Tackling ambiguities

I have recently published a paper with a rather lengthy and abstract title. I wanted to enlighten in this entry a little bit what is going on.

The paper is actually on a problem which occupies me by now since more than a decade. And this is the problem how to really define what we mean when we talk about gluons. The reason for this problem is a certain ambiguity. This ambiguity arises because it is often much more convenient to have auxiliary additional stuff around to make calculations simple. But then you have to deal with this additional stuff. In a paper last year I noted that the amount of stuff is much larger than originally anticipated. So you have to deal with more stuff.

The aim of the research leading to the paper was to make progress with that.

So what did I do? To understand this, it is first necessary to say a few words about how we describe gluons. We describe them by mathematical functions. The simplest such mathematical functions makes, loosely speaking, a statement about how probable it is that a gluon moves from one point to another. Since a fancy word for moving is propagating, this function is called a propagator.

So the first question I posed was whether the ambiguity in dealing with the stuff affects this. You may ask whether this should happen at all. Is a gluon not a particle? Should this not be free of ambiguities? Well, yes and no. A particle which we actually detect should be free of ambiguities. But gluons are not detected. Gluons are, in fact, never seen directly. They are confined. This is a very peculiar feature of the strong force. And one which is not satisfactorily fully understood. But it is experimentally well established.

Since therefore something happens to gluons before we can observe them, there is now a way out. If the gluon is ambiguous, then this ambiguity has to be canceled by whatever happens to it. Then whatever we detect is not ambiguous. But cancellations are fickle things. If you are not careful in your calculations, something is left uncanceled. And then your results become ambiguous. This has to be avoided. Of course, this is purely a problem for us theoreticians. The experimentalists never have this problem. A long time ago I actually already wrote together with a few other people a paper on this, showing how it may proceed.

So, the natural first step is to figure out what you have to cancel. And therefore to map the ambiguity in its full extent. The possibilities discussed since decades look roughly like this:

As you see, at short distances there is (essentially) no ambiguity. This is actually quite well understood. It is a feature very deeply embedded in the strong interaction. It has to do with the fact that, despite its name, the strong interaction makes itself less known the shorter the distance. But for weak effects we have very precise tools, and we therefore understand it.

On the other hand at long distances - well, there we knew for a long time not even qualitatively what is going on for sure. But, finally, over the decades, we were able to constrain the behavior at least partly. Now, I tested a large part of the remaining range of ambiguities. In the end, it indeed mattered little. There is almost no effect left of the ambiguity on the behavior of the gluon. So, it seems we have this under control.

Or do we? One of the important things in research is that it is never sufficient to confirm your result just by looking at a single thing. Either your explanation fits everything we see and measure, or it cannot be the full story. Or may even be wrong and the agreement with part of the observations is just a lucky coincidence. Well, actually not lucky. Rather terrible, since this misguides you.

Of course, doing all in one go is a horrendous amount of work, and so you work on a few at the time. Preferably, you first work on those where the most problems are expected. It is just ultimately that you need to have covered everything. But you cannot stop and claim victory before you did.

So I did, and looked in the paper at a handful of other quantities. And indeed, in some of them there remain effects. Especially, if you look at how strong the strong interaction is, depending on the distance where you measure it, something remains:

The effects of the ambiguity are thus not qualitative. So it does not change our qualitative understanding of how the strong force works. But there remains some quantitative effect, which we need to take into account.

There is one more important side effect. When I calculated the effects of the ambiguity, I learned also to control how the ambiguity manifests. This does not alter that there is an ambiguity, nor that it has consequences. But it allows others to reproduce how I controlled the ambiguity. This is important because now two results from different sources can be put together, and when using the same control they will fit such that for experimental observables the ambiguity cancels. And thus we have achieved the goal.

To be fair, however, this is currently at the level of an operative control. It is not yet a mathematically well-defined and proven procedure. As with so many cases, this still needs to be developed. But having operative control allows to develop the rigorous control easier than starting without it. So, progress has been made.

Redundant ghosts

A recurring topic in our research are the joys and sorrows of the redundancies in our description. As I have discussed several times introducing these redundancies makes live much easier. But this can turn against you, if you need to make approximations. Which, unfortunately, is usually the case. Still their benefits outweighs the troubles.

One of the remarkable consequences of these redundancies is that they even affect our description of the most fundamental particles in our theories. Here, I will concentrate on the gluons of the strong interactions (or QCD). On the one hand because they play a very central role in many phenomena. But, more importantly, because they are the simplest particles exhibiting the problem. This follows essentially the old strategy of divide and conquer. Solve it for the simplest problem first, and continue from there.

Still, even the simplest case is not easy. The reason is that the redundancies introduced auxiliary quantities. These act like some imaginary particles. These phantom particles are called also ghosts, because, just like ghosts, they actually do not really exist, they are only there in our imagination. Actually, they are called Faddeev-Popov ghosts, honoring those two people who have introduced them for the very first time.

Thus, whenever we calculate quantities we can actually observe, we do not see any traces of these ghosts. But directly computing an observable quantity is often hard, especially when you want to use eraser-and-pencil-type calculations. So we work stepwise. And in such intermediate steps ghosts do show up. But because they only encode information differently, but not add information, their presence affects also the description of 'real' particles in these intermediate stages. Only at the very end they would drop out. If we could do the calculations exactly.

Understanding how this turns out quantitatively is something I have been working on since almost a decade, with the last previous results available almost a year ago. Now, I made a little bit progress. But making progress is for this problem rather though. Therefore there are usually no big breakthroughs. It is much like grinding in an MMO. You need to accumulate little bits of information, to perhaps, eventually, understand what is going on. And this is once more the case.

I have presented the results of the latest steps recently at a conference. A summary of this report is freely available in a write-up for the proceedings of this conference.

I found a few new bits of information. One was that we certainly underestimated the seriousness of the problem. That is mainly due to the fact that most such investigations have so far been done using numerical simulations. Even though we want to do in the end rather the eraser-and-pencil type calculations, ensuring that they work is easier done using numerical simulations.

However, the numerical simulations are expensive, and therefore one is limited in them. I have extended the effort, and was able to get a glimpse of the size of the problem. I did this by simulating not only the gluons, but also simulated the extent to which we can probe the problem. By seeing how the problem depends on our perception of the problem, I could estimate, how big it will become at least, eventually.

Actually, the result was somewhat unsettling, even though it is not hopeless. One of the reason, why it is not hopeless is the way how it affects everything. And there it turned out that the aforementioned ghosts actually carry the brunt of the problem. This is good, as they will cancel out in the end. Thus, even if we cannot solve the problem completely, it will not have as horrible an impact as was imaginable. Thus, we can have a little bit more confidence that what we do makes actually sense, especially when we calculate something observable.

