Lately I added a bunch of sine functions and I wondered what the maximum was. And to be honest I had no idea, in math that is pretty normal otherwise you would not search for such answers. The questions you can answer instantly are often much more boring and often don’t add much value or insights. So what was I looking at? Well take the sine function, lets write it as sin(t). Make a timelag of one unit or if you want a translation and that’s sin(t – 1). Proceed in taking time lags like sin(t – 2), sin(t – 3) and so on and so on and add them all up. The question is: Can you say something about the maximum value that this sum can take? And no, I had no idea about how to approach this problem.
The interesting detail is of course that this sum of sines does not seem to converge or diverge in any significant way. You can check that for yourself in for example the DESMOS package, just type the word sum and you get the sigma symbol for a summation. I like the package and in case you have never seen it, here is a link: https://www.desmos.com/calculator?lang=en.
As far as I know this problem has no or little math meaning, it is just some recreational stuff. But if you in your life had the honor of calculating a bunch of Fourier coefficients again and again, you know that the summation of sines and or cosines and or complex exponentials can have very tricky convergence questions. Now with my little sum of sine time lags we don’t have any convergence at all, the funny thing is it also does not diverge.
This post was meant to be short but as so often it grew to five pictures long and on top of that there are three extra figures added to the mix. It’s all pretty simple and not deep complicated math that as so often is very hard for human brains to digest. Have fun reading it.
Figure 1: Yep, this is not a periodic function.Figure 2: Positive interference leads to larger amplitudes.Figure 3: Oh oh some stupid typo’s with the number 50…
At the end of this post I want to remark that I framed the question for some finite sum of sines. That is because I wanted to avoid all things related to taking a supremum and stuff like that. Look at Example 1 above, here of course the maximum value of the two sines is not 2 because there is no real solution to this, but of course the supremum is 2 because you can come arbitrarily close to 2. Of course in Example 3 I wanted to know the sup and the inf of the amplitudes, but I framed the question in a finite sum of sines anyway.
Ok that was it for this unclassified post. If you want you can think a bit about sums of sines and if you get bored of that you can try to figure out what an electron pair is if the Pauli exclusion principle says it must have opposite spins… Thanks for your attention and see you in the next post.
Now I’ve seen a lot of relatively boring videos the last years with electron beams and magnetic fields. And the only thing they often show is just the Lorentz force that is perpendicular to both the magnetic field and the direction of the electrons. Never ever do they jump to the conclusion you can do your own ‘Stern-Gerlach experiment’ by trying to separate the electron beam into two. As such those guys, it’s almost always guys, often do nothing more as holding the magnetic field perpendicular to the electron beam. And no matter how hard I shout and curse at youtube on my television, they never listen… But serious, today I came across a video of a teacher who tried to make the magnetic field as parallel to the electron beam as possible. In the past I have done a similar thing and I still have photo’s from that. But the way we had set up these experiments is rather dual to each other.
The way Francis-Jones does it in the video: His magnetic field is wide, he uses those Helmholtz coils and one steady electron beam.
Back in the time I could still buy an old black and white television that still works to this present day. Because it’s a black and white television it only has one electron beam that constantly covers the entire television glass tube. So my electrons were spread out and my magnets was more a point like thing because it was a stack of neodymium magnets.
If you look at such experiments as ‘wide’ against ‘narrow’ there are two other possibilities this way: 1) A Helmholtz coil against a television screen, I don’t think you will get interesting results but you never know. 2) A stack of magnets against one steady electron beam, I expect a central point on the screen for the middle of the electron beam and a vague ring around it from the electrons that get repelled.
Anyway the reason that still today I think electrons are in fact magnetic monopoles was simple: My own simple and cheap experiment could absolutely not disprove that electrons are not tiny magnets but monopoles. All that stuff from quantum theory that for some mumbo jumbo reason the dipole magnetic field of the electron will anti-align with external magnetic fields, it is just fucking bullshit. It is so fucking stupid in say the electron pair we know from chemical bondings and also from super conductivity, why the hell should those tiny magnets anti align? A few months back I made a picture for what the official version of an electron pair is, of course this madness should also have an south pole to south pole variant, but here is that nonsense once more:
Really true: Maxwell’s little demon holds all electrons in place…
Let me stop ranting and lets turn to the video. At one point in time Francis turns the electron beam a little bit and there is where the next screen shot comes from. It is at 7.50 minutes into the video:
It could be some light reflection but is it still one electron beam?
