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.

Bosons

The first type of particles are bosons bosons. Those are these having integer spin. In the standard model, there is the Higgs particle, which has spin zero, and the photons, the W and Z bosons, and the gluons, which all have spin one.

Particles with spin zero are also called scalar particles. Since their spin is zero, the properties of such particles are the simplest when changing to a different coordinate system: They just look the same.

Particles with spin one are also called vector particles. Such vector particles are described like photons. The name vector stems from the fact that under a coordinate transformation the fields describing a vector particle changes in the same way as a line which connects the origin of a coordinate system and an event. The latter line is also called a vector, and hence the name for particles of spin one.

There is actually also a hypothetical particle with spin two, the graviton. Such a particle is also called a tensor particle. Tensors are generalizations of vectors when it comes to coordinate transformations, and fields of spin two particles transform in the same way as such tensors. In general, tensors are rectangular collections of numbers, where the columns transform like a vector under coordinate transformation.

Elementary particles with higher spin are not known. However, particles made up from elementary particles add their spin together (though not necessarily in the sense 1+1=2 - it can also be subtracted, 1-1=0, and everything in between), and can thus have higher spins.

Furthermore, to each such type of bosons, there exists a so-called pseudo bosons, i. e. a pseudo scalar, a pseudo vector (sometimes for historical reasons also called an axial vector), and a pseudo tensor. The difference between a boson and a pseudo boson is what happenes if you reflect the world in a mirror (a parity transformation). Ordinary bosons just become bosons once more. In contrast, the fields of pseudo bosons are multiplied by minus one.

Ok, after all this classification and name stuff, what is special about bosons? The most striking feature is that you can pile them upon each other. That is different from the small balls one often uses to imagine elementary particles: We can stack such balls next to each other, but never ever can two of these balls be at the same place. But bosons can. That is very hard to get in line with our ideas of how things work, and it shows just how quantum bosons are: they behave in a way which is just unexpected.

This is, of course, only true, if the bosons do not repel each other by some force. For example, if you would have two electrically same-name charged bosons, you would have a hard time to bring them together. But if they have oppositely named charges then they would just love to sit at exactly the same place.

In fact, if bosons do not repel each other because of a force acting between them, they have a tendency to lump together - two bosons rather prefer to be at the same place than being apart. This phenomenon is again a pure quantum effect: If you would have two balls, which are not talking to each other, they ignore each other very consequently. The reason for this different behavior is encoded in what physicists call statistics. In case of the boson this statistics is called Bose-Einstein statistics, in contrast to the classical statistics of the balls. Statistics describes how particles distribute themselves. Classical statistics is essentially randomly distributed, but bosons with Bose-Einstein statistics are not entirely randomly distributed but tend to get together.

This property also pertains to a different thing: The energies the particles have. While classical particles have just their energy, independent of every other particle, as long as they do not interact, bosons tend to have the same energy.

The extreme case of getting together is occurring when a sizable fraction of all available bosons are involved, and all of them have the lowest possible energy. That is what is called a Bose-Einstein condensate. This type of stuff is a state of matter similar to being liquid or being solid. But it only occurs under rather extreme conditions, in particular at very low temperatures. On Earth, there is no naturally occurring case of such a condensate. But it was possible to create such condensates in the laboratory using atoms.

In particle physics, such condensates play a central role. The Higgs effect was associated with a condensate of Higgs particles: It is just such a Bose-Einstein condensate. The same applies to the mass generation from the strong force, though in this case it is not the quarks that form a condensate. Since they are fermions, as will be discussed next, this is not directly possible. But states made up from two quarks (or a quark and an anti-quark) can condense. Since spin adds, such states have either spin zero or one, and thus behave like a boson, if one is not looking too closely. And these effective bosons are, loosely speaking, condensing to a Bose-Einstein condensate in this case.

These are only some examples, but such condensates play very often a role, from superconductors to the interiors of neutron stars. Thus bosons, with their strange properties, are very important to physics, and especially particle physics.

Spin

One of the most intriguing and most important properties of an elementary particle is its spin. At the same time, spin is one of the conceptually most problematic quantities, and has led to an enormous amount of misunderstandings.

The reason for this is that there is something in classical physics, which is very closely related to the concept of spin. But this relation is in spirit, rather than literally, and this has led to a lot of confusion. This analogue is angular momentum.

So, first, what is angular momentum? Angular momentum is connected with any kind of rotation of a particle around some center. Formally, it is a product involving the radius of the rotation and the speed along the path of the object. In classical physics, without friction, it is conserved, and it is what keeps the planets' orbits in their respective plane. It is likely also responsible for the fact that all the orbits are more or less in the same plane, or that the milky way has the over-all form of a discus (neglecting the spiral arms). In essence, it is just a reformulation of the ordinary speed, mixed with the mass of a particle. Essentially a kinematic quantity, despite its importance.

If an object just rotates, e.g. a ball, then each of the elements of the ball rotates. This can be described by giving the ball as such an angular momentum. Since the geometry of the ball is known and fixed, it is possible to defer from this total angular momentum the angular momentum of every piece of the ball.