You may say that we could use experiments to check our approximations. It appears easier. After all, this is what we want to describe - or is it? Well, this is certainly true, when we are thinking about the standard model. But fundamental physics is more geared towards the unknown nowadays. And as a theoretician, I try to predict also the unknown. But if my predictions are invalidated by my approximations, what good can they be? Knowing therefore that they are not quite as affected as they could be is more than valuable. It is necessary. I can then tell the experimentalists with more confidence the places they should look, with at least some justified hope that I do not lead them on a wild geese chase.

Searching for structure

This time I want to report on a new bachelor thesis, which I supervise. In this project we try to understand a little better the foundations of so-called gauge symmetries. In particular we address some of the ground work we have to lay for understanding our theories.

Let me briefly outline the problem: Most of the theories in particle physics include some kind of redundancy I.e., there are more things in it then we actually see in experiments. The surplus stuff is actually not real. It is just a kind of mathematical device to make calculations simpler. It is like a ladder, which we bring to climb a wall. We come, use the ladder, and are on top. The ladder we take again with us, and the wall remains as it was. The ladder made live simpler. Of course, we could have climbed the wall without it. But it would have been more painful.

Unfortunately, theories are more complicated than wall climbing.

One of the problems is that we usually cannot solve problems exactly. And as noted before, this can mess up the removal of the surplus stuff.

The project the bachelor student and I am working on has the following basic idea: If we can account for all of the surplus stuff, we should be able to know whether our approximations did something wrong. It is like preparing an engine. If something is left afterwards it is usually not a good sign. Unfortunately, things are again more complicated. For the engine, we just have to look through our workspace to see whether anything is left. But how to do so for our theories? And this is precisely the project.

So, the project is essentially about listing stuff. We start out with something we know is real and important. For this, we take the most simplest thing imaginable: Nothing. Nothing means in this case just an empty universe, no particles, no reactions, no nothing. That is certainly a real thing, and one we want to include in our calculations.

Of this nothing, there are also versions where some of the surplus stuff appears. Like some ghost image of particles. We actually know how to add small amounts of ghost stuff. Like a single particle in a whole universe. But these situations are not so very interesting, as we know how to deal with them. No, the really interesting stuff happens if well fill the whole universe with ghost images. With surplus stuff which we add just to make life simpler. At least originally. And the question is now: How can we add this stuff systematically? As the ghost stuff is not real, we know it must fulfill special mathematical equations.

Now we do something, which is very often done in theoretical physics: We use an analogy. The equations in question are not unique to the problem at hand, but appear also in quite different circumstances, although with a completely different meaning. In fact, the same equations describe how in quantum physics one particle is bound to each other. In quantum physics, depending on the system at hand, there may be one or more different ways how this binding occurs. You can count the number, and there is a set which one can label by whole numbers. Incidentally, this feature is where the name quantum originates from.

Returning to our original problem, we do the following analogy: Enumerating the ghost stuff can be cast into the same form as enumerating the possibilities of binding two particles together in quantum mechanics. The actual problem is only to find the correct quantum system which is the precise analogous one to our original problem. Finding this is still a complicated mathematical problem. Finding only one solution for one example is the aim of this bachelor thesis. But already finding one would be a huge step forward, as so far we do not have one at all. Having it will probably be like having a first stepping stone for crossing a river. From understanding it, we should be able to understand how to generate more. Hopefully, we will eventually understand how to create arbitrary such examples. And thus solve our enumeration problem. But this is still in the future. For the moment, we do the first step.

Being formal

One of the topics I am working on is about basic properties of so-called gauge symmetries. I just have published a new paper on it, and here I want to describe what it is about.

A gauge symmetry is, very roughly stated, a useful tool for which we pay the price of a very redundant description. Pictorially speaking, we can say the same thing with very many different words. This may sound awful. However, in practice, it seems to work just like a charm. So what is there still to investigate?

Well, knowing how to use something in one way, and understanding it fully are two very different things. And actually, we are not, on a very strict and formal level, absolutely sure that we know how to use gauge symmetry. Though this is likely the case. But the situation is nonetheless for two reasons not really satisfactory.

The first is more a question of approach. When we use something, we would really like to know what we are actually doing. The second is that if we would understand it better, there may very well be ways to use it much better than we currently do. So there are reasons for understanding it better.

But what is it what I actually want to do?

It all starts when going back to the meaning of symmetries. Symmetries introduce redundant directions, meaning that when you have a symmetry you have more directions to point then there are actually. That is in general very helpful on a technical level.

But here enters the problem. If we have directions, we should be able to say 'go in this direction'. To do so, we introduce coordinate systems. Now comes the catch: For a local symmetry this is easier said than done, especially when it comes to the gauge symmetries of the standard model.

The problem is somewhat abstract. When you think about directions, then usually you think about left, right, up or down, and so on. This is true if you think about our usual space around us. But not everything has the same geometry. Especially, symmetries can also have a direction of bending. This is still not a major issue. But, there are some symmetries where the bending of directions becomes so strong, that some directions bend back on themselves or meet others, when going too far. And this is a problem. If they bend back, or even worse, bend on a different direction, what is direction anyway? I can start walking in one direction, and then I am in another direction. Sounds like a catastrophe, right?

Well, the reason it sounds like that is that we insisted to define directions once and for all. This is what we are used to. A direction is a direction is a direction. Unfortunately, not everything is so straight and something, especially gauge symmetries, have additional directions which are warped, and can intersect each other. The problem then arises how to orient oneself, if directions change. The answer to this is that it is necessary to give up directions which are always the same. Rather, you need to define directions only in some area around you, and when you move, you may need to change them.

The aforementioned paper now investigates this bending of directions. In a sense, it tries to map how far it is possible to go in a fixed direction, before this direction changes. Finally, it attempts to draw a map of where these directional changes occur. That sounds now pretty graphical, but the reality is once more mathematically involved. But in the end, this map hopefully will help to setup useful collections of coordinate systems, and a dictionary telling you where to use which coordinate system to get your directions.

The details are pretty involved. But the rough outline of what I did was to put myself at many points, create there coordinate systems, follow the directions they give and check when they started to make no more sense - when they hit other directions or themselves. Then I got a list of collisions, and where they occurred. And from this I could get a map of collisions. What I did not yet do is to make something useful out of the map. That comes next.

Blessing and bane: Redundancy

We have just recently published a new paper. It is part of my research on the foundations of theoretical particle physics. To fully appreciate its topic, it is necessary to say a few words on an important technical tool: Redundancy.

Most people have heard the term already when it comes to technology. If you have a redundant system, you have two or more times the same system. If the first one fails, the second takes over, and you have time to do repairs. Redundancy in theoretical physics is a little bit different. But it serves the same ends: To make life easier.

