Search FQXi


If you are aware of an interesting new academic paper (that has been published in a peer-reviewed journal or has appeared on the arXiv), a conference talk (at an official professional scientific meeting), an external blog post (by a professional scientist) or a news item (in the mainstream news media), which you think might make an interesting topic for an FQXi blog post, then please contact us at forums@fqxi.org with a link to the original source and a sentence about why you think that the work is worthy of discussion. Please note that we receive many such suggestions and while we endeavour to respond to them, we may not be able to reply to all suggestions.

Please also note that we do not accept unsolicited posts and we cannot review, or open new threads for, unsolicited articles or papers. Requests to review or post such materials will not be answered. If you have your own novel physics theory or model, which you would like to post for further discussion among then FQXi community, then please add them directly to the "Alternative Models of Reality" thread, or to the "Alternative Models of Cosmology" thread. Thank you.

Contests Home

Current Essay Contest


Contest Partner: The Peter and Patricia Gruber Fnd.

Previous Contests

Trick or Truth: The Mysterious Connection Between Physics and Mathematics
Contest Partners: Nanotronics Imaging, The Peter and Patricia Gruber Foundation, and The John Templeton Foundation
Media Partner: Scientific American

read/discusswinners

How Should Humanity Steer the Future?
January 9, 2014 - August 31, 2014
Contest Partners: Jaan Tallinn, The Peter and Patricia Gruber Foundation, The John Templeton Foundation, and Scientific American
read/discusswinners

It From Bit or Bit From It
March 25 - June 28, 2013
Contest Partners: The Gruber Foundation, J. Templeton Foundation, and Scientific American
read/discusswinners

Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?
May 24 - August 31, 2012
Contest Partners: The Peter and Patricia Gruber Foundation, SubMeta, and Scientific American
read/discusswinners

Is Reality Digital or Analog?
November 2010 - February 2011
Contest Partners: The Peter and Patricia Gruber Foundation and Scientific American
read/discusswinners

What's Ultimately Possible in Physics?
May - October 2009
Contest Partners: Astrid and Bruce McWilliams
read/discusswinners

The Nature of Time
August - December 2008
read/discusswinners

Forum Home
Introduction
Terms of Use

Order posts by:
 chronological order
 most recent first

Posts by the author are highlighted in orange; posts by FQXi Members are highlighted in blue.

By using the FQXi Forum, you acknowledge reading and agree to abide by the Terms of Use

 RSS feed | RSS help
RECENT POSTS IN THIS TOPIC

Colin Walker: on 3/7/17 at 21:53pm UTC, wrote Dear Joe, You am everywhere on an infinite sphere. I am too. Cheers, ...

Colin Walker: on 3/7/17 at 21:52pm UTC, wrote Dear SNP. Gupta You ask about quaternion singularities. Just like real or...

Satyavarapu Gupta: on 3/7/17 at 20:04pm UTC, wrote Nice essay Walker, Your ideas and thinking are excellent on Complex...

Joe Fisher: on 3/7/17 at 17:04pm UTC, wrote Dear Colin Walker, Please excuse me for I have no intention of...

Colin Walker: on 3/6/17 at 16:30pm UTC, wrote I mistakenly used a made-up word "analycity" in place of the correct word...

Colin Walker: on 3/3/17 at 20:22pm UTC, wrote Hi Branko, 48 numbered equations in my essay is waaay too much math for...

Colin Walker: on 3/3/17 at 20:18pm UTC, wrote Hi Gary. I think I see now what you are getting at with your equation 3.1 -...

Gary Simpson: on 3/3/17 at 14:25pm UTC, wrote Colin. Yes, I was generally referring to Eq 5 from the previous essay. ...


RECENT FORUM POSTS

Eckard Blumschein: "Gary Simpson, While PHALLUS is an abreviantion, a phallus is a Willy, a..." in FQXi Essay Contest 2016:...

jay grey: "Thanks for the link, I believe it will be very useful for personal..." in Santa Barbara Gravity...

Lorraine Ford: "Good one Gary! Regards, Lorraine" in FQXi Essay Contest 2016:...

Joe Fisher: "Attempting to account for complex finite invisible particles will not help..." in Bohemian Reality:...

jany watson: "At we have a long list of similar http://www.essayhelperuk.co.uk/law-essay..." in Koalas, Quantum Mechanics...

