Looking inside the standard model

The tools of the trade

2012-01-24T01:15:00.000-08:00

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

2012-01-19T01:49:00.000-08:00

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

2012-01-12T07:58:00.000-08:00

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

2012-01-11T03:28:00.000-08:00

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

2012-01-09T06:32:00.001-08:00

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

2011-11-29T07:45:00.001-08:00

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

2011-10-26T09:15:00.000-07:00

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.

Mass from the strong force

2011-10-10T08:55:00.000-07:00

Quite some time ago, I have discussed the Higgs effect, and how it gives the matter particles in the standard model their mass. However, if we look around us, it turns out that most mass we see is actually not due to the Higgs effect.

If we knock on a table, or look at us, then most of this is made up out of atoms. As you might remember, atoms are made up out of nuclei and electrons. The electrons actually get their mass from the Higgs, so that is alright. But they make up less than 0.05% of the mass of the atoms. Thus, one can forget about them for this purpose. Then there are the nuclei. They are made up out of protons and neutrons. These in turn consist out of quarks. But the quarks are rather light, and make up not more than one percent of the mass of the protons and neutrons, and thus of the nuclei. So where does all the remaining mass comes from?

Well, this comes this time from the strong nuclear force, QCD, which has been presented here and here. I have already indicated there that it is a pretty strong force. It is this strength which, indirectly, creates all the mass we are yet missing.

How it works is actually quite complicated in detail, but when being a bit fuzzy about the details, it can be illustrated quite nicely. Looking through such fuzzy glasses, it actually looks like a repetition of the Higgs effect. Remember, the Higgs effect worked by letting the Higgs particles condense. The interaction with this condensate slows particle down, and therefore they behave as having a mass.

Now, how does this proceed in the case of the strong force? The first observation is that the strong force is attractive between quarks, i.e. quarks are attracted to each other. As a consequence, the quarks can form all these nice things like protons. However the force is also attractive between the quarks and their so-called anti-particles. What an anti-particle is I will discuss next time. This time, it is just sufficient to say that it behaves like a quark, but has opposite charges and the same mass.

The strong force now pulls also the quarks and the antiquarks together. The combination of two such particles is then neutral, as all the charges are opposite. It behaves therefore very similar to a Higgs particle. In very much the same way, but this time because of gluons rather than due to the Higgs interacting with itself, this creates also a condensate. That is just like for the Higgs particles. Thus, the universe is filled with quarks and anti-quarks, bound together, and condensed by the strong force. The strong interaction of quarks with this condensate, and the corresponding slowing down, is what provides the remainder mass for the protons and neutrons, and thus for the nuclei. Again, because all the charges cancel, we do not see this condensate, as photons due not directly interact with it.

Putting in numbers, the contribution of the strong force to the mass of the nuclei is much larger than the one due to the Higgs effect. However, the calculation of this was rather challenging. Hence, most of the mass stored in the atoms in the universe is due to the strong force.

It is also said that the strong force provides all the luminous mass in the universe. Here, luminous means actually not only all stars which emit light by themselves, but also everything which reflects light, like planets, interstellar gas clouds, and asteroids. Actually, the latter also emit a kind of light by themselves, but our eyes are not sensitive for the wave-lengths they emit, and therefore we do not see it. The distinction is necessary, because we have indirect evidence that there is also more than just this type of mass in the universe. In fact, we expect that there is about five times more non-luminous, or dark, matter in the universe, than luminous matter. What the origin of mass is in this case is not known.

There is another thing you may wonder about. The quarks have all very different masses due to the Higgs effect. Is the contribution due to the strong interaction also very different for the different quarks? The answer to this is actually no, the contribution from the strong force is about the same for all quarks. Thus, it makes up about 99% of the mass of the light quarks, but less than half a percent for the heaviest one. Thus, while the Higgs makes a difference between the different quark (and lepton) species, the strong force does not. Why this is the case is also yet unknown, and one of the bigger mysteries. Since the different quarks and leptons are also called different flavors of quarks and leptons, it is said that the strong force is flavor-blind, it makes no difference between different flavors. On the other hand, the Higgs makes a different between different flavors.

Finally, it should be noted that the generation of mass can be traced back to a symmetry, though I will not detail this now. This is the so-called chiral symmetry. A thing is called chiral, if it makes a difference between left and right. The associated symmetry in the standard model is a local symmetry. The Higgs effect breaks, loosely spoken, this symmetry to a global one. The strong force then breaks this global symmetry. It is possible, but mathematically involved, to show that these breakings correspond to the existence of mass. Hence, both the Higgs effect and the strong force produce mass. But the above explanations are somewhat more illuminating, I think, though mathematically both views are essentially equivalent. Thus, it is said that mass is created by chiral symmetry breaking, a notion I will right now not dwell on anymore.

