# What IS an electron?

When I was a student I got kicked out of a professor's office for having the gall to say that an electron was nothing more than a theory. I still believe that. It is an amazing and awesome theory, and the more we learn the more wonderous the theory becomes.

The word electron first appeard in 1891 long after electricity had been in use. In 1897 the electron was discovered by J.J. Thompson who proved it was a sub atomic particle. By this time we already had electric lights. In fact Edison and Tesla were fighting over how to light them.

While it was the early 20th century that saw the discovery of relativity, Maxwell's equations were already relativistically correct. Maxwell combined the laws of Gauss, Ampere and Faraday into a single form. Using $A^\mu$ as 4 potential and $J^\mu$ as charge and current we write Maxwell's equations in relativistic tensor notation as$$\square A^\alpha = \mu_0 J^\alpha$$ The electric and magnetic fields are derived from the potentials via$$F_{\alpha \beta}=\partial_{[\alpha}A_{\beta]}$$For those unfamiliar with this notation, the square brackets mean "anti-symmetric": $F_{\alpha\beta}=\partial_\alpha A_\beta - \partial_\beta A_\alpha$. Physicists like to be lazy.

The electron is a source of charge in Maxwell's equations. For an external force $F^\mu$ the relativistic equations of motion are $$m a^{\mu}=\frac{e}{c}F^{\mu\nu}v_{\nu}+F^{\mu} +\frac{2}{3}\frac{e^2}{c^3}\left(\dot a^{\mu} - \frac{a^{\lambda}a_{\lambda}v^{\mu}}{c^2}\right)$$ where $v^{\mu}$ is the 4 velocity, $a^{\mu}$ is the 4 acceleration and $\dot a^{\mu}$ is the time derivative of the 4 acceleration. From this we see that steady moving charges generate magnetic fields and accelrating charges give rise to electromagnetic fields. Not so easy to see maybe, but the time derivative of acceleration is key to radiation.

Once the electron was seen as subatomic, it was clear the electron had to revolve around the nucleus. This orbital motion is an acceration, so the electron should radiate all its energy and fall into the nucleus. Obviously, it doesn't do that! To solve this problem we "discovered" quantum mechanics. The electron has properties that make it a wave so it "fits" around the nucleus.

This also explained the excitation colors of heated atoms. The quantum theory was amazingly accurate. But then people started looking at the sprectral lines when the atoms were in a magnetic field, and they noticed fine structure. To solve this problem they gave the electron a magnetic moment. Since the electron is a moving charge, they called this "spin".

Once relativity was well established, the connection between quantum mechanics and relativity was attempted by Dirac. His relativistic description of an electron has 4 components: $$\left(\beta m c^2 + c \left(\sum_{n=1}^3{\alpha_n p_n}\right)\right)\psi(x,t)=i\hbar \frac{\partial \psi(x,t)}{\partial t}$$ where $p_n$ is momentum, $\beta$ and $\alpha_n$ are each 4 x 4 matricies:$$\begin{array}{c c}\beta =\left\lgroup\matrix{1 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0& -1 & 0\cr 0 & 0 & 0 & -1}\right\rgroup\\\alpha_x=\left\lgroup\matrix{0 & 0 & 0 & 1\cr0 & 0 & 1 & 0\cr0 & 1 & 0 & 0\cr1 & 0 & 0 & 0}\right\rgroup\\\alpha_y=\left\lgroup\matrix{0 & 0 & 0 & -i\cr0 & 0 & -i & 0\cr0 & -i & 0 & 0\cr-i & 0 & 0 & 0}\right\rgroup\\\alpha_z=\left\lgroup\matrix{0 & 0 & 1 & 0\cr0 & 0 & 0 & 1\cr1 & 0 & 0 & 0\cr0 & 1 & 0 & 0}\right\rgroup \end{array}$$ This equation accounts for spin up and spin down of electrons but gives rise to positrons - anti-matter!

The concept of anti-matter was bizzare at the time, but today we know that thunder clouds generate positrons. The discovery led to more questions and soon we began accelerating particles to higher and higher energies. Electrons look like perfect points at relativisitic velocities. But around nuclei they are waves, which don't radiate even though they move in orbits which should include acceleration.

One of the things that has always amazed me is that the orbital magnetic moment should be orders of magnitude larger than the intrinsic magnetic moment of an electron. If electrons actually move in their orbit, they should 1) radiate and 2) create a huge magnetic field. They do neither. The only thing that makes sense to me is that the electron does not actually move!

In fact, electrons have relativistic velocities which was the whole point of Dirac's formulation of his quantum description. Permanent magnets keep their magnetic fields because the magnetic moments of the electrons in $d$ orbitals all align in one direction and just stay there. If the orbitals aligned, the strength of the magnets would be 100 times greater. This makes no sense to me - how can the spins align but the motion around the nucleus does not?

In the early 20th century Langmuir studied electrons and ionized gases. He found that electrons reach equilibrium about $10^{16}$ times faster than collision theory could account for. At the time it was called "Langmuir's paradox" because simply assuming the electron gas was always in thermal equilibrium makes the mathematics much easier. Over the last century we have learned that quantum particles can exchange places if they have the same spin direction - that is - their magnetic moments align. This is called the "exchange - correlation" force. This is a uniquely quantum behaviour which allows electrons to change places with other electrons. In a plasma, this takes about 3 femtoseconds which we now know because we have lasers that can produce pulses in attoseconds.

In solids the theory that has gained a lot of support in the past 50 years is called Density Functional Theory. It combines the quantum effects so that the whole system of electrons that make up a solid can be computed as a single function. This has to include the exchange correlation force as well as spin. In this theory electrons are waves not particles but they give rise to particle behaviour with "holes" and "free electrons".

And then there are superconductors. In a superconductor, electrons bind to each other and move through the atomic lattice without friction. At this point in time we are really not sure how electrons do that. It makes sense that it has to be cold, the thermal energy can easily break this bonding between two similarly charged particles - they should repel each other after all! But here we describe the pair of electrons as a single particle!

So what we have is a classic "it depends" description: an electron is a perfect point at high speed which spreads out into a large volume when slow, acts like a wave and a particle, repels and attracts itself, can change places with other similar particles so they are all in equilibrium essentially instantaneously and its mass and fields change at relativistic velocity but its charge is invariant.

We have $10^{25}$ of them which hold us together. The electrical signals which make us function are no different than the signals we use to build robots and communications systems. The more we learn, the more questions we raise. Asking more questions helps us refine our theory and advance our technology.

Just what IS this thing? It is an amazing and magical theory of matter. Chances are, if you really belive you know what an electron is, you are wrong.

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Number Theory for Codes

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Dealing With Fixed Point Fractions

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