Physics and quaternions

[Chris Hillman]

You can do everything with quaternions, as far as I can see.

Not true at all. In particular, if you think of quaternions as a generalization from complex numbers--- which form a two dimensional real associative linear algebra, in fact a Cayley-Klein algebra--- to a four dimensional real associative linear algebra, in fact a Cayley-Klein algebra, one natural thing to look for is a theory of "quaternionic analysis" parallel to undergraduate "complex analysis", e.g. with analogs of Cauchy integral theorem and so on. But very soon after you write down a quaternionic analog of the Cauchy-Riemann equations you will find that you get stuck!

Now, there is a large literature from such mathematical luminaries as Lars Ahlfors on getting around these difficulties, but all authors agree that the best definitions are far from clear and that most attempts to make a viable theory of "quaternionic analysis" lead to failure.

I prefer pictures, even linguistic.

We can see that, and I would characterize your claims as also belonging to the genre of "simple physics" (see my recent post in another ATM thread).

As others have already pointed out, to someone who knows something about either quaternions or gtr it is quickly apparent that you have little knowledge of the relevant math or physics.

So let me explain my idea about the principle of relativity (oops, I've wrote equivalence principle) with a picture.
Imagine a piece of spacetime. It's four-dimensional, but that is hard to imagine. But it's possible to 'cheat' a bit. We could leave out one dimension and make it three-dimensional and and encode time into rotation.

You should stop and study something like a Wiki article I wrote, "Lorentz group"; the last version I edited is at
http://en.wikipedia.org/w/index.php?...oldid=42316189

Do you see now why you are already in trouble here?

So now think about a cube of spacetime of some extension.
The principle of relativity now says, that relative to any such path you draw through spacetime, the laws of physics are the same.

As someone already told you, that is a very serious misunderstanding.

So for all kinds of free-falling bodies the universe looks like the universe looks to us.

Do you really believe that this statement makes any sense?

I'd say it's a good example of a statement which is "not even wrong".

But if anyone cares: when you compare the optical experience of different families of inertial observers in such standard models as the Schwarzschild vacuum, an exact vacuum solution to the Einstein field equation which lies at the heart of general relativity,. you find that in fact different observers have very different physical and/or optical experiences, depending upon the details of their kinematic (motion) history.

(Here, by "physical experience" I mean such things as tidal forces, while by "optical experience" I mean such things as strong-field light bending and gravitational lensing. For example, close to the event horizon of a black hole, light rays can very easily wind around several times before "escaping to infinity", and for this reason, an observer hovering just over the event horizon of a Schwarzschild hole sees a picture of "the night sky" which is very different from that seen by a distant observer!)

Space is what we make of our own way through spacetime. So time and space are to some extend interchangeable.

Likewise.

I guess you mean vectors. But that is a critical thing to use in general relativity. I use intervals, that represent not form, but some more abstract relation what is defined by
ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
If you'd take the square root of that and name the components (w,x,y,z) that's it. The indices do not mean coordinates. They have a relation to coordinates I want to show.

This is almost comically muddled, but FWIW I think Heger is trying to refer to the definition of a Cayley-Klein algebra, in which we assume that we have a real associative algebra with units whose squares are either -1,0,1. So for example the complex unit has square -1, as do the three quaterionic units. Then by linear extension we obtain the squared norm of a linear combination of unity plus these "adjoined units", i.e. the squared norm of a Cayley-Klein number. In the same way, one can follow Dirac by trying to factor something like the wave operator, following the model of how one "factors" the two-dimensional Laplace operator in undergraduate complex analysis. That eventually leads to the notion of spinors. But this appears unrelated to Heger's muddled claims.

I try to avoid special theory of relativity and use only general theory of relativity as basis. But I don't like tensors and prefer a quaternion alternative.
SRT is somehow not what I want, because it compares two observations instead of kicking out the observer.