In the world of particles, this angular momentum is reappearing whenever there is something having some kind of relative motion. E. g. in an atom, it is possible to assign the electrons an angular momentum, which is then often called orbital angular momentum (a somewhat complicated name). However, the electrons are not actually small spheres orbiting around the nucleus, bur rather smeared out over the whole of the atom. What this precisely means, I will discuss later. The important thing is that this smeared out something has a kind of orbital movement (the whole object 'rotates' in a certain sense), and can therefore be assigned such an orbital angular momentum.

It is a remarkable observation in quantum physics that angular momentum cannot take any value it likes. It is quantized. The reason for this quantization is the inherent relation between angular momentum and speed, and then speed and energy. Because energy is quantized this implies that angular momentum is quantized.

As orbital angular momentum depends on the momentum, and thus on the speed, its numerical value changes when we as the observer are changing our movement. This does not change the path of the rotating objects, just our perception of it, of course. Therefore, this change of values is closely tied to our change of our coordinate system, when we move.

Now enter spin: It was very early on recognized in quantum physics that elementary particles have a property which changes in the same way as the angular momentum of the ball when we change our coordinate system. This was an intrinsic property of the particles, unchangeable. However, the elementary particles are point-like, at least to the extent we can resolve them. Thus, they cannot rotate in any way, as they do not have any extension. In fact, if this would be an ordinary angular momentum, and the elementary particles would have a small extension, then within our experimental knowledge about the upper limit of this extension, their surface would need to rotate much faster than the speed of light.

Thus, this property got its own name: Spin. This is still inspired by the similarity to (orbital) angular momentum under a change of coordinate system, but by keeping strictly the difference in name, it can always be distinguished from it. However, from time to time its is useful to refer to them both together, and in this case they are called total angular momentum, which is in principle somewhat a misnomer.

Now spin is also quantized, and there exist both half-integer and integer values for it (when choosing appropriate units). This is different from ordinary angular momentum for two reasons. First, there is no simple explanation for the quantization like for angular momentum. There is indeed a complicated explanation, which shows that for the space-time structure which we have, these are the only two possibilities consistent with this type of change under a change of the coordinate system. Second, angular momentum, when measured in the same units, can have only integer values.

The latter is an intriguing difference. It has a very important consequence: Particles having integer spin behave very different from those having a half-integer spin. Therefore, these two types of particles received different names: The former are called bosons, and the latter are called fermions. This distinction is of fundamental importance to particle physics, and therefore the next two entries will discuss both types of particles in more detail. Also, none of these types behave in the same way as an ordinary small ball. But it turns out that if one takes the classical (long-distance) limit, both behave in the same way, and like small balls: Classically fermions and bosons can not be distinguished, their existence is a pure quantum effect, which is intricately linked to the structure of space and time. That is one of the reasons why some people believe that the quantum effect of spin and gravity may be related at a deeper level, but we are very far from understanding whether this suspicion is correct.

Chiral - or why left and right is not always just a mirror image of each other

One of the things we observe in everyday life is that things have a distinct left and right. The simplest case is just the hands of a human: Obviously, the left hand and the right hand are different from each other. That is a very general thing in nature that things can be 'like a left hand' or 'like a right hand'. Of course, they do not need to be so. A ball has obviously no distinct left or right. But things can have. This fact is known in science as chirality, originating from a Greek word for hand.

Left and right are actually not that different. If you take a mirror, and look at a left hand in the mirror, it looks light a right hand. Such a process, which turns something behaving like a left hand into something like a right hand, is called a parity transformation in particle physics.

So far, so good, and some fancy names. Why should this matter? Indeed, it does matter quite a bit. In biology, molecules can also be chiral. And then it turns out that a certain handedness is nutritious for us, while the opposite handedness is at best useless and at worst toxic. Our body has a preference for a certain hand, it is chiral. The fact that the left-handed version of the molecule and the right-handed version of the molecule have different consequences implies that looking through the mirror is not always just a mirror image, but can be something entirely different. Parity is not just a change of perspective: The mirror image in this case is broken, and therefore one tends to say that parity, the property that something becomes just the mirror image without further changes, is broken.

So, what has this to do with particle physics? Well, also some elementary particles have a handedness. This handedness is an intrinsic property of such particles, such as a color for a billiard ball. This is especially important for the quarks and leptons of the standard model. Of each of them two exists: A left-handed one and a right-handed one.

When it comes to the strong interactions or to electromagnetism, this actually does not matter. For these two forces, both types of particles look exactly the same, and thus neither of these forces can actually distinguish between between left and right. These forces are also said to be parity invariant.

This changes when it comes to the weak interactions. The weak interactions are very special, and they distinguish between both types of particles. In fact, they are very extreme in this respect: The only act on the left-handed particles, but completely ignore the right-handed particles. It is said that the weak force is parity violating, or simply it is said that the weak interaction is chiral.

