What the strong interactions, temperature, and density have to do with each other

After the preparation with the last entry, I can now move forward to my next research topic.

Let me just collect what we had so far about the strong interactions, QCD. QCD is the theory that describes the interaction between quarks and gluons. It tells you how these make up the hadrons, like the proton and the neutron. It also describes how these combine to the atomic nuclei. The quarks and gluons cannot be seen alone: The strong force confines them. At the same time, the strong force makes the quarks condense, and thereby creates the illusion of mass. Well, this is what one can call a complicated theory.

Now imagine for a while what happens, if you heat up matter. Really make it hot, much hotter than in the interior of stars. Heat is something like energy. You can imagine that the hotter something is, the faster the movement of particles. The faster they go, he hotter they are. But the higher the energy, the smaller the things are which get involved. Hence, if things get very hot, the quarks and gluons are the one to really feel it.

When this happens, two things occur. One is that the condensate of quarks melts. As a consequence, the quarks can move much more freely. The other is that you can convert some of the energy from the heat to particles. I will come back to the mechanism behind this later. For now, it is enough that this is possible. It is nothing special to the heat case. Anyway, you can convert a small fraction of the heat to particles. If things get hot enough, a small fraction is actually quite a large number. And if things are really hot, you have created so many particles that they do not fit anymore in the space you have available. Then they start to overlap. At that point, even if you have confinement, you can no longer tell the hadrons apart. The things just overlap and you can start to swap quarks and gluons from one to another. You see, when you put enough heat into matter, things start to look very different.

Why is this interesting? I said it must be much hotter than in a star, and actually much hotter even than in a supernova. It does not appear that there is anything in nature being so hot. However, we can create tiny amounts of matter in experiments which is so hot. And also, very far in the past, such super-hot matter existed. Right after the big bang, the whole universe was very densely packed, and the temperature was so high. This was only a fraction of a second after the big bang. So, is this relevant? Well, yes. If we want to look back even further, we have to understand what happened in the transition of the strong force at that time. Since we cannot create universes to study it, we need to extrapolate back in time. And for this, we need to understand each step of going back in time. And hence, we need to understand how QCD works at very high temperatures.

This is not the only case where we need to understand QCD in extreme conditions. The other case is the interior of neutron stars. In these stars, matter is packed incredibly densely. It is very cold there, at least when comparing to the early universe. But it is so dense that hadrons again begin to overlap. Thus, you would expect that you can again exchange quarks and gluons freely between hadrons. But because it is cold, you will keep your quark condensate. In fact, there are quite a lot of speculation, whether you can create also condensates of quarks with different properties than the one usually known. To really understand how neutron stars work, and eventually also how black holes form, we have to understand how QCD works when things become dense.

When you take both cases together, and permit for good measure to also include all possible combinations of dense and hot, you end up with the QCD phase diagram. This phase diagram answers the question: When I have this temperature and that density, how does QCD behave? Determining this phase diagram has been a topic of research ever since these questions were first posed in the 1970s. Very important for this task have been numerical simulations of matter at high temperatures. With them, we have become confident that we start to understand what happens at very small densities and rather high temperature. We understood from this that the universe has undergone a non-violent transition when QCD changed. We can therefore now extrapolate further back in the history of the universe. But we are not yet finished for this case. Right now, we leaned what happens when we are at fixed temperature and density. But these two quantities changed, and we have not yet fully understood how this proceeds.

Things are a lot worse when we turn to the neutron stars. We have not been able to develop efficient programs to simulate the interior of a neutron star. We even suspect that it is not possible at all, for rather fundamental reasons concerning conventional computers. Progress has therefore been mainly made in two ways. One was to rely heavily on (very) simplified models, and more recently using functional methods. Another one was to study artificial theories, which are similar to QCD, but for which efficient programs can be written. From the experience with them we try to indirectly infer what happens really in QCD.

I am involved in the determination of this QCD phase diagram with two angles. One is to develop the functional methods further, such that they become a powerful tool to address these questions. Another one is to learn something from the stand-in theories. Especially in the latter case we just made a breakthrough, on which I will comment in a later entry.

