New roots of unity (the 3D complex ones) & The rain theorem.

3D complex numbers, Exponential circle

I just finished brewing the 100-th batch of a beer under the names Dark Matter and Spin 1/2 beer. All in all that is an amazing amount of beer; in the past I brewed 35 liter per batch but now it is 40 liter per batch so all in all an amount of something between 3500 and 4000 liters… So ein prosit my dear reader.

The ‘new’ roots of unity aren’t that new, this post is a re-editing of something I posted on 05 Jan 2014 on the other website. Later that year I started this website. Actually these roots of unity are just over five years old. In mathematical terms that is still very young so in that sense they are still new.

Recall the roots of unity in the complex plane are solutions to z^n = 1 and as such these roots are found on the exponential circle (the complex exponential) in the complex plane. As you have found the exponential circle or exponential curve in some space, from that you can always make new roots of unity. That is hardly a mathematical achievement because it is so simple to do once you have found your exponential circle or curve. But in the diverse spaces these new roots of unity behave very different, for example in this post we will add them all up but unlike in the complex plane they do not add up to zero. That is caused by the fact that in the complex 3D space the number alpha is at the center of the exponential circle, as such if we add n roots of unity in 3D space the result is n times alpha. Last year we studied the space of 4D complex numbers and if you would craft new roots of unity in that space it will behave much more like those in the complex plane because in the 4D complex numbers we have 0 as the center of the exponential curve. (For dimensions above 3 the exponential curve always lies in a hyperplane so it can never be a circle.) It always amazes me that you have all those physics people who study string theory but as far as I know never use exponential curves…

Life is beautiful, because how can you do string theory without math like that? But in physics almost everything is beautiful, for example if they explain the outcome of the Stern-Gerlach experiment always 50% of unpaired electrons align with the applied magnetic field and the other 50% for some mysterious reason do the anti-align thing. And if one hour later the same physics professor explains how a permanent magnet can attract some piece of iron, all of a sudden 100% of the unpaired electrons align and all that talk of 50/50 suddenly is not observed… Life inside the science of physics is wonderful; all you have to do is a bit of blah blah blah and if people complain this is not logical at all you simply say: Quantum physics is such that if you think you understand it, you don’t understand it… How wonderful is the life of physics professors; talk some blah blah blah and if people complain you blame them for ‘not understanding quantum things’. For sure that is a beautiful form of life.

But enough of the talking, somewhere in the next seven pictures I did forget to insert a graph of the determinant. Yet I showed you the structure of the non-invertible numbers so often, I think I post it with that fault included. After all why should life be perfect? If life would be perfect you would have no way of improvement and likely that is the moment you die: no more possibility of improvement. As usual the pictures are 550×775 pixels but I had to make the first one a tiny bit longer. Good luck with digesting it & have a bit of fun in the process.

End of the pictures.

For myself speaking it was fun to read my own two proof for the rain theorem again after five years. Please do not forget that new roots of unity on other spaces can be very different in behavior, after all they are always part of the exponential circle or curve in that space so they will derive their math properties from that. Till updates.

Short intro to the rain theorem.

3D complex numbers, Quantum mechanics

At first I wanted the next post to be about the so called Bell experiment because with this experiment comes the so called Bell inequalities and weirdly enough these are a perfect for the determinant of 4D complex numbers. Everything just looked perfect: the Bell inequality has maximum breaching when a bunch of correlations takes on the value of two time the square root of two. But that value of two time the square root of two is also what makes a 4D complex number on the unit sphere in 4D space noninvertible… So everything looked perfect but in the end it did not work because I could only find solutions with correlations above 1 (or below -1) and we all know that is not possible.

