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.

Methods. United, they are strong.

In the last few postings, I have collected a number of methods: perturbation theory, simulations, and the abstract equations of motion. I have furthermore gave you a bit of a taste of one of our most important strategies: divide and conquer. Or, more bluntly, if the original problem is too complicated, first try a simpler one, which resembles it. This lead us to a stack of models, which bit by bit always included more details of the world.

This list is by no means complete. Over the years, decades, and centuries, physicists have developed many methods. I could probably fill a blog all by its own just by giving a brief introduction to each of them. I will not do this here. Since the man purpose of this blog is to write about my own research, I will just contend myself with this list of methods. These are, right now, those which I use myself.

You may now ask, why do I use more than one method? What is the advantage in this? To answer this, lets have a look at my work-flow. Well, actually this is similar to what many people in theoretical particle physics do, but with some variations on the choice of methods and topics.

The ultimate goal of my work is to understand the physics encoded in the standard model of particle physics, and to get a glimpse of what else may be out there. Not an easy task at all. One, which many people work on, many hundreds, probably even thousands nowadays. And not something to be done in an afternoon, not at all. We know the standard model, more or less, since about forty years at the time of this writing. We think essentially as long as it exists about what else there might be in particle physics.

Thus, the first thing I do is to make the things more manageable. I do this, by making a simpler model of particles. I will give some examples of these simpler models in the next few entries. For now, lets say, I just keep a few of the particles, and one or two of their interactions, not more. This looks much more like something I can deal with. Ok, so now I have to treat this chunk of particles happily playing around with each other.

To get a first idea of what I am facing, I usually start off with perturbation theory, if no one else did this before me. This gives me an idea of what is going on, when the interactions are weak. This hides much of the interesting stuff, but it gives me a starting point. Also, very many insights of perturbation theory can be gained with a sheet of paper an a pencil (and many erasers), and probably a good table of mathematical formulas. Thus, I can be reasonably sure that what I do is right. Thus, whatever I will do next, it has to reduce to what I just did now when the interactions become weak.

Now I turn to the things, which really interest me. What happens, when the interactions are not weak? When they are strong? To get an idea of this, the next step is to perform some simulations of the theory. This will give me a rough idea, of what is going on. How the theory behaves. What kind of interesting phenomena will occur. Armed with this knowledge, I have already gained quite a lot of understanding of the model. I usually know then what are the typical way the particles arrange themselves. How their interaction changes, when looking at it from different directions. What the fate of the symmetries is. And lot more of details.

With this, I stand at a crossroad. I can either go on, and deepen my understanding by improving my simulations. Or, I can make use of the equations of motion to understand the internal workings a bit better. What usually decides for the latter is then that many questions about how a theory works can be best answered when going to extremes. Going to very long or very short distances when poking the particles. Looking a very light or very heavy particles. Simulations cannot do this with an affordable amount of computing time. So I formulate my equations. Then I have to make approximations, as they are usually too complicated. For this, I use the knowledge gained from the simulations. And then I solve the equations, thereby learning more about how the model works.

When I am done to my satisfaction, then I can either enlarge the model somewhat, by adding some more particles or interactions, or go a different model. Hopefully, at the end I arrive at the standard model.

What sounds so very nice and straightforward up to here is not. The process I describe is an ideal. Even if it should work out like this, I am talking about the several years of work. But usually it does not. I run across all kind of difficulties. It could turn out that my approximations for the equations of motion have been too bold, and I can get no sensible solution. Then I have to do more simulations, to improve the approximations. Or the calculations with the equations of motion tell me that I was looking at the wrong thing in my simulations. That the thing I was looking at was deceiving me, and gave me a wrong idea about what is going on. Or it can turn out that the model cannot be simulated efficiently enough, and I would have to wait a couple of decades to get a result. Then, I have to learn more about my model. Possibly, I even have to change it, and start from a different model. This often requires quite a detour to get back to the original model. This may even take many years of work. And then, it may happen that the different method give different results, and I have to figure out, what is going on, and what to improve.

You see, working on a problem means for me to go over the problem many times, comparing the different results. Eventually, it is the fact that the different methods have to agree in the end what guides my progress. Thus, a combination of different methods, each with their specific strengths and weaknesses, is what permits me to make progress. In the end, reliability is what counts. And with this nothing cuts it like a set of methods all pointing to the same answer.