Why colors cannot be seen

Before I can continue to my next research topic, I have to introduce yet another fascinating feature of the strong interactions, QCD. As you may remember, QCD had three charges: Red, green, and blue. There was also anti-matter with the anti-charges anti-red, anti-green, and anti-blue. To have total charge zero, one needed either a charge and an anti-charge, or one of each of the charges (or anti-charges). Total charge zero is then often also called white, just to keep with the analogy.

Now comes the fascinating fact: However hard we tried, and we did try very hard, we were never able to find something with either of the charges alone. Whenever we saw something with, say, red charge, we could be certain that enough other charges have been very close by to make the total charge within a very tiny part of space-time again zero. And tiny means here much less than the size of a proton! That is totally different from electromagnetism. There we had the electric charge and the anti-charge. We can separate such electric charges easily. Every time you move with plastic shoes over carpet and then touch something made from metal, you do so, albeit rather. unpleasantly. In fact, the screen where you read this blog entry is based on this: Without being able to separate the electric charges to very large distances (at least from the perspective of an electron), it would not work. Even the fact that you can see at all is based on this separation. In the nerves of your eyes and your brain, electric charges are separated and joined together when you see something

So why is QCD different? That is indeed a very, very good question. In fact, it is not even simple to find a mathematical way to state what is going on. The general phenomena: "We can not pull the charges apart" is commonly referred to as confinement. The charges are what is confined, and somehow the strong force confines it. That is already a bit strange. The force confines the things on which it itself acts. Not necessarily a simple thing to ponder. It seems to be somehow self-related in a bizarre way.

But it is really not that strange. Think of an atom. It is held together by the electromagnetic force between the electron(s) and the atomic nucleus. The total atom is electrically neutral. But because electromagnetism is not so strong, we can pull the components apart from each other, if we just invest enough force. The reason we can do this is that the force pulling electrons and the nucleus together becomes weaker the farther apart we move the electrons and the nucleus.

The strong force is now, precisely, stronger. In fact, the force between things with color charge is not diminishing with distance. It stays constant. Thus, we cannot really tear anything apart. As soon, as we stop forcing it, it gets back together immediately. So no way we get it apart. That seems to be odd. Indeed, when you look at the equations describing QCD, you will see no trace of this behavior. Only when you solve them, this becomes different. The solutions describing the actual dynamics of QCD show this. But solving them is very hard. Thus, back when QCD was developed, people could not solve them. Hence, this behavior in experiments seemed to appear out of the blue, and made it hard for many people to believe in QCD. And actually, even today we can only solve the equations of QCD approximately. But good enough that we can convince ourselves that this type of behavior is indeed an integral part of QCD. Confinement is there.

As so very often, I am right now dropping quite a number of subtleties. One of them is that I did not say anything about gluons. For them, very similar things apply as for quarks. The only thing is that they have different colors than the quarks, and you have to juggle around with now eight different ones rather than three. A bit more messy. But that is essentially all.

More severe is that white things actually can break apart. That may seem to look like a contradiction to what I said above. However, it is not. The subtlety with this is that they break into more white things, and not into their colorful constituents. For example, you can try to break a proton apart. A proton consists out of three quarks, one of each color. If you try to break it apart, at some point you will end up with a proton and a meson. A meson is something which consist of one quark with a color and an anti-quark with the corresponding anti-color. You may be irritated where I have the mass from. That is something different, and has nothing to do with confinement, and I will come back to this later. For now, just accept that this can happen. Anyway, you just do not get that proton apart, you just get more particles.

You see that this confinement has quite striking consequences. It is still something we have neither fully understood, nor do we have yet fully appreciated what it means. It is and remains something to understand for us. We have made great progress in this, but we are still lacking some basic notions of what is actually really going on. In fact, sometimes there are heated debates about what is actually a part of confinement, and what is something else, because we do not yet have a full grasp of what it means.

Irrespective of that, we have this phenomenon. We observed it experimentally. And we are able to get from the equations describing QCD its presence, even if we do not yet fully understand what it means and how it works. And it is this confinement what plays an important role in the next research topic of mine.

