Complex Quaternions & Physical Reality
Representing Plane Waves in Three Dimensional Space

"We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra."
“To sum up: it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. ... the life of that man is one long sin against mankind." (William Clifford)

This page has a summary of my little journey of discovering quaternions, then realising that combined with a wave structure of matter you can explain numerous thing in physics (it is a very elegant mathematics that is important since it has the same three dimensional structure as space - which is why Hamilton invented them!)

I do need help from mathematicians with knowledge in this area, so hopefully a few of you reading this are doing so since I sent you a nice letter and can help develop this.

So let's start with an interesting question about quaternions and physical reality that occurred to me several weeks ago and now I am very curious to know the answer - I am hoping you can answer it!

Question - What is the mathematical representation for plane waves flowing in all directions through three dimensional space, and are there solutions to the phase arrangements that cause the transverse components of these waves to exactly cancel leaving us with a scalar spherical standing wave?

My Prediction - You will find four solutions (two pairs of two solutions). i.e. The intersection of plane waves creates two opposite phase spherical standing waves (electron and positron) and there are two solutions (phase arrangements) that can create each phase standing wave, explaining the two spin states of the electron and positron. To me this seems to be the simple obvious explanation for how fundamental matter 'particles' (leptons) with correct spin can be created from 'empty space'.

Here is a short background – it explains what led me to posing this question – hopefully it will fill in the gaps and make some sense to you.

I am a natural philosopher. i.e. I live in Nature, read good minds and think and observe the world around me trying to understand it – to know the truth. Thirteen years ago, from reading Feynman, Lorentz and Einstein it occurred to me that matter could be formed from a spherical standing wave in space, where the wave-center is the ‘particle’. A year later I found the same spherical wave structure of matter (WSM) in the work of Milo Wolff who had deduced a number of things, most notably that when we consider the Doppler shifts of the spherical in and out waves with relative motion we find the de Broglie wave and the relativistic mass (frequency / energy) increase appear in the same wave equation. This seemed a very positive confirmation and motivated me to continue my research. (Since then I have read a great deal and many more things have been understood – it is easy to show that WSM is not only reasonable and scientific, but also simple, sensible and logical).

However, I was never completely satisfied with Milo's explanation of spin. His view is that the out waves of all the other matter (wave-center ‘particles’) in the observable universe create the spherical in waves of our matter (i.e. Huygens principle, and this does cleverly explain Mach’s principle too!). Further, he assumes the spherical in-waves are flowing in towards a point and must do a spherical rotation to be transformed into spherical out-waves. This does generate the correct spinor maths, so naturally you would think it a very clever and true solution (but I have learned how tricky and deceptive nature can be, so I try to keep an open mind!).
I had two problems with this view that gnawed away at my mind for years;

i) I do not think waves flow into a point, they simply flow through space and through other waves (albeit non-linearly, with velocity changing with wave amplitude, as required for energy exchanges / matter interactions). Thus to me it made more sense that waves just flow in and out through the wave center creating a scalar (longitudinal) spherical standing wave about a central point.

ii) I wondered what kept electrons and positrons locked in their opposite phase across the universe (matter and anti-matter are opposite phase standing waves). The only conclusion I could come to is that each wave center must be locked into a certain phase relationship with the fixed wave pattern of background space.

From these considerations I imagined reality (waves in space) could also be described in terms of just plane waves (spherical waves tend to planar at large distances) flowing through space in all directions. Based on this, several years ago I made up some wave diagrams to show these two alternative (though similar) views. See;

This shows how other matter's spherical Out-Waves form our matter's In-Waves (Huygens' Principle). Four important points!

i) In reality there are about 1080 other wave center 'particles' whose spherical out-waves form into our spherical in-waves.

ii) They are obviously not all the same distance away, but distributed throughout the space of our finite spherical observable Universe (Hubble Sphere) within infinite eternal Space.

iii) These other wave-center 'particles' around us are also formed from the matter waves around them, and this process extends to infinity.

iv) The diagram is misleading in showing the waves starting. In reality the waves are continually flowing out from other matter around us. i.e. The system is perpetual, the in and out waves are always being shared between electron wave-centers wherever you are in infinite space.

So this is a very simplistic diagram of what is really going on. Electrons are very large complex wave structures of the universe (this is true for humans too!).


Below is a simple diagram representing these intersecting plane waves. When you combine the left eight diagrams you get the image on the right. You can clearly see the spherical effects starting to form - and importantly you also see opposite phases (positrons and electrons) being formed.