Well you can judge for yourself but the problem with looking at such video’s is that they just never ever try to split the electron beam in two… So it is hard to say if here are two electron streams or that we are looking at some light reflection. So I cannot use this video for making my point it is stupid to view electrons as tiny magnets since their magnetism is just like their electric field properties: Monopole and permanent.
After having said that, let me show you once more a photo of the old black and white television. And a miracle happened: Not only did my experimental setup succeed into two classes of electrons with regard to their monopole magnetic charge. It also turns the old black and white television into a color television!
Please note the small white region, it’s circular but you can’t see it on the photo.
Yeah yeah, that small circular region behind the magnet is what gave me a bit of confidence years ago. These electrons are magnetic monopoles and not tiny magnets or whatever what. But the professional physics people much more like to talk about stuff like “Spin orbit coupling” or other mysterious sounding stuff.
I have no idea what that teddy bear is doing there.
At the end I want to remark my total costs were 12€ for the old black and white television and about 50€ for the stack of neodymium magnets. But this Francis guy says the tube is about 500 pounds, so likely Francis is from the UK. So shall I buy me one of those things for myself? No of course not, I am not interested in writing a publication that could be read by professional physics people. Why should I? In case electrons are the long sought magnetic monopoles, it is obvious you won’t get much published into such lines of thinking.
Lets leave it with that while noting it was fun for me to write a new post on magnetics.
Updated two days later: Today, that was 06 March so actually yesterday, I realized that if you have access to one of those beautiful cathode ray tubes, you can also use two stacks of those strong magnets.
Since the goal is to make the beam split in two, you must use the north pole of the one stack and the south pole of the other stack. If you have never worked with these kinds of magnets, practise first before you hold them near the glass. If the magnetic fields are strong enough and the electron beam splits in two, what does that mean for if electrons are magnetic monopoles or bipolar tiny magnets? Well if you view the electrons as magnetic monopoles, it is logical from the energy point of view that the beam splits: Both kinds of magnetic charges only try to lower their potential energy.
And suppose that electrons are tiny magnets, in that case the electrons that align themselves with the applied magnetic field will lower their potential energy. And if you believe that electrons anti-align where does the energy come from that makes them do this? All that anti-align stuff of electrons is rather mysterious and I think that is important for the physics people. If you are interested in quantum mechanics you likely have heard the next phrase of saying a few times:
If you think you understand quantum mechancis, you do not understand quantum mechanics.
Well that is an interesting point of view but you can also think: If I get crazy results with thinking that electrons can anti-align, may be there is something wrong with my theory? But you never see physics professors talking that way, after all talking out of your neck is a shared habit amongst them.
Now the idea of using two stacks of magnets must be executed carefully as you see in the next picture:
There are many ways to introduce CR-equations for higher dimensional complex and circular numbers. For example you could remark that if you have a function, say f(X), defined on a higher dimensional number space, it’s Jacobian matrix should nicely follow the matrix representation of that particular higher dimensional number space. I didn’t do that, I tried to formulate in what I name CR-equations chain rule style. A long time ago and I did not remember what text it was but it was an old text from Riemann and it occured he wrote the equations also chain rule style. That was very refreshing to me and it showed also that I am still not 100% crazy…;) Even if you know nothing or almost nothing about say 3D complex numbers and you only have a bit of math knowledge about the complex plane, the way Riemann wrote it is very easy to understand. Say you have a function f(z) defined on the complex plane and as usual we write z = x + iy for the complex number, likely you know that the derivative f'(z) is found by a partial differentiation to the real variable x. But what happens if you take the partial differential to the variable y? That is how Rieman formulated it in that old text: you get f'(z) times i. And that is of course just a simple application of the chain rule that you know from the real line. And that is also the way I mostly wrote it because if you express it only in the diverse partial differentials, that is a lot of work in my Latex math typing environment and for you as a reader it is hard to read and understand what is going on. In the case of 3D complex or circular numbers you already have 9 partial differentials that fall apart into three groups of three differentials each. In this post I tried much more to hang on to how differentiation was orginally formulated, of course I don’t do it in the ways Newton and Leibniz did it with infitesimals and so on but in a good old limit. And in order to formulate it in limits I constantly need to divide by vectors from higher dimensional real spaces like 3D, 4D or now in the general case n-dimensional numbers. That should serve as an antidote to what a lot of math professors think: You cannot divide by a vector. Well may be they can’t but I can and I am very satisfied with it. Apperently for the math professors it is too difficult to define multiplications on higher dimensional spaces that do the trick. (Don’t try to do that with say Clifford algebra’s, they are indeed higher dimensional but as always professional math professors turn the stuff into crap and indeed on Clifford algebra’s you can’t divide most of the time.)