What is strong and what is weak?

2011-09-16T02:29:00.000-07:00

Given the previous discussion that things we measure depend on the energy at which we measure them, one could reasonably ask whether names like a strong force or a weak force are justified, or whether the strength of a force may not also depend on the energy.

Actually, it is precisely like this. An this is something which is at the same time a blessing and a bane to theoretical calculations.

When we measure the strength of an interaction, what we do is actually rather indirect. We start out with two particles. These we accelerate by some means, e.g. a particle accelerator like the LHC at CERN. Then we aim them at each other and let them collide. Afterwards, we measure what is left of them, and whether something new has been created in the process.

A practical complication is actually quantum physics. The latter tells us that when we perform an experiment more than once, we cannot perform it such that it will produce precisely the same result. However, this does not mean that physics is arbitrary. Lets make the experiment many times. Then we get an average result. Furthermore, if we build a second experiment to make the same collision, on the average it will yield the same result. In this sense it is reproducible. However, for many things we have to make very many collisions such that we get a precise enough average. That is one of the reasons why we not can just make one experiment at LHC and have immediately the answer whether the Higgs is there or not. We have to perform our averages in this case very precisely.

Well, lets return from this detour. Just keep in mind that when I talk here about an experimental result, it is actually more than the simple picture I will draw.

So, we measured what is left after the two particles collide. Then we may either get the result that the two particles after the collision are essentially unscathed, maybe a bit deflected, but that is it. Or, we may end up with new particles, and a serious redistribution of energy among them. Obviously, in the first case the interaction is not so very strong between the particles, wile it is rather strong in the later case. When looking at the details, it is not that simple, but this should illustrate the point that we can learn something about the strength of an interaction when we let particles collide.

We did this, and what we found is that the strength of the forces is indeed dependent on the energy. These findings are rather remarkably, and where of the following nature:

When we did this with the electromagnetic force, we found that the electromagnetic interaction becomes stronger and stronger the higher the energy. However, the increase is very, very slow. But we were able to measure it very accurately, and it also agrees with the theory to an extremely good precision. The increase of the strength of the electromagnetic force is such that by increasing the energy by a factor of 100, we get only an effect of a few percent. Thus, no man-made experiment is just yet able to test how strong the electromagnetic force is really becoming before we run into the problem that the standard model is just a http://axelmaas.blogspot.com/2011/08/limits-of-standard-model-ii.html.

The opposite turns out to be the case with the strong nuclear force. At low energies it is actually very strong, the strongest force we know so far, making all the nuclear physics, and keeping our protons and neutrons together in the nuclei, and the quarks even more deeply hidden. However, when increasing the energy, the strength of this force quickly decreases. If we would be able to look at arbitrary large energies, it would become arbitrarily small. This feature is called asymptotic freedom. As a consequence, the strong force becomes the simpler to deal with in theory the higher the energies. Therefore, our best tests of our theory of the strong force comes actually from high energies rather then from nuclear physics, as one would expect.

The weak force is actually very similar to the strong force, and it becomes weaker the higher the energy. The difference is that it starts out already rather weak, at least superficially, and thus its consequences are not nearly as spectacular as the ones of the strong force.

And there is the last remaining set of forces in the standard model. These are the ones associated with the Higgs, which provide the mass to all particles. This is a force we could not yet measure reliably in experiment, since we were not yet able to find the Higgs. Thus, we can make only a statement based on indirect evidence, and theoretical expectations. If they are true, then we expect that the interactions due to the Higgs will, like the electromagnetic interactions, also strengthen with energy. However, it turns out that the question of when these forces become strong depends very much on the mass of the Higgs. If the Higgs is as light as currently still permitted by the experiments, the corresponding interactions will rise very slowly, just like for electromagnetism. Then this rise can be ignored also for most purposes. If the Higgs would be only ten times heavier then the situation would be completely different. We would then expect to see relatively strong interactions between the Higgs and the other particles in the standard model at the LHC. That we did not see them yet does not mean they are not there - it is still hard to produce a Higgs which could interact strongly. Thus such potentially strong interactions are very rare even at such an experiment like the LHC.

Thus, the names of the forces are what they are just because at the low energies, where we first encountered them historically, they were weak and strong. And then these names just stuck, as they always do.

Why mass depends on energy

2011-09-01T00:27:00.000-07:00

One of the fascinating topics one is confronted with in particle physics is the fact that quantities depend on our perspective. For example, masses and the strengths, with which particles interact, depend on the energy we use to probe them. That may sound strange at first. However, it is not so surprising that such quantities may depend on our way of looking at them.