Now to move along to the discovery and conclusion (which you may well have already guessed since you understand complex quaternion wave functions!).
A year ago I came upon quaternions and read about Hamilton's discovery while he was walking, the carving of the equation in the stone bridge - and this was after 13 years of Hamilton thinking about how to represent the rotation of objects in 3D space, so you can understand his excitement (I resonated strongly with the story since I share this trait of walking and thinking - and thinking about things for years till you solve them!).
I took his view seriously that this was important maths because its three dimensional structure was in harmony with the three dimensional space we experience and which underpins the WSM. (But with WSM we can now understand his connection of space and time into a 4D continuum – the fourth dimension of time is really motion as the wave motion of space.)
However, at the time I was busy (I have two young daughters!) and was meaning to return to quaternions for more study. Then about two months ago I received a letter from a reader of my philosophy / physics website asking why Mendel Sach's has not received a Nobel prize. I had a quick look at his work and was immediately inspired to re-examine quaternions (and biquaternion wave functions) more closely.

When I was multiplying a quaternion by its complex conjugate and seeing how all the complex orthogonal vectors cancelled and left us with just the scalar terms in a 4D continuum (identical to the form of a sphere) the idea occurred to me that this was happening to real waves in space (physical reality). i.e. All the vector transverse waves (waves on the complex orthogonal planes) were canceling one another leaving just a scalar spherical standing wave. And now I think I can end this summary, repeat my question, then discuss some of the details! (And I consider this a very important question - after 20 years of study of 2,500 years of knowledge).

Question - What is the mathematical representation for plane waves flowing in all directions through three dimensional space, and are there solutions to the phase arrangements that cause the vector transverse components of these waves to exactly cancel leaving us with a scalar spherical standing wave?

As above, I think you will find four solutions. Two opposite phase spherical standing waves are created (electron and positron) and there are two solutions for each phase standing wave, explaining the two spin states of the electron and positron.

Basically it seems that space can actually vibrate in two quite different ways, as background space (the quantum field) where it vibrates in all directions, and as matter ‘particles’ where it vibrates radially around a point, forming a scalar spherical standing wave. This provides a simple obvious explanation for how fundamental matter 'particles' (leptons) with correct spin can be created from 'empty space'.

Let us now consider this in more detail.

We are determining what wave patterns naturally arise if we assume this three dimensional space around us does in fact exist and has plane waves flowing through it in all directions.
Firstly we can answer four fundamental questions that are deduced from only one substance, space, existing.
Space is necessarily;
i) Infinite (not bounded by a second substance).
ii) Eternal (not created by a second substance).
iii) Continuous (does not contain a second substance, thus there are no discrete particles).
iv) Thus the plane waves in infinite space are also infinite as true plane waves must be (and we begin with the simplest assumption that the plane waves are scalar / longitudinal rather than vector transverse).

Space itself must be very rigid (slightly elastic) to propagate such high velocity waves (velocity of light c). I assume the wavelength is somewhere between the Compton wavelength and nuclear size (10^-12 to 10^-16 m) so the frequency is between 10^20 and 10^24 Hz.

However, we still have the “in all directions” issue to deal with. If we allow this to be an infinite number of directions within three dimensional space then we would have infinite plane waves and infinite energy in each region of space (the same infinity problem is found in quantum field theory, they renormalise the equations to remove it and put in the correct value – while it works it shows the mathematical structure of QED is not completely correct).
The most logical approach is to put in the value that represents the number of other wave-center ‘particles’ in the observable universe, roughly 10^80, since they can be treated as sources of spherical out waves, where the surface of a spherical wave tends to a plane with large distances.

So now we can calculate how space vibrates given these 10^80 plane waves flowing through space in (nearly) all directions. Generally, because there are so many plane waves from trillions upon trillions of directions it means that there will invariably be a plane wave traveling in the opposite direction to any plane wave so we are actually looking at planar standing waves in background space. As we have assumed them to be longitudinal plane waves thus any point in space will vibrate in all 10^80 directions (in some complex pattern of motion through three dimensions – a mix of longitudinal scalar and transverse vector motions when viewed in any direction?)