Modelling reality

Ever wondered why it is called the standard model of particle physics? And what a physicist has in mind, when she talks about models?

Models are the basic ingredient of what a theoretical physicist is doing. The problem is that we do not know the answer, we do not know the fundamental theory of everything. Thus, the best we can do is take what we know, and make a guess. The result of such a guess is a model. Such a model should describe what we see. Thus, the standard model of particle physics is the one model what we know about particle physics right now, as incomplete as it may be. It is called the standard one, because it is our best effort to describe nature so far, to model nature in terms of mathematics. There are also other standard models. We have one for how a sun functions, the standard model of the sun, or how the universe evolved, the standard model of cosmology.

Now, when I say, it is our best guess this implies that it is not necessarily right. Well, actually it is, in a sense. It was made the standard model, because it describes (or, if you read this in a couple of years, perhaps has described) our experiments as good as we can wish for. That means, we have found no substantial evidence against this model within the domain accessible in the experiment. This sentence has two important warning signs attached.

The one is about the domain. We do not know what is the final theory. But what we do know is the models. And any decent model will tell us, what it can describe, and what not. This also applies to the standard model. It tells us: 'Sorry guys, I cannot tell what is happening at very large energies, and on the matter of gravitation, well I stay away from this entirely.' This means that this standard model will only remain the standard model until we have figured out what is going on elsewhere. At higher energies, or what is up with gravitation. However, this does not mean that the standard model will be completely useless once we managed that. As with many standard models in the past, it likely will just become part of the large picture, and remain a well-trusted companion, at least in some area of physics. Happened to Newton's law, which was superseded by special relativity, and later by general relativity. Happened to Maxwell's theory of electromagnetism, which was superseded by Quantumelectrodynamics, and later by the standard model. Of course, there is once more no guarantee, and it may happen that we have to replace the standard model entirely, once we see the bigger picture. But this seems right now unlikely.

The other thing was about the experiment. Models are created to describe experiments (or observations, when we think about the universe). Their justification rests on describing experiments. We can have some experimental result, and cook up a model to explain it. Then we do a prediction, and make an experiment to test it. Either it works, and we go on. Or it does not, and then we discard the model. While people developed the standard model, this was a long, painful process during which many models have been developed, proposed, checked, and finally discarded. Only one winner remained, the model which we now call the standard model.

Ok, nice and cozy, and that how science works. But I was talking about methods the last couple of times, so what has this to do with it? Well, this should just prepare you for an entirely different type of models, to avoid confusion. Hopefully. Now the standard model is the model of particle physics. But, honestly, it is a monster. Just writing it down during a lecture requires something like fifteen minutes, two blackboards, and two months of preparation to explain all the symbols, abbreviations and notions involved to write it in such a brief version. I know, I have done it. If you want to solve it, things go often from bad to worse. That is where models come in once more.

Think of the following: You want to describe how electric current flows inside a block of, say, Aluminum. In principle, this is explained by the standard model. The nuclei of Aluminum come from the strong force, and the electrons from the electromagnetic one, and both are decorated with some weak interaction effects. If you really wanted to try describing this phenomena using the standard model, you would be very brave indeed. No physicist has yet tried to undertake such an endeavor. The reason is that the description using the standard model is very, very complicated, and actually most of it turns out to be completely irrelevant for the electric current in Aluminum. To manage complexity, therefore, physicists investigating aluminum do not use the standard model of particle physics in its full glory, but reduce it very, very much, and end up with a much simpler theory. This models Aluminum, but has forgotten essentially everything about particle physics. This is then a model of Aluminum. And it works nice and well for Aluminum. Applying it to, say, copper, will not work, as Aluminum nuclei have been put into it as elementary entities, to avoid the strong interactions. You would need a different model for copper then, or at least different parameters.

So, we threw away almost all of the power of the standard model. For what? Actually, for a price worth the loss: The final model of Aluminum is sufficiently simple to solve it. Most of our understanding of materials, technology, chemistry, biology (all described by the standard model of particle physics, in principle) rests on such simplified models. With only the standard model, we would not be able to accomplish anything useful for these topics, even knowing so much about particles. In fact, historically, the development was even the other way around. We started with simple models, describing few things, and generalized bit by bit.