James Putnam: ""The math time = 1/nu defines time ... " Nu is cycles/second. 1/nu is..." in Alternative Models of...

Steve Agnew: "Your work has shown how normal physics like S&Z mass, length, and time have..." in Alternative Models of...

kab: "The word is enternal magic. Hope all secret never has answer will be out..." in Does Quantum Weirdness...


RECENT ARTICLES
click titles to read articles

Bohemian Reality: Searching for a Quantum Connection to Consciousness
Is there are sweet spot where artificial intelligence systems could have the maximum amount of consciousness while retaining powerful quantum properties?

Quantum Replicants: Should future androids dream of quantum sheep?
To build the ultimate artificial mimics of real life systems, we may need to use quantum memory.

Painting a QBist Picture of Reality
A radical interpretation of physics makes quantum theory more personal.

The Spacetime Revolutionary
Carlo Rovelli describes how black holes may transition to "white holes," according to loop quantum gravity, a radical rewrite of fundamental physics.

Riding the Rogue Quantum Waves
Could giant sea swells help explain how the macroscopic world emerges from the quantum microworld? (Image credit: MIT News)


FQXi FORUM
March 28, 2017

CATEGORY: Wandering Towards a Goal Essay Contest (2016-2017) [back]
TOPIC: Seeking the Analytic Quaternion by Colin Walker [refresh]
Bookmark and Share
Login or create account to post reply or comment.

This essay's rating: Community = 3.3; Public = 2.0


Author Colin Walker wrote on Mar. 2, 2017 @ 19:56 GMT
Essay Abstract

By combining the complex analytic Cauchy-Riemann derivative with the Cayley-Dickson construction of a quaternion, possible formulations of a quaternion derivative are explored with the goal of finding an analytic quaternion derivative having conjugate symmetry. Two such analytic derivatives can be found. This unanticipated finding may have significance in areas of quantum mechanics where quaternions are fundamental, especially regarding the enigmatic phenomenon of complementarity, where a quantum process seems to present two essential aspects.

Author Bio

Most of my career involved academic research. Relevant to this essay, I assisted research into spectrum estimation using complex analysis in the Department of Geophysics and Astronomy at the University of British Columbia. I received a BMath from the University of Waterloo in 1970.

Download Essay PDF File




Gary D. Simpson wrote on Mar. 3, 2017 @ 01:03 GMT
Colin,

Glad you could make the party!

This is an interesting essay to me. I am not familiar with the theorems that you reference or the terms analytic or non-analytic although the definitions you provide are satisfactory.

I think that the individual differential terms that you use are the various partial derivatives that populate the matrices that I presented in my last essay.

All in all, this is a very informative essay for me ... of course, I am a big believer in the quaternions and octonions.

How would you interpret the following:

(1 + complex i)x(1 + vector u)=1+(complex i)+(vector u)+(complex i)x(vector u)

I'll give you a rating after the haters give you a one bomb.

Best Regards and Good Luck,

Gary Simpson

report post as inappropriate

Author Colin Walker replied on Mar. 3, 2017 @ 06:01 GMT
Hi Gary. Good to be here. (Good to be anywhere!)

Are you referring to Eq.5 in your essay Calculus 2.0 from last year, and my Eqs.(16,18-20)? That looks close to your quaternion multiplication, but I can't see the signs matching up, even allowing conjugates. One thing I found is that a complex derivative is different from a vector derivative, and a quaternion derivative is different from a matrix derivative. As I mentioned in your blog, your essay from last year with quaternion derivative is what got me started.

I find the role of the imaginary, i, in quaternions a bit strange. It seems to blend in as one of three imaginaries, or stand out in the matrix representation as something on equal footing with the real. It is interesting that a polynomial with real coefficients can have complex roots. That seems to me to be more consistent with the look of the matrix synthesis, where the real and i are almost interchangeable.

If the vector u you refer to is 3-dimensional, and 1 + vector u is a quaternion, these would be the components from the essay you submitted this year. The product is like an octonion, but I really do not know. Geometric algebra might be helpful, but the only way I can learn something is to use it over and over, constantly refreshing. Otherwise it fades to the point of having to start all over. So it's low hanging fruit for me, whenever I can find it.