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.

What am I doing?

I have swamped you with a lot of methods, being the important tools I use for research. The natural question is: What do I do with them?

Well, this changes over time. This is quite normal. If you do research, you do not know what lies ahead. As you make progress, things you know change. This may be due to you own research, or by results from other people's research. Some topics may become less interesting as they appeared some time ago. Others become more interesting, and more relevant. This is a slow, but continuous evolution. It is the normal way things go: The most important characteristic of research is that you never know in advance where it will take you. Only when you get there, you will finally know.

Thus, what I will describe now and in the next few entries, is not set in stone. It is, what I am currently working on. In fact, when I started this blog, I was only investigating the standard model. And within it, I was mostly concentrating on the strong interactions, QCD. That was some time ago. By now, the topics I am looking at have became somewhat broader. I have also started to look at topics which are more mathematical, and also at some which are beyond the standard model. Anyway, I will not change the title of the blog: What today is beyond the standard model may be the standard model of tomorrow, you never know. I will also, from time to time, give you an update, when a new topic becomes interesting.

Let me now concentrate on the topics of my current research. I will list them in an order which is not indicating importance. It is rather my, somewhat subjective, view from what is well-founded to what is more speculative.

The first topic is the most mathematical one. As you remember, local symmetries are very important in the standard model. As any new standard model will contain the standard model as a special case, so will local symmetries continue to play an important role. These symmetries have been recognized to be important in particle physics already before the 1940s. At first sight, one may expect that everything is known about them, what there is to know. However, the combination of local symmetry and quantization is very complicated. It becomes especially problematic when you look at an interaction like QCD, which is very strong and complex. For many purposes, we have this problems under control. We can calculate what happens in QCD when we can use perturbation theory. We can also deal with the local symmetry when we calculate something like the proton mass. But in both cases, this more operational then a full solution. Beyond this, things become very complicated, as I will discuss in the next entry.

The second topic is QCD. QCD is a very rich topic, and just saying QCD is not very specific. Let me be more specific. One of nowadays great challenges in QCD is to understand what happens if we have many, many quarks and gluons. In this case, they may be a very dense, and possibly hot, soup. Describing this soup is dubbed 'determining the phase diagram of QCD'. This phase diagram is of great importance. Such a soup is expected to have existed in the early universe. It likely exist today within the core of neutron stars. Thus, in this topic elementary particle physics meets with astrophysics. I will discuss this in detail later.

The third topic is about the Higgs. Or rather about the interplay between the weak force and the Higgs. I am not really interested in what they actually precisely measure at the LHC. At least, not yet. Right now, I am trying to understand some rather basic questions about Higgs physics. These are closely related to the fundamental questions I am working on about local symmetries. However, just recently, this may actually lead to some predictions about what the may measure. You will see more about this in an upcoming entry.

The last topic is rather speculative. I try to understand what may be there beyond the standard model. Right now, we have few hints what may be the next big thing after the standard model. Thus, you are rather free in your speculation. Again, I am more interested in how the things, whatever they are, may function. Nonetheless, I have to be somewhat specific at least about the rough shape of the topic I am considering. Right now, I am most interested in what is called technicolor. I will explain this scenario (and other ideas for beyond-the-standard-model physics) in an upcoming entry. For now, it suffices to say that it has both local symmetry and strong interactions. This makes it rather messy, and therefore interesting. Let me give you the details later, in another entry.

As you see, I am interested in a number of rather different questions. But there is a common theme to all of them. What I want to understand is the mix of local symmetry and strong interactions. What does this mix imply for the physics? What is going on with these two concepts in the standard model, and beyond? How can we mathematically deal with these theories? Do we really understand what they mean? And what possibilities does this combination hold, once we get a full and firm grip on it? These are the questions I want to know the answers to. These are the things which may shake up what we know. Or, they may be just fixing some minor detail in the end. What of both will be the case, I cannot say yet. I will only know it, when I understood it. And to follow these questions is very interesting, very fascinating, and, yes, very exciting. I am very curious about where my research leads me. Let me take you along for the ride.