Ok, you may say. You see the worth of simplified models for practical applications. But, you may ask, you surely do not simplify in particle physics? Well, unfortunately, we have to, yes. Even when only describing particles, the standard model is so complicated that we are not really able to solve it. So we very often make models only describing part of it. Most what we know about the strong interactions has been learned by throwing away most of the weak interactions, to have a simpler model. When talking about nuclear physics, we even reduce further. Also, when we talk about physics beyond the standard model, we often first create very simple-minded models, and in fact neglect the standard model part. Only, if we start to do experiments, we start to incorporate some parts of the standard model.

Again, we do this for the sake of manageability. Only by first solving simpler models, we understand how to deal with the big picture. In particle physics the careful selection of simplified models was what drove our insight since decades. And it will continue to do so. This strategy is called divide and conquer. It is a central concept in physics, but also in many other areas where you have to solve complicated problems.

Of course, there is always a risk. The risk is that we simplify the model too much. That we loose something important on the way. We try to avoid that, but it has happened, and will happen again. Therefore, one has to be careful with such simplifications, and double-check. Often, it turns out that a model makes very reliable predictions for some quantities, but fails utterly for others. Often, our intuition and experience tells us ahead what is a sensible question for a given model. But sometimes, we are wrong. Then experiment is one of the things which puts us back on track. Or that we are actually able to calculate something in the full standard model, and find a discrepancy compared to the simple model.

In the past, such simplified models were created by very general intuition, and including some of the symmetries of the original theory. Over time, we have also learned how to construct, more or less systematically, models. This systematic approach is referred to as effective field theory. This name comes about as it creates a (field) theory which is an effective (thus manageable) version of a more complicated field theory in a certain special case, e.g. low energies.

Thus, you see that models are in fact a versatile part of our tool kit. But they are only to some extent a method - we have still to specify how we perform calculations in them. And that will lead us then to the important concept of combining methods next time.

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.

Simulating a universe

One of the things which has changed science drastically in the last few decades has been the computer. With nowadays computational resources it is feasible to simulate even complex physical systems inside a computer with an accuracy which is useful. This is done in all areas of physics, and of course also in particle physics. In the latter context, the numerical simulations of particles is known mostly known as lattice gauge theory, due to its predominant version.

So how does such a simulation proceed? And what can we actually learn from it? Well, the rules of quantum physics forbid us to just use a number of marbles and simulate their behavior in the computer. With such an approach, we would be able to do classical physics (as is even done with the marbles representing entire galaxies), but not quantum effects. Quantum effects require us to simulate the whole universe, such that all the quanta can interact with each other. Thus, what we do in quantum physics, is indeed to simulate many possible histories of a universe, and from that infer the quantum behavior of a system of elementary particles.

Stop! You may say. The universe is really, really big. Even with the most powerful computer today, how can we ever dream of simulating it completely? And you are right to say so, it is not possible, not with all computers on the world. But it is also not necessary. For most questions we are interested in, most of the universe only contributes negligible, and what is really interesting happens only inside a small part of it. What we want to know in particle physics is how the particles interact and how they form bound states, but not much larger than, say, a nucleus. But in an atom the electrons are so far away from the nucleus with so much space in between that the electron plays little role for how the nucleus is build from the quarks. We can thus just look at a universe which is just a little, maybe a factor ten or so, larger than the nucleus we are interested in, and we should capture already almost everything. At least as much as we can expect to experimentally verify within the foreseeable future. Thus, we are permitted to just simulate a very small universe to answer all the questions of relevance in particle physics.

But this is not yet sufficient to make a meaningful simulation. As you may remember, the standard model is just a low-energy approximation of whatever the theory is at some higher energy scale. If I just try to simulate now this universe, the simulation will not work, because we are lacking the knowledge of what happens at very short distances. As a consequence, our simulation would just produce either zeros or infinities, which is not too helpful. To deal with that we have to include the limits of our knowledge. Since what we do not know is the physics at very short distances, this is most easily done by not simulating at very short distances. The simplest solution to do this is not to take every space-time point of the universe you want to simulate, but only a finite number of them, which have finite distances in between. Technically, this is usually realized by arranging this finite number of points on a lattice, and thus the name lattice theory. The still missing word gauge just stems from the fact that the standard model is a gauge theory. Thus lattice gauge theory.