When one thinks about a theory in particle physics, one thinks about the particles it describes. But if we would write down a theory only using the particles which we can observe in experiment, these theories would become very quickly very complicated. Too complicated, in fact, in most cases. Thus people have very early on found a trick. If you add artificially something more to the theory, it becomes simpler. Of course, we cannot just simply add it really, because otherwise we would have a different theory. What we really do is, we start with the original theory. Then we add something additional. We make our calculations. And from the final result we remove then what we added. In this sense, we added a redundancy to our theory. It is a mathematical trick, nothing more. We imagine a theory with more particles, and by removing in the end everything too much, we end up with the result for our original problem.

Modern particle physics would not be imaginable without such tricks. It is one of the first things we learn when we start particle physics, the power of redundancy. A particular powerful case is to add additional particles. Another one is to add something external to the system. Like opening a door. It is the latter kind with which we had to deal.

Now, what has this to do with our work? Well, redundancies are a powerful tool. But one has to be careful with them nonetheless. As I have written, we remove at the end everything we added too much. The question is, can this be done? Or becomes everything so entwined that this is no longer possible? We have looked at especially was such a question.

To do this, we regarded a theory of only gluons, the carrier of the strong force. There has been a rather long debate in the scientific community how such gluons move from one place to another. A consensus has only recently started to emerge. One of the puzzling things were that you could prove mathematical certain properties of their movement. Surprisingly, numerical simulations did not agree with this proof. So what was wrong?

It was an example of reading the fine-print carefully enough. The proof made some assumptions. Making assumptions is not bad. It is often the only way of making progress: make an assumption, and see whether everything fits together. Here it did not. When studying the assumptions, it turned out that one had to do with such redundancies.

What was done, was essentially adding an artificial sea of such gluons to the theory. At the end, this sea was made to vanish, to get the original result. The assumption was that the sea could be removed without affecting how the gluons move. What we found in our research was that this is not correct. When removing the sea, the gluons cling to it in a way that for any sea, no matter how small, they still moved differently. Thus, removing the sea little by little is not the same as starting without the sea in the first place. Thus, the introduction of the sea was not permissible, and hence we found the discrepancy. There have been a number of further results along the way, where we learned a lot more about the theory, and about gluons, but this was the essential result.

This may seem a bit strange. Why should an extremely tiny sea have such a strong influence? I am talking here about a difference of principle, not just a number.

The reason for this can be found in a very strange property of the strong force, which is called confinement: A gluon cannot be observed individually. When the sea is introduced, it offers the gluons the possibility to escape into the sea, a loophole of confinement. It is then a question of principle: Any sea, no matter how small, provides such a loophole. Thus, there is always an escape for the gluons, and they can therefore move differently. At the same time, if there is no sea to begin with, the gluons remain confined. Unfortunately, this loophole was buried deep into the mathematical formalism, and we had to first find it.

This taught us an important lesson that, while redundancies are a great tool, one has to be careful with them. If you do not introduce your redundancies carefully enough, you may alter the system in a way too substantial to be undone. We now know what to avoid, and can go on, making further progress.

Hiding our ignorance

The radiative corrections discussed last time have another important aspect. For this, it is useful to recall the entry on Einstein's famous relation E=m*c*c. This relation told us that you can convert energy to mass, and thus to particles.

Now, quantum physics is a cheater. Always was, always will be. One of the most basic things it cheats about is knowledge. It tells you that certain pairs exist of which you cannot know both at the same time with certainty. If you know one very precisely, you can have only little knowledge about the other. The most important and fundamental such pair is position and speed. If you know the position of a particle well, you cannot know its speed very well. And the other way around. This is an observation of nature, which has been confirmed in numerous experiments. We cannot yet really explain why this is so, and have to accept it for the time being as an experimental fact. What we can do is derive an enormous amount of knowledge from this fact.

Among this is that a very similar relation holds for energy and time. If we know time very precisely, we do not know the energy very precisely. If you combine this with Einstein's formula, you get a very interesting consequence: For very short periods of time, energy is not very well defined, and may be much larger than assumed. Since this energy is equivalent to mass, this means that for very short periods of time you can have particles pop out of nowhere and vanish again. To be precise, you can have a pair of a particle and an anti-particle for very brief moments in time.

This seem to be almost unbelievable: Something hops into and out of existence, just like this. However, you can measure actually this effect, and it has been experimental confirmed very well. Also, it should not be taken too literally. What really happens is that quantum physics does something, and in our mathematical description it appears like you would have these pairs.

So what does this have to do with the radiative corrections? Radiative corrections are quantum corrections. As such they involve precisely this type of process: Something hoping out of the vacuum. It then briefly interacts with whatever you are actually looking at. Then it vanishes again. Therefore, radiative corrections include all the possible interactions of some particle with all other possible particles. Now comes the real boon of this: In reality this happens with all particles, not only those we know of. This has been used in the past to predict new particles, like the top quark, some of the neutrinos, and, yes, also the Higgs.

Great, so I can get everything from it! you may say. Unfortunately, it is not that simple. The heavier the particles, the less their contributions to radiative corrections, and thus the more precise an experiment has to be to detect their influence. As a consequence, the Higgs was the last particle for which we had strong such indirect evidence. And this was already experimentally challenging.

But it is much more troublesome for theory. Since we do not actually know what is there, our calculations have a problem. We create at very short times a lot of energy, but we do not know where to put it, since we do not know all the particles. Our theories thus lack something. And this something haunts us as failures of our theories, when we try to calculate radiative corrections. This was a very big problem for theories for a while, but we finally managed it. The key concept was named 'renormalization', which is again somewhat of a misnomer. Anyway, it gives a name to the process of hiding our ignorance. In fact, what we do is that we introduce in our theories placeholders for all these unknown particles. These placeholders are designed on purpose to remove all the problems we have. The way we designed them they can never described something of nature, but they absorb all the problems we encounter with our ignorance.

Since we know that we have these problems, it also tells us that the standard model cannot be the end - or for that matter any theory having such problems. They only describe our world at (relatively) low energies: The standard model is a low-energy effective theory, as was briefly indicated before. Here, you now have a better view of what the reason for the infinities encountered back then is: That we do not know what particles may appear in our radiative corrections, and thus that we do not know where to direct our energy to. And that the parameters used back then just mock up the unknown particles.

You may wonder whether this is a generic sickness of quantum theories. This is very hard to tell for a realistic theory. Of course, we assume that if we would know the theory of everything, it should not have these problems. We can indeed construct toy theories of toy worlds, which do not have these problems, so we think it is possible. Whether this is true in the end or not, we cannot say yet - perhaps we will need in the end a whole new theoretical concept to deal with the real world. For now, renormalization prevents us from the need to know everything already. This permits us to discover nature step by step.

Groundwork

The first topic of my research is both very fundamental and very abstract. It has something to do with coordinate systems coordinate systems, but not the ones to describe where event takes place in space and time. A while ago, I have discussed local symmetries. Saying something has a local symmetry means that I can change things at different places in different ways. Back then, I used a grid of billiard balls. I could rotate each of the balls differently, but because the balls are perfect spheres (we over-painted the numbers), this did not change the way the grid looked.

