs a set of four 7-vectors that have locally Euclidean metric...
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Hello Tom - sorry I haven't gotten around to leaving a note on your essay! I'll also have a look at the attachment that you point out.
As Joy also cautioned, my thought of looking for time in his model may not be a valid one. Yes, he does write about spin orientations { n_1, ..., n_4 } at that section of his paper, and he constructs a set of four 7-vectors that have locally Euclidean metric on the 7-sphere. But only because that part of it resembles the locally Euclidean metric of our familiar 3-space doesn't mean that you could interpret it as some form of observer space, and find an independent observer time such that we have a description of the quantum system in a frame of reference in the sense of special relativity. Joy hints at a hypersurface to S7 to get this done, which would leave pure octonions. That would be ugly, as we all agree.
But then again, coming from the other side and supposing that Joy's model is a special case in that all four spins are modeled at the exact same observer time, for all observers (as there is no observer time in Joy's model whatsoever), then that necessitates that all four spins are at the same point in space. Rather than modeling time as an extraneous construct, through hypersurfaces or other bolt-on constructs, I would be interested in twisting those four spins against one another. With that I would start with only two spins, model them as Joy does, and then "pull them apart". As you know, once you create a distance between any two objects, in the sense of special relativity, you must define your frame of reference that then gives you a dynamic but distinct observer time in that reference frame. What may appear as four co-located spins { n_1, ..., n_4 } in Joy's ansatz would become the extreme case of total co-location; where pulling these spins apart would appear not as pretty, as the result of being forced to define an observer frame of reference. While the formulas would start to look ugly, it would be conceptually simple.
It might not work. Worth a try, though, I think.
Best wishes, Jens
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Yuri Danoyan wrote on Sep. 14, 2012 @ 18:05 GMT
Hi Jens
After reading your essay i would like to send my observation
http://vixra.org/abs/0907.0014
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Thomas Howard Ray wrote on Sep. 18, 2012 @ 23:22 GMT
Hi Jens,
I wish I could find some young students who really do have a dialogue like that. :-) Maybe I shop in the wrong stores.
Delightful essay! Going to spark a lot of conversation here, I predict -- and certainly most relevant to how physics and mathematics intersect. I hope to have something of substance to say later; a little pressed right now.
Best wishes in the contest -- hope you get a chance to visit my own essay site.
Tom
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Author Jens Koeplinger replied on Sep. 19, 2012 @ 01:25 GMT
Thank you, Thomas! Will do. Octonions are around since many generations, we should not need to be in a hurry even if they seem to become more fashionable again.
Hoang cao Hai wrote on Sep. 26, 2012 @ 07:43 GMT
Dear Jens Koeplinger and John Shuster
Very strange!like a fable.
If two you can give such a direction, why not boldly gives a new theory such is sketched using the E8 lattice (whether a expected)?
It looks like you a little lack of confidence in yourself?
Kind Regards !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
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Author Jens Koeplinger replied on Sep. 26, 2012 @ 17:41 GMT
Dear Caohoàng,
Thank you for leaving your thought provoking note. You suggest that it might be lack of confidence that caused us not to boldly sketch a new theory on the E8 lattice. To me it is more a reality check that assures me that most all suggestions for new physics that we can come up with, at any time, are wrong. This assurance comes from the mere number of possibilities out there. In order to keep this essay both entertaining for a wide audience, but also to make a strong point nevertheless, we focused on what it most important to us: To sketch a sense of naturalness, beauty, and simplicity as motivating some of our current and future work elsewhere. We did that at the expense of actually proposing a model, granted, but isn't it amazing with how few and simple assumptions you necessarily arrive at the E8 lattice? Repeat, invert, closure; and self-duality of the space under addition and multiplication. Lattices in 1, 2, 4, and 8 dimensions satisfy these very simple assumptions; and only in these dimensions. The E8 lattice is of course an immensely complicated construct if you attempt to understand its properties and automorphisms - something I will never succeed in fully. But conceptually it is *that* simple. Pure math at its finest, we feel, and that's what we want to convey to the reader.
