Finally I have some time left to update this website. I would like to proceed with another part into the introduction to the four dimensional complex numbers. The previous part 20 dates back to 02 Feb 2019 and that too was about this structure of the non-invertible numbers in four dimensional complex space.
When I was reading back a few of my own old writings like part 20 from 02 Feb it struck me that those non-invertible numbers are all just linear combinations of the so called imitators of i. Imitators of i live always in dimensions higher than 2 and they mimic the behaviour of the number i from the complex plane. For example in the spaces of the 3D complex and circular numbers those imitators are not capable of squaring to minus one but they do a pretty good job at rotating stuff by 90 degrees if you multiply by them. In the 4D complex space there are two of such imitators and they do square up to minus one. Of course this is related to the fact you can find two copies of the 2D complex plane in the 4D complex space. So in that regard the 4D complex numbers are a bit different compared to the 4D quaternions that exist of 3 copies of the complex plane (but those do not commute and as such you cannot differentiate or integrate stuff).
Another interesting detail is that in the 4D complex number system the set on non-invertible numbers consists of just two lines that are perpendicular to each other. That is very different from the 3D situation where the set of non-invertibles is always a plane combined with a perpendicular line through zero. The reason that in 4D complex space the set is so small lies of course in the matrix representation and the determinant. On the 4D complex space the determinant is non-negative, just like the determinant is non-negative on the complex plane. Every 4D complex number has four eigenvalues and they come in conjugate pairs, so the product of these four eigenvalues gives the determinant hence the determinant cannot be a negative real number.
And say for yourself: aren’t the eigenvalue functions a very handy thing? If you want to find the eigenvalues of let’s say the 4D complex number Z = 1 + 2l + 3l^2 + 4l^3, that is often a horrible mathematical exercise. But once you have these four eigenvalue functions, you simply plug in any Z and voila: there are your 4 eigenvalues. This post is seven pictures long, as usual in the 550×775 pixel size.
Remark det(Z) = 0 does not show up in a ray tracing method.Oops, did I forget the eigenvalues of the number tau?
Ok, that was it for this post. Till updates my dear reader.
I estimate that most of my readers are familiar with emission and absorbtion spectra as used in astronomy. Light is produced when electrons fall in to a lower energy state in atoms, but that same light (the photons so to say) can also exite another electron in another atom and as such the photon gets absorbed.
Another example: why is glass transparent? Well the photons in the visible range have energies that do not interfere with the electrons in the glass. That is why these photons simply pass through and we can use stuff like glass for the windows in our homes.
So an element, say atomic hydrogen, is capable of emitting light at particular frequencies and at the same time that atomic hydrogen can absorb the same frequencies.
Now we go to the famous 21 cm wavelength: the standard explanation for the source of this em radiation is that in atomic hydrogen you find that both the proton and the electron have the same spin. The spins are aligned so to say and that state has a tiny bit more energy compared to the situation where the electron spins are not aligned. If the spins are aligned (that can be both up or both a down spin) there is a tiny probability that the electron spin flips. That releases a photon of 21 cm wavelength. It is never explained as why it is the electron spin that should flip, after all if the proton spin would flip this should give rise to the emission of a 21 cm photon also…
It is not much of a secret that I think that electrons are not magnetic dipoles but magnetic monopoles. Electrons get accelerated into the direction of the applied magnetic field, but if electrons were magnetic dipoles they would be neutral to external magnetic fields. Ok ok, professional physics professors come up with non homeogenous magnetic fields that should do the acceleration but if I do an easy estimate I find crazy gradients are needed. Something like 100 thousand Tesla per meter or so. It is important to remark that all those people doing the blah blah thing about inhomeogenous magnetic fields only do the blah blah thing: they never show a calculation that supports the blah blah. And yes, they also have a Hamiltonian kind of thing, but in the Hamiltonian the size of the electron is not incorperated. But the smaller in size a magnetic dipole is, the less it will get accelerated by such magnetic fields.
Another example that is hard to believe is the deflection of the solar wind by the earth magnetic field. Not only is the earth magnetic field very weak out there in space, it is hard to believe it has a serious gradient there in outer space. It must be very constant. Yet the solar wind gets deflected by the weak magnetic field of the earth. In my view this can only be done if electrons and protons are magnetic monopoles.
Here is an old ‘picture of the day’ from December 18, 1996 ‘A sky full of hydrogen’.
Why doesn’t the 21 cm radiation get absorbed?
The spin flip that ’causes’ the 21 cm radiation seems to be a seldom thing; about once in 10 million years. And it is always mentioned that it is spontaneous. In the next picture from a wiki you see how this supposedly works. Link: Hydrogen line https://en.wikipedia.org/wiki/Hydrogen_line
Again: Why no absorbtion?