Now, lets assume that for some obscure purpose you would like to keep track of what is going on with the balls. That you want to know the position on each ball in some way. You can do this by introducing coordinates on every ball. Think of painting a point on every ball (actually, you will need two, which are not lying on opposite points on the balls). These points destroy the local symmetry. You can now keep track of all rotations by looking at the points. In exchange for loosing the symmetry you now always know when the balls are rotated. Especially, you can now always talk about a ball pointing in some direction, because you can use the points on the ball to identify a direction.

If you now go to a particle physics theory, you also have such local symmetries. A consequence of these local symmetries was that the amount of numbers you needed to describe a photon was very small. But is was very inconvenient to use this minimal amount. At least that was what I said. If you go to something more complicated than the photons, like the gluons, it becomes even more inconvenient to use the smallest possible number.

Ok, wait a minute. What does these two things have to do with coordinate systems, you my say. And the answer is, actually, quite a lot. The procedure of using more numbers than necessary requires that you use a coordinate system to measure these additional numbers. That is something which we call introducing a gauge or fixing a gauge in particle physics. We have to know the size and the direction of the additional elements we bring into the discussion. That sounds awfully technical, and indeed it is. But at the same time it is very often very convenient. I fear, here I have to ask you to take this statement on faith. Even the simplest case where one sees how powerful this is fills a page with formulas. In fact, most of the page would be filled with the calculations which try to not use the additional coordinates. And only a few lines at the bottom would use the additional coordinates. In calculations with the full standard model, the reduction becomes incredibly large. Thus, we do it often. Only for very few calculations, in particular those done by a computer in a simulation, we can afford to do without.

This was so far only the prelude to my actual research topic. As I have warned, it is very abstract. The topic has now to do with how such coordinate systems can be chosen. So far, it seemed to be quite straightforward to do so. For the balls, we just make points on them, one ball at a time. That is, what we call a local coordinate system. This means, the choice of the coordinates can be done independently at every place. In this case, by working on each ball separately.

But the theories used for particle physics are strange. Imagine a hideous, malevolent demon. Whenever you make a point on one ball, he sneaks behind your back and changes the points you have already made, depending on where you make right now a point. How would you fight such a creature? You would take many, many pencils, and then construct a sophisticated device such that you can make the points on every ball at the same time. With this you trick the demon. What you have done was choosing your coordinate systems all at once everywhere. Since you have constructed the machine, you have no longer the possibility to make an individual choice on each ball just when you pick it up. The choice is fixed for all balls by the machine. That is what we call a global choice of coordinate systems.

Though we do not have demons running around the particles, at least, as far as we know, we have a very similar problem in particle physics. The mathematical structure of the theories in the standard model requires us to make global choices when introducing our additional coordinates. Seems to be not so complicated at first sight, but it is. Constructing appropriate machines is very complicated. Especially it turns out to be very complicated to construct a machine which works wells with every method. But when you want to combine methods, you must use the same coordinates. And thus build a machine which works with more than one method. And this is baffling complicated. In fact, so complicated that we have been struggling with it ever since the problems has been recognized. And that was in the late 1970s.

Why does this not stop us completely in our tracks? The reason is that in perturbation theory the global choice becomes again local. That means, whenever you can do perturbation theory, you can make a much simpler local choice. That is something we can do perfectly. And since perturbation theory is so helpful in many cases, this problem is not a show stopper. The reason is that perturbation theory only admits small changes. And small changes means that we never go far away from a single point we made on the ball. Thus, the demon can never sneak up on us.

Furthermore, when we do simulations, we know how to evade the problem. However, the price we pay is that things become obscure. Everything is just a black box, and in the end numbers come out. But for many problems, this is quite satisfactory, so this is also often fine.

But then there remain some problems where we just cannot evade the demon. We have to fight it. and that is where I enter the scene. One of my research topics is to understand how to make the same global choice with different methods. That is something I have been working on since 2008, and it proved and proves to be a formidable challenge. So far, the current state is that I have made some proposals for machines working with more than one method. Now, I have to understand whether these proposals make sense, and if yes, if they are simple enough to be used. It is and remains a persistent question, and one which will accompany me probably for the rest of my scientific life. But sometimes you just have to bite it, and do things like that. Its pure technical, almost mathematical. There is no physics in it - coordinate systems are choices of humans and not of nature. It is the type of ground work you have to do occasionally in science. It is part of building the tools, which you use then later for doing exciting physics, learning how nature works.

The equations that describe the world

Ever since mathematics has been introduced into the description of physics we have striven to describe reality in terms of equations. One of the arguably most know equations is Newton's law that the acceleration of an object is given by the ratio of the force acting upon this object divided by its mass. These equations should not be taken as everlasting truths. For this law of Newton we know that its not fully satisfied if we try to describe a quantum object or if the speed of the object is close to the one of light. However, this makes the equation nonetheless useful, as there are many cases where neither is the case. The best known example is the movement of the planets around the sun. This equation is, however, not yet complete. It states everything that is to known about the particle, but nothing about the force. It is what is called a kinematic equation.

We need to supplement it by an equation for the force. In case of the movement of a planet around the sun, it is Newton's law of gravitation: The force is given by a constant, which has to be measured, times the mass of the sun times the mass of the planet, divided by the square of the distance between the sun and the planet. With this, we know enough to solve the equation, and find after some tedious calculations (to be done by every first semester student of physics) that the planet moves on an elliptical orbit around the sun. With the force given, the equation therefore describes the motion of the planet. It is thus called an equation of motion. Generically, if we can formulate the equations of motion for a theory, we have everything at our disposal to describe the solutions of the theory. However, in general we have to supplement the equations with the situation we want to actually describe with the theory. In the case of the planet, we have to add where the planet was and where it moved to at a certain instance of time. Otherwise the equation of motion would give us the solutions for all possible initial positions and velocities of the planet, and thus an infinite number of possible solutions to the theory. Such additional information are called boundary conditions. They select out of any possible kind of behavior described by a theory the particular one which is compatible with the state a system is in.

This concept now sounds at first like something which is very much tied to Newton's law. In fact, it is not. Already before 1900 we have known how to write down the equations of motion for all kinds of non-quantum physics happening at a speed much less than the one of light. Unfortunately, knowing how to write down the equations is not the same as being able to solve them. For example, we know very well the equations of motions describing how a river flows. But as soon as it flows quickly over rough grounds, such that it becomes turbulent, we are no longer able to solve the equations. In such cases we are often forced to revert to the simulation methods discussed previously.

