Essay Abstract

Operator quantum mechanics on non-associative algebra aims at modeling existing forces of nature on a genuinely non-associative background. We highlight select findings, and discuss time (and space) as emergent parameters. Key characteristics of quantum mechanics will be put in question, namely time asymmetry, indeterminism, and uncertainty. In this essay, these are hypothesized to result from an unnatural, human-centric desire to ask for the result of a measurement outcome as a function of time. Supporting arguments come from findings on octonionic algebras: One may model unobservables (as opposed to "hidden variables"), relate Heisenberg uncertainty and the light cone geometrically, and describe four dimensional Euclidean operator quantum gravity. A new unified treatment of isospin and space-time symmetries is envisioned, where observables cannot be distinguished anymore from the parameters which describe them. Time would emerge as a parameter only when asking a quantum system for information, whereas a speculated master symmetry would not be built on a space-time (or higher dimensional) manifold anymore. - Acknlowledging the speculative nature of some of these claims, we supply a good dose of self-irony as well.

Author Bio

Jens Koeplinger received a "Diplom" (M. Sci.) in physics at the University of Heidelberg, Germany, in 1999. At daytime hours, he works as IT Systems Analyst for AT&T in Greensboro, NC, USA. Vladimir Dzhunushaliev is Professor for Theoretical Physics at the Kyrgyz-Russian Slavic University in Bishkek, Kyrgyz Republic. Merab Gogberashvili is Professor for Theoretical Physics at the Andronikashvili Institute of Physics and Javakhishvili State University in Tbilisi, Georgia. Both (V.D. and M.G.) are Senior Associates of the Abdus Salam ICTP.

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Hello Mr. Köplinger

loved the clock! how'd he do that?

very fun. :-)

thanks,

matt kolasinski

Hello Matt - thanks for asking! I took a public-domain version from Wikipedia, and used The Gimp to modify it. The font for the digits is Minisystem. Thanks, Jens

There's a wrong reference pointer in the submission; on page 7, it should read "(this and the following sections per [7])" instead of pointing to ref [6]. Apologies for any inconvenience. Thanks, Jens

Dear Jens et. al.

Very interesting and entertaining work! Nice presentation of octonions. I definitely agree that quaternions, octonions, and Clifford algebras must play a role at a fundamental level.

Good luck with your research.

Cristi Stoica

“Flowing with a Frozen River”,

http://fqxi.org/community/forum/topic/322

hello Jens,

re:

"I took a public-domain version from Wikipedia, and used The Gimp to modify it. The font for the digits is Minisystem."

i've just recently started exploring Gimp.

i'll have to remember that one next time someone asks what time it is.

thanks for your reply.

:-)

matt.

Dear Cristi - thank you for your kind words, and your well-wishing. I'll pass it on.

Dear Matt - :))

Thanks, Jens

I loved this essay. I am curious if the generalized uncertainty principle can come from your method or did you just focused on the Heisenberg uncertainty principle? Thank you for leaving consciousness out and discussing only the physics. Stay humble physicists and leave that stuff for psychology and neuroscience.

I prefer the apple instead of the dead cat. In Newtonian physics we can only bang our heads against the apples. In quantum physics we get to taste the forbidden fruit.

Enjoyed your essay Jens, et al,

Also have followed your work on Muses and hypernumbers,

and sorry Brian, consciousness enters the picture with this.

(even though this essay does not draw attention to it).

And Wolfgang Pauli was one of the main advocates of the

necessary inclusion of psychology in the fundamental foundations of physics. Many of the problems discussed in these essays are because of the exclusion of consciousness. Another major proponent of this connection is Jacob Needleman, considering it humble to admit the need for understanding how physics is related to our inner nature. All for "now"!

Dear Brian - thanks for leaving your notes! You also name exactly the two reasons why it just had to be an apple.

You ask about the uncertainty principle; Merab Gogberashvili showed that Heisenberg uncertainty appears on the same geometrical footing as the maximum speed of light, on split-octonions, and we generally tried to remain with algebraic properties. The goal here was to advertise physics on non-associative algebra as serious field of research, demonstrate how proven laws emerge from it, yet are embedded in a formulation that allows new ways of manipulating them that may not have been tried before. We are not attempting a more general interpretation, as e.g. in "Free will, undecidability, and the problem of time in quantum gravity" by Rodolfo Gambini here in this contest.

On a sidenote, the "not-so-serious" summary should not be overextended. When proposing physics on non-associative algebra, it may appear difficult to exactly define the domain these models affect: There seems to be so much impact across the laws of nature we know, that already work very well. The "humbleness" note at the end is really pure humor, and we're not talking down on anyone. Very subjectively, and imcompletely, I did enjoy reading articles outside my domain, so to speak, for example "Time, Consciousness, and the Subjective Universe" by Daegene Song, or "Ultimate Reality and Nonmaterial Origin Of Universe" by Prem Nath Tiwari.