So let’s put the stuff on the shelf and wait a few years… The Bell experiment is one of those crazy quantum experiments and I am very interested in it because the original proposition as done by Mr. Bell was done with electrons and positrons. Yet results with electrons and positrons have never been published, all there is are experiments with photons and all those experiments seem to violate the Bell inequality…

Anyway a few days back I came across some old work I had written and that contained stuff like the anihilation theorem and the rain theorem, I did absolutely not remember what it all was about. And oh yes, it was that time that I wrote about a new set of so called roots of unity. And I remembered that I more or less hoped all those years ago about some kind of reaction from the math professionals. Of course there was zero reaction one more year; in those long lost years I still had to learn that university math professors are all shit. That is a uniform property of those people; they are all perfumed princes in relatively high paid permanent jobs. And perfumed princes do what the average perfumed prince think is important: we do perfumed prince stuff like the Langlands program and oh oh oh how smart we are. Luckily we have no dealings with those dirty peasants that live in the mud and cannot afford our exclusive perfumes. Tax payers should be happy they can finance us because what is a modern economy without smart math professors?

Once more I was stupid to the bone: The fact that you can easily find 10 videos with math professionals stating that the roots of unity are so very wonderful does not mean if you throw in a new set of unity roots there will be a healthy response…

I will leave the new post more or less like the old one, I only change the title and will do some editing to make it more readable. For example in the past I used the matrix environment for multi-line calculations while at present day I use the align command in the Latex typesetting program.

Here is the old file from five years back:

The Song of Omega Reloaded
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#05Jan2014

Ok, a small screen shot of the new roots of unity. Actually they are already five years old but the perfumed math professors have of course better things to do. They are so smart…

As you see it could use a bit typo improvement, in the meantime let it rain perfume.

End of the intro to the rain theorem.

Ok, I have done editing and decided to make a very simple teaser picture containing a simple calculation that indeed shows that if you square the opposite point of 1 on the exponential circle you get +1.
As such we have three solutions to the equation X^2 = 1 in the complex 3D space: the usual X = +1 and -1 and the third solutions lies on the exponential circle. Here is the teaser picture and likely later this week I will hang in the rest on this website.

Ok, it is not an advanced & fancy calculation but it is still 1.

Till updates.

On the acceleration of electrons in time-constant magnetic fields.

Magnetism, Quantum mechanics

This post is a continuation of the 01 May post on magnetism where we estimated that it is totally impossible that actual spinning of the electron would cause it’s magnetic properties. In the 01 May post I told you I had never seen how in physics they think electrons get accelerated in an inhomogeneous (and constant in time) magnetic fields. But when I finally tried to do some internet searching it was terrible easy to find. The offcial view is that they have an expression for the potential energy and the force is simply the gradient of the potential energy. But in order to explain the splitting in a beam in two in the Stern-Gerlach experiment something very strange has to happen: half of the electrons go into a somewhat lower energy and the other half in a somewhat higher state.

By all standards this is strange. Compare it for example to the next: You are standing on top of a building or a mountain and you start throwing rocks. Half of those rocks start falling to the ground as as such they are lowering their potential energy. The other half start flying up and as such gain potential energy like they feel anti-gravity. By all standards this is strange…

In my version of electrons where they are not magnetic dipoles but carry magnetic charge, you do not have this strange energy behavior because all electrons simply will follow their magnetic charge and as such all will go to lower energy levels.

And if the official version was true, that is half of the unpaired electrons turn into a lower energy and the other half in a higher energy spin state, that instantly brings problems when it comes to explaining permanent magnet behavior. If I grab a permanent magnet and stick it to a piece of iron, if half of the unpaired electrons would have spin up and the other half spin down, the magnet would never stick… Basically the official version of explaining the SG experiment is that you get those separation in unpaired electron spin states while when you stick a permanent magnet to a piece of iron all unpaired electrons will align to the magnetic field of the permanent magnet… That is highly contradictionary!

When just over five years back I found out the results of the SG experiment for the first time, my understanding of using an inhomogeneous magnetic field has always been that the electron feels tiny different forces with it’s north and south pole by the applied extermal magnetic field. And because the electron is so tiny, how could one unpaired electron pull an entire silver atom in two different directions?

Anyway this post is 8 pictures long, I had to made two of the a larger the rest is of the usual 550×775 pixel size.

