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FQXi FORUM

March 17, 2018

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]

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

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

Hi Colin,

You wrote a remarkable and enjoyable Essay. I consider quaternions as very intriguing and fascinating objects. I think that your work could really have important consequences on quantum mechanics and the mysterious complementarity which is important also in the framework of my recent research in black hole quantum physics. In order to help you to spread your results within the scientific community I will give you the highest score.

Congrats and good luck in the contest.

Cheers, Ch.

report post as inappropriate

You wrote a remarkable and enjoyable Essay. I consider quaternions as very intriguing and fascinating objects. I think that your work could really have important consequences on quantum mechanics and the mysterious complementarity which is important also in the framework of my recent research in black hole quantum physics. In order to help you to spread your results within the scientific community I will give you the highest score.

Congrats and good luck in the contest.

Cheers, Ch.

report post as inappropriate

Dear Colin,

With great interest I read your essay, which of course is worthy of the highest praise.

I'm glad that you have

**«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.»**

Your assumptions are very close to me

**«about quaternion singularities. Just like real or complex numbers, division by zero is not allowed.» **

«no black holes, no dark energy, no big bang.»

«Given the success of LIGO, it seems strange that LATOR has not been supported.»

You might also like reading my essay , where the fractal principle of the device of matter is substantiate.

I wish you success in the contest.

Kind regards,

Vladimir

report post as inappropriate

With great interest I read your essay, which of course is worthy of the highest praise.

I'm glad that you have

Your assumptions are very close to me

«no black holes, no dark energy, no big bang.»

«Given the success of LIGO, it seems strange that LATOR has not been supported.»

You might also like reading my essay , where the fractal principle of the device of matter is substantiate.

I wish you success in the contest.

Kind regards,

Vladimir

report post as inappropriate

Hello Colin,

I had to go at this 3 times to read it through. And I did a little research in between. But once I did, I found it easy to follow the Maths used in support of your thesis. I am not certain, but I think you are discovering here that there is an interior and exterior quaternion derivative component (perhaps inner and outer would be better terminology). I am no expert, but I suspect that you are re-discovering things known in the field of exterior derivatives and differential forms.

However; this could be a step forward for Physics folks, in terms of having a missing piece of our understanding filled in. You appear to be revealing that there are symmetric and anti-symmetric components - under the Cayley-Dickson construction. An interesting romp through some fun Maths. It was not all obviously. or completely, pertinent to the assigned topic throughout, but was largely germane considering the result and its interpretation. So I give you kudos and high marks.

All the Best,

Jonathan

report post as inappropriate

I had to go at this 3 times to read it through. And I did a little research in between. But once I did, I found it easy to follow the Maths used in support of your thesis. I am not certain, but I think you are discovering here that there is an interior and exterior quaternion derivative component (perhaps inner and outer would be better terminology). I am no expert, but I suspect that you are re-discovering things known in the field of exterior derivatives and differential forms.

However; this could be a step forward for Physics folks, in terms of having a missing piece of our understanding filled in. You appear to be revealing that there are symmetric and anti-symmetric components - under the Cayley-Dickson construction. An interesting romp through some fun Maths. It was not all obviously. or completely, pertinent to the assigned topic throughout, but was largely germane considering the result and its interpretation. So I give you kudos and high marks.

All the Best,

Jonathan

report post as inappropriate

I'm probably bearing old news but...

The quaternions are isomorphic to parallelized S3, so examples can be geometrized and algebratized as is needed. Sometimes; it is easier to see contrasting cases as geometric examples, but I am not sure what the exact equivalency is here, in the case of the two analytic derivatives. So I would hesitate, and need to think further, before making an association.

Regards,

Jonathan

report post as inappropriate

The quaternions are isomorphic to parallelized S3, so examples can be geometrized and algebratized as is needed. Sometimes; it is easier to see contrasting cases as geometric examples, but I am not sure what the exact equivalency is here, in the case of the two analytic derivatives. So I would hesitate, and need to think further, before making an association.

Regards,

Jonathan

report post as inappropriate

Hi Jonathan,

I am glad you put in the effort to get a good understanding. Getting to a symmetric balance in the derivatives is the main thing. All else falls from that. This is quite obscure stuff which can get complicated, but ultimately it comes down to ordinary real derivatives.

I am fairly sure this is a new take on quaternion derivatives and analyticity, but it would not surprise me to have reinvented the wheel in some other context. Actually, it would be great if a correspondence to the quaternion derivatives could be found in some existing formulation, possibly linking the two.

Your point about geometric examples is a good one. There must be a way to interpret the derivatives in terms of well-known geometric algebra, but I will have to set this project aside for a while.