Thanks, Jens

Dear Michael - thanks for leaving your note! Thanks, too, for mentioning my personal interest (trying to establish clear defining relations for the rather vague "hypernumbers" after C. Muses). You must have posted when I wrote my previous note, please let me know if there's anything else that you'd like to see addressed. Thanks, Jens

Enjoyed reading your essay -- octonions rule!

A couple of comments:

While the norm on the octonions is indeed Euclidean, it is well-known how to use them to describe Minkowski space (in 9+1 dimensions).

And I'm afraid I find the unification h and c to be somewhat artificial. Put differently, I'd like to see a discussion of the naturalness of your unification of position and momentum.

Jens, Thanks for mentioning this, "Merab Gogberashvili showed that Heisenberg uncertainty appears on the same geometrical footing as the maximum speed of light, on split-octonions, ..." (Can you give a reference, for clues to a "natural" view?) This is fascinating, and I'll note this for further study. Arthur Young (for those unfamiliar with this, Arthur Young collaborated with Charles Muses on several projects) showed how close Sir Arthur Eddington was to recognizing pi as a key to the relationship between the uncertainty of quantum theory and the curvature of relativity. He also suggested complex numbers were sufficient to explain asymmetry. Perhaps a secret to ponder here is pi written in its complex form (Nahin).

You've significantly advanced the vague, intuitive ideas, and worked the mathematical relationships out in much greater detail. Your latest paper "Quantum of area from gravitation on complex octonions" shows you've made a lot of progress from "conic sedenions" and "time permitting" hope to study this further too.

Okay Jens, found the references:

"Observable Algebra" Merab Gogberashvili arXiv:hep-th/0212251v3

"On Gravity and the Uncertainty Principle"

Ronald J. Adler, David I. Santiago

arXiv:gr-qc/9904026v2

Dear Dr Dray,

Thank you so much for leaving your comments, they're well-taken.

We were happy to reference your and Dr Manogue's "Quaternionic spin" paper in the essay [9], in which you discuss dimensional reduction from 10 to 4 in detail, using octonionic spinors. Important works on octonions, e.g. by M. Günaydin, F. Gürsey, S. K. Adler, S. Okubo, F. Toppan, and many more could not be referenced, due to the constraints of the essay contest (5000 words, enjoyable, less technical than a review article, original content). We had to leave out octonion treatment of the strong force, supersymmetry, Yang-Mills instantons, and octonion uses in GUTs and String Theory, and instead focus on space and time operators, what a non-associative background infers for the time evolution of a general operator, and how extraction of information over time is somewhat "unnatural" to a quantum system.

Whether the proposal in this essay is any more "natural" than what is put in question, remains speculative, and I hope that we have left appropriate wording at the right places. "Guilty until proven innocent", I would respond to your second comment, and agree that further investigation is needed. However, I do find Dr Gogberashvili's treatment of the light cone convincing, as e.g. in "Octonionic Electrodynamics" arXiv:hep-th/0512258, and "Octonionic Geometry" arXiv:hep-th/0409173. Together with Dr Dzhunushaliev's treatment of unobservables ("Hidden structures in quantum mechanics" arXiv:0805.3221, and "Non-associativity, supersymmetry and hidden variables" arXiv:0712.1647), it creates the appearance as if quantum theory on octonions would not care to distinguish parameters from properties.

Thanks again for writing, and also my congratulations to you and Dr Manogue for winning the 2008 FQXi grant, for work on octonions! I am eager to see your "Octonions, E6, and Particle Physics" conference proceeding from September this year, to show in print (is a preprint version available?).

Best wishes,

Jens

M. Shrebon mentions sedenions. Since you run out of division algebra it would appear best if one is to consider higher ordered systems to consider sporadic groups. Can you really do much with sedenions?

Lawrence B. Crowell

Dear Michael - sorry for my belated reply! Thank you so much for your encouraging note. Also, PI certainly is an extraordinary value, as you contemplate in your interesting essay. Regarding the references you asked for, you've already found one, and two more should be in my reply to Dr Dray right above; just let me know if you need anything else.

Dear Lawrence - thanks for writing. Just a couple of remarks: I don't think division algebra is exhausted with the octonions; you could contemplate algebras without unit element, some Banach algebra, or any kind of space with a reasonable topology. You would need to show how (why) that would be required. Regarding sedenions (from Cayley-Dickson construction), one could even argue that it only becomes interesting *from* here on :) - Notably, Robert de Marrais has done extensive work on zero-divisor subspaces that show only from the sedenions on: de Marrais in arXiv.

Thanks, Jens

PS: Lawrence, if you meant "normed division algebra", then you're of course correct: Octonions are as wide of an algebra as you can get. Thanks, Jens

Higher algebraic systems are sporadic groups, which are usually lattice associations of E_8 and octonions. Yet it appears difficult to define an algebra by moving further up the Hopf fibration. I tend to think that working with Golay codes on Leech lattice constructions to be more fruitful.

Lawrence B. Crowell

Dear Lawrence - thanks for clarifying! I'll read up on the areas you mention. Jens