Oh oh am I now shaking in fear because of the above photo as found on the preprint archive? If true, that would smash my idea of electrons carrying magnetic charge because if they carry magnetic charge it would not make much of a difference if the applied external magnetic field is inhomogeneous or not. A constant magnetic field simply would do.
Ok, for the time being is this the end of this post.

3D numbers: Decomposition of all numbers into two non-invertible numbers.

3D complex numbers, Quantum mechanics

I got the idea for this post already a couple of years back but I shelved it because I would like to have some application for it. But I still haven’t found a killer application yet anyway I decided to write this rather simple post about the projections you get when you multiply any 3D complex and circular numbers with the number alpha. If you need a refreshment on the importance of the number alpha (or what it actually is) please use the search function of this website and search for ‘Seven properties of the number alpha’.

Now two months back I observed some guy in a video explaining the math you need for quantum physics (yes I have a very boring life) and he was explaining you also need projectors P such that P^2 = P meaning that if P is some measuring operator, if you measure it twice you get the same result. And ha, now I write this post down I realize I did not prove that for both operators in the pictures below so that is something you can do for yourself if you want that.

Basically it goes as next: Pick any 3D number X, circular or complex, and multiply it by the number alpha. The result is a number on the main axis of non-invertible numbers (and as such an entire 2D plane gets projected on each of the main axis non invertible numbers). The other operator is (1 – alpha) and if you multiply any 3D number X by that, it gets projected on the plane of non-invertible numbers (and as such a line gets projected on a point of that plane).

All in all it is very basic, but ha ha ha I am doing this stuff now for years on a row and may be for the average reader it is all not so basic. This post is easier to grasp if you understand the shape of the non-invertible numbers: it is a plane and perpendicular on that plane the main axis and both the plane and main axis go through zero. In this post I skipped all things eigenvalue, but in 3D space we have 3 eigenvalues per capita number so unlike in 4D space we cannot have eigenvalue pairs only. In 3D space it has to be different and that explains more or less the shape of the non-invertible numbers.

This post is five pictures long, as usual all 550×775 pixels and I really hope it is not that hardcore this time.











That is caused by the eigenvalues of a number.

Before we split, on the other website I posted reason number 73 as why electrons cannot be magnetic dipoles. I was that lucky to come across an old 1971 translation of some stuff of the Goudsmit & Uhlenbeck guys. I always suspected there had been some very sloppy physics going on back in the time at the local Leiden university. The translation confirms that more or less (anyway in my view it does). Even after reading the 1971 translation for a third time I kept on falling from one amazement into the other. Have fun reading it, here is a link:

09 May 2019: Reason 73: In his own words; S. Goudsmit on the discovery of electron spin.

Ok, that was it for this post. Thanks for your attention (even if you are one of those sleazeballs from the Leiden university).

A classic: Imitation of the Pauli calculation given the results of the SG-experiment.

Magnetism

Often you observe people stating that the magnetic dipole moment of the electron cannot be explained by actual spinning of the electron. Because for that to happen, even if all electric charge was located on the equator of that spinning electron, it had to spin faster than the speed of light.

If memory serves, it was also Mr. Pauli (from the Pauli matrices describing electron spin stuff) that calculated this. So finally I did that calculation for myself, it takes a few minutes to collect the constants needed like the electron mass (I always forget that number). But within two or three minutes I arrived at a result of something like 15c or 15 times the speed of light.

And I started writing the text for the five pictures below and I don’t trust it and are there no errors or so? Yes there is a dumb typo on my Casio fx-82 made; it was not 15c but 15 thousand c… May be I made more dumb errors I haven’t found yet but that is all rather irrelevant because the beef of this post is not if all details of the simple calculation are correct but much more about when you can apply math in physics and when not.

