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**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. ...

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FQXi FORUM

March 28, 2017

CATEGORY:
Wandering Towards a Goal Essay Contest (2016-2017)
[back]

TOPIC: Seeking the Analytic Quaternion by Colin Walker [refresh]

TOPIC: Seeking the Analytic Quaternion by Colin Walker [refresh]

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

**RATE THIS ESSAY**

This essay is a candidate in FQXi's current Essay Contest,*Wandering Towards a Goal.*

If you would like to rate this essay: After logging in, you'll be able to rate this essay as well as other submissions in the current Essay Contest.

Would you like to rate this essay?

This essay is a candidate in FQXi's current Essay Contest,

If you would like to rate this essay: After logging in, you'll be able to rate this essay as well as other submissions in the current Essay Contest.

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.

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.

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

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

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

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

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

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

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

Best to you,

Colin

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

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

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

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

I mistakenly used a made-up word "analycity" in place of the correct word "analyticity" in the essay.

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

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

Dear Joe,

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

Cheers,

Colin

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

Cheers,

Colin

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

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

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

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

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