May be I should have given more examples or work them out a bit more but the text was already rather long. It is six pictures and picture size is 550×1100 so that is relatively long but I used a somehow larger font so it should read a bit faster.
Of course the most important feature of the CR-equations is that in case a function defined on a higher dimensional space obeys them, you can differentiate just like you do on the real line. Just like we say that on the complex plane the derivative of f(z) = z^2 is given by f'(z) = 2z. Basically all functions that are analytic on the real line can be expanded into arbitrary dimension, for example the sine and cosine funtions live in every dimension. Not that math professors have only an infitesimal amount of interest into stuff like that, but I like it. Here are the six pictures that compose this post, I hope it is comprihensible enough and more or less typo free:
Ok that was it, thanks for your attention and I hope that in some point in your future life you have some value to this kind of math.
This post is very simlilar to a few back when we calculated the results on 4 different spaces. This time I hardly pen down any calculation but only give the results so we can compare them a little bit. The way most professional math professors tell the story of complex numbers it goes a bit like this: We have the real number line, the complex plane and on top of that a genius named Hamilton found the quaternions. On top of that there are a bunch of so called Clifford algebra’s and oh we math professors are just so good. There is no comparison to us, we are the smartest professionals in the world!
Well that is very interesting because it is well known these so called ‘professionals’ could not find the 3D complex numbers for about 150 years. So how come they all say we have this and that (complex plane and quaternions) and that’s enough, we are just perfect! Why they keep on saying rubbish like that is the so called Dunning-Kruger effect. That’s something from psychology and it says that people who lack understanding of some complicated stuff also lack the insight that they are stupid to the bone when it comes to that particular complicated stuff. So the views of professional math professors is very interesting but can be neglected one 100 percent, it’s just Dunning-Kruger effect…
If you look at the seven results of the ‘Speed = the Square’ equations, the solutions form a strickt pattern that only depends of the number of dimensions and if it is the complex or the circular multiplication. So every time a math professor goes from the complex plane to the wonderful world of quaternions you now know you are listening to a weirdo.
I said I only give results but since I have never ever introduced the 4D circular numbers I just extrapolated the other six spaces to the solution that lives in that beautiful space. So the last example is a bit longer.
Anyway although the math depth of this post is not that very deep (solving a differential equation that wants the derivative to be the square of what you differentiate), it clearly demonstrates solutions of all 7 different spaces look strikingly similar. But because of the Dunning-Kruger effect likely the math professors will keep on telling total crap when it comes to complex numbers. Why am I wasting my time on explaining math professor behaviour? Better go to the five pictures of our post. Here we go & bye bye math professors.
May be I should write some posts about general complex number theory on spaces of arbitrary dimension. On the other hand I found the 3D complex numbers back in the year 1990. So if after all those years I will once more try to write some general theory one thing will be clear: Math professors will keep on trying to convince you of the beauty of quaternions or that garbage from the Clifford algebra’s.
Why, as a society, do we keep on wating tax payer money on math professors? Ok, they do not everything wrong but all in all it is not a great science or so where the participants are capable of weeding the faults out and grow more of the good stuff. Let me end this post and thank you for your attention.
With ‘speed = square’ I simply mean that the speed is a vector made up of the square of where you are. The four spaces are: 1) The real line, 2) The complex plane (2D complex numbers), 3) The 3D circular numbers and 4) The 3D complex numbers.