However, the most interesting thing is that space can also vibrate radially / spherically around a central point forming what we call matter ‘particles’. It seems the quaternions are telling us that there are four unique phase arrangements of the plane waves where all the transverse vector components cancel one another (as represented by qq*) creating the scalar spherical in-wave (this is still Huygens principle, but from plane wave fronts rather than spherical).
The spherical in-wave then flows in and out through its wave center (no spherical rotation, the spherical in-wave simply becomes a spherical out-wave on the other side of the wave center) and then undergoes another phase transformation on the way out from the wave center that causes the vector transverse waves to re-appear (mirror image of in-wave transformation, chirality?).
Thus space vibrates radially around a central point as a scalar spherical standing wave.
This, to me, seems to explain both the creation of 'particles' from background space and the spin of the 'particles'.

Significantly this also solves the renormalisation problem that Feynman gets because he used spherical vector waves (advanced and retarded) that have infinite field strengths as r tends to zero, but this is not a problem for scalar waves which have a finite wave amplitude at the wave center. Thus the above formalism solves both infinities found in QED / QFT.

Some Speculation on the Higgs Mechanism and Virtual Particles

Having only thought about this new ‘quaternion’ way of seeing the wave structure of matter for a few weeks, my mind is having to adjust quite quickly, but already I can speculate about spontaneous symmetry braking and the Higgs mechanism.
The quaternion equations give paired solutions, meaning that if there is an alignment of phase of planar waves that creates an electron, then at the same time this wave phase adjustment will necessarily create the opposite phase spherical standing wave the positron.
These high wave amplitude spherical standing waves create a non-linear region of space where the wave velocity is higher (the cause of all electrical forces / charge). This changes the in wave velocity of the plane waves flowing between the two ‘particles’ which causes the electron and positron to reposition towards one another and annihilate (destructive wave interference). This must be happening everywhere in space all the time given the number of waves we are considering (in the order of 10^100 waves flow through any region of space each second).
This also leads me to think that to create enduring positrons and electrons from this system they must be given enough kinetic energy to escape annihilation, and it is this energy that we define as being the electron’s energy, which is really its electromagnetic potential energy (due to its non-linear high wave amplitude wave center). This suggests that we have ignored the true energy of the electron, its standing wave energy, a possible source of dark matter / dark energy?

I assume to solve the above you need to use a biquaternion wave function multiplied by its complex conjugate - to essentially find the factors of a spherical standing wave (or its energy equation). However there are two different conjugates, and eight solutions (rather than four as there are for quaternions), so I am confused as exactly what the extra four represent (Lanczos’ work deduced these four extra terms that were later identified as Isospin of protons and neutrons, perhaps it is saying that these heavy ‘particles’ are made from positron electron pairs (positronium)).
If this wave structure of matter is correct then I assume you should get an equation similar to the Lanczos equation, a more general form of the Dirac equation. I am also curious how this WSM quaternion maths relates to the Klein Gordon wave equation, which I understand represents energy of a particle but is in the form of a plane wave (is this correct?).

Are you able to do this?
Are you interested in doing this?
If not can you recommend others who would be?

I must admit to eagerly awaiting your reply!
Geoff Haselhurst
(May, 2011)

PS - I also wrote this letter to academic philosophers about metaphysics, physics and quaternions - it is useful!

Below is some further discussion about the philosophy and metaphysics of three dimensional space.

1. Is it Scientifically Valid to Assume that Space Exists as a 3D Wave Medium

This is reasonable science as we all experience space indirectly via matter-energy interactions in space-time, this is the most simple foundation as only one substance exists, Space, (i.e. Occam's razor), and this then most importantly brings science into harmony with metaphysics and the very beginning of human reason (ancient Greek, Indian and Chinese philosophy) that the world was a dynamic unity to explain the interconnected change of all things.
Further, there are many things you can deduce and explain with a (spherical standing) wave structure of matter, e.g. antimatter is just the opposite phase standing wave, the relativistic mass increase and the non-local de Broglie phase wave are deduced from Doppler shifts with relative motion (see Milo Wolff), the spherical wave structures change shape ellipsoidally when accelerated (foundation of general relativity and Riemann geometry), the wave center is non-linear explaining how wave interactions and energy exchanges occur, the spherical in out waves explain how matter is in continual two way communication with all other matter in the space around it / the universe, .. there are many reasons to take a wave structure of matter seriously, thus to take a space wave medium seriously, i.e. waves need a continuous medium (so does mathematics and reality as I see things!)

However, it seems that mathematical physicists have difficulty comprehending that space can exist! Below are a couple of comments from a relatively famous physicist who works with quaternions.