For example, it is very simple math to show that the electron must spin over the speed of light and you can conclude this is not going to happen. That is more or less an allowed way of applying math in physics. Now it is not a secret that I think it is impossible that electrons are magnetic dipoles and as such I often frown over the use of the Gauss law for magnetism. If we use the variant of the Gauss law that uses a closed surface (magnetic flux through a closed surface like a spere always adds up to zero) to an electron, can we conclude the electron is a magnetic dipole because the Gauss law says so? Of course not, you absolutely need experimental evidence for such claims and only after that you can say: The Gauss law for magnetism also holds for electrons.

Not to mention you can accelerate electrons via inhomogeneous magnetic fields, you never hear about a calculation for that kind of miracle…

Ok, enough of the blah blah blah. This post is five pictures long, all of the usual size 550×775 pixels.



Ok, may be this post is just a giant mess if I read it again in a few years of time. But the huge number like 15 thousand times the speed of light can be tempered a little bit by using the so called ´Classical electron radius´ and that classical radius is far bigger compared to an electron diameter of 10^-16 meter. Here is one of those weird wiki’s:

Classical electron radius https://en.wikipedia.org/wiki/Classical_electron_radius

It looks like we are at the end of this post.

Added on 05 May 2019: A small appendix showing the difference in magnetic behavior of a single electron. I really do not know if it is possible the measure magnetic field strength of just a single electron. But at present day there are detectors that can detect just one photon, ok ok not any photon, it has to have enough energy to be detected. But anyway, if my view on electron magnetism is correct the strength should fall off in a 1/r^2 kind of law and if it is a magnetic dipole it is all very different.

Of course I cannot do such an experiment that shows how the magnetic field falls down, I do not know if such an experiment is possible. But if it is possible that experiment would likely make chopped meat of the idea the electrons are magnetic dipoles. The appendix is just one picture long, I had to enlarge it a little bit so it has size 550×850 pixels:

Again; no idea if such an experiment is possible…

And now you are really at the end of this post.

No no, you are not at the end of this post because on 30 June I also place a funny video from the Youtube channel science asylum below. I would not hold my hand in the fire for the physics in it, after all I think that it is impossible for electrons to be magnetic dipoles. I think it is magnetic charge and as such it is reasonable to talk about spin 1 particles (so basically stuff with two unpaired electrons in it). A Stern Gerlach experiment done on a spin 1 particle gives results like in the next picture and once more this validates again that electron spin is only magnetic charge…

Magnetic charge looks more reasonable to me.

Here is the video from that funny Youtube channel:

https://www.youtube.com/watch?v=sB1EPGmpzyg

Ok, that was it.

Using the Cayley-Hamilton theorem to find ‘all’ multiplications in 3D space.

3D complex numbers, CR equations, Matrix representation, Uncategorized

It is a bit vague what exactly a multiplication is, but I always use things that ‘rotate over the dimensions’. For example on the 3D complex space the imiginary unit is written as j and the powers of j simply rotate over the dimensions because:

j = (0, 1, 0)
j^2 = (0, 0, 1) and
j^3 = (-1, 0, 0). Etc, the period becomes 6 in this way because after the sixth power everything repeats.

In this post we will look at a more general formulation of what the third power of j is. The Cayley-Hamilton theorem says that you can write the third power of 3 by 3 matrices always as some linear combination of the lower powers.

That is what we do in this post; we take a look at j^3 = a + bj + cj^2. Here the a, b and c are real numbers. The allowed values that j^3 can take is what I call the ‘parameter space’. This parameter space is rather big, it is almost 3D real space but if you want the 3D Cauchy-Riemann equations to fly it has to be that a is always non zero. There is nothing mysterious about that demand of being non zero: if the constant a = 0, the imaginary unit is no longer invertible and that is the root cause of a whole lot of trouble and we want to avoid that.