Take an apple. It has a certain mass, say 100 gram. However, if you put yourself inside the apple, the amount of apple you see before your (if you look towards the pit) is less than the whole apple, and its mass is thus less. In a very cartoon idea, you could also say that this depends on the energy: If you ride a (rather slow) bullet, its penetration depth will depend on its initial energy, and thus the higher the energy, the less of the apple mass remains in front of you.

Of course, things can get not only less, but also more. Talk a light bulb (or LED, if you like it more modern). Put veils upon it. The amount of light you perceive depends then on the number of veils between you and the bulb. The more veils are already behind you, and the less before you, the brighter the bulb. Of course, the idea with the bullet applies here, too. Just avoid hitting the bulb.

Well, for particles you do not have veils or outer shells of an apple. So what is going on there? What is going on is that the vacuum around a particle is actually a rather thick soup than really empty. This seems surprising at first - after all, we have been thinking that the space in, e.g., an atom is essentially empty. The reason for this apparent contradiction is quantum physics. In quantum physics, we got used to the fact that we cannot really say what is going on, and everything becomes fuzzy. In particular, we cannot say whether a portion of the vacuum is really empty or contains particles, as long as we do not make a very precise measurement. In the head of theoretical physicists, this observation of nature has formed the picture of so-called virtual particles.

Virtual particles are particles which appear and disappear all the time. They may either be emitted by a source, like another particle, or may even pop out randomly (though necessarily at least in pairs) from the vacuum. They only exist for a very brief glimpse, and are then reabsorbed by the source, or annihilate again into nothing. Only, if we look close enough, i.e. short distances and thus high energies, we can check whether such particles are there or not.

In particular, if we want to look at one particle, or the interaction of two or more particles, they are surrounded by a cloud of such virtual particles. Only if we get close enough to them, and that means at high energies, we can dive through this cloud. However, to really see the pure and unaltered particle (or process), we would need to resolve it with a wave-length of its size, but this is zero. Hence, this would require infinite energy, because we want to measure them at zero distance. So we cannot.

However, the more energy we invest, the deeper we get into the cloud, and the more we see of the particle's or process' properties. Measuring these accurately, we can even extrapolate to the real properties of the particles. Then we find that, e.g., the masses of particles become less and less the closer we get and thus most of their mass is made up by this cloud of virtual particles. Also, we find that some of the interactions become less, and others become stronger. Since these quantities change with energy, a physicists also calls them running quantities. Running means here that they change comparatively quickly with a change of energy. We also know the concept of walking quantities which change slowly, but we do not know an example of such theories (yet) in nature.

When thinking about such things, we should always keep in mind that the standard model of particle physics is, as discussed previously, just a low-energy effective description. Hence, when we try to extrapolate, we use our theory as input, meaning that our extrapolation necessary will fail at some higher energy, and we do not even know precisely when. So this running is telling us just how things change in a certain range of energies we can test. However, this is actually useful: By looking for deviations from what we think should happen, we can find something new.

The limits of the standard model II

2011-08-15T02:30:00.000-07:00

The last entry focused on the low-energy, or long-distance limit, of the standard model. This time, lets have a look at the opposite limit, the one of very short distances, or, as discussed previously, the one of very high energies.

If we go to smaller and smaller distances, we try to look deeper and deeper into something. Just like with the ocean: First, we just see the essentially plain waters. When we go nearer, we see the large movements of very large waves. When we go closer, we see that on the large waves there are small waves, and even deeper, we see ripples on all of the smaller waves. However, if we would go even closer, we would see that the water is made up of water molecules, out of discrete things. Wow. We just made a jump from one description - a continuous amount of water - to another one - that of water molecules. This means, when we look at shorter and shorter distances, we can learn how the things work in the interior, and in detail. Therefore, by looking at smaller and smaller distances, or higher and higher energies, we learn something about the nature of things.

In terms of theories, physicists like to speak of the description in terms of the water molecules as the 'underlying theory'. The description in terms of water as a fluid is called the 'low-energy effective theory', i. e., a theory which describes the relevant features of the underlying theory if we are looking on distances where we cannot distinguish the individual constituents of the underlying theory anymore.

In doing so, we actually notice something: As we have discussed, molecules are made up of atoms and the atoms are made up out of even smaller particles, and these smaller particles are described by another theory, the standard model. Hence, the theory of water molecules is out of a sudden no longer an underlying theory, but a low-energy effective theory for the standard model. Thus, a theory can be both, an underlying theory, and a low-energy effective theory. It is just a question of whether we look at it from larger or smaller distances than the characteristic scale of it.