As someone else already pointed out, this is ridiculous on the face of things: special relativity is incorporated into general relativity, in the sense that spacetime models in gtr are Lorentzian manifolds, and the tangent space to any event in a Lorentzian manifold has the geometry of Minkowski spacetime, by definition.

In fact, nothing Heger has said here has anything to do with general relativity, or even with physics. He has merely alluded to some ideas in pure mathematics, but his references are so muddled that it seems clear to me that he doesn't understand what he is trying to talk about.

So a worldline that is straight to one observer, could be curve for another....
Intervals are the basis of spacetime and don't mean length. So spacetime is not geometric in the sense of a vector. It's based on multiplications, that connect neighboring 'points'. Those points are the quaternions, but they are not defined by their position, but by their interval to other such 'points'. A fundamental process could way easier deal with intervals than with vectors, because a vector needs an observer, but those processes don't.
If spacetime would be treated *really* relativistic, than all those strange mixed-mode models would vanish. Amongst those are things like black-holes...

It is clear that Heger has no idea what such terms as "spacetime", "straight", or "quaternion" mean in mainstream math/physics.

(An other btw.: the same argument would exclude black holes. I see it like this: there is a terrible mistake made for a long time and by many: see above. I talked about free-falling bodies. That is with reason. You HAVE to center the reference frame at the observer. If you did, the black hole in the center of our galaxy would be gone and seen from there it would be here.)

It's perfectly clear Heger has no idea whatever what "black hole" means in the context of relativistic physics.

It seems to me that he is trying to hide behind his own private language (his own private meaning of "black hole", "quaternion", "spacetime", and so on) to hide the fact that he really has nothing meaningful to say.

[thomheg]

I cannot access this with either of the browsers at my disposal. Could you please use something standard like a PDF? And I mean a real PDF, not links to a web site masquerading as PDF like the Aether Physics Model people have done.

Thanks in advance.

I have to setup my own website. Not yet done, so please click on
http://docs.google.com/Present?docid...&skipauth=true
There is a 'print slides' menu. One point is 'print as pdf'. That would download the text to you as pdf.
If that doesn't work I mail the pdf to you.

I have to excuse, that it's not really finished. Some stuff is missing and there minor and major faults. But I'd like to hear comments.

[thomheg]

But very soon after you write down a quaternionic analog of the Cauchy-Riemann equations you will find that you get stuck!

Now, there is a large literature from such mathematical luminaries as Lars Ahlfors on getting around these difficulties, but all authors agree that the best definitions are far from clear and that most attempts to make a viable theory of "quaternionic analysis" lead to failure.

Thanks. Now you have just answered a few of my questions.

We can see that, and I would characterize your claims as also belonging to the genre of "simple physics"

That's true, since I had initially called my paper 'a simple theory'.

As others have already pointed out, to someone who knows something about either quaternions or gtr it is quickly apparent that you have little knowledge of the relevant math or physics.

That's true, too. But I guess for different reasons, than you'd think.
My plan was this: I find myself in front of a huge pile of different theories, knowledge and math. Really huge. Enormous!!
How to go through this monstrous pile? Read everything? Impossible! Studying math for a few years? Too long!
I decided to try it stepwise. Choose a good starting point and proceed from there. To find the primary assumption took me two years. I looked at various forums, read a lot about quantum-physics and relativity. The math was assumed as known by someone and I left that aside. I concentrated on the possible starting point and chose quaternions. Than I decided to know nothing. (What was in a way the case). From that starting assumption I made my way back to known physics and started with GR. GR with quaternions was done various times, so it is quite easy. But I thought: what do these quaternions do? What are they actually good for and how does that look like. I found after some time, that would look like quantum-physics under certain assumptions. So I pluged that assumptions into my model and proceeded.
That process was performed mainly in the usenet. I read that stuff and started threads, too. The way of finding things out is: formulate a statement, that you think is possible and then try to defend it. In this process you'd learn a lot, get new information and could check your statement. This leads to new questions and so forth.
So you have always and every time someone, that calls you stupid. But in a way he tells why, and that's what I wanted to find out.