It is well known that sir Hamilton tried to find the 3D complex numbers for about a full decade. Because he wanted this 3D complex number space as some extension of the complex plane, he failed in this detail and instead found the quaternions… But if the 3D numbers were some extension of the 2D complex plane, there should be at least one number X in 3D such that it squares to minus one. At the end I give a simple proof why the equation X^2 = -1 cannot be solved in 3D space for all allowed parameters. So although we have a 3D ocean of parameters and as such an infinite amount of different multiplications, none of them contains a number that squares to minus one…

I gave a small theorem covering the impossibility of solving X^2 = -1 a relative harsh name: Trashing the Hamilton approach for 3D complex numbers. This should not be viewed as some emotional statement about the Hamilton guy. It is just what it says: trashing that kind of approach…

This post is 7 pictures long, each of the usual size of 550×775 pixels.

Test picture, does jpg upload again?












Sorry for the test picture, but the seven jpg pictures refused to upload. And that is strange because they are just seven clean jpg’s. Now it is repaired although I do not understand this strange error.

Anyway have a cool summer. Till updates.

The supersized electron, can you accelerate one cubic cm of this stuff with a magnetic field?

Magnetism

This post is a continuation of Reason number 50 as why electrons cannot be magnetic dipoles as found on the other website. I published nr 50 in 2017 on 14 Oct. In that nr 50 Reason I tried to estimate the gradient of an inhomogeneous (non constant) magnetic field. It was just a rough estimation so you can have all kinds of critisism on it, but the gradient needed in the magnetic field was so huge that we safely can conclude that electrons cannot be accelerated by non constant magnetic fields.

Back in the year 2017 I more or less stated that universities are never very helpful. I joked that the word cooporation was not found in their dictionary. So now about 18 months later this seems to be true, why is that? Well all universities are relatively formal structures, most things go along some kind of protocol. For example when I would try to get a research job for the study of magnetic domains (because I think magnetic domains have surplusses of either one of the spin variants, so every magnetic domain is a magnetic ‘monopole’ on the domain level), that likely would not be possible. Because everything goes in such a formal manner likely I have to start as a first year student of physics, slowly climb the ranks and that’s it because ‘we cannot make an exception’. And from the university this is rather logical; if they give in to one weirdo that think that electrons cannot be magnetic dipoles, next comes along another crazy person that wants to study more homeopathic medicines or whatever what.

Ok, what are we looking at in this post? I simply view the electron as some kind of massive small sphere with a diameter of 10^-16 meter and as such estimate the density or the mass per cubic meter.

Without trying any kind of calculation, try to accelerate such an object with an inhomogeneous magnetic field…

It is three pictures long so this is a short post although the numbers are impressive. Picture sizes all 550×775 pixels.

Ok, this is what I had to say on the impossibility of accelerating magnetic dipole electrons in any meaningful amount while using inhomogeneous magnetic fields… See you in the next post.

The Cayley-Hamilton theorem neglected for 25 years?

3D complex numbers, Matrix representation, Uncategorized

That is strange, if you don’t know the Cayley-Hamilton theorem; it is the finding that every square matrix A, if you calculate the characteristic polynomial for the matrix A it is always zero. At first this is a very surprising result, but it is easy to prove. It’s importance lies in the fact that in this way you can always break down higher powers of the matrix A in lower powers. In the study of higher dimensional complex and circular numbers we do this all the time. If in 3D space I say that the third power of the imaginary component is minus one, j^3 = -1, we only write the third power as a multiple of the zero’th power…

In this post I will give two simple proofs of the Cayley-Hamilton theorem and although in my brain this is just a one line proof, if you write it down it always gets longer than anticipated.

At the end I show you an old video from the year 1986 from the London Mathematical Society where it is claimed that the CH theorem was neglected for 25 years. Now Hamilton is also famous for having sought the 3D complex numbers for about a full decade before he gave up. And I still do not understand why Hamilton tried this for so long but likely he wanted to include the imaginary unit i from the complex plane in it and that is impossible. Or may be he wanted a 3D complex number system that is also a field (in a field all elements or numbers that are non-zero have an inverse, in algebra wordings; there are no divisors of zero). A 3D field is also impossible and in this post I included a small proof for that.