The characteristic scale, which I have introduced here without warning, is actually a rather sloppy term: It means essentially distances in which the typical behavior of the objects in a theory show themselves. In the case of the waves, this scale is of the order of kilometers down to micrometers, the water-theory with water as molecules then takes over until one reaches the domain of femtometers, where the standard model comes into play. To not give a scale range, one usually uses one intermediate scale to indicate such a characteristic scale. For the theory of water molecules this is typically some Angstrom (about 0.0000000001 meters). For the standard model, it depends on the sector: about 0.000000000000001 meter for the strong interactions, and about 0.000000000000000001 meter for the weak sector, about one-thousand times smaller.

But wait: How can we be sure that the standard model is the underlying theory? The answer is we cannot. In fact, we firmly believe that it is not. The reason for thinking so is the following: On the one hand, it lacks gravity. We think, that the last theory in such a hierarchy of effective theories should include both quantum physics with the standard model as well as gravity. We just do not see right now any logical possibility that these two should be and remain separate.

On the other hand, there is a technical problem, which shows that the standard model is incomplete. If you do calculations in the standard model, then it turns out that for doing everything mathematically consistently you have to consider arbitrarily large energies. However, if you do this, the results are infinite, and thus at first glance meaningless. If you, however, assume that the standard model is just a low-energy effective theory, it is possible to remove these infinities by defining a small number of parameters appropriately. This is called renormalization, and the proof that this is possible for the standard model, at least to some extent, has been awarded with a noble prize. In a way, the standard model is telling us: "Hey, I m not the final answer, but you can parametrize your ignorance such that I still make sense, if you just do not poke me with too large energies."

Ok, all well and fine. But what is the underlying theory to the standard model? We do not know right now. And to figure this out, we have to look at physics a ever shorter distance scales and thus ever higher energies to get an answer to this. That is the reason we built and use the LHC and its predecessors. There is also the possibility to indirectly interfere the very high energy behavior by making very precise measurements. You could imagine this in the following way: You would also figure out that water is made out of molecules when you would weigh water very carefully. Then you would notice that it is not possible to have an arbitrary weight of water, but only discrete portions. And similarly we try to infer the high-energy behavior by very precise measurements.

Anyway, this is the current goal: To see what is the underlying theory of the standard model. This process of identifying the next underlying theory has been driven physics since centuries. Will it ever terminate? That is a good question,a and one we cannot answer (yet). The only thing sure right now is that it did not terminate with the standard model. And that we do not even yet fully understand the standard model, though this is necessary to answer whether something we observe is genuinely a signal of the underlying theory, or just a feature of the standard model. A difficult question indeed.

The limits of the standard model I

2011-06-21T03:36:00.000-07:00

After the rather technical discussion in the last few entries let us return this time to a more mundane topic: What is the validity of the standard model. For that purpose assume for the sake of the argument that the Higgs particle will eventually be found.

The question can be paraphrased differently: What is the lowest and the highest energy at which the standard model can be used? This question can also be formulated even more differently: An energy can be associated with a distance. That is very similar to what has been discussed previously in the entry on "Fields, waves, particles, and all that". If you have a very large energy, movement is essentially very rapid. In particular, the fields associated with the particle oscillate very quickly, and thus the distance between the crests of its waves is very small. Hence, changes on very small distances can be sensed by the particle, and thus high energies can be associated with small distances. In the opposite extreme, this means that low energies can be associated with large distances.

Let us then start with the more simple of both limits, the lowest energy. Since the standard model is a quantum theory, this can be also posed as the question when do we no longer observe things, which are distinctively quantum. A quantum theory means associating particles with a field. Thus take again the picture of waves, and let us go again back to the picture of the ocean. If you hover a short distance above it, you can see the individual waves. If you then zoom out, at some point everything blurs together, and you have the impression that only a - more or less - flat surface is there. At this point you do no longer realize the individual particle (wave), but only all of the particles (waves) together, in the form of the ocean as such. Similarly, if you zoom out of the standard model up to, say, the level of your desk, you do not note anymore the particles, but only the surface of the desk.

This is not yet telling you that the standard model is not applicable anymore, just that your are no longer able to distinguish its parts. It is therefore actually a very complicate question, whether the standard model is only valid up to a certain distance scale, because it becomes so hard to see its content. People have tried very hard to see the consequences of the standard model at ever larger distances, but, depending on the part of the standard model you look at, it becomes very hard to make a statement. Once leaving the size of a few times a nuclei, it is essentially only the electromagnetic force we can still test. For that part of the standard model we know that it works at least on the order of our own galaxy, and we have evidence, though far less rigid, that its seems to work rather well even at much larger cosmic distances. Still, answering the question to which distance we can observe the standard model is thus tricky and a persisting challenge. Perhaps even our understanding of the universe would be altered, if we someday would figure out that the standard model is not a suitable description at long distances.