Best wishes, Jens
Sergey G Fedosin wrote on Oct. 4, 2012 @ 05:56 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
and
was the quantity of people which gave you ratings. Then you have
of points. After it anyone give you
of points so you have
of points and
is the common quantity of the people which gave you ratings. At the same time you will have
of points. From here, if you want to be R2 > R1 there must be:
or
or
In other words if you want to increase rating of anyone you must give him more points
then the participant`s rating
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
Sergey Fedosin
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Author Jens Koeplinger replied on Oct. 5, 2012 @ 02:25 GMT
Hello Sergey - yeah, the ratings ... with almost 300 submissions I estimate that any community vote will favor known authors or known topics, or both. This is not out of boredom or disinterest or dishonesty of the voters; but stems from mere statistics when overwhelming a decision engine with too many choices to evaluate. Forced to employ some kind of efficiency scheme, the likely pattern of an interested reader (and voter) would be to review topics that sound the most interesting - which in turn include a disproportional amount of known authors and topics. I projected that if I were to ring the advertising bell really loud here and attempt to create more visibility, people would still not really have the time to read and evaluate the essay - instead, we would receive a well-meaning "7" at best, which would give us no chance of reaching the top 12% (to arrive in the first 35 essays that are planned to be considered for an ordinary prize). But all of that is OK - for one there's always the off-chance for a special commendation prize. But much more so, we are very satisfied to have communicated our research vision to the few people who we wanted to reach. Small group work has been my preference always, which makes this essay contest so valuable: For one we reached the handful of people who expressed interest in our work; and for the other we reach the other handful who might be interested but isn't quite keen to show face yet. Imagine you're working on something as remote as we are, and there are 10 people who actually care! To me, that is a big achievement. Best wishes, Jens
Member Benjamin F. Dribus wrote on Oct. 5, 2012 @ 04:41 GMT
Dear Jens and John,
What a lot of profound topics you weave into your story! The following thoughts come to mind:
1. I guess the “loss of information” involved in addition is a very general aspect of “noninvertible morphisms;” for example, maps that aren’t injective (one-to-one). It’s interesting to regard this as a foundational problem and a viewpoint I hadn’t considered in this explicit way! After all, the superposition principle is an example of this, and superposition occurs even for classical waves. But your analysis goes much deeper than this…
2. For the logarithm function (and other similar functions), the usual way of dealing with this in complex analysis is of course to use Riemann surfaces; this was one of the ways in which such objects were first introduced. These play a striking role already in quantum information theory, but this seems like a new possible application.
3. The suggestion of introducing internal structure (in this case for purposes of distinguishability) is embodied in a cutting-edge area of abstract algebra that hasn’t yet been properly applied to physics. This is the theory of “categorification,” in which elements are elevated to objects; for instance, lattices. I have written about this near the end of my essay
here; it might interest you.
4. This specific use of root systems of exceptional Lie groups is something I have not seen before. It’s a good idea, regardless of its ultimate scope of applicability.
Yours is one of the few submissions that earns a solid “10” from me. Thanks for the great read! Take care,
Ben Dribus
P.S., Regarding your previous comment, I'm sorry you didn't "evangelize" more actively... your title didn't stand out to me, and I read your essay only because I read them all. Hence, you nearly missed out on a thoroughly deserved top rating. Yours is about the 245th I've read, and it's one of the best in the contest.
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Author Jens Koeplinger replied on Oct. 8, 2012 @ 15:12 GMT
Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric...
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Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric hypothesis, but let me first respond to your note.