I will write the solutions always as dependend on time, so on the real line a solution is written as x(t), on the complex plane as z(t) and on both 3D number spaces as X(t). And because it looks rather compact I also use the Newtonian dot notation for the derivative with respect to time. It has to be remarked that Newton often used this notation for natural objects with some kind of speed (didn’t he name it flux or so?). Anyway this post has nothing to do with physics, here we just perform an interesting mathematical ecercise: We look at what happens when points always have a speed that is the square of their position.
On every space I give only one solution, that is a curve with a specific initital value, mostly the first imaginary component on that space. Of course on the real line the initial condition must be a real number because it lacks imaginary stuff.
If you go through the seven pictures of this post, ask in the back of your mind question as why is this all working? Well that is because the time domains we are using are made of real numbers and, that is important, the real line is also a part of the complex and circular number systems. The other way you can argue that the geometric series stuff we use can also be extended from the real line to the three other spaces. To be precise: we don’t use the geometric series but the fractional function that represents it.
Ok, lets go to the seven pictures:
That Newton dot notation just looks so cute…The words ‘Analytic continuation’ are not completely correct…
Remark: This post is not deep mathematics or so. We start every time with a function we know that if you differentiate it you will get the square. After that we look at it’s coordinate functions and shout in bewilderment: Wow that gives the square, it is a God given miracle!
No these are not God given miracles but I did an internet search on the next phrase of Latex code: \dot{z} = z^2. To my surprise nothing of interest popped up in the Google search results. So I wonder if this is just one more case of low hanging math fruits that are not plucked by math professors? Who knows?
A happy new year by the way, it is now 3 Jan over here so it is not too late to wish you that. So be happy if you can ask a physics professor or teacher as why there is no experimental proof at all that electrons are tiny magnets. And if the answer is not satisfactory, just chop the head of while being happy…;)
But serious, I selected the first video because the guy from the Science Asylum channel gives are very tiny estimated upper bound for the possible size of the electron: 10 to the -18 power meter as diameter. That is very very small, it is a nano nano meter.
Lets construct a so called ‘toy model’ for imitating in a simple manner how the electron is supposed to be a tiny magnet: Take two pointsize magnetic monopoles, a north and a south one and place them 10 to the power -18 meter apart. Lets name this distance d. An important feature of such a dipole is that it’s magnetic field declines inversely with the third power of d.
Let me give you an example: Take a line through the north and south pole of our toy electron and go out a distance of say 10d above the north pole. So the distance of our point on that line is 10d to the north pole and 11d to the south pole. The magnetic forces or field strength if you want is now proportional to 1/10^2 and 1/11^2. But north and south pole have opposite workings so we are looking at the difference: 1/10^2 – 1/11^2 and that is something of the order 1/1000.
If the electron diameter is indeed at most this distance d, in that case the two overlapping magnetic fields cancel each other almost out. If all that tiny magnet stuff is true, in that case the electron should be magnetically neutral. In a constant magnetic field that does not vary in space, by definition this tiny magnet electron should be neutral (if it all was true).
Let me show you two screen shots from the video from the Science Asylum. The first simple shows you the claim the electron has at most this size d.
On a nano nano scale this should be magnetically neutral…
A long time ago I estimated the result in next picture too but I always used an electron diameter of 10 to the -16 power, so one hundred time as big as the Asylum guys claims. Anyway there is nothing spinning over there because it must rotate a huge multiple of the speed of light. Now we can honestly say that Albert Einstein did not understand much about electron spin, but we can safely conclude that electron spin is not related to rotation of a spherical charged body the size of d.
One million times the speed of light…
Ok, let me hang in the video where we have once more the implicit claim that magnetism is always a magnetic dipole without one iota of experimental proof for that claim:
In my view the most misleading name is spin, it sets your brain totally wrong.
In the next video you see a guy at work showing that the oxygen in the air you breathe is magnetic. The magnetic properties of oxygen are truly breathtaking because it has to do with a so called ‘non-binding’ electron pair. In chemistry a non-binding electron pair is a pair with the same electron spin. Weirdly enough the physics professors keep their mouth shut: All electron pairs obey the Pauli exclusion principle! Until it doesn’t like in molecular oxygen.