Thus, to the best of our current knowledge, the standard model works (though we have a hard time seeing it) at the largest distance scales, and thus at the lowest energies, we can observe and test. However, it is a technical problem to check whether this is actually true or not: We need very sensitive experiments to check this, and the observation of true quantum effects is up to now limited to very small sizes, like in a Bose-Einstein condensate of atoms. The size of the latter is currently at best below some centimeters. Only some very specific quantum effects can be observed using photons at larger distances, like when using a fiber or making the famous double-slit experiment. But photons are only a very restricted part of the standard model.

The situation will change at high energies. There is also a technical problem, but in addition also a conceptual problem.

Internal and external space(s)

2011-05-06T08:21:00.001-07:00

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

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

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

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

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

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

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

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

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

2011-02-17T07:56:00.000-08:00

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

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

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

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

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

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

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

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

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

Fields, waves, particles, and all that

2011-01-24T03:50:00.000-08:00

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

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

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

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

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

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

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

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

Electromagnetism, photons, and symmetry

2010-10-21T09:33:00.001-07:00

After this rather abstract enumeration, it is time to take a closer look at a particular example. The simplest sector embedding a local symmetry in the standard model is electromagnetism. Classically, electromagnetism describes electric and magnetic fields, and thus also light, X-rays, and every other form of electromagnetic waves. As their names already hints, electric and magnetic fields are so-called fields. Fields in physics are something which associate with each point in space and with each instance in time a quantity. In case of electromagnetism this is a quantity describing the electric and magnetic properties at this point. Each of these two properties turn out to have a strength and a direction. Thus the electric and magnetic fields associate with each point in space and time an electric and a magnetic magnitude and a direction. For a magnetic field this is well known from daily experience. Go around with a compass. As you move, the magnetic needle will arrange itself in response to the geomagnetic field. Thus, this demonstrates that there is a direction involved with magnetism. That there is also a strength involved you can see when moving two magnets closer and closer together. How much they pull at each other depends on where they are relative to each other. Thus there is also a magnitude associated with each point. The same actually applies to electric fields, but this is not as directly testable with common elements. Ok, so it is now clear that electric and magnetic fields have a direction and a magnitude. Thus, at each point in space and time six numbers are needed to describe them: two magnitudes and two angles each to determine a direction.

When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as Maxwell equations in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with quantum electro dynamics - QED.

So, this appeared as a big step forward in describing numerically electromagnetism. However, when looking deeper into the mathematical concepts, it turned out to be technically rather complicated to describe all electric and magnetic phenomena with just these two properties of the photon. It was then that people noticed that including a certain redundancy things became much simpler. An ideal solution was found to describe electromagnetism with four numbers at each space-time point, instead of two. These can then not be independent, of course. And it is here where the symmetry comes into play: It is a symmetry concept which connects these numbers.

First, here is a simple example of how it works. Take someone walking only along the circumference of a circle. Then you can either describe her position by the height and width from the center of the circle. Or you can use the angle around the circle's circumference. Both is equally valid. Hence, the two numbers of the first choice are uniquely connected to the second choice: Changing the angle will change both height and width simultaneously! And because this connection comes from the fact that the circle is rotationally symmetric, it is this symmetry. And the symmetry of a circle is called U(1). Now, the relation between the four convenient numbers and the two important ones is quite in analogy to this case, and is therefore also a U(1) symmetry. That is how the symmetry becomes associated with electromagnetism. This tells us that if we change the four numbers by, so to say, moving them around on the circle, we do not change the two numbers describing the photon (or the six describing the electric and magnetic field). Only when we move away from the circumference, the two (and six) numbers change. In this way the symmetry is only helping us in a mathematical description, but is not influencing what we can measure. It is therefore also called a gauge symmetry. It is actually a local gauge symmetry, because these are fields, and we can do this at every point.

The symmetries of the standard model

2010-09-09T04:58:00.000-07:00

With the previous couple of entries a number of basic concepts have been introduced. It is now about time to make use of them in terms of the standard model.

The standard model from the theoreticians point of view is a set of local and global symmetries, which constraint the overall form of the theory. This skeleton is then fleshed out by adding to the symmetries particles such that they respect the symmetries. Furthermore, interactions between the particles are added, which superficially respect the at least the local symmetries, i.e. they do not break them explicitly. This then gives the set-up of the standard model (the procedure is quite similar if one is looking for a theory beyond the standard model, though there is not (yet) coercive experimental guidance how to choose the ingredients). And then...we let the system run, and see what comes out. This may actually break some of the symmetries, there may appear interactions which have not been there before, or we can observe new particles, which are somehow constructed from those we have put in. The proton is an example of the latter case.

So what are the symmetries in the standard model?

First, there are three local symmetries, which are at the heart of the theory. Each of them is associated with an interaction.