[worldcruiser]

If you think your method is a short cut to knowledge you are really on a long lasting nonsense trip.

[agingjb]

Around page 200 of "Road to Reality" by Roger Penrose there is a useful discussion of why quaternions, despite their uniqueness and interest, are not as useful in physics as might have been hoped.

(BTW, the book is probably required reading for anyone seriously proposing an ATM physical theory.)

[thomheg]

If you think your method is a short cut to knowledge you are really on a long lasting nonsense trip.

Again and again. What I think, guess or know is completely irrelevant.
I'm setting up a model. That's what I want to do

[thomheg]

Around page 200 of "Road to Reality" by Roger Penrose there is a useful discussion of why quaternions, despite their uniqueness and interest, are not as useful in physics as might have been hoped.

(BTW, the book is probably required reading for anyone seriously proposing an ATM physical theory.)

I have not yet, but will. I recommend
Bertrand Russel's Abc of relativity, what is a very good book, that I've found just recently

[Celestial Mechanic]

I finally got through to the Google docs version of your theory, but the Print Slides button does not work on either of the browsers/machines that I have access to. Not really a problem, since your document appears to be a 118 slide Powerpoint-like presentation. Can you please put together a more text-oriented version, more like a formal paper that you could submit to a peer-reviewed journal, and again, in standard PDF format, NOT Microsoft Word format, NOT Google docs. Thank you.

[Disinfo Agent]

The way of finding things out is: formulate a statement, that you think is possible and then try to defend it.

A more prudent approach is trying to attack it before everyone else does.

[thomheg]

I finally got through to the Google docs version of your theory, but the Print Slides button does not work on either of the browsers/machines that I have access to.

I guess, Java Script is shut off on your computer, since that button uses it.

Not really a problem, since your document appears to be a 118 slide Powerpoint-like presentation.

Has grown a bit. That version was not intended to be submitted. It's more like a pile of notices and something to discuss.

Can you please put together a more text-oriented version, more like a formal paper that you could submit to a peer-reviewed journal, and again, in standard PDF format, NOT Microsoft Word format, NOT Google docs. Thank you.

How many pages would you suggest?

[Chris Hillman]

My plan was this...

Heger appears to be beyond my help, since he has simply dismissed out of hand the good advice offered by myself and others.

But I imagine there might be some college students reading this thread who have (like Heger) reacted with horror to the prospect of years of study with no guarantee of success. For those students I offer a few comments, off the top of my head, concerning why "Heger's four year plan" is such a disaster, and why the "standard four year plan" (earn a degree) is by no means an impossible task.

Heger says

I looked at various forums, read a lot about quantum-physics and relativity. The math was assumed as known by someone and I left that aside.

Terrible choice. I like to define mathematics as "the art of reliable reasoning about simple phenomena without getting confused". And I'd define theoretical physics as "the art of cutting through the chaff to get to the simplest way of thinking about some physical phenomenon, in order to construct a conceptual framework for making simple but effective mathematical models of suitable idealizations of the phenomenon of interest". Mathematical reasoning is an essential part of formulating, studying, comprehending, evaluating, and revising a developing theory. You simply don't stand a chance of getting anywhere without it. Any physicist will tell you that.

When I say "simple", students should bear in mind a genuine quotation from Einstein, to the effect that a physical theory should be "as simple as possible, but no simpler". He said that about a long-forgotten failed theory which he regarded as too simple. Being Einstein, he had a mathematical criterion in mind: he argued, mathematically of course, that a good theory of the phenomenon of interest should have a solution space in which each solution should require a certain number of functions of so many variables--- but the theory Einstein was critiquing boasted a far smaller solution space. Thus, it was "too simple". Einstein's own theory was "as simple as possible, but no simpler" in the mathematically precise sense he had in mind. So are infinitely many other theories, which is where experiment and observation enter the picture in seperating "viable theories" from the ones which are proven not to work sufficiently well.