But I digress, the reason I selected this video can be found at 3.40 minutes into it: The guy ‘explains’ the behavior of the oxygen by stating that the two electrons in the non-binding pair align their magnetic dipole to the applied magnetic field. The problem with this kind of ‘explanation’ is that it does not explain as why the electrons get accelerated. As said above; if electrons are tiny magnetic dipoles, they are basically magnetically neutral. And we are to believe that the oxygen molecules get accelerated by the applied magnetic field because two little electrons ‘align their dipole magnetic moment’. Give me a break: that is crap and the next stuff look much more logical and observable: Electrons are not magnetic dipoles but magnetic monopoles.
Here is the second video:
The reason for posting this second video is that I often obverve people from physics thinking that the alignment or for that matter the anti-alignment explains the acceleration and forces involved.
After seven years into this stuff I only wonder:
Why do the physics professionals like teachers and professors not see they are telling utter crap? Why are they so fucking stupid all of the time? End of this post. Once more: A happy new year.
Quantum computing looks like a good idea when it comes to simulation of quantum stuff like chemical reactions. But if your basic assumption of electrons being ‘tiny magnets’ it will all run from the rails if in the future it is found out that electrons are magnetic monopoles just like they are electric monopoles. Lately I have been joking that the only place in the universe where electron spin gets flipped is inside the heads of our professional physics professors. I think there are two kinds of electrons, one with a magnetic north charge and the other with a magnetic south charge. For the time being I think this magnetic charge is permanent so there is no way or mechanism that turns a south charge electron in a north charged and vice versa. Doing chemistry on a computer on the level of individual electrons and nuclei consumes an awful lot of computer resources. To focus the mind say you have a molecule with 100 to 150 electrons and some nuclei. The Coulomb forces alone are hard to simulate. But the way the magnetic forces must be done with the ‘tiny magnet’ model for electron spin is even much more horrible; in principle you have to calculate 100 to 150 vectors representing the bipolar magnetism of the electron. So this is all horribly complicated. Yet if electrons are magnetic monopoles, calculating a simulation for the magnetic forces should be of the same order of computer recourses as the Coulomb forces. That still is not very appealing but it is less worse as the Google engineers do it for the most simple atom there is in our universe: The hydrogen molecule made of two protons and one electron pair (the pair has opposite magnetic charges my dear reader, that is more logical as ‘opposite spins’ or for that matter ‘quantum numbers’). The horror is that physics but also chemistry professors seem to think that in the electron pair every particle is in a super position of spin up and spin down. That is where that stuff like “If I separate the two quantum particles and I take one to the Andromeda galaxy and measure it’s spin in the vertical direction, instantantly the spin of the particle left behind will be the opposite“. And why don’t they never say that for a hydrogen atom? After all these are two quantum particles to but are these two particles that are both in a super position of their positive and negative electrical charges? Is the mass of both particles not defined but is either the proton mass or the electron mass? Most people will say that is mumbo jumbo. The next picture is from a video I will show you below, likely all these Google people think that electrons are tiny magnets. So that is an amzing amount of salary costs wasted year in year out.
The |10> and |01> refer to the super position of magnetic states.
The Google view on electron magnetism is very different from mine where I like to keep it simple by stating electrons are magnetic monopoles. But they can make it much more complicated without any reason at all, in the next picture you see one of those singlet states and the chemistry folks have found out that an electron pair is ‘non binding’ if the two electrons have the same spin. If they have the same spin they don’t bond? And non of the chemistry weirdo’s remark that in a permanent magnet the spins must point in the same direction to make some magnetic bonding… Why does nobody see this is all not very logical and that this ‘tiny magnet’ stuff is just not true because it leads to all kinds of contradictions? Why are they so fucking stupid?
Let’s proceed with the second screen shot ensemble:
The thing with the minus sign is the anti-bonding pair. These people are crazy because a monopole magnetic charge is far more simple and much more logical.
Lets try to hang in the Google video.
So much money and so low in brain capacity.
So far for this Google stuff. Luckily IBM is also very good at donating a coin of quantum wisdom. Lately we had a Noble prize in physics for faster than light quantum teleportation of quantum properties like photon properties… I am not saying all this Bell inequality stuff is impossible, all I am saying is an electron cannot be in a superposition of spin up and spin down because electrons carry a permanent magnetic charge.