There is first a very simple symmetry, called electromagnetic or U(1) symmetry, which is associated with electromagnetism and the photon. It tells us that we can modify the electromagnetic field locally to some extent without altering the physics.

The next in line is the one associated with the strong interactions, the gluons, and the quarks, the so-called color symmetry or SU(3). It tells us that the interaction among quarks and gluons can locally be changed to some extent, again without changing anything measurable.

Finally, there is the one associated with the weak force, the so-called weak symmetry or SU(2). Except for the gluons, everything in the standard model is in one way or the other associated with this symmetry. This implies we can change a lot of how the standard model looks without changing the measurements.

These three symmetries, also called together SU(3)xSU(2)xU(1), are at the very heart of standard model. Everything else is build around it. However, the interactions change this structure considerable, and when looking just at measurements, it appears at first sight that the weak local symmetry is gone. However, in fact it is still there, but very well hidden by the interactions. I will come back to this in the future.

Then there are a number of global symmetries. First, there is a so-called chiral symmetry associated with the quarks and leptons. I.e., there is a special relation between particles spinning in direction of their movement and those spinning in the opposite direction. Because you can visualize them with either left or right hand, this is associated with the word chiral, which in a loose sense means handedness (precisely, it means hand). This symmetry is not left intact by the interactions, and this can be associated with how the particles become a mass. The second is that the number of each type of quarks and leptons are individually conserved. Also this symmetry is not surviving when interactions are turned on. However, the total number of quarks and leptons is actually almost conserved, and their change in number is, at the current time, essentially negligible. For a quark to turn into a lepton, experiments found that this needs at least 10000000000000000000000000000000000 years. The next symmetry counts the total number of quarks and leptons. This number is conserved in the standard model. Finally, there is also a rather obscure symmetry, which relates things which have a very distinct property when looking at them or at their mirror image, called axial symmetry. Again, this symmetry is broken. In contrast to the previous cases, this symmetry is actually not broken by the interactions, but enforcing the theory to describe quantum effects. Because that is so different from the rest, this is called an anomalous breaking, and the effect itself is called an anomaly.

On top of these local and global symmetries, there are three more symmetries, which have to do with fundamental properties of a physical system. One is related about what happens if you look at things and then again look at them in a mirror. That is called parity. The next connects to what happens when you replace every particle by its anti-particle and vice versa. This is called (charge) conjugation. And the last one is a statement what happens if you reverse all movements, and thus is called time reversal. All the three individual symmetries are broken by the interactions. However, if you combine all three together, this is a single symmetry, and this is still obeyed.

So, you see, the standard model is essentially a zoo of symmetries, and they again become very much modified by interactions. This is one of the reasons which yields many technical problems when one tries to answer even simple questions in the standard model.

Global and local symmetries

2010-08-04T05:29:00.000-07:00

An important distinction in physics is global and local.

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

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

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

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

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

Symmetries

2010-05-27T04:34:00.000-07:00

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

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

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

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

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

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

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

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

Invariance

2010-04-27T06:22:00.001-07:00

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

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

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

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

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

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

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

Coordinate Systems

2010-03-18T06:26:00.001-07:00

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

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

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

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

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

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

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

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

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

The Higgs effect

2010-03-05T09:10:00.001-08:00

As has been discussed previously, the weak interactions make a difference between left and right. This has very profound consequences for particle physics, since we do not know how to formulate a theory which at the same time is in agreement with this asymmetry, experiments, and has quarks and leptons with an intrinsic mass. So, it seems that everything build up so far is not very stable. Fortunately, there is a way out. And this way is to let the mass of a particle not be a fixed property but to make it an acquired one. Something, which happens dynamically, and is not static.

We know a vivid example of how such a thing could happen from everyday experience. If we move a spoon through honey, it moves much slower than it would if we use the same force to move it through water. It feels, as if we dragging a much larger mass. So, the environment can give us the illusion of a larger mass than there actually is. It is essentially the same concept, though a bit more sophisticated, which is invoked in particle physics to provide mass to the particles.

Actually, there is not only one concept, but many, which can provide this feature. For the standard model of particle physics, we have settled so far to the most simple one. We are not yet quite sure whether it is the correct one, since we have no experimental confirmation of its main actor. This main actor is the so-called Higgs particle. The search for it is something which many experiments, most notably the Tevatron and the LHC, pursue at the time of writing. Yet without success, and with every passing month it becomes more likely that we need a different concept. But for now, let us remain with the simplest one.

This simplest one foresees this Higgs particle. And the idea now is that this particle condenses, very much like vapor condenses into water. The so-formed condensate fills all of space. Since the Higgs particle interacts with quarks and leptons, they start to stick to this condensate while moving through it. By this, the illusion of their mass is created. The same holds true for the W-bosons and Z-boson of the weak interaction. Only photons and gluons can escape this effect, and remain massless. Even the mass of a single Higgs particle itself is modified by the condensate of all the other Higgs particles, because it can also interact with itself.