Ok, now we are ready to go. We just take a small box with a finite number of points in it, arranged on a (usually square) lattice, and let the computer run. And this works rather well. With this we can calculate the mass of the proton or how strongly the Higgs and the W boson interact, and so on.

Great, you say, so that is it. We just do simulations, and that will be all we will ever need. Unfortunately, it is again not that simple. There are two drawbacks to this approach.

One is that we still have limited resources at our disposal. The result is that the volume can be still rather small, so that we cannot cover all processes we like. Also, If we increase the volume and do not increase the number of space-time points, their spacing will increase. At some point, this spacing may become so large that we can simply not resolve some part of physics anymore. It falls through the cracks (or the lattice), figuratively speaking. And adding more points increases the run time. Thus, we are not yet able to simulate at the same time, say, a proton and a Higgs. It will take much, much more computing time to do so. We do not expect that so much computing power will be available in any reachable future. So there are things we cannot answer with simulations. There are also technical reasons. We have not yet been able to develop algorithms with which we could simulate very light fermions or parity violations. We are currently also stuck when it comes to having many fermions, like it is the case in the interior of say, a large nucleus or a neutron star. Thus, we are yet just limited in simulating by the power of our computers and algorithms.

The second thing is that a simulation will always create a number. Good, you may say, an experiment does so as well. So you can compare with experiment, and find out whether your theory is correct. True. But this is not yet satisfactory. Just because you can compare the speed a car drives and you can simulate the car and see that the simulation produces the same speed is not yet telling you how the car does it. You know that your description of the car is right, but you do not know how the different parts are really interacting with themselves, or what they mean. For a car you can then just go on and look at the details. But in quantum physics that is not possible, because the laws of quantum physics dictate that everything interacts with everything to a certain degree. Thus, disassembling is not entirely simple, or even possible. An alternative is to have expressions in which you could see for each knob what it turns. That cannot be delivered by any numerical calculations.

Make no mistake here. This is not making numerical simulations useless. In fact, some of the greatest discoveries in the past decades were only possible using computers. But it is not all, and therefore we need more. If the theory is only weakly interacting, perturbation theory is doing the job nicely. But what if it is not? Well, then we have further possibilities which I will discuss next.

As a final remark, I should add that I use heavily computer simulations in my work. Right while I am typing this entry many, many CPUs do work for me. But as you see there are limitations. This is why I use also some of the further methods I will discuss next.

Perturbation theory

Now, let us start with a look at the different methods in more detail. The first is the mainstay of theoretical physics, and the first thing everyone tries when encountering or designing a new theory: Perturbation theory, often abbreviated simply by PT.

The basic idea of perturbation theory is rather direct. When we have a theory, it is very often the case that we can solve a simpler version of it exactly. For example, if we have QED then we can solve exactly the case where the electromagnetic charge would be zero, because the particles then do not interact with each other. And free, non-interacting particles is something we can do very well. Of course, this is not what QED is really like. Otherwise, we could not see, as the electrons in our eyes would not react to the light made out of photons. To capture this, perturbation theory assumes that the interactions between electrons and photons is only a small alteration to the picture of free particle: A perturbation, and hence the name. Of course, finding a useful split depends on the theory in question, and many different types are actually in use.

Once such a setup is available, we have created powerful mathematical tools how to calculate then anything we want under this assumption. The most important principle is that we can reformulate what we mean by perturbation mathematically by stating that some quantity is small What this precisely means depends on the theory in question. In the above example of QED, it would be the electric charge.

We can then organize perturbation theory systematically by counting how often the small quantity appears in an expression. We then speak of the order of perturbation theory. If it appears the lowest possible number of times, which may be zero, we call this tree level. The reason for this name is that the mathematical expressions can be generated in the way a tree grows, i.e., in the form of starting somewhere and then moving on. In fact, in general this is often equivalent to a classical theory. This means that we treat the particles as quantum particles, but the interaction between them like a classical interaction, without additional quantum effects.

We can now increase the number of times the quantity appears. We also say that we calculate higher orders in the quantity, where order counts the number of times the quantity appears. If we calculate the contribution with the second-least number of times the quantity appears, we call this leading order correction. Since such a correction only appears in the quantum theory, we also call it a quantum correction. The contribution with third-least appearance is called next-to-leading order. If we further increase the order, we just add the corresponding number of times next-to in front, e.g. next-to-next-to-next-to-leading order. This seems to become quickly awkward, but no fear, no too high orders are often calculated.