The consequences of this is quite profound, though not obvious. Take for example an atom with a nucleus which is unstable, and decays by emitting so-called beta radiation, i.e. electrons. If you suspend such an atom in a magnetic field, it turns out that the electrons emitted move in a preferential directions. This occurs, because the weak interactions are chiral. If they would not be, this would not happen. Nonetheless, this example shows that it requires something of sophistication to observe this.

Still, this chirality in the standard model is quite important. From a mathematical point of view, it is very restricting for the structure of the standard model. It has also quite important implications for each and every of our attempts to extend the standard model. Furthermore, in actual calculations it is quite a nuisance.

However, after all, we do not know why the weak interaction, but not the other two, are chiral. It is something we observe, and it is one of the bigger mysteries in particle physics. Therefore, looking for modifications of chiral properties is also a big chance to find something new. Since we have either perfect parity or not at all in the standard model, anything else would be new. Also, because we are so completely baffled by it, we think that whatever kind of observation is unexpected in context with a parity violation will very quickly leads us to a glimpse of whatever there is beyond the standard model.

Always the opposite: Anti-matter

The last time, I made a brief remark about anti-particles. It is about time to illustrate this rather obscure notion.

What is meant, when we talk about anti-particles? Well, just from the experimental point of view, it is found that for every particle there exists another particle, which has (within experimental certainty) exactly the same mass. It has also the same properties when it comes to the way it spins, the so-called spin. This spin is also something I will explain sometimes else, what this mysterious property is.

However, considering everything else, it is exactly the opposite: If the particle has a negative electric charge, the anti-particle has a positive electric charge. If the particle has color red, the anti-particle has an opposite charge, which is called for the lack of a better name anti-red. And so on. The only exception to this rule are those particles which have, except for mass and spin, no other properties. An example is the photon, which is completely uncharged. In this case, the particle is its own anti-particle.

Now, these are rather surprising objects, but we have very good experimental proof that they exist. In fact, we know anti-matter so well that some experiments, like the old LEP at CERN, use matter and anti-matter routinely as a starting point: At LEP electrons and their anti-particles, the positrons, have been collided.

Matter and anti-matter show a very spectacular effect: Because one plus minus one is zero, it is possible for matter and anti-matter to annihilate each other when they are colliding into something else. For example photons. Or other particles. That happens very easily. Hence you may ask, why we do not annihilate whenever we touch something. The answer is surprisingly simple: Because everything around us is made from matter. If we want to use anti-matter or study it, we have to create it artificially. That is not simple, and we can only create very tiny amounts efficiently. Large amounts become rapidly very expensive, mostly because it is not simple to keep it away from matter, so that is not annihilating with it.

That seems a simple enough answer, but the real question baffling physicist is: Why is this so? If they are so equal, why do we not have the same amount of both (and thus vanish in a big photon cloud)? That is another of the questions we do not yet have a real answer to. Irritatingly, the problem is actually not that we do not know how this can be realized. In fact, in the standard model of particle physics, there is a very, very slight preference for matter over antimatter when it comes to the weak force. This implies that though matter and anti-matter are essentially equal, the forces make a difference between them. However, this effect is by far too small to explain why there is so a fantastically little amount of antimatter around us.

Ok, so you might say: Let us forget for the moment about the experimental evidence, and ask, do we really need anti-mater. Could this simple explanation just be a misinterpretation of the experiments, and what we think is anti-matter is really something else? Well, if this should be the case, we would have to rethink our complete view of how the standard model is described theoretically as well. Indeed, the mathematical structure of the standard model requires the existence of an anti-particle for each particle to work properly. If we would remove the anti-particles from the theory, the consequence would be dramatic. It would even be possible to obtain effects without cause or causes without having effects. This is not what we observe, but what we observe is described by the standard model with particles and anti-particles. Thus we take the experimental results as evidence for anti-particles, and everything fits together when we calculate something.

Of course, this means that we essentially double the number of particles. Up to exceptions like the photon, all particles are now accompanied by their anti-particles. And to every charge comes an anti-charge. However, this also provides new options for new phenomena. The last time last time, this gave us the option of a condensate of quarks and anti-quarks. Also, their are bound states of quarks and anti-quarks, the so-called mesons. The most famous and lightest of them are the so-called pions, of which there are three: One is uncharged, and there is one positively charged and one negatively charged. Th neutral one is again its own anti-particle. The reason is that it is made up out of a quark and the corresponding anti-quark. Thus replacing constituent particle by constituent anti-particle gives again the same bound state. The charged ones are each others anti-particle, because they contain an up and an anti-down quark and an anti-up and a down quark, respectively. Exchange particles by anti-particles yields an exchange of both bound states. So, one can have a lot of fun with building things out of particles and anti-particles.

You can also take a hydrogen atom, and exchanges its nucleus, a proton, by the anti-particle of the electron, a positron. Because the positron has the same electric charge as the proton, you get even something looking very much like an atom. This is called positronium, known for a very long time. Recently, it has also been possible to create true anti-atoms, made from an anti-nucleus and positrons. These are very important to test, whether we really have understood everything about anti-mater. If we have, they should behave in the same way as ordinary atoms. And whether this is the case the experimentalists right now try to find out.