And by this mechanism all the particles get their mass. So, all around us the space is filled with the condensate. We can see through it, because the photons do not become slowed down. But the rest is, and so we feel a mass, including our own.

In a sense, the Higgs particle is thus a kind of a fifth force, since it not only forms the condensate, but is also exchanged between the condensate and other particles. At the same time, it is also affected by the other forces, so it is also a bit like the quarks and leptons. Therefore it is commonly not regarded as a force of its own. The theory of the Higgs particle is usually refereed to as the Higgs sector of the standard model. Our quantum theory of it is actually downright ugly, since we need a lot of very special assumptions about the properties of the Higgs to make it compatible with the world around us, and still cannot predict how massive itself is, and if and how we can see it directly with contemporary experiments. That is also one of the reasons for the great popularity of alternative explanations, which nonetheless all boil down to replace this Higgs effect by something else, having essentially the same effect and provide mass for the particles.

With this Higgs particle and its interactions, the last of the players in the standard model have been introduced. The next step is then to think about how describing their physics.

The forces of nature IV - The weak force

2010-02-23T07:52:00.001-08:00

The last force of nature is the weak force. Actually, its name is a bit misleading, but this will be explained later. This force is something which is even less accessible to our daily experience, but it is as much of relevance to our very existence as the other forces. The most direct evidence of it is already a nuclear process, the so-called beta decay. In such a decay, a neutron transforms, or decays, into a proton, an electron, and an electron-anti-neutrino. Precisely such decays, or the inverse process of neutron formation, is of quite central importance for suns. In the interior of suns, this process is very much involved in the generation of heat and light, which makes live on earth possible. But it is also of relevance in the formation of heavy elements, like lead, during supernova explosions. So, much of the things we see around us exist only due to the weak force. The details of how this works in suns and novas is very intricate, and complicated to model. These are central questions in the field of nuclear astrophysics. However, this has little connection to the research illustrated in this blog, so I will skip it. Many excellent resources on these questions can be found elsewhere on the web.

For my purposes, let me delve deeper into what happens during beta decays. The proton and neutron are themselves build from quarks, and it is actually not the neutron which transforms, but one of the quarks. The neutron consists out of two down quarks and one up quark, while the proton has two up quarks and one down quark. Thus, to get from a neutron to a proton, one down quark has to be exchanged for an up quark. This is exactly what the weak force is doing. For this to happen one of the agents of the weak force needs to become involved. This agent is called the W boson. There are actually two of these, which differ by their electric charge, one having a positive charge like the proton, the other a negative charge like the electron. What happens is that the down quark emits a negatively charged W boson - the down quark itself has a third of the charge of the electron - and by this transforms into an up quark, which has two thirds of the charge of the proton. This transforms the neutron into a proton. The emitted W boson then decays into the electron, which carries the electric charge, and the electron-anti-neutrino, which are observed.

So that is how the weak force acts. But why are the W bosons not itself detected like the photons. Are they strongly bound like the gluons? The answer is no. The reason is rather different. The W bosons have a large mass, which is actually just about half the one of the top quark. They are therefore some of the heaviest particles found so far. Hence, it is favorable for them to decay into lighter particles after traveling a short distance. That is then also the reason why the force appears so weak: It can only act over a very short distance, before its agents decay, and the resulting particles start to act differently.

When investigating this phenomenon more in detail, it turns out that the there is another agent of it, the so-called Z boson. This is electromagnetically neutral, and about ten percent heavier than the W bosons. It is thus the second-heaviest elementary particle we know so far. Because of its properties, it turns out that it often acts very much like a very heavy copy of the photon. Indeed, upon closer inspection it is found that the photon and Z boson are not two particles apart, but mix quite heavily with each other: At long distances what looks like the photon is more of a Z boson at short distances. This is because of the mass of the Z boson, which is so much heavier than the photon and can therefore not travel far without decaying. But at short distances it looks more like a Z boson.

Therefore, and because of the charges of the W bosons, the description of both interactions - the weak and the electromagnetic ones - rest on a common theory, the so-called electroweak theory. QED, described earlier, is actually only its long-distance face. At short distances, reached in modern particle physics experiment, the unification of both forces into one is very evident.

There are two things odd about the weak part of this interaction. First, it acts actually not directly on the particles described earlier, say the up quark, but only on certain combinations of them. Therfore, saying before that the down quark decays is not quite right. It is more like that a combination of the down quark and a strange quark which appears as a quantum fluctuation inside the neutron act together to produce the W boson, and by this change into the up quark and a quantum fluctuation of a charm quark. These quantum fluctuations inside the neutron and proton are not strange - that is something which is natural in quantum physics that things just pop up and vanish here and there. That will be looked at in detail later. The strange thing is that such combinations are necessary. Why this is so is one of the big questions of the theory.