Ok, what is my version of events? Very simple: Suppose there is an hydrogen atom in outer space that has it’s proton and electron carry the same magnetic charge. So both proton and electron have a north or a south magnetic charge. Suppose it is a north hydrogen atom. This atom simply bumps into a single electron having the south charge. Under the right conditions (bump not to slow and not to fast) this leads to a hydrogen atom with a north proton and a south electron. The north electron gets ejected because like magnetic charges repel. And the radiation emmited simply has wavelength of about 21 cm.
The 21 cm em radiation is also used in atomic clocks like those in the EU Galileo global satellite positioning system. In an apparatus named maser the hydrogen atoms get separated due to their magnetic charge and released in a resonance chamber. The point I want to make is that such a resonance chamber must have a very special coating on the inside otherwise the hydrogen atoms all ‘spin flip’ much to fast. So the coating must be a material without any unpaired electrons in it’s outer shell and of course it can’t be a metal because metals often have an electron sea that just sloshes around. From the ESA, here is a scetch of a hydrogen maser:
This was more or less what I had to say on this magnetic subject. If you start thinking about if it is possible that electrons are magnetic dipoles always you will find it is not logical. For example if it is true that the electron goes round the proton in atomic hydrogen, why the hell would the stuff get anti-aligned? If magnetism is just a vector pointing into some direction, if the electron goes round and round the up & down state would lead to precisely the same hydrogen atom…
All you need is a bit of logic.
Ok, we are at the end of this post. All I want to share with you is a teaser picture for the next post. The math of the next post is more or less finished but I still have to turn the stuff into the standard jpg pictures. The next post is more or less Part 21 into the basic introduction to the 4D complex numbers. I stopped those intro files back in Feb 2019, at this point in time I do not know if I will proceed but at least in a couple of days I will add post number 21 into that 4D number system. It is mainly about the so called ‘imitators of i‘, these imitators mimic the number i from the complex plane. Here is the teaser picture:
Here is an internal link to Part 20 intro to the basics of the 4D complex numbers:
I found an old video (what is ‘old’, it is from Jan 2019) and I decided to hang it in the website because it has such a beautiful introduction. The title of the video is The Secret of the Seventh Row. Seldom you see such a perfect introduction and I hope you will be intrigued too when you for the first time see the secret of the seventh row…
Now before I started brewing beer I often made wine. That was a hobby that started when I was a student. In the past it was much more easy to buy fruit juice that was more or less unprocessed, like 100% grape juice for 50 cents a liter. And with some extra sugar and of course yeast in a relatively short time you have your fresh batch of 20 liters wine. And somewhere from the back of my mind it came floating above that I had seen such irregularities arising from wine bottles if you stack them horizontal. But I never knew it had a solution like shown in the video.
Video title: The secret of the 7th row – visually explained
The next video is from Alexander Unzicker, the vid is only five minutes long. First I want to remark that I like Alexander a lot because he more or less tries to attack the entire standard model of physics. That not only is a giant task but you also must have some alternative that is better. For example when I talk or write about electrons not being magnetic dipoles, I never end in some shouting match but I just apply logic.
Let me apply some logic: In the Stern Gerlach experiment a beam of silver atoms is split in two by an inhomeogenous magnetic field. The magnetic field is stronger at one side and weaker at the other. One of the beams goes to the stronger side while the other goes to the weaker side of the applied magnetic field. But the logical consequence of this is that the stream silver atoms going to the weaker side gains potential energy. This is not logical. If you go outside and you throw a few stones horizontal, they always will fall to the earth and there is the lowest potential energy. The stones never fly up and accelerate until they are in space. In order to gain the logical point it is enough to assume that electrons are magnetic monopoles and that is what makes one half of the beam of silver atoms go to the weaker side of the applied magnetic field. If the electrons come in two varieties, either monopole north or monopole south, both streams do what the rest of nature does: striving for the lowest energy state.
Talking about energy states: Did you know that the brain of math professors is just always in the lowest energy state possible?
But back to the video: Alexander is always stating that often when progress is made in physics, all in all things become more easy to understand. That also goes for electrons, all that stuff about electrons being magnetic dipoles is just very hard to understand; why do they gain potential energy?