Regarding Riemann surfaces, as you know of course they work great when modelling physical systems where the configuration space is locally Euclidean. Physicists also believe they're great for locally pseudo-Euclidean spaces; all conventional description of fundamental physical law is built on such after all. When using lattices as configuration space, in contrast, we need new mathematical tools to do analysis: Lattices are made from countably infinite objects, and when comparing lattices against one another you need to embed them into some kind of encompassing space. Taking the E8 lattice, a natural embedding would be the eight dimensional Euclidean space over the reals, R8, and you could then shift and rotate instances of such lattice against one another in R8 and do math. You know all of that, of course, I am just summarizing. The whole thought on lattices came last year when I began studying a mathematical concept developed by Prof Mark Burgin (UCLA), which he calls "hypernumbers and extrafunctions". He generalizes the concept of "number" to infinite sequences, and defines rules for comparison, arithmetic, differentiation, and integration. Using his concept, I played with making an exponential function where "1 ^ (1/n)" would have all those "n" points on the unit circle as their solution set where the point taken to its "n-th" power would be 1. Such solution set of "n" points should, in turn, be a single "number". Burgin's hypernumbers do the trick, since he allows alternating or even divergent sequences to be understood as a single number. The set of convergence points for the sequence is what he calls the "spectrum" of a number, and with that you can model a single number where the spectrum of "1 ^ (1/n)" indeed are a set of "n" points on the unit circle. The amazing result - to me at least - is that for "n --> infinity" the spectrum of "1 ^ (1/n)" becomes the topologically closed unit circle in the complexes!
Currently I am writing a paper about this, but I am very slow in math so don't expect things to turn up quickly. All of the actual material is scattered out across my little online group (e.g. the 10th topic post, "Burgin 10", contains a summary http://tech.dir.groups.yahoo.com/group/hypercomplex/message/
1101 but also note the follow-ups with corrections ... I really do need to write this into a self-contained, intelligible paper, sorry for not having anything better at hand right now). Being able to make such a generalized exponentiation of an expression (a/b) ^ (c/d) where a, b, c, d are positive nonzero integers, this gave us a lot of hope since lattices in any finite dimension are made from countably infinite points as well; and Burgin sequences can always be enlarged by more (countably infinite) subsequences without leaving their realm. With that, Burgin hypernumbers appear powerful enough for modeling generalized octonionic arithmetic on the E8 lattice. That gave us the needed motivation that such a thing could even exist. So, here's what'll happen next: I do my homework and write a little paper on that, put it in the arXiv, contact Prof Burgin and other mathematicians for comment, and then go from there. Estimated time to completion: Whenever it's ready ... this is "open-source research" after all, contributors and claims of ownership welcome :)
You mention "categorification", which is an interesting field in mathematics to be looked at for use in physics, as you wrote. John and I had looked at category theory a bit a couple of years ago, but found that it may not be a good fit. There is an associativity requirement on morphisms that seems too restrictive for what we want to do, and in turn when dropping associativity from categories you end up with even less mathematical structure. Categories are already so general, so wide in what all they could encompass; we didn't want to explore weakening its definitions even more. So we opted for working on specific examples of number systems, rather than attempting to understand what more general framework they would fall into. One of our works, "W space", can indeed be restated in compact form as a primitive 2-category. Here's a preprint version of our work: http://www.jenskoeplinger.com/P/PaperShusterKoepl_WSpace.pdf
-- If we would describe it as a 2-category, then the paper would just be one page! So obviously, categories can be helpful. There was another reason for making such a chatty paper, namely that we connected to rather colorful prior research and felt we needed to spend the extra time to put things on sound feet first, by themselves. A second number system that we defined, "PQ space", cannot be described as a category ( http://www.jenskoeplinger.com/P/PaperShusterKoepl-PQSpace.pd
f ) since the functor on the morphisms "+" and "x" would not be associative. That ended our interest in category theory, for now at least.
Regarding your causal metric hypothesis, let me think that through and then post on your essay thread. Thanks again for writing! And best wishes, Jens ( & John)
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Eckard Blumschein wrote on Nov. 4, 2012 @ 16:35 GMT
Dear authors,
While my
conclusions are rather contrary to yours, I agree with you on that some fundamentals of mathematics may play a crucial role in physics. Please feel challenged to factually object to my arguments.
The style of your essay reminds me of a book by Detlef Spalt: "Vom Mythos der Mathematischen Vernunft" Wiss. Buchgemeinschaft: Darmstadt 1987.
May I ask you to comment on Spalt's opinions too?
Eckard
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Eckard Blumschein replied on Nov. 4, 2012 @ 16:37 GMT
Author Jens Koeplinger replied on Nov. 4, 2012 @ 22:23 GMT
Dear Eckard, thank you for leaving your note, and for referring to your essay and the work of Spalt. I'll have a look. Of course, until proven right or wrong, everything is opinion. Best wishes, Jens
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