Furthermore in this post at some point may be you read the words ‘total incompetents’ and ‘local university’. You must not view that as some emotional wording, on the contrary it is a cold clinical description of how math goes over there. So you must not think I am some kind of frustrated person, for me it is enough that I know how for example to craft a 3D complex number system. If they don’t want to do that, be my guest. After all this is a free country and we also have this concept of ‘academic freedom’ where the high shot math professors can do what they want.

And what is this ‘academic freedom’ anyway? If for example unpaired electrons are never magnetically neutral but electron pairs always are magnetically neutral, can the physical reality be that electrons are magnetic dipoles? Of course not, that is a crazy idea to begin with. But 97 years of academic freedom since the Stern-Gerlach experiment have never ever brought any meaningful understanding of the magnetic properties of the electron. If it acts as a magnetic charge and you say it is not a charge it is easy to understand how you can fool yourself for about one century of time.

This post is seven pictures long although the last picture is empty.
The two proofs of the Cayley/Hamilton theorem is how I would prove such a thing but good theorems always have many proofs. All pictures are of the size 550×775 pixels.













Why is the seventh picture without math?

Here is the old video from 1986 where it is claimed the Cayley-Hamilton theorem was neglected for about 25 years. Oh oh oh what a deep crime. But the human mind is not made to produce or understand math, so in my view 25 years is a short period of time if in the good old days math professors were equally smart as the present day math professors. The title of the video is The Rise and Fall of Matrices.

Matrices saved my life from crazy math professors.

Ok let me leave it with that an not post a link to the top wiki on the Cayley-Hamilton theorem where all kinds of interesting proofs are given. Till updates my dear reader.

What would a quantum measurement on Paul his IQ yield?

Magnetism

Today I came across one of those video’s where people try to explain how permanent magnets work. And originally I thought of a title like ‘Idiot of the day observed’ but soon I changed my mind because Paul Sutter does not do it on purpose; what he says is more or less the general accepted version of permanent magnetism…

In general there are two lines of reasoning when it comes to permanent magnets: One line of reasoning is that the magnetic domains get aligned, the other way is that the electron spin of all unpaired electrons align.

Paul Sutter goes for the second line of reasoning; the spin of all unpaired electrons align giving rise to a permanent magnet. Just like everything else in the video it is just wrong; in my view where the electrons carry magnetic charge, it is the placement of the unpaired electrons in the inner shells of for example an iron atom that makes the global permanent magnet emerge.

If it was just electrons having all their dipole magnetic moments point in the same direction, in that case with a strong magnet you could always change or invert the magnetic direction of a weak magnet. In practice this just does not happen; last spring I even made a simple experiment with this: I took my stack of the most strong magnets I have and placed them over 24 hours against the two most weak magnets I have. And, like expected, there was no change at all in the weak magnets indicating there is some kind of threshold at work. The threshold is of course that it is hard to remove the magnetically charged electrons from the inner shells of the iron atoms…

Here are two pictures of the simple experiment from 08 March 2018; the permanent but very weak magnets on the left were exposed to the stack of neodymium magnets for just over 24 hours and just nothing changed in the behavior of the weak magnets. If electron magnetic moment alignment were a significant factor in permanent magnetism, the stronger permanent magnets should alter the magnetic properties of the weak magnets. It just does not happen…

Weak at the left, strong at the right.
After 24+ hours of waiting zero change observed in the weak magnets.

A link to what I wrote one year back on this very simple experiment is:
08 March 2018: Reason 56: This experiment shows zero spin torque transfer. http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#08March2018

It is lovely to see so many of the wrong stuff bound together in just one short video: For example when Paul explains why electron pairs are magnetically neutral while the unpaired electrons are not. In my view if it is true that electrons are magnetic dipoles, they would be magnetically neutral. They are not and in my view this shows electrons are magnetic monopoles. How does Paul explain it? Very simple: The electron pair is magnetically neutral because one of the electrons has spin up while the other has spin down and that cancels each other out.