The IBM video was so bad that it became funny and some kind of parody. Here is another superposition of screen shots.
Last week I finally found out after seven years that there is indeed at least one repeated Stern-Gerlach experiment. It is well known in quantum mechanics that the Pauli matrices can be used to calculate the probabilities for finding electrons into a particular spin state. And in a repeated SG-experiment, if you turn the magnetic field 90 degrees the Pauli stuff says it is 50/50 divided. If you example you first applied a vertical magnetic field and after that some horizontal magnetic field, you should get 50% of the electrons having spin left and 50% spin right.
But if you try to do a search on a term like “Experimental proof for the Pauli matrices” or just “Repeated Stern-Gerlach experiment” never ever serious popped up in the last seven years.
Seven years ago I arrived at the conclusion that it is impossible that electrons are “tiny magnets” or for that matter have a bipolar magnetic field. A lot of things can be explained much better and more logical compared to mystifications like the Pauli exclusion principle. If electrons are magnetic monopoles, in that case it is logical that if they form pairs they must have opposite magnetic charges. And with the electron pair we already have a detail where the ususal model of electrons as “tiny magnets” fails; two macroscopic magnets are attracking only if their magnetic fields are aligned. If two macro magnets are anti-aligned, they repel. So how the hell is it possible that two electrons only form a pair if they have opposite spins, only if they anti-align? What I still don’t understand is why people like Pauli, Einstein, Feynman etc etc never remarked that it is nonsense to suppose that electrons are tiny magnets. Remark there is zero experimental proof for the assumption that electrons are tiny magnets. They just projected the Gauss law for magnetism on electrons without ever remarking you must have some fucking experimental proof. In the next picture you can see the experimental setup; you see two Stern-Gerlach experiments and in the middle is a inner rotation chamber where they try to flip the spin of the electrons.
Einstein proposed the use of the hot wire…
So Einstein must have given it a thought, this SG-experiment and never realized the impossibility of the Gauss law for magnetism for electrons.
Last week I found a nice pdf upon the Frisch-Segrè experiment and I would like to quote a few hilarious things from it:
“The physical mechanism responsible for the alignment of the silver atoms remained and remains a mystery” and quoting Feynman, “… instead of trying to give you a theoretical explanation, we will just say that you are stuck with the result of this experiment … ”
This is also the first time that I see this ‘problem’ actually stated; how is it possible that a tiny thing like an electron anti-aligns it’s spin with the applied external magnetic field? That is very very strange, for example water molecules are tiny electric dipoles and if they meet an electric field the only thing they want to do is to align themselves with that electric field. Why do electrons gain potential energy in a magnetic field?
To understand how crazy this is: If you go outside and throw away a bunch of rocks, do half of those fall to earth and the other half flies into space? Nope, in the end all rocks try to get at the state of minimal potential energy.
But if you view electrons as magnetic monopoles this weird detail of climbing in potential energy is’n there any longer: an electron with say a north pole magnetic charge will always go from the north pole to the south pole of a macroscopic magnetic field. And vice versa for an electron with a south pole magnetic charge. The weird energy problem isn’t there any longer. You can compare that to a bunch of electrons and protons entering an electric field; they feel opposite forces and that is how they both lower their potential energy.
At last let me give you the pdf. This pdf is not very useful because it is written by one of those weirdo’s that keep on believing that electrons are tiny magnets…
Once more I want to remark that if you see a physics professor doing his or her blah blah blah thing on electron spin, they just don’t have any serious experimental proof that electrons actually have two magnetic poles. Furthermore, none of them has a problem with that. So why are we funding these weirdo’s with tax payer money?
Ok, that was it for this post. Thanks for your attention.
To be honest I like the Unzicker guy; he is from Germany I believe and he alsways attacks the standard model for particles. According to him there are zillions of problems with the standard model and likely he is right with that. But he fully buys the crap that electrons must be magnetic dipoles without any experimental confirmation at all. So that I post a video of him talking weird stuff about electrons is not a way to rediculize him. On the contrary, because he always tries to attack the idea’s inside the stadard model he in itself is a perfect example as why the physics community swallows all those weird explanations upon electron spin.
