This way of doing the complex multiplication keeps on drawing my attention because of the funny property that i^3 = -1. As such it has interesting parallels to the 3D complex number. For example the eigenvalues of this defining imaginary unit is the third root out of -1 (and it’s conjugate). That is in line with the results from the 3D numbers although over there 3D numbers have 3 eigenvalues and not 2.
In this post I want to show you a way to find the logarithm of this imaginary unit i via integrating the inverse from 1 to i. Just like on the real line if you integrate 1/x from 1 to say some positive a, you get log a. It is important to remark there are more methods to find such logarithms. For example you can diagonalize the multiplication and take the log of the eigenvalues and as such you can find the log of the imaginary unit.
Anyway back in the time I did craft my first complex exponential for the 3D complex numbers this way (using the integral of the inverse) so for me it is a bit of a walk down memory lane. You always get integrals that are hard to crack but if you use the WolframAlpha website it’s easy to find. Remarkably enough the two values for the integrals we will find below are also found in 3D and even the 6D complex numbers. So for me that was something new.
For myself speaking I loved the way the inverse of a complex number based on i^2 = -1 + i looks. You have to divide by the determinant once more proving that norms do not have very much to do with it. (In standard lessons on complex numbers it is always told that the norm of the product is the product of the norms, but that’s only so for the complex plane and the quaternions. So if you keep on trying such idea’s you won’t come very far…)
This post is five pictures long, lets go:
May be this parametrization is the next post.
Ok, that was it more or less for this post. Since we are now getting more and more posts on two dimensional complex (and split complex) numbers may be I will open a new category for those posts. On the other hand you must not open a new category every time you things that are a bit different from what you usually do…
Last year the Nobel prize in physics went to a bunch of people that did experiments that gave rise to a so called violation of the Bell inequalities. As a consequence we are told we are living in a so called ‘non local’ universe and if you measure the quantum state of a particle here, this particle can be entangled with a particle in another galaxy and instantly that entangled particle will change or jump into another quantum state. So the idea is that entangled particles can influence each other at a speed that is infinite so information travel is faster as the speed of light. Well that is very interesting but when it comes to electrons and their spin state I just don’t buy that kind of crap. Why the Nobel prize committee thinks this is science worthy of their famous prize is unknown to me.
Now what is a ‘violation of the Bell inequalities’? Informally said there is too much correlation observed that cannot be explained by so called ‘hidden variables’ that are unknown. To focus the mind let me give you a simple example:
Two electrons in an electron pair get separated, one electron stays here on earth while the physics professors transport the other electron to another galaxy. It is claimed that if you measure the spin state of the electron here on earth into a particular direction, in that case the spin state of the electron that was transported by the physics professors to another galaxy instantly jumps into the other spin state. And the Nobel prize committee handed out a Nobel prize for that.
Well that is all very interesting but I think electrons carry a monopole magnetic charge and as such it is impossible to flip the spin. Take for example any chemical stuff based on binding via electron pairs, say your own body. Now if we put a magnetic field through your body tissue, does any electron flip it’s spin state? Do half of your electron pairs turn from a binding pair into a non-binding pair? No that never happens, electron spin state is not a fragile thing, it is permanent and cannot be altered.
And that is where all those experiments where they try to violate the Bell inequalities go wrong: They all assume that the photons they produce are coming from an electron that is in a superposition of spin up and spin down. But that is never the case if it is true that electron monopole charge is a permanent feature.
I have a video for you and the preprint pdf from 24 Aug 2015 that was published by the TU Delft group that did this loophole free Bell test. The video is easy and it even has music, what more do you want? The pdf is hard to read and it takes some time to grasp what is going on. I made four pictures from screenshots with a bit comment from me in it. After that the video and the pdf paper.
I do not know if they used the same electrons over and over again. Likely not because in quantum mechanics there is also the idea that if you measure some quantum property of a particle, after that in stays in that quantum state.
If electrons have a permanent magnetic charge, this must have a profound effect on the photons they produce. To be precise, the magnetic phase will be shifted 180 degrees if you compare the two different kind of photons. Without good solid proof I always assume that is why we have left and right circular photons. But information on that important detail is hard to find for years and years.
It has been a while since I updated this website with a post on magnetism. But some months back all of a sudden I would have more posts on magnetism compared to the main category of this website: The 3D complex numbers. So that is why in the last couple of months I did post nothing about magnetism over here but only on the other website.
At the end let me link you to the pdf from the preprint archive
At last I want to remark that even if you think it is very likely there is something wrong with the official version of electron spin (the tiny bipolar magnet model). In that case you must not think that for example the Nobel prize committee will come out saying they were wrong on electron spin in say the next 3 to 5 years. That’s not going to happen and that in itself is an interesting social phenomenon. Well thanks for your attention and see you in the next post.