"I don't believe that we do experience space at all, or that it is a 'thing' that can propagate anything. I believe that what we do experience is our interaction with matter." (MS)

My first thought is that we should be cautious scientists who realise that our reasoning is only as good as our premises and that history shows that we are often misled by the beliefs of our times thus the safest solution is to just test the wave theory in space and see if it works.

But it is an odd reply from a physicist that agrees that matter and space (and energy and time) are one thing (holism, dynamic unity of reality) and that matter is spatially extended as continuous wave fields. Thus we experience space whenever we experience matter. (We also experience 3D space when we close our eyes and lean in different directions and experience a sense of balance - and the three orthogonal ear canals confirm that evolution works in 3D too.)

It is also worth noting that most of the physicists in the 19th century believed in a luminiferous aether to propagate light waves, most physicists in the 20th century think it is naive to talk about absolute space. Most importantly, the conception of the aether was childishly simplistic and naive (I think they imagined it to be made of tinier particles than the matter particles - and they maintained the concept of the matter particle with this aether which is silly). Likewise, the reasons for rejecting absolute space are also incorrect. e.g. MM experiment and special relativity were both explained by Lorentz in his Theory of the Electron (1905) with the Lorentz ether theory (LET) and Einstein agreed that there must be something that existed in 'empty space' to create the space-time metric.

"I see space (and time) as the words of a language that we use to facilitate an expression of the laws of matter. The waves we talk about, associated with "particles of matter" are the basic representation of this matter. We see that the language of the laws of matter in general relativity entail a quaternion formalism. This is indicated by the irreducible representations of the symmetry group of general relativity theory. The geometrical logic of this language is associated with a (Riemannian) curved spacetime." (MS)

These opening comments are pretty much text book postmodernism (see Kant quotes below), I have read similar words many times from many people. The thing I always notice about them is they are stated as beliefs, never deduced from first principles. And in 20 years study of physics, philosophy and metaphysics I have yet to find one valid argument for the non-existence of space (or the non-existence of absolute objective truths - and I think you will find the former causes the latter as continuous space is the source of continuity / causal connection which is the source of truth.).

It is also interesting that the quaternion mathematics you use was deliberately designed by Hamilton to describe things (rotations) in three dimensional space.

"Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." (William Rowan Hamilton)

And Clifford, from his work on quaternions, went further and came to the conclusion matter was of a wave nature that propagated through a space medium. Both these people were smart - both developed the maths you use, both believed space was three dimensional and thus this mathematics was special in its ability to explain physical phenomena. Strangely you agree with everything except their starting point - the existence of three dimensional space!

In philosophy Kant is most famous for deducing that space and time are mere representations of the mind, and this has profoundly influenced the evolution of human thinking in the direction of critical idealism (the world exists, but the we can only know our ideas of the thing, never the thing in itself). Yet even here if you make the effort to read Kant's reasoning you find it is simply not true. Kant begins by stating space is a priori for science to exist (empirical senses of external world).

"Natural science contains in itself synthetical judgments a priori, as principles. Space then is a necessary representation a priori, which serves for the foundation of all external intuitions." (Immanuel Kant, Critique of Pure Reason, 1781)

But being subtly influenced by Newton's Mechanics and thus a particle conception of matter moving about in an absolute space and time (which really is naive) he then wrote;

"Here I shall add that the concept of change, and with it the concept of motion, as change of place, is possible only through and in the representation of time. Motion, for example, presupposes the perception of something movable. But space considered in itself contains nothing movable; consequently motion must be something which is found in space only through experience -in other words, is an empirical datum." (Kant, 1781)

From wave structure of matter (WSM) foundations the correct answer is that Space in itself must have Properties. i.e. That Space exists with the properties of a wave-medium and thus contains wave-motions which ultimately cause not only time, but also matter and its forces. (Kant made the common mistake of only considering Motion of matter particles and not the (wave) Motion of Space itself!) From this we must then conclude that Space and (wave) Motion are a priori and first necessary for us to experience the world. This then explains the current confusion of modern physics due to the incorrect conception of the Motion of matter 'particles' in Space and Time, rather than the spherical wave-motion of Space causing both matter 'particles' and time. Significantly, Kant realized the importance of this problem of synthetic a priori knowledge as the foundation of the Sciences and thus of certainty of knowledge.