In his video Alexander gives a bad space as example where a so called three sphere is located. On the quaternions you cannot differentiate nor integrate, they are handy when it comes to rotations but that’s more or less all there is. So Alexander I don’t think you will make much progress in physics if you start to study the quaternions. And by the way don’t all physics people get exited when they can talk about ‘phase shifts’? They use it all the time and explain a wide variety of things with it. I lately observed Sabine Hossenfelder explaining the downbreak of quantum super positions into the pure ground states (the decoherence) as done by a bunch of phase shifts that make all probabilities of super positions go to zero. Well, the 4D complex numbers have a so called exponential curve and voila; with that thing you can phase shift your stuff anyway you want…
Video title: Simplicity in Physics and How I became a Mathematician
Yesterday I started working on the next post. It is all not extremely difficult but ha ha ha may be I over estimate my average reader. After all it is about the non-invertible numbers in the space of four dimensional complex numbers. The stuff that physics and math professors could not find for centuries… So you will never hear people like Alexander Unzicker talk about stuff like that, they only talk in easy to understand common places like the quaternions. And when I come along with my period of now about 18 years completely jobless, of course I understand the high lords of all the universities have more important things to do. All these professors are just soooooo important, they truly cannot react on social slime that is unemployed for decades. I understand that, but I also understand that if such high ranked people try to advance physics with the study of quaternions, the likelihood of success is infinetisimal small…
Anyway, here is a teaser picture for an easy to understand problem: if two squares are equal, say A^2 = B^2, does that always mean that either A = B or that A = -B?
In another development for decades I always avoided portraits and photo’s of myself on the internet at all costs. Of course after 911 that was the most wise strategy: you stay online but nobody know how you look. But over the years this strategy has completely eroded, if for example I just take a walk at some silly beach about 30 km away people clearly recognize me. So I more or less surrender, likely I will still try to prevent my head being on some glossy and contacts with journalists in general will also be avoided for decades to come. But in the present times why not post a selfportrait with a mask?
The upper half of the picture below is modified in the ‘The Scream’ style and the lower half is modified with something known as ‘vertical lines’.
Is the glass half full or half empty? You can argue that it is half full because the so called ‘pure tones’ as introduced in the previous post work perfectly for making a four dimensional Fourier series based on the 4D complex numbers. The glass is half empty because I started this Fourier stuff more or less in order to get some real world applications done, but 1 dimensional signals like a sound fragment do not reconstruct properly.
Why do they not reconstruct properly? Well often you need to take the sum or the difference of a 4D complex number Z and it’s conjugate that I write as Z* (because I cannot do ‘overline’ in this text mode). But the sum or the difference of such 4D numbers removes only the real part or the second imaginary part. The first and third imaginary parts stay in this sum or difference, this stuff is what makes the reconstruction of a signal s(t) going wrong.
Yet I was not crazy, as far as the reconstruction works it does it more or less as expected only you get only half of the signal reconstructed. That is not that worse but the garbage that enters the reconstruction is what makes this kind of making a 4D Fourier series something that will never have any practical benefit. But again does that mean the glass is half empty?
I remember that a long time ago in something like 1991 or 1992 I had found the product of a 3D complex number X and it’s conjugate X*. My naive idea was that this should only give the unit sphere in 3D space, but this product that also two imaginary components that I considered garbage at the time. Back in the time, it was just before the internet era, I could not know that this ‘garbage’ was actually the equation of a cone. And if you intersect this cone with the unit sphere in the space of 3D complex numbers, you get the 3D exponential circle. So it wasn’t garbage, it was the main prize in 3D complex number theory. Back in the time in 1991 it was stuped from me to expect the 3D complex numbers would behave ‘just like’ the two dimensional complex numbers from the complex plane. May be in this year 2020 I am making the same mistake again by expecting a 4D kind of Fourier series must behave ‘the same’ as those defined on the real line (the original Joseph Fourier proposal) and the more advanced version from the complex plane.
With the 4D Dirichlet kernel just like with the 2D Dirichlet kernel from the complex plane, you must take the difference of a number and it’s conjugate. In the complex plane this makes the real component zero and this difference is just an imaginary number. The 2D Dirichlet kernel is the quotient of two such imaginary numbers and as such it is always a real number. For the 4D Dirichlet kernel stuff is not that easy but for me it was surprising that you can show relatively easy the 4D Dirichlet kernel has to be a ‘self conjugate’ number. That means Z = Z* (on the complex plane when you have a number z such that z = z* it means it is a real number).
This post is 8 pictures long, 7 of them have the standard size of 55×775 pixels but I had to make one picture both a bit more broad and higher in order to get the math fitting in it. Ok, let’s upload the math pictures with the stuff around the 4D Dirichlet kernel.
Do not fear if the 4D kernel looks a bit complicated, just take your time…Yes trouble on the road, but it sure looks cute!
Ok, credits have to go to where that is deserved. I remember that back in the time like in 1990 I found it relatively hard to calculate the 2D Dirichlet kernel. It took me over 15 or 20 minutes but again: that was before the internet era. Yet at present day I was all so simple and why was that? That is because there is a nice Youtube video doing the easy stuff, it is from ‘Flameable math’:
May be at the end I can say the glass is half full because now this reconstruction stuff does not work properly, luckily I do not have to construct the Fejer kernel for 4D complex numbers… Ok, let’s call it a day and let me end this post.