The reason is that perturbation theory at higher order becomes rather complicated just from an organizational point of view. Quickly, perturbative expressions fill hundreds and thousands of pages with expressions, which have to be evaluated. The final end result will only fill a very few pages, if more than one at all. Over the time, we have developed very powerful methods to deal with this complexity. If you ever heard of Feynman diagrams, given that these have made their way even into some obscure corners of pop culture, then this is one of these tools. Its a very powerful graphical technique to organize perturbation theory in a very efficient way. And this is quite important. There are furthermore other ingenious methods to reduce the amount of calculations necessary. Nonetheless, in the end the expressions remain rather long, and it requires computers to evaluate them. This is in general straightforward to tell the computer what to do. But it is very challenging to do it in a way that the computer is not occupied for the next couple of years, but only a few days or less.

With these methods, we have went to order ten in QED, and for some quantities to order four in the standard model. This seems little, but because of the complexity required many people and decades of time. Still, some times experiments are so precise that the accuracy achieved by these calculations is not sufficient. But of course, there are also many cases where it is the other way around. In a way, it is a kind of arms race between theoreticians and experimentalists.

In the end, however, perturbation theory will not give you the full answer. You can mathematically prove that certain phenomena cannot be calculated using perturbation theory. You may be lucky, and using a different starting point, this can be circumvented for a certain quantity, but then other quantities will not be possible to access. Furthermore, we know that perturbation theory cannot be pursued to arbitrary order, but will collapse at a certain point, for mathematical reasons. Though also here progress has been made, we know that perturbation theory cannot provide the full answer to any question. Already as simple a quantity as the mass of the proton cannot be calculated in perturbation theory. Nonetheless, much of what we measure in experiments, say at LHC, can be very well and very accurately calculated with perturbation theory. Thus, perturbation theory remains to be one of the main tools in particle physics, and for very good reasons so.

The tools of the trade

By now, I have collected and presented you quite a number of the basic ingredients of the standard model (and beyond). You should be now well equipped to get a good understanding of what I am doing. Therefore, I can come back to the original idea of this blog, and can discuss some aspects of my own research. At times, and when need be, I will add further more general entries.

Before I can enter the subjects of my research, I have to present another important part of the work of a theoretical physicist: The methods she or he is using. Each methods has its distinct advantages and drawbacks. As a result, a given problem can often be addressed by multiple methods. If this is the case, it is also possible to combine the different methods.

The latter is of particular importance because of an insight of singular importance in physics: Any problem of fundamental interest in particle physics so far is so complicated that we were not (yet) able to find an exact solution. At first, this appears like a very depressing insight. It is usually a cultural shock for students when they enter research, as up to then one is usually only exposed to simple problems which an be solved exactly, for reasons of a pedagogical and manageable presentation. At times, one acquires the insight that this horrible complexity of real problems is just a natural consequence of the richness of physics, even of the very elementary particles which lie at the heart of our current understanding of the universe. Nonetheless, physicists strive for getting better and better and ultimately exact solutions, and perhaps this holy grail of a theoretician can be reached someday. For now, however, this is not the case, and we have to live with the fact that despite our methods working often exceptionally well, they can never give you the full answer. But for some questions they can provide answers, which are ten or more digits precise. And this is quite encouraging.

For the topics I am interested in such enormously good results have not been achieved. The reason for this is that problems become simpler the weaker the interactions are. The method perfectly suited for this is perturbation theory, the first method I will be introducing shortly.

However, if the interaction is weak not so much interesting is happening. Particles ignore each other most of the time, and if they meet, they, well interact weakly, and just scatter a bit off each other. If the interactions become stronger, interesting things start to happen. Bound states form, particles condense, and much more. That is where my interest lies.

The downside of this is that if the interactions between particles become strong, it becomes very hard to find a mathematical handle to treat them. That is the challenge, and the reason why rather few exact results are available. One solution is then to use brute force and just simulate the physics using a sufficiently large computer. That has provided us with very deep insights, and has become an invaluable tool in modern theoretical physics. For the type of problems I am most interested in such simulation methods are called lattice gauge theory, for reason I will explain later.