Heger says he "read a lot" but elsewhere he gives as an example a popular book on relativity published some fifty years ago. This just doesn't cut it. The old saw "a little knowledge can be worse than none" applies. Popular books are intended as light entertainment, not as a substitute for textbooks, and there are some very good reason why that is the case. I'll warrant that any "old hand" on the public physics forums will tell you that we see every day more evidence supporting that hypothesis that popular science books only confuse people who seek genuine understanding. Relativistic physics (like any other branch of physics) is full of subtle issues which are almost impossible to spot, much less to deal with effectively, without the use of appropriate mathematical tools. You'd be very foolish to decline to take advantage of a hundred years of progress in understanding relativistic physics (by declining to study some of the dozens of excellent modern textbooks on gtr), especially if your only reason for doing so is that you would need to employ mathematics and to spend a few years getting up to speed.

Heger says

I concentrated on the possible starting point and chose quaternions. Than I decided to know nothing.

I trust it is as obvious to students as it is to those of us who have already mastered large chunks of math/physics that trying to build upon a state of complete ignorance about the alleged foundation (quaternions, in this subject) is a reliable recipe for disaster.

Picking up a point someone else already mentioned: Heger appears to be unaware of the vast amount of work put in by one of the leading figures in 20th century mathematical physics, Roger Penrose, and dozens of students and colleagues, in developing twistor theory, which makes use of quaternions (and many other things) in an attempt to reformulate classical relativistic physics, including gtr, in an ultimately more transparent and powerful way. Despite the development of some beautiful mathematics, most observers would probably agree that so far twistor theory has not realized Penrose's initial hopes. Now, Penrose has offered in various places his own assessment of his efforts. Ignoring what he has to say is exceptionally foolish, both because his comments ought to prompt a re-evaluation of Heger's choice (we all agree, it seems, that this choice was naive) to try to found physics upon quaternions alone, and because they provide invaluable insights for anyone hoping to provide some kind of "post-twistor theory".

Those are just two reasons why Heger's plan has no chance of success. Turning to the reasons why the standard plan is a sensible alternative, Heger asks:

My plan was this: I find myself in front of a huge pile of different theories, knowledge and math. Really huge. Enormous!!
How to go through this monstrous pile? Read everything? Impossible!

But of course, it is not at all impossible--- witness the thousand of physics Ph.D.s and tens of thousands of undergraduate degrees with a concentration in physics which are awarded each year around the world by respectable universities. One of the most salient advantages of taking formal courses and studying good textbooks is that experts have gone to much trouble to smooth your way, by providing a well worn and proven path to mastery of the most essential current knowledge, a path which takes you at maximal speed to the frontiers of knowledge, and which maximizes opportunity for intellectual growth in addition to mastery of required technical skills.

An important point worth keeping mind is that the reasons why mathematics is so hard to master (the multiplicity of representations, formalisms, and conceptual approaches to any topic, the tight interconnections between seemingly unrelated topics, and so on) are also the reasons for its extraordinary power. We speak of "a steep learning curve", but that steepness should -encourage- students, because we are saying "mathematics is the most efficient way to rev your brain up to full speed". Or in other words: past a certain point, you will find that it becomes exponentially easier to learn even more math (and thus, even more physics); you will be able to leverage what you already know to rapidly acquire new skills and new insights.

As for the lack of guarantees of ultimate success if you follow the standard four year plan, well, can't help you there, but as you will probably agree, the same could be said about any hard endeavor. At some point you just have to trust your luck and hope for the best.

Summing up: math is your most important ally in the struggle toward understanding. And formal coursework and textbooks are gateways to knowledge, not artificial barriers. Make use of the opportunities they provide!

[thomheg]

Heger appears to be beyond my help, since he has simply dismissed out of hand the good advice offered by myself and others.