Now, what happens, when things get fast or quantum? Well, when things get fast, not much changes. The equations just look a bit different, and are much nastier, but that is more or less all. When things go quantum, it becomes more weird. Since in a quantum world we have this problem with being either wave or particle, it is no longer really possible to talk about moving objects anymore. Nonetheless, people have been able to formulate something which is in spirit close to the equations of motions, the so-called quantum equations of motions (or some times called Dyson-Schwinger or Schwinger-Dyson equations, honoring those people who have developed them). These equations describe, in a way, the average behavior of particles in a quantum theory. Nonetheless, supplemented once more by boundary conditions, they describe the contents of a theory completely. Thus, they are powerful indeed. But as with anything powerful, it gets complicated. Thus, only for very, very simple theories it is possible to solve these equations exactly. For theories like the standard model, one has to introduce severe approximations (often called truncations) to be able to solve them. If these approximations are made wisely and with insight, these approximations are such that still questions we have to the theory can be answered correctly. But it takes often very long to understand how to do approximations right.

The way these quantum equations of motions (or also the ones for non-quantum physics) look is by no means unique. We are free to do mathematical reformulations of them. These leave us always with the same physics of course, but the equations look rather different. This is often very helpful, as the different formulations have very different properties, and very different advantages and disadvantages when it comes to doing calculations. Thus, by exploiting the different reformulations in a wise way, one can go a long way in solving the equations.

In case of the quantum equations of motions a particular useful reformulation is given by the so-called functional renormalization group equations. That sounds like a awful big thing, but the idea behind it is rather straightforward. The idea behind this reformulation is not to swallow the whole theory as one big thing, but chop it off in simpler bites, taking one after the other. Technically, it is realized by slicing the energy which particles are allowed to have, and only include particles with a particular range of energy values in each single step. Building up the whole reality is then done by adding up the particles with different energies one after the other. Though also this cannot be done exactly for most theories, it a very useful complementary way of solving the equations, with great successes.

Both approaches together are often collected under the common name of functional methods. Functional here stands for the fact that on a mathematical level both are strictly speaking not dealing with ordinary functions. Rather, they deal with functions of functions, so-called functionals. This sounds awful and is in fact as awful as it sounds. But it is the price one has to pay when one wants to venture into quantum physics mathematically. Nonetheless, this name is nowadays attached to a collection of different formulations of the quantum equations of motions. These are a great help in describing and understanding physics in every detail. In contrast to the lattice methods, it is easy to disassembled the equations, and to understand what each every part is doing. Though, while very complicated to solve, they are a vital part of the physicists tool box in one way or the other, and thus remain a thing I am working with on a daily basis.

Wave functions and fields, once more

In the discussion about fermions, the concept of a wave function appeared, to explain what makes fermions so very strange under a change of coordinate systems. The analogy of particles with waves and oceans has been made also already quite a bit back. It is about time to be just a bit more precise about what a wave function and a field is for a theoretical physicists.

Go back to the idea that particles emerge a some waves at a particular point on an ocean. Two particles would then be just two such waves at two different points. Now the underlying concept appears just to be the ocean, rather than the waves. And indeed, the waves can very well be identical.

That is the underlying idea also in theoretical physics - not only particle physics, but this permeates many ares of theoretical physics: The basic object is the ocean. In the context of particle physics, this ocean is then called a field. Such a field is now existing at every point in space and at every instance in time. In the very literally meaning of the word, it fills up all of the universe. If there is nothing of interest around, this is because the size of the field at this point in space and time is small or even vanishing. However, if there is a spike at some point in the field then just as in the picture of the ocean there sits a particle. If there is a second spike somewhere else, then there is another particle, and so on. Since all the spikes belong to the same field, they describe the same type of particle, say an electron. The spikes may move with different speeds, so the electrons appear to have different speeds, but they are still electrons. That is the reason why all electrons are the same: They are just spikes in the same field. Such a spike is often called an excitation of the field, and this excitation is the electron.

Then what is about the other types of particles? The quarks, the gluons, the Higgs? Well, these belong just to other fields. That is, our universe is filled up with many fields, all existing simultaneously at every point in space and time.

You may be wondering how this should work, and if this is not a bit crowded. But you know already that fields are mathematical concepts. For example, you can associate with every point in space and time a temperature, and thus create a temperature field. At the same time, there is an atmospheric pressure field. Both can happily exist simultaneously. But they are not ignoring each other. As you know, both a related with each other: If either changes this indicates a change of the other as well. Though this analogy is not exactly the same as the particle physics fields, and there are more things involved, the basic idea is the same.

Also the particle physics fields interact, and thus not ignore each other. Their interaction can be more or less translated once more from the analogy with the waves, which has been discussed earlier. So, in this way, everything is realized we see in particle physics. There are fields for every type of particle, which may interact. We are then 'just' a very complicated, combined, and correlated simultaneous excitation of all of these fields, as is your desk or your computer.

Now, what are the wave-functions? Well, in the beginning, quantum physics was formulated not taking into account the effect of large speeds, i.e. of special relativity, something I will explain in more detail later. In this case, the concept of fields can be reduced to instead describing only the waves making up a single particle. In principle, you isolate each wave describing a particle, and discuss it alone. These mathematical quantities describing these single particles are then called wave functions. So wave functions can be thought of as the slow-speed limit of the fields, when all particles are treated separately. Mathematically, this is not quite precise, but should give a rough idea.

Now it is possible to come back to fermions. When you rotate the coordinate system once, it is this wave function (or the field), which change not directly back to the original, but only after a second rotation. Of course, nothing you can actually measure (or experience) changes when rotating your coordinate system once fully. That is because the wave function or the fields cannot be directly measured, just things we can derive from them. However, the underlying fact that you have this obscure change influences the properties of fermions, and leads, e.g., to the Pauli exclusion principle.

Internal and external space(s)

I have repeatedly discussed symmetries, and often made examples where one imagines some object, and how it looks from different perspectives. It seems surprising at first that something like a symmetry, which is looking like something belonging to the deepest properties of a system, should be so readily visible as an ordinary object. How so?

The reason for this is rather mundane, though far from obvious: There is not such a big difference between symmetries and the world around us. As a physicist, I refer to this fact as an internal and an external space.

An external space is just the world around us - length, width, height, time. It is the arena, in which physics takes place. At the same time, it exhibits symmetries. You can rotate things, and if they are symmetric, they look the same. You can choose a coordinate system, and describe things, but what happens is independent of the coordinate system. That is also a kind of symmetry: Physics is independent of the coordinate system, looking from any coordinate system everything happens in the same way. This is called a space-time symmetry. Physicists have also a more complicated name for it: They call it a diffeomorphism invariance.

Now, how is all of this related to the symmetry, say, of electromagnetism? Well, go back to the four numbers describing electromagnetism, and forget for a while that they change at different places. Then the four numbers can also be taken to describe four directions, four new coordinates, with which I can describe things. Since these coordinates are not the usual ones, it is said that these coordinates describe an internal space. Now, in these new coordinates, we can also choose a coordinate system, and physics is again the same, irrespective of our choice of coordinates. However, with this coordinate system we do not measure lengths or times, but we measure electromagnetism.