Another project has captured my attention. Rob McEachern put forward the idea (with a computer program as a demonstration) that quantum correlations are a result of sampling a process that can provide, at most, one bit of information per sample. I stumbled upon a faster way to produce the correlations using a similar Monte Carlo method, which also requires a large number of trials to accumulate statistics. It is easily verified that McEachern's one-bit criterion is satisfied. Bell correlations are produced matching the theoretical sinusoid to about 1% accuracy. Second paper will be posted on vixra hopefully by the end of the month. It turns out that there is a way to speed the procedure up by replacing Monte Carlo statistics by easily calculated probabilities. The CHSH test verifies that the procedure is classical, but there is a simple way to nearly reproduce the quantum expectation for CHSH. Here is my original paper on the subject, which includes links to Rob's work. I have a feeling that that this is fundamentally related to QM, but it is also interesting for its potential in simulations.

You have an impressive base of knowledge, Jonathan, and I really appreciate your feedback.

All the best to you,

Colin

I am glad you put in the effort to get a good understanding. Getting to a symmetric balance in the derivatives is the main thing. All else falls from that. This is quite obscure stuff which can get complicated, but ultimately it comes down to ordinary real derivatives.

I am fairly sure this is a new take on quaternion derivatives and analyticity, but it would not surprise me to have reinvented the wheel in some other context. Actually, it would be great if a correspondence to the quaternion derivatives could be found in some existing formulation, possibly linking the two.

Your point about geometric examples is a good one. There must be a way to interpret the derivatives in terms of well-known geometric algebra, but I will have to set this project aside for a while.

Another project has captured my attention. Rob McEachern put forward the idea (with a computer program as a demonstration) that quantum correlations are a result of sampling a process that can provide, at most, one bit of information per sample. I stumbled upon a faster way to produce the correlations using a similar Monte Carlo method, which also requires a large number of trials to accumulate statistics. It is easily verified that McEachern's one-bit criterion is satisfied. Bell correlations are produced matching the theoretical sinusoid to about 1% accuracy. Second paper will be posted on vixra hopefully by the end of the month. It turns out that there is a way to speed the procedure up by replacing Monte Carlo statistics by easily calculated probabilities. The CHSH test verifies that the procedure is classical, but there is a simple way to nearly reproduce the quantum expectation for CHSH. Here is my original paper on the subject, which includes links to Rob's work. I have a feeling that that this is fundamentally related to QM, but it is also interesting for its potential in simulations.

You have an impressive base of knowledge, Jonathan, and I really appreciate your feedback.

All the best to you,

Colin

Wow Colin!

Your idea above sounds impressive actually. Quantum information theory asserts that simplicity of calculation is what sets the likelihood for a process to run or outcome to occur. I'll have more to say, and hope we can stay in touch after.

All the Best,

Jonathan

report post as inappropriate

Your idea above sounds impressive actually. Quantum information theory asserts that simplicity of calculation is what sets the likelihood for a process to run or outcome to occur. I'll have more to say, and hope we can stay in touch after.

All the Best,

Jonathan

report post as inappropriate

Dear Sirs!

Physics of Descartes, which existed prior to the physics of Newton returned as the New Cartesian Physic and promises to be a theory of everything. To tell you this good news I use «spam».

New Cartesian Physic based on the identity of space and matter. It showed that the formula of mass-energy equivalence comes from the pressure of the Universe, the flow of force which on the corpuscle is equal to the product of Planck's constant to the speed of light.

New Cartesian Physic has great potential for understanding the world. To show it, I ventured to give "materialistic explanations of the paranormal and supernatural" is the title of my essay.

Visit my essay, you will find there the New Cartesian Physic and make a short entry: "I believe that space is a matter" I will answer you in return. Can put me 1.

Sincerely,

Dizhechko Boris

report post as inappropriate

Physics of Descartes, which existed prior to the physics of Newton returned as the New Cartesian Physic and promises to be a theory of everything. To tell you this good news I use «spam».

New Cartesian Physic based on the identity of space and matter. It showed that the formula of mass-energy equivalence comes from the pressure of the Universe, the flow of force which on the corpuscle is equal to the product of Planck's constant to the speed of light.

New Cartesian Physic has great potential for understanding the world. To show it, I ventured to give "materialistic explanations of the paranormal and supernatural" is the title of my essay.

Visit my essay, you will find there the New Cartesian Physic and make a short entry: "I believe that space is a matter" I will answer you in return. Can put me 1.

Sincerely,

Dizhechko Boris

report post as inappropriate

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