To make it clear: I regard your post as rude personal offending. That is because you demand me to follow the advices you give. But you have no right at all to demand anything. I do what I do and you do what you do. Agree on that?
It's a personal attack, because you criticize my method and my knowledge. But I'm responsible to nobody, especially not to you. You didn't mention any of my ideas, so you don't criticize them but me.
I would be able to find a lot of words, to comment that, but that would be against the rules.

[slang]

What is so rude about it? If you dismiss Chris Hillman's help, you are by definition beyond it, meaning he cannot help you. That seems like a factual statement to me. It seems less rude than dismissing mainstream geology experts on the assertion that the earth is like a melon.

It's a personal attack, because you criticize my method

If he is attacking your method, how is he attacking your person? *sighs*

[thomheg]

What is so rude about it? If you dismiss Chris Hillman's help, you are by definition beyond it, meaning he cannot help you. *sighs*

He did not. He asked me to stop musing around. It would be help, if I wanted to become a physicist, what was not the case. Some other kind of help would be some hints, how to enhance the model or maybe to completely change it, but asked me to dismiss it. That's no help at all.
I wrote a kind of short story about it:

THE FOX
A man had the idea to visit a fox-hunt. He thought, that would be nice to see all those hunters on their horses, all dressed in pink.
So he was looking for a date of such a hunt and planned to go there. But he was quite late on that day and when he finally was at the gathering, all hunters were gone and all the horses. He was quite dissapointed and was wandering what to do. He said to himself, well I have no horse and I have no gun. So what can I do? Those man are way faster than me, because they are hunters and they have horses. I can't catch up.
After a while he though, that he really wanted to see the hunters in their red jackets and that he waited so long for this. So he said, well they seem to hunt a fox. I heard some people talking about foxes in an area that is possible to reach by foot. So I just go there and there should be the hunters, since they want to hunt that fox.
So he made his way through deep forest and finally reached a clearing and saw a fox. He didn't want to shoot him, and he had no gun. But there were still no hunters. So he decide to move all the way back to tell somebody of his finding. He found a hunter and told him about the fox. The hunter replied:
"Well, man. how can you do that? That is nothing for you, you aren't supposed to find foxes so we can't trust you, since we are the hunters and you don't even have a horse!"

[papageno]

He did not. He asked me to stop musing around. It would be help, if I wanted to become a physicist, what was not the case.

Weird... I see the word "Physics" in the thread title...

Some other kind of help would be some hints, how to enhance the model or maybe to completely change it, but asked me to dismiss it. That's no help at all.

It is if your model is utterly worthless.

Have you checked the list of abstracts tusenfem linked to?

Hundreds of papers in the last seventy years, published in mainstream journals, with titles like "Quaternionic quantum field theory", "Quaternion quantum mechanics: Second quantization and gauge fields", "Foundations of Quaternion Quantum Mechanics", "Quaternionic Formulation of Dirac Theory in Special and General Relativity", "Quaternionic Weinberg-Salam Theory", "Quaternions in Relativity"...

[thomheg]

Have you checked the list of abstracts tusenfem linked to?

I couldn't, because I can't access that server. You could because, I guess, because you belong to a university.

[tusenfem]

I wrote a kind of short story about it:

THE FOX
A man had the idea to visit a fox-hunt. He thought, that would be nice to see all those hunters on their horses, all dressed in pink.
[snip]
After a while he though, that he really wanted to see the hunters in their red jackets
[snip]

So, even in a little story you cannot get your facts right!

To talk about your method, now. Somehow, you wanted to do some physics, and looked through everything there is and then, instead of thinking: "wow, there are a lot of books written about physics, maybe I should start with some college book on general physics", you came up with: "wow, this very complicated mathematical structure called quaternion, of which I know nothing, looks like a great starting point to take a look a physics." And instead of reading literature about quaternions, you decided to just "take a stand" and let other people attack your idea, and maybe something beautiful will grow out of it.