If you then combine the internal and external space, you have the total space. Each point is now characterized by eight numbers: The four conventional coordinates, and the four internal coordinates of the photon field.

The fact that we can change the internal coordinate system freely is the reason why we have four numbers, though physics only depends on two numbers: The symmetry permits to make a coordinate system choice, and this does not matter. If there would be no symmetry, there would be just one coordinate system permitted, and we could not change it.

However, even if there is a symmetry, we are not permitted to make any coordinate system choice. For example, we could in the real world, the external space, not make a choice of coordinates such that time were finite, or would make a loop. Similarly, in the internal space, one cannot make always an arbitrary choice. In fact, in the internal space of electromagnetism only coordinate systems where all coordinates do make a loop are permitted. That is one of the big differences between space and time and electromagnetism. Indeed, all the symmetries of the standard model have symmetries, which have only coordinate systems, which have loops. In fact, how one can choose a coordinate system is very hard to understand for the strong and weak force, and we actually only know for sure how to make a choice close to the point where we look at at some instance. How to make a descent choice far away from where we are right now looking is a complicated problem, and actually one of my research topics.

However, for this tourist guide, the most important point to remember is that symmetries and coordinate systems are closely related, and that the coordinate systems of the internal spaces are not so much different from that of the external space.

Pointing in space and time or why one needs four numbers for a photon

In the previous discussion it was described how photons are described by fields, and that the fields are somehow like the surface of an ocean. The truth is, unfortunately a bit more complex. This can already be seen from the magnetic field. If you have a magnet, you cannot only feel its field in the same plane as where the magnet is, but also above and below it. Thus, the field is something which not only is like the surface of an ocean, but which is more like the ocean itself, it is above and below and all around. Well, this is not yet a problem, since one can imagine that, say, a subsurface explosion also can make a wave which has volume, and the analogy is only a bit more harder to imagine because of the third direction.

But things become still a bit more messy. Take the magnet and take a pretty hot flame, and place it under the magnet, not too close. If you now measure the magnetic field at some point in the space surrounding the magnet, you will notice that the magnetic field decreases over time. That is because when you heat a magnet sufficiently (a couple of hundred degrees), it will loose its magnetic properties. Thus, the field is not static, it changes with time, and can even vanish. Of course, you could have noticed the same feature by just moving the magnet far away, but then you could bring it back again. Thus, a field is something that tells something about a direction and a strength at some point in space and time.

But these seems a bit odd. To identify a position, you need four numbers, four coordinates. But the direction of the magnetic field you can enumerate with just three, two for the direction, one for its strength. There is nothing like a time direction to the magnetic field. Indeed, electric and magnetic fields are peculiar in this sense. As said before, they can be derived from a quantity which had four numbers, as the four coordinates just needed to characterize the evolution of the magnetic field. It is about time to tell what the four numbers are.

Indeed, it turns out that a field which describes a particle has four components, each of which depends on the space-time point one is looking at. So what is this fourth number? In a sense it is the direction of the field in time. That sounds a bit peculiar, and in fact it is. The reason for this is the arena in which physics takes place.

If one goes back to ones experience of reality, then there is the space with its three dimensions, and there is time, which appears to be just flowing along in the background. But in fact space and time are connected, and are not two independent entities. That has been an observation which has actually been made very early on in physics. However, it took a while to note that the structure is peculiar, but this will be discussed at a different time.

Again, it helps to make an analogy. Take a flat cylinder. Put in the cylinder a disc, which fits perfectly in it. Now, if you elevate the disc at a constant rate than everything on the disc can move freely on the disc, but there is a constant change in height, just as time changes constantly. In our world, the disc has one dimension more, and the changing height is the changing time, but otherwise it is the same concept. Somebody on the disc could even measure time by measuring height, because it is lifted constantly.

Now, of course, it is possible to give a direction which is entirely on the disc. But for us, which can see the cylinder as a whole, we can also give a direction which points upwards or downwards from the disc. In contrast to someone living on the disc, we need one quantity more to specify a direction. But if someone on the disc is very clever, he will notice that his space is larger, and then she can invent, at least as a mathematical concept, a direction off the disc, which will agree with our idea of direction. However, since she only knows the disc she has no intuition of what means 'off the disc', but has a mathematical grasp of it.

And so it is the case for us with time. We can mathematical describe our cylinder (though it actually looks very much different from a cylinder), and we can describe a direction off our three-dimensional world by giving it a direction in both time and space. Then, we notice that the field that describes a particle is actually requiring to have such an additional direction, and this is the reason why the photon field has four numbers at every space-time point: a magnitude and a direction in space and time. And the electric and magnetic field with only a direction in space are something like shadows of this object in time and space in a purely spatial world, in which we can move freely.

Of course, these four numbers are not independent, but this is because of the symmetry. Without the symmetry, they would be. The symmetry is something additional, and has nothing to do with space and time.

Fields, waves, particles, and all that

So, there has been quite a bit of talk about fields but then there also appeared a particle, the photon. And both have been associated with electromagnetism. But what is it, really?

Well, this question baffled scientists in the early 20th century. There was a lot of talk about a particle-wave-duality and things like that, which are still used as a simple explanation that things are either like a wave of like a particle, depending on the circumstances. And wave is connected to field, because a field is like an ocean: The height of the water at each point is also a kind of field. And like an ocean, there can be waves on it.

All that sounds a bit confusing? Indeed, people have made up their minds by now. And despite the usefulness of the picture of something which can be either particle or wave it is rather that it is both simultaneously. And the thing connecting it is the field.

Go back to the analogy with the ocean. Imagine that your field is an ocean. If the ocean is totally flat, there are no ripples and nothing else, so you could say that there is nothing happening. That is what people call a vacuum when they talk about fields: Just a field where each point looks exactly the same as everywhere else, and there is no change from one point to another.

Now, imagine, something is happening. Whenever something happens in an ocean, it makes ripples and finally waves. That is what people call an excitation of a field. Something is moving. Now, when you are very close by, then you just see the waves around you, and they do not have much of a structure. They are just waves. On the other side, if you are very far away then what happens just looks like a point, or a flat ball. That is exactly the analogy to the question whether it is particle or wave. If what happens (the 'excitation') is very far away, you do not see an internal structure to it, it is like a point. If this would be beneath the surface, it would look like a ball. And that is what you are usually refereeing to as a particle. If you go closer and closer, then the internal structure becomes apparent, and you see that the thing is much more like a wave again, rather than a particle.

Of course, this analogy can only be approximate. Just think of a moving particle: That would be like all the waves stay together and move at as a whole. You usually do not see this on an ocean - that what was originally a particle dissolves into waves, never to reunite again. That is different for the fields in the standard model. They can keep together, and even come together again if they have resolved earlier. One should keep these limits in mind when working with such analogies that they have their limits.