"Upon the solution of this problem, or upon sufficient proof of the impossibility of synthetical knowledge a priori, depends the existence or downfall of metaphysics." (Kant, 1781)

Kant's error leads him to conclude;

"If we take away the (human) subject, or even only the subjective constitution of our senses in general, then not only the nature and relations of objects in space and time, but even space and time themselves disappear; and that these, as appearances, cannot exist in themselves, but only in us. What may be the nature of objects considered as things in themselves and without reference to the receptivity of our sensibility is quite unknown to us. .... not only are the raindrops mere appearances, but even their circular form, nay, the space itself through which they fall , is nothing in itself, but both are mere modifications or fundamental dispositions of our sensible intuition, whilst the transcendental object remains for us utterly unknown." (Immanuel Kant, Critique of Pure Reason, 1781)

Two final points - we know science exists and that we have both logical and empirical knowledge - and we know mathematical logic exists and relates to physical reality. Thus you can ask yourself what properties does this infer on reality, and does a wave structure of matter in space satisfy these? (i.e. Mathematics does not exist in some magic fantasy land - it exists in physical reality and infers certain things about that reality.)

We find that a wave structure of matter satisfies both of the above, it explains how mathematics can exist due to necessarily interconnected wave patterns which behave logically and can be quantified (the source of all logic) and the spherical in out wave structure explains how we can have empirical knowledge due to this two way sharing of waves between wave-center 'particles' across the universe.

Perhaps in concluding this I could say that humans are genetically programmed to be culturally programmed - and once programmed to a certain belief it is almost impossible to change. This is true for you, me, every human. This is why philosophers realise that it is a rare person that can rise above the customs and beliefs of their time and sincerely seek truth.
I hope that we can!

Geoff Haselhurst

PS - And once you confirm for yourselves this is true - please send it on to others for comments - thanks!

And a few quotes ...

"Someday we'll understand the whole thing as one single marvelous vision that will seem so overwhelmingly simple and beautiful that we may say to each other; 'Oh, how could we have been so stupid for so long? How could it have been otherwise!'" (J. A. Wheeler)

"We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.” (William Clifford)

"The danger to society is not merely that it should believe wrong things, though that is great enough; but that it should become credulous, and lose the habit of testing things and inquiring into them; for then it must sink back into savagery. ... To sum up: it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call into question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it--the life of that man is one long sin against mankind. ...
Inquiry into the evidence of a doctrine is not to be made once for all, and then taken as finally settled. It is never lawful to stifle a doubt; for either it can be honestly answered by means of the inquiry already made, or else it proves that the inquiry was not complete.
"But," says one, "I am a busy man; I have no time for the long course of study which would be necessary to make me in any degree a competent judge of certain questions, or even able to understand the nature of the arguments."
Then he should have no time to believe." (William Clifford)

"What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). ... The world is given to me only once, not one existing and one perceived. Subject and object are only one. The barrier between them cannot be said to have broken down as a result of recent experience in the physical sciences, for this barrier does not exist. ... The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists." (Erwin Schrodinger)

"The notion that all these fragments is separately existent is evidently an illusion, and this illusion cannot do other than lead to endless conflict and confusion. Indeed, the attempt to live according to the notion that the fragments are really separate is, in essence, what has led to the growing series of extremely urgent crises that is confronting us today. Thus, as is now well known, this way of life has brought about pollution, destruction of the balance of nature, over-population, world-wide economic and political disorder and the creation of an overall environment that is neither physically nor mentally healthy for most of the people who live in it. Individually there has developed a widespread feeling of helplessness and despair, in the face of what seems to be an overwhelming mass of disparate social forces, going beyond the control and even the comprehension of the human beings who are caught up in it." (David Bohm, Wholeness and the Implicate Order, 1980)

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Biography: Geoffrey Haselhurst, Philosopher of Science, Theoretical Physics, Metaphysics, Evolution. Our world is in great trouble due to human behaviour founded on myths and customs that are causing the destruction of Nature and climate change. We can now deduce the most simple science theory of reality - the wave structure of matter in space. By understanding how we and everything around us are interconnected in Space we can then deduce solutions to the fundamental problems of human knowledge in physics, philosophy, metaphysics, theology, education, health, evolution and ecology, politics and society.

This is the profound new way of thinking that Einstein realised, that we exist as spatially extended structures of the universe - the discrete and separate body an illusion. This simply confirms the intuitions of the ancient philosophers and mystics.

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