There are two things that I would like to comment here:
1. It is not wrong to be drawn to an interesting word. Believe it or not, I went to Utrecht University because they studied there "plasma-astrophysics." At the end of high school, I had no idea what that meant, but I decided that I wanted to study anyway because it sounded somehow fascinating (and never regretted it, although it was a lot of work). So, it is okay to chose something and stick to it. However, you need to put some work in it. From what you have shown us up till now, there has been no real effort in doing that.
2. Put up an idea and have people go at it, then changing it later with all the critique is a very unfair way of doing science. You basically let all other people do the work, and you put it together, new idea, attacks, put together, and in the end you will present your Hegerian Theory of Quaternion Quantum Physics, without having done any of the work, except collecting, without having the foggiest what it is all about. That is not how science works, unfortunately, it is how a wiki works (sort of) and see what kind of nonsense is written there in some of the entries.

[tusenfem]

I couldn't, because I can't access that server. You could because, I guess, because you belong to a university.

That is rediculous, ANYONE can access ADS. It may be slow sometimes.


HERE is another link to ADS, with a list of 195 articles on quaternions in peer reviewed journals.

If you cannot load it again, then just go to ADS yourself, the page has the following address: http://esoads.eso.org/ads_abstracts.html

have fun

[thomheg]

That is rediculous, ANYONE can access ADS. It may be slow sometimes.

ADS is possible, of cause. But when I want to read the article, I get a SpringerLink and are asked to sign in, what I can't. So those links are useless to me.

[01101001]

ADS is possible, of cause. But when I want to read the article, I get a SpringerLink and are asked to sign in, what I can't. So those links are useless to me.

You can lead a horse to water...

[thomheg]

2. Put up an idea and have people go at it, then changing it later with all the critique is a very unfair way of doing science. You basically let all other people do the work, and you put it together, new idea, attacks, put together, and in the end you will present your Hegerian Theory of Quaternion Quantum Physics, without having done any of the work, except collecting, without having the foggiest what it is all about. That is not how science works, unfortunately, it is how a wiki works (sort of) and see what kind of nonsense is written there in some of the entries.

That would be unfair, indeed. But I didn't work that way. I developed my idea totally separated from what I'm discussing. The model is developing and I get some insights from discussions. But I'm usually not discussing my idea, but i.e. earthquakes. I regard that as a fair way, because it covers a topic of interest to all kind of people. I discussed i.e. Euler angles. But I had to decide, what I read (btw. I have read a lot) and what I discuss and how I make use of it. My 'book' was started as a collection of posts, because I wanted to send them to a friend. I later started to turn that into a theory and now I can discuss that, because it's finished and I know in fact, what it's about, because I've written it.

[tusenfem]

ADS is possible, of cause. But when I want to read the article, I get a SpringerLink and are asked to sign in, what I can't. So those links are useless to me.

Then say that and do not say the ADS server cannot be accessed.

And again and again it is written here on the board that if you cannot get an article yourself, you can ask in the thread whether someone with access can download it for you.

At least you can read the abstracts of all the papers, and that is already something and often (like immediately 8url=http://esoads.eso.org/abs/2008IJTP...47.1497B]the first paper in my list[/url]) there is a direct link (top left of the page) to the arxiv version of the paper. So, don't complain about having no access.

[Celestial Mechanic]

[Snip!] there is a direct link (top left of the page) to the arxiv version of the paper. So, don't complain about having no access.

Speaking of arxiv.org, trying their experimental full-text search for the term quaternions came up with a list of 1,068 papers. It might be wise to add a few other keywords to narrow it down a bit, but that is still impressive. And all of these papers may be downloaded in PDF format for FREE.

[thomheg]

Speaking of arxiv.org, trying their experimental full-text search for the term quaternions came up with a list of 1,068 papers. It might be wise to add a few other keywords to narrow it down a bit, but that is still impressive. And all of these papers may be downloaded in PDF format for FREE.