Here is the Youtube video of Paul Sutter, please don’t think that Paul is a dumb person or so. This is just the view of professional physics folks that have studied magnetism for centuries…

Link: https://www.youtube.com/watch?v=6uwjqy2HCgY

May be at last I am getting a little bit sarcastic: They have studied that for centuries… But if you let go that dumb and unproven Gauss law for magnetism, just try to think about our physical reality as electrons also carry magnetic charge beside their electric charge, a lot of things become better to understand. After all, why do we only observe electron pairs? Why never something else like an electron triplet?

Ok, let’s leave it with that. Till updates my dear reader.

On Schrödingers cat & an example known as the envelope problem.

Quantum mechanics

Today the Youtube channel SciShow had one more video out on quantum mechanics and as such the famous cat of the Schrödinger cat in a box problem comes along once more.

As usual we are told the cat can be in a super position of being alive and dead at the same time. I wonder why people think that this can be true, as far as I know history the Schrödinger guy came up with this example as an antidote as being everything into a super position…

I suppose you already know what the cat in the box setup is. The cat dies if just one radioactive atom decays yes or no. If you are outside of the box it makes sense to use a probalistic model of the situation, but does this mean that in reality inside the box the cat is dead and alive at the same time? After all the cat will be the very first to observe if radio active decay has happened because as soon as it does the state of the cat goes from alive to dead. So inside the box there is at least one observer present and as such all quantum states we are interested in (radio active decay yes or no) is constantly measured all of the time.

For myself speaking I use the fact that a cat cannot be in a super position of being alive and dead as an example that an individual atom cannot be in a state where radio active decay has passed yes or no.

That does not mean quantum particles cannot be in super positions, for example photons behave often like they took all possible paths to arrive somewhere. But as soon as there are all kinds of different energy levels involved this becomes more and more problematic. For example can a particle be in a super position of being a neutron and a proton? Can a particle be in a super position of being an electron and a positron? Can a particle be in a super position of being a hydrogen ion (a proton) and a plutonium atom?

Energy is at the heart of the quantum measurement problem: In order to measure a quantum particle some kind of interaction with the particle must be there. This interaction changes (or not) the state of the particle. It is a bit like this: Suppose I am sitting in my home country and I have to measure the length of some grassfield in Germany or Belgium but I can only use atom bombs for that. No matter how smart I craft my grass length measuring device, the giant explosions from the atom bomb will bring a great uncertainty in the outcome of the measurements… Here is the video:

The cat is also an observer…

Ok, now for the lesser known but rather interesting envelope exchange problem. In a nutshell it goes as next:

You can choose one of two closed invelopes and they contain money. The only thing you are told is that the amount in one of the envelopes is double that of the other envelope.

Now you play the game and you choose one of the envelopes, let’s say it contains 100€. You are asked by the quiz master if you want to keep those 100€ or that you want to change your choice and go for the other envelope.

You think about that for a few seconds and you figure out: If this envelope has 100€ and given the rules of the game, the other envelope contains 50€ or 200€ with equal probability of 50%. Suppose I want to swap to the other envelope, what is my expectation for the amount of money? That is simple, both 50€ and 200€ have 50% probability so the expectation of swapping becomes 0.5*50 + 0.5*200 = 125€. Therefore it makes sense to swap and choose the other envelope.

But hey, whatever envelope you choose at first and you find X money in it, isn’t it weird to swap that always? If you would have chosen the other envelope you would also swap…

This envelope swap problem or paradox has a relative simple solution: You assume equal 50% probabilities for having double or half the amount of money you found in the first envelope. But in that case the whole thing crashes because you are now calculating with three outcomes: the 100€ from the first envelope and two other amounts 50 and 200 Euro while there are only two enveloples. It is unwise to calculate the expectation values because the 50€ and 200€ exclude each other: if the outcome 50€ is observed all of the time the 200€ was non existant. And as such the expectation value makes no sense for an individual experiment.

Ok, let me end this post with a standard wiki around the two envelope thing: Two enveloples problem. https://en.wikipedia.org/wiki/Two_envelopes_problem

End of this post.