Anyway, sticking with the analogy, it is possible to see another important concept. If you are far away than the average distance between two peaks of the waves is very small compared to your distance. On the other hand, when you are close, the distance between two peaks is of similar order as your distance. This tells you that the relative sizes are important if you want to resolve the internal workings of something. You need to have something which is of the same size as the internal structure of the thing you want to analyze.

Particles are very tiny (the proton is of size 0,00000000000001 meters, the electron to the best of our knowledge smaller than 0,000000000000000000001 meters!). If you want to investigate their inner workings, you will need something which is even smaller. The only thing which is smaller than a particle is another particle. And there is also something else, which comes to help - it is possible to make a particle effectively small by making it faster. That sounds a bit weird, but it is not so far off. Think of the following: Take a parking car. Mark its beginning and end by going first to the front, and place a marker. Then walk to the end of the car, and when you reach it, put another marker. Measure the distance between both markers. Now try the same when the car moves. If you walk with the same speed, you will not get as far as when the car stood still, because it moves. It appears shorter, smaller. Now that may appear as cheating, and in a sense it is. But the laws of nature actually make this cheating true, by a much more subtle mechanism, called special relativity. This is a topic of its known, to which I will return in due time. For the moment, the only important thing is that if you want to probe a particle with another particle of the same kind, you need to make the probe particle move very fast compared to the particle you wish to analyze. That is the reason to build particle accelerators: Their only purpose is to get very fast particles to probe very short distances. And this is in fact not a simple task, and requires the most modern technology available to date.

Global and local symmetries

An important distinction in physics is global and local.

A global property is something which is inherent to a system as a whole. A local property is something attached to a particular point in space and time. Assume for the moment that the earth would be a perfect sphere, which it is to a rather good approximation. Then the rate at which the earth's surface bends under one's feet is a global property, because it is the same on the whole planet. On the other hand, whether there is water and land under the feet is a local property, and depends on where on the earth one stands.

So far, this is a static situation, which permits to divide between global and local properties. Even more important in physics is the difference between local and global changes. A local change modifies something at a given place. E. g., the property whether there is land or water below one's feet is changed locally by the tides. A local change is not limited to a certain point, but it can affect many (or all) points at the same time, but something different may go on at every point. The tides all over the world are an example of a local change, which let the water rise at some point and removes it at another point. A global change is then a special case of a local change in that it makes the same change at each and every point. For example covering the earth's surface everywhere by a meter of sand would be a global change.

This leads back to symmetries. It is now possible to divide between a global and a local symmetry. A global symmetry is something inherent to the system as a whole. A global symmetry transformation would then be a symmetry transformation applied to every point which leaves the system unchanged.

A local symmetry transformation is much more complicated to visualize. Take a rectangular grid of the billiard balls from the last post, say ten times ten. Each ball is spherical symmetric, and thus invariant under a rotation. The system now has a global and a local symmetry. A global symmetry transformation would rotate each ball by the same amount in the same direction, leaving the system unchanged. A local symmetry transformation would rotate each ball about a different amount and around a different axis, still leaving the system to the eye unchanged. The system has also an additional global symmetry. Moving the whole grid to the left or to the right leaves the grid unchanged. However, no such local symmetry exists: Moving only one ball will destroy the grid's structure.

Such global and local symmetries play an important role in physics. The global symmetries are found to be associated with properties of particles, e. g., whether they are matter or antimatter, whether they carry electric charge, and so on. Local symmetries are found to be associated with forces. In fact, all the fundamental forces of nature are associated with very special local symmetries. For example, the weak force is actually associated in a very intricate way with local rotations of a four-dimensional sphere. The reason is that, invisible to the eye, everything charged under the weak force can be characterized by a arrow pointing from the center to the surface of such a four-dimensional sphere. This arrow can be rotated in a certain way and at every individual point, without changing anything which can be measured. It is thus a local symmetry. This will become more clearer over time, as at the moment of first encounter this appears to be very strange indeed.

Symmetries

A concept very closely related to invariance is symmetry. In fact, symmetries are what currently guides us most in the construction of theories of elementary particles.

A symmetry is in the beginning the fact that something looks similar when viewed from different perspectives. Take a ball, like a snooker ball, but paint it only in a single color with no markers. Then, no matter from which direction you look at the ball, it always looks the same. Or, you can turn it as you like, it always looks the same. The ball is just the same from all directions, a perfect sphere. Thus, it is called to be symmetric under a rotation. Therefore, this symmetry is called rotational symmetry. With this already the link to invariance comes in: The ball looks the same from all direction, it is invariant under the position of the one looking at it. There is always an invariance when there is a symmetry.

If you start looking around, you will find symmetries to be a rather general concept. If you take a blank sheet of paper, its front and back look the same: It is symmetric under flipping it from front to back. Or take a snow-flake. When looking closely, it has a structure with six rays. Thus, if you rotate it by a sixth of it circumference, it looks like without rotating. Both these examples are so-called discrete symmetries. For the ball, we could rotate it arbitrarily little, and it still looks the same. Not so the snow flake. If we would rotate, say, by a tenth of its circumference, it would be obvious that someone rotated it. It only looks the same when rotating it by a sixth of its circumference. There is only a finite number of things we can do to it to make it look the same, while there is an infinite number of things we can do to the ball.

To find another example of a symmetry like the rotational symmetry, which is also called a continuous symmetry in contrast to the discrete symmetry of the snow flake, imagine empty space. If there are no stars or galaxies or so, then you could move a step to the left, right, front, or whatever, or half a step, and whatever you do, it always looks the same. This is the so-called translational symmetry. Moving you in another direction just gives the same result. You could also rotate yourself in space, without changing anything. Thus, you can combine the rotations and the translations to a bigger symmetry, a so-called product symmetry.

What is, if there are two people in outer space? Now you cannot move alone, and everything is the same again, because the other did not move. However, if both of you take a step of the same length in the same direction, nothing appears to be changed. In this case, one says that the symmetry is only applying to the complete system: When always moved together, the two of you form a system, which is symmetric under common translations and rotations.

Another important concept with symmetries is that of an approximate symmetry. Take a person. The left-hand side and the right-hand side of her face look at first symmetric. You could just mirror them, and it would look the same. This appears to be a discrete symmetry, actually a mirror symmetry. However, if you look closely than the person might have a slightly different shade of eye color on the left than on the right. Thus, though it looks almost as if there is a symmetry, it is actually not there, but almost. This is an approximate symmetry. If, for example, the person would have painted her face on one side blue, then the symmetry is not even approximately there, it is just different. In this case, one also calls it a broken symmetry, broken by some external effect, here the painting. Symmetries which are not flawed in either of these ways are called exact. The snooker ball had an exact rotational symmetry. Would we have left the number on it, the symmetry would have been broken.