I usually use that or try to find associated websites. Some interesting papers from arxiv.org
About four-vector analysis (i was looking for that)
http://arxiv.org/PS_cache/arxiv/pdf/...711.3220v1.pdf

quote:
"The mathematical structure of quaternions
has always been considered as more appropriate
than the simple vectors to represent
the real physical variables. Nevertheless,
the quaternions were dismissed for the
difficulties and complications produced by
their quaternion product."


http://arxiv.org/PS_cache/quant-ph/p.../0609161v1.pdf
http://arxiv.org/PS_cache/nlin/pdf/0607/0607020v2.pdf
http://arxiv.org/PS_cache/astro-ph/p.../0311122v2.pdf
http://arxiv.org/PS_cache/math-ph/pd.../0309061v1.pdf
http://arxiv.org/ftp/physics/papers/0308/0308017.pdf
http://arxiv.org/ftp/physics/papers/0305/0305024.pdf
http://arxiv.org/PS_cache/hep-th/pdf/9508/9508011v1.pdf
http://arxiv.org/PS_cache/hep-th/pdf/9406/9406040v1.pdf

Quantum field theory for non-specialists:

http://www.awa.tohoku.ac.jp/~yhitosh.../particle.html
http://www.awa.tohoku.ac.jp/~yhitosh...b/partic2.html

[thomheg]

The Four-vectors in Relativity seem to be quaternions by another name.

I think, it's allmost like that.
Quote from my text:
"Now I merge QM and GR by exchanging their time.
This step is a bit surprising, but useful. I exchange the definition of time in spacetime and space. This turns intervals into four vectors and pinpoints the observer. After this exchange we see. that intervals are in fact just the same thing as four vectors and the vectors used in QM.
So spacetime and space are in fact the same thing and we could use any of their coordinates as time, but only one at a time, since the observer has to do this and is always somewhere. "

[DrRocket]

Hi

this is my first post here.
I've written a text about quaternions in physics.
It has grown a bit. I would like to hear some comments about this text.
The topic is quite interesting. You can do everything with quaternions, as far as I can see.
The text is a google.doc presentation. It is quite annoying, but google. docs have benefits, i.e. there is a build in chat system, that you can access if you have a google account.


[Thomas Heger

If I recall accurately the quaternion approach has been tried, but with limited success. It has since been abandoned.

Also, your quaternions don't quite look like what I recall. The matrix version in your book, has a problem. The matrix that you list as s1 does not have the stated property that s1 * s1 = 1, but rather s1 * s1 = s1. s1 is not even invertible. It is in fact projection from the y axis onto the line y=x in 2-space.

The usual view of the quaternions is that they are a real (non-Abelian)division algebra. With your matrix definition s1(s1 + s2) = 0, yet neither factor is 0.

I think something is amiss.

[thomheg]

If I recall accurately the quaternion approach has been tried, but with limited success. It has since been abandoned.

Also, your quaternions don't quite look like what I recall. The matrix version in your book, has a problem. The matrix that you list as s1 does not have the stated property that s1 * s1 = 1, but rather s1 * s1 = s1. s1 is not even invertible. It is in fact projection from the y axis onto the line y=x in 2-space.

Could you give me some hints, where you have found that. I can't remember, having used any matrix.
As far as I know now, I didn't use quaternions but four-vectors, but named it contravariant form of spacetime coordinates.

[DrRocket]

Could you give me some hints, where you have found that. I can't remember, having used any matrix.
As far as I know now, I didn't use quaternions but four-vectors, but named it contravariant form of spacetime coordinates.

It was in the section where you represented the quaternions in terms of Pauli spin matrices.

[thomheg]

It was in the section where you represented the quaternions in terms of Pauli spin matrices.

You think, that we are talking about the same document?
My text is found here. It is still in this stupid google.doc format. But, well, I change that later.
http://docs.google.com/Present?docid...&skipauth=true
'Pauli' was not found in my text. Could you give a hint in form of a page or a short quote?

[agingjb]

I assume the reference to Pauli is in the first document linked by thomheg in post 54.