Cheers, Colin



Gary D. Simpson replied on Mar. 3, 2017 @ 14:25 GMT
Colin.

Yes, I was generally referring to Eq 5 from the previous essay.

The relation

(1 + complex i)x(1 + vector u)=1+(complex i)+(vector u)+(complex i)x(vector u)

from above is basically your Equation 7. The only question is do you think there are two i's (i.e., complex i and vector i) or is there only one i?

Your Equation 31 is very similar to my Equation 3.1.

We are thinking a lot alike.

Best Regards and Good Luck,

Gary Simpson

report post as inappropriate

Author Colin Walker replied on Mar. 3, 2017 @ 20:18 GMT
Hi Gary. I think I see now what you are getting at with your equation 3.1 - it is a combination of two quaternions but having complex coefficients instead of real in the real quaternion basis. I recall that what you would get is a biquaternion. It turns out that a biquaternion 2x2 complex matrix is not a quaternion, and any 2x2 complex matrix can be expressed in biquaternion form. Like an octonion, a biquaternion has 8 independent real variables. A quaternion is a biquaternion with a specific combination of symmetries that allows only 4 independent real variables. I cannot see getting an octonion unless your 'complex i' was the octonion 'l' in the sequence of seven octonion imaginaries i,j,k,l,m,n,o - and then it would have to be checked for (or arranged to have) the appropriate symmetry. I will post this in your blog too.

Best to you,

Colin




Branko L Zivlak wrote on Mar. 3, 2017 @ 07:42 GMT
Hi Colin,

I like essays with math. But I did not learn quaternions and octonions. Yet can I say? I feel that the solution to the biggest problems in physics (Alpha constant) is in the imaginary part. What do you think about that? As I understand your essay deserves the top rating it will be done at the right time.

Best regards,

Branko Zivlak

report post as inappropriate

Author Colin Walker replied on Mar. 3, 2017 @ 20:22 GMT
Hi Branko,

48 numbered equations in my essay is waaay too much math for the average reader, and probably too much even for those who like math!

The fine structure constant has been a continuing source of curiosity since Eddington made a rash conjecture. Wikipedia has a collection of formulas relating it to different physical constants. I would say trust your intuition.

Best to you,

Colin




Author Colin Walker wrote on Mar. 6, 2017 @ 16:30 GMT
I mistakenly used a made-up word "analycity" in place of the correct word "analyticity" in the essay.




Joe Fisher wrote on Mar. 7, 2017 @ 17:04 GMT
Dear Colin Walker,



Please excuse me for I have no intention of disparaging in any way any part of your essay.

I merely wish to point out that “Everything should be made as simple as possible, but not simpler.” Albert Einstein (1879 - 1955) Physicist & Nobel Laureate.

Only nature could produce a reality so simple, a single cell amoeba could deal with it.

The real Universe must consist only of one unified visible infinite physical surface occurring in one infinite dimension, that am always illuminated by infinite non-surface light.

A more detailed explanation of natural reality can be found in my essay, SCORE ONE FOR SIMPLICITY. I do hope that you will read my essay and perhaps comment on its merit.

Joe Fisher, Realist

report post as inappropriate

Author Colin Walker replied on Mar. 7, 2017 @ 21:53 GMT
Dear Joe,

You am everywhere on an infinite sphere. I am too.

Cheers,

Colin




Satyavarapu Naga Parameswara Gupta wrote on Mar. 7, 2017 @ 20:04 GMT
Nice essay Walker,

Your ideas and thinking are excellent on Complex analysis of complex analytic Cauchy-Riemann derivative with the Cayley-Dickson construction of a quaternion…

Very good useful application like….”Ultimately, it is mathematics that allows us to entertain the notion of understanding quantum mechanics.” …………………….. Hope you will clarify me, I am...

view entire post


report post as inappropriate

Author Colin Walker replied on Mar. 7, 2017 @ 21:52 GMT
Dear SNP. Gupta

You ask about quaternion singularities. Just like real or complex numbers, division by zero is not allowed.

I have come to many of the same conclusions as you: no black holes, no dark energy, no big bang.

Best regards,

Colin




Login or create account to post reply or comment.

Please enter your e-mail address:

And select the letter between 'X' and 'Z':


Note: Joining the FQXi mailing list does not give you a login account or constitute membership in the organization.