There are two major alternatives to such brute force simulations. One is the use of models and the other are so-called functional methods. In both cases the idea is to simplify the problem while capturing everything of interest.

Models, a term which I use here in a very broad sense, underlies the idea to find a simplified version of the theory at hand, sufficiently simplified to be easier to handle. Such theories than have often a very narrow range of applicability (for very similar reasons as the standard model itself ). However, if they are constructed very carefully such models very often help to understand not only broad features but often even quantitatively what is going on.

Functional methods are a different approach. The basic feature of theses methods are a set of equations which are in principle exact. Unfortunately, this set is often infinite, and in general approximations are needed to find solutions to them. If the approximations are good, it is possible to describe very much successfully with these equations and at the same time get deeper insight. Also, the approximations can be improved step-by-step, and thus permit eventually a full solution to the theory. I.e., at least in principle.

There are, of course, many other methods available, but these are the most important ones for my own research, and, except for models, I use them essentially on a day-by-day basis. The important methodological aspect in this is the combination of all the methods, and this results in something which is much more than just the sum of its parts.

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.

Fermions

You thought bosons were strange? Well, wait, now comes the really strange quantum stuff - fermions.

At first sight, fermions are innocently looking and differing from bosons by the fact that they have half-integer spin. In the standard model, all quarks and leptons are fermions, and have spin one half. No elementary particle is known (though some hypothesized) which are fermions and have a larger spin than one half. But again, some particles made up from several elementary particles may look from afar like having a larger half-integer spin. E.g. the Delta, a heavier cousin of the proton and made up also from three quarks, has spin three halves.

In contrast to bosons, fermions dislike being at the same place. In fact, they can never take the same position, much like the classical balls. But there is a difference to the classical balls. For fermions, this not only applies to position, but also to all quantum numbers and energies. As a consequence, there can never be two fermions being having the same energy. This is the famous Pauli exclusion principle.

This principle has very fundamental consequences: It is responsible for the stability of all matter. If your desk would be made out of bosons, only electromagnetic repulsion would prevent it from collapsing to a pile of bosons. But because it is made out of fermions - all the quarks and electrons - it could never collapse to a single point. Because the fermions can just not get so near to each other. That is the fundamental reason which prevents a white dwarf or a neutron star from collapsing.

Very similar, it also prevents the electrons in an atom, which are attracted by the nucleus by electric forces, from collapsing into the lowest energy level, or into the nucleus outright. All of chemistry works the way it works because the electrons, since they are fermions, cannot all go into the lowest energy level. Otherwise, our chemistry, and thus our biology, would be very different, indeed.

But this is not the only strange thing about fermions. Fermions are also very strange in many other respects. As a consequence of the Pauli principle they obey again a different statistics, the so-called Fermi-Dirac statistics. The consequence of this are at the heart of why there are electric insulators.

But fermions are also strange in the sense that when you turn your coordinate system by 360 degrees, i.e. once fully around, everything is unchanged. Only the fermions do not play along: You have to turn your coordinate system twice around so that they look again the same (or, more precisely, their wave-function explained next time, looks the same). That is so mind-boggling that it is hard to believe it is true, and one cannot really intuitively understand this. It is a very deep combination of our space-time structure and quantum physics. There is no classical objects which behaves like this.

The mathematical consequences of these properties are little less strange. Fermions are the only objects which we cannot describe by ordinary numbers. Theoretical physicists had to invent a whole new type of numbers (well, actually borrow them from your friendly mathematician next door) to describe fermions - so-called Grassmann numbers. These are really strange. If you multiply an ordinary number with itself, you get a new number. If you multiply a Grassmann number with itself, you always get zero. That is the mathematical realization of the Pauli principle. This feature makes fermions very hard to handle in actual calculations, and they have been a bane especially to numerical simulations.

Nonetheless, they are there, and we are bound to live with them, as we are bound to live with bosons. Though - you can always combine two fermions to make something which looks from afar like a boson. But you can never combine two bosons such that they look from afar like a fermion. This fact has been found to be exploited very often by nature, as already described last time. And it lies at the heart of some ideas, so-called technicolor scenarios, to get rid of the Higgs with all its annoying properties: In such proposed extensions of the standard model, the Higgs is just a combinations of two new particles, so-called techniquarks.