This is already a long number of different types of symmetries. There have been continuous and discrete symmetries, the symmetry of a system and the individual symmetry, product symmetry, an exact, approximate, and broken symmetry. If you go around, you will easily spot more of them. A sausage shows a symmetry when rotating it about its length, a leaf of a tree has a mirror symmetry like a face, and so on.

In elementary particle physics, it turns out that symmetries are deeply connected to the properties of particles. For example, each force can be connected to a symmetry. The fact that we have mass can be traced back to a broken symmetry, as that there is more matter than anti-matter. And this is just a short excerpt. However, to really understand these, it requires another concept, the difference between local and global.

Invariance

Last time we have defined coordinate systems. We also made the statement that for two people to agree about something measured with the coordinate system, they had to agree where to position the origin, and how to orient the coordinate system. The latter could e.g. be done by making one of its axis point north and the other point east and the third perpendicular in the heavens. An interesting question is now why we had to agree about orientation and origin. Obviously, a player on the field will not care about how we locate him and how we discuss about his location (I neglect here the possibility of markers on the field for the purpose of playing a game. Just assume that they are not necessary and the rules of the game do not need them). She will just keep on playing, no matter how often we change our agreement or how extreme our conventions are.

With this, we have a first example of a feature which is very central to our understanding of how we can describe physics. This is the concept of invariance. It means essentially that nature is not caring about how we describe it, and whatever we do, we have to respect this. In particular, nothing can depend on us. We are just observers. That seems to be an innocent enough statement, and moreover a pretty obvious one. It is actually not.

First, nothing dictates nature to be that way. There is no reason that nature should not depend on who it observes how. Though this would quite ruin our current understanding of how nature works, it is just an empirical fact, and one which we can not (yet) explain. It is a law of nature, so far.

The second is that as innocent as the statement looks, it has become one of our most powerful tools to devise a description of nature. Lets get back to the players on the field. Given the just said, the numbers which with we describe the position of a player on the field are not of importance. The player is not even aware of them. Things start to change when we add a second player. Also she is not aware of which numbers we assign to her to keep track of her position. What both players are very much aware of, however, is where the other one is, and how far she is away. That is something we can also quantify with our coordinate systems. If the first player is at the origin, say, and the second player is at the next grid point at the first tick in the direction of one axis, their distance is the distance of the tick marks, say one meter. Hence, their distance is one meter.

What happens now if we change our coordinate system? Well, lets flip it somehow, and move the origin to the sun. But this does not change the distance of the two players, it is still one meter. Hence, their distance is (so-called) invariant under a change of the coordinate system! That is a first example of how actually an invariance pops up. Hence, if we try to describe how the two players behave, the numbers of the coordinate system will not matter, but their distance will. So, we know now that a theory describing the players (e.g. to determine the rules of the game) will not make use of the coordinate system, but only of the distance of the two players. Thus, invariance has given us a first tool how to describe the behavior of the players.

This could also be formulated differently (and very popular). The players do not care about the coordinate system we put on the field, despite this having a universe-wide particular point of reference, its origin. They only care about the distance with respect to each other. That is, the absolute frame given by the coordinate system does not matter. Only the relative position of the two players matters. Thus, it is only relative quantities which do matter. The popular phrase made from this fact is that "everything is relative". Here, we have seen that this phrase embodies the principle of invariance under a change of description.

Is the coordinate system now of complete uselessness after we have introduced and bargained about it so much? No, it is still very useful. We can still use it to describe the two players on the field. This makes life much simpler. However, we know now that of the numbers associated with each player only the ones giving their distance will enter the rules of the game, the description of nature, and the remaining ones only serve us to provide a clear picture. It is this possibility to have a clear picture to the human mind, which lets us keep the additional coordinate system when we describe something in most cases.

Coordinate Systems

With the players now on the field, it is about time to say something about the field itself.

One thing quite necessary when one wants to talk about the field in a reproducible way, a central requirement for scientific investigations, is to be able to denote a point on the field. If it would indeed be a field, one could just lay a grid with regular squares of length, say, one meter each, over the field. A position on the field is then just given by denoting a certain square. Or? Well, there are two points which have to be added.

The first is that a square of one meter extension in both directions is rather vague when it comes to an object the size of a cherry, though it may be sufficient to locate a player rather well. So, it is necessary to make the grid finer for a cherry. That can be done by taking each square and subdivide it further in squares of, e.g., one centimeter extension. That should be sufficient for a cherry, but would not be for a bacteria. Then, we would have to subdivide it further into micrometer. And for an atom or a nuclei or a quark even much further. Therefore, such a grid should have a resolution of the field in useful units, such that everything can be located as good as necessary.

The second thing is that it is still very hard to agree on where a player is. The reason is that we have not yet fixed our grid, and two different observers could slide it differently over the field. We therefore need a reference point. For example that a certain square has its lower-left corner in the middle of the field. But this is not enough. Besides sliding the grid, there is also the possibility to rotate the grid. Therefore, we have to have a reference orientation. For example, if the lower left corner of a given square is at the center of the field, we could agree that then the edge which connects it to its upper left corner should point in the direction of the magnetic north-pole. Now, we have a well-defined grid.

Actually, we have already made another choice. We decided to have a grid of squares. We could also have chosen, say, a rectangular grid. Or a circular. Or something more twisted. We just have to specify it.

So, altogether, to be able to locate something on the field requires us to fix a grid with a certain geometry of elementary grid patches, like the squares, having a certain resolution, associate a particular patch with a particular point - this is called the origin of the grid - and its orientation. All these information together define a coordinate system for the field.

We could now go on, and add also a further direction, say, up in the sky, so we can not only talk about where on the field, but also in which height above the field. By this additional direction, we have added a further coordinate axis to the coordinate system. We have tacitly assumed that it has the same patch geometry and resolution, and given it an orientation. Again, we need to fix the point where it touches the field, which is usually then the origin of the grid on the field. With this step, we have promoted our flat coordinate system on the field to one with height and volume: We have added another dimension to it. Originally, we had two directions on the field - depth and width. These are two dimensions. By adding one, we gained another dimension, a third one, the height. We could go on, and add another one measuring (invisibly) the time, so we can specify where and when and how far above the field something happened. These four information are then the coordinates of this something, of this event. It is such a four-dimensional grid, which is usually used to describe things happening in our world in physics.

An important insight is that what we did to set the origin, orientation, and resolution has been arbitrary. If somebody would want to have the origin a bit more to the left, and it direction pointing towards the south-pole, it could have done so as well, and would also be able to specify an event on the field. The important thing is that if we know how he has chosen his coordinate system relative to ours - a bit more to the left and the direction towards south - we are able to translate his coordinates into ours. Hence, though we need the coordinate system to make a definite statement where and when something happens, it is not unique. We could chose any coordinate system, as long, as we know how to relate it to all others.

This is an important idea in the description of physics in general and in elementary particle physics in particular. We can chose an adequate coordinate system for a problem to make things simple, as long, as we keep in mind how to translate it to other coordinate systems.