What I do in my Master Thesis

All the matter around is is made up from atoms. The atoms are really small, about $10^{-8} \, \mathrm{cm}$. That is $0.000\,000\,01 \, \mathrm{cm}$. Each atom consists of a nucleus and electrons. The movement of the electrons around the nucleus is described by quantum mechanics. The image I drew below is not very accurate; the accurate thing is really hard to draw. The atomic nucleus is just a ten-thousandths of the size of the whole atom; the atom is mostly empty! Inside the nucleus, there are protons and neutrons, those are called nucleons. Each of the protons and neutrons consist of three quarks. The quarks are bound together by the strong force which is mediated by gluons.

The hierarchy of (1) matter, (2) atoms, (3) atomic nucleus, and (4) nucleons.

The size of the quarks and gluons is ridiculously tiny, it is $10^{-13} \, \mathrm{cm}$ as indicated in the drawing. This is the scale which is covered in my thesis.

Quarks have a so called "color charge" which is like the electric charge just that it comes in three colors. There is no actual color, the behavior can just be pictured well with red, green, and blue light. No color is black, all three colors is white. The opposite of a color is its complementary color. The gluons interact only with colored objects, they do not "see" objects which are black or white. One characteristic of the strong force is that the quarks are never seen alone. They always come in bundles of two or three quarks that are black or white together. You can see that the proton consists of each color each, it is a white object together.

There are two major complications which set the strong force apart from the other forces:

• The force is strong, so it has a major effect on everything. Compare this to gravity: All the objects around us fall to the ground, but they do so in one piece. It is not like gravity would tear everything apart. Also things to not attract each other measurably, they just fall down towards earth. The strong force is like sticky glue, everything strongly sticks together. One cannot look at a quark or a gluon without having to look at all the things that also stick to it.

• The force carrier particles also interact with each other. This is very hard to imagine. With light, we know from experience that there is no noteworthy interaction between light particles. If one takes two flashlights and crosses them, the beams of light pass through each other. If one could make flashlights which have beams of strong force, it would look very different: The beams would stick to each other and form a blob of light, like a ball of wool.

These two complications mean that one cannot do computations with the normal tools that a theoretical physicist has. This is an example for a process that holds two parts in a proton together:

The force carriers create new particles from the vacuum (quantum fluctuations) which get annihilated again quickly. The strange thing is that more complicated processes are more likely to occur! This means that in order to compute what actually happens, one needs to include arbitrary complex processes. Of course this is not feasible, and there is nowhere one can start working. One needs a different approach.

One approach is to just "ask" nature what happens and make a particle physics experiment. Experimentalists can "weigh" the proton (which can occur on its own because it is "white") and obtain its mass. They cannot measure the quarks because they have color and stick to all the gluons and quantum fluctuations. The theory one has about the strong interaction in principle predicts what the mass of the proton is when one would know the masses of the quarks. Besides the complexity that one cannot weigh them directly, the theory is so hard to evaluate, that this prediction cannot be made with an analytic computation.

Another approach is to use a computer and simulate the quantum fluctuations of the vacuum inside some box. One cannot simulate the whole universe but taking a box which is large enough to contain a whole proton is the best approximation one can do. The computer cannot simulate to infinite precision inside the box, so one has to limit oneself to a couple of points inside that box. Effectively one reduces time and space to a grid, just like on graph paper. There the number of possibilities is limited to a finite number of integrals. This number is still huge, for 64 lattice points we have $536\,870\,912$ integrals to solve. Solving an integrals takes at least 100 points, so we have to look at $53\,687\,091\,200$ values. Luckily, most of these are almost zero, so we can safely ignore them. Using a couple of tricks, one can just look at the most important ones.

Unfortunately, it still takes a lot of computing power to simulate this. One step that occurs often is the solution of a system of equations that in the largest simulations has $40\,532\,396\,646\,334\,464$ unknowns. Here one needs additional tricks, but it will stay horrible complex. In order to get some results, one needs a supercomputer. They do not make programs magically faster, one has to work hard as a programmer to utilize the power that the computer actually has. Actually, it is not one computer, but many computers with a powerful network. Each of the computer has multiple CPUs (processors) and GPUs (graphics cards). Each CPU has multiple cores, on each core there can be multiple concurrent threads. In each thread one can cram four similar operations at a time. The GPU has lots of cores, each core can do 32 similar operations at a time. Each unit has its own memory (RAM). When writing a program, one has to think really hard about splitting the problem into small but similar tasks that can be distributed among all those computing resources.

On a computer which has around $10\,000$ processor cores, it still takes days or weeks to run a simulations. It means that one has to have a massive budget of computing time. Running that simulation on a laptop is just not feasible.

There are multiple projects out there for doing those kind of simulations. They usually have between $300\,000$ to $700\,000$ lines of programming code. Printing that would give up to $10\,000$ pages. That is more than two times the whole "Harry Potter" series! Converting the printed code to soccer fields, it would be around 0.03 to 0.09 soccer fields filled with paper.

Either way, this is not something that a single person can write in a year. In my thesis work, I extend existing code with more physics and try to make things faster than they already are.

• When converting that into soccer fields, I noticed that the soccer field is not even standardized, it can vary between $10\,800 \, \mathrm{m^2}$ and $4050 \, \mathrm{m^2}$. The FIFA/UEFA suggestion is $7140 \, \mathrm{m^2}$. I have used that.

  I have used that A4 paper has an area of 1/16th square meter. On a soccer field you can therefore fit $114\,240$ pages of A4 paper. On a page of A4 paper you might get 70 lines of code. That makes $7\,996\,800$ lines of code on a soccer field.