For myself speaking I think that electrons don’t have their spins ‘up’ or ‘down’. I don’t think that they are tiny magnets with two magnetic poles but in itself they are magnetic monopoles that come with only one magnetic charge… My estimate is that this magnetic charge is a permanent charge, that means there is no such thing as spin flip of an individual electron.
In the Unzicker video Alexander asks for help about differentiation on the quaternions or so. Well have I done my utmost best to craft all kinds of spaces where you can integate and differentiate, stuff like 3D complex numbers, 4D complex numbers etc, comes a weirdo along asking about the quaternions… On quaternions differentiating is a true horror and that is caused by the property that in general the quaternions don’t commute. I wrote a one picture long explanation for that. The problem is that differentiation on say the square function on the quaternions destroys information. That is why there is no so called ‘Complex analysis on the quaternions’, it just doesn’t exist. Ok, lets go to the first video. It is not that very good because he constantly throws in a lot of terms like SO2 and SO3, but for an audience like physics people that is allowed of course.
Because it is still the year 2022, it is still one hundred years back that the Stern-Gerlach experiment was done. The next short video is relatively good in it’s kind; there are a lot of videos’s out there about the SG experiment and most are worse. In this video from some German at least there are some more explanation like it is not the Lorentz force because these are silver atoms. But as always in all explanations out there it misses as why exactly electrons do anti-align themselves with the applied external magnetic field. For example water molecules are a tiny electric dipole, if you apply an electric field to clean water, all these tiny electric dipoles for 100% align with the electric field. So why do electrons not do that?
As always: electrons being magnetic monopoles is a far better explanation for what we observe. But all these physics people, one hundred percent of them have no problem at all when there is no experimental evidence that electrons are indeed ‘tiny magnets’. That is what I still don’t understand: Why don’t they see that their official explanations are not very logical when you start thinking on these explanations? Why this weird behavior?
Ok, lets hang in why differentiation on the quaternions is a total horror.
Hasta la vista baby!
The last video is a short interview with John Wheeler where he explains the concept of positrons being electrons that travel back in time. At some point John talks about an electron and positron meeting and anihilate each other. Well it has to be remarked that this doesn’t always happen. They can scatter too and why could that be? Well it fits with my simple model as electrons being magnetic monopoles. Positrons and electrons only kill each other if they have also the opposite magnetic charge…
Ok, that was it for this post. Thanks for your attention.
This is a problem I found about thirty years ago and I was never ever able to solve it. The problem as I formulate it is about finding a so called ‘composition root’ to the exponential function. Just keep it simple, say the composition ‘square root’. If we denote that as r(x) what I mean is that this function if composed with itself gives the good old exponential function: r(r(x) = f(x) = e^x. There are many interesting aspects to this problem. For example take a piece of paper and a pencil and draw the graph of the exponential function and the identity function. It is now very easy for every point on your graph of the exponential function to find the graph of the double composition f(f(x) = e^(e^x)). But, as far as I know, you cannot go back and given the function f(x) find it’s composition root r(x). It is very well possible that this problem is solved in the theory of dynamical systems. If memory serves we once had a lesson in when a family of functions could be interpolated but that was 30 years back and what I want is explicit expressions and formula’s and not only a vague existencial proof without a way to find an explicit answer.
Back in time before the logarithm was invented, the people of those long lost centuries had a similar problem understanding what exponential behaviour was. And you can go a long way in understanding exponential behaviour but say for yourself; without knowledge of the logarithm that kind of knowledge is far from being optimized.
In this post I only talk about the composition square root but of course any n-th root should be possible and as such giving rise to the idea that you can iterate or compose the exponential function also a real number amount of times. I have to admit I also have no proofs for the solution to this all being unique, but you should be able to differentiate all stuff found and it should still be coherent so my gut feeling says the solution is unique. My guess is there is only one ‘composition square root’ r(x) that is as smooth as f(x) itself…
This post is only two pictures long so here we go:
And it is also the end of this post. Give it a thought and if you are able to make some inroads on this that would be great. But all in all I think we do not have the math tools to crack this old old problem.