The other thing strange is that the weak interactions makes a difference between left and right. In fact, it prefers left over right as much as possible. This has directly observable consequences. For example, if a decaying neutron is put into a magnetic field, the emitted electron has a preferred direction with respect to the field. Neither the strong nor the electromagnetic force has such a preference. This may be seen as an oddity of nature, at first sight. However, it has very profound consequences for our understanding of nature, as will be discussed next.

The forces of nature III - The strong force (Part II)

2010-01-29T05:14:00.000-08:00

As has been told, the hadrons are made up out of quarks. But there is something peculiar about this. When one looks at the constituents of, say, an atom - the nuclei and the electrons - then one can observe all of these also as individual particles. However, this is not applying to quarks. It has not been possible to isolate a quark experimentally. The only thing which can be seen are the hadrons.

When analyzing the structure of the hadrons in search for the reason, it turns out that one can assign to a quark a new charge, the so-called color charge. The name is just fancy and the charge has nothing to do with color. This color charge comes in six types. There are three 'positive' charges, called red, green, and blue (some people occasionally exchange one of the names for yellow), thus carrying the metaphor further. They are like the positive electric charge. Then there are three 'negative' charges, anti-red, anti-green, and anti-blue. They are like negative electric charge. As for electric charges, a negative and a positive color (say anti-green and green) neutralize each other. The amazing difference compared to electric charges is that also three different colors of either type neutralize each other. For example, a red, a green, and a blue quark together are neutral with respect to the color charge.

It is then found that all hadrons are always color-neutral. The mesons are made from one quark with color and one with anti-color, and the baryons are made from three quarks, each carrying one of the colors. In fact, the ones we observe around us are all made of three quarks carrying color. Those which carry anti-color are actually anti-matter, which will be discussed later.

Now, this gives an idea why quarks and hadrons are different. But it does not explain why quarks are not observed. This is now du to the force acting between two color charges. In contrast to all other forces, this force is not getting weaker with distance, but stronger instead. And it gets so quickly stronger that it is not possible to tear a hadron apart into quarks. At least that is what it looks like at the surface. The truth is somewhat more subtle, and not fully understood, and part of my research. Therefore, I will come back to this question many times in the future.

Irrespective of this, the force between colored objects is mediated by gluons. In contrast to photons gluons carry themselves color charges, though they are of a different type than those of quarks, and there are eight different ones, not usually given a name. As a consequence, the enormous strength of the force also binds gluons, and they cannot be observed as freely roaming particles either. In fact, at least in principle it is possible that gluons alone form bound states, much like hadrons. These are called glueballs, but are up to now only hypothetical constructs which have not been observed in nature, though some observations may hint at them. It is an ongoing experimental endeavor to find them.

The justified question is, if the force is so strong, why do we know about quarks and gluons? And why can it still bind the nucleons to nuclei if the nucleons are color-neutral?

Well, the force is not strong at all distances. Indeed, it grows quickly with distance, but on the other hand it diminishes as quickly with shorter and shorter distances. To the best of our knowledge it even ceases completely if one would be able to reduce the distance to zero. That is called asymptotic freedom. Therefore, if one can send a probe close to a quark, then one can identify its existence. For this purpose it helps very much that a quark is not only carrying color charge, but also electric charge. Therefore, it can be registered more easily by hitting it with a photon or an electron. That is somewhat indirect, but that is one of the main sources of experimental information on the quarks. The gluons are even more complicated, since they only carry color charge. Therefore, our evidence for them is rather indirect.

This is then also how nucleons feel each other by the strong force. When they come close to each other, they start to see each others quarks, which then can interact by the strong force. This is a comparatively weak effect, since it is, vastly simplifying spoken, just a bit of penetration what makes them feel each other. Nonetheless, this remainder of the force is so much stronger than the electric force that it makes the nuclei about 100000-times smaller than an atom. This should give an idea of how very strong this force must be that even such a small glimpse of it has such far-reaching consequences.

It is this strong force to which I will return repeatedly, as it is and has been for a long time my major focus of research. The one reason is that our understanding of this force is not very good is because many approaches just have to give up when faced with such an enormous strong force. Only at very short distances we have reliable control over it. Hence, the theory of this force, which is called quantum chromo dynamics (for the Greek word chromos for color and short QCD), is very hard to tackle. There is a simpler version of it, which only deals with the gluons and neglects the quarks (and is therefore not a picture of nature). It is called Yang-Mills theory. Because it contains already many essential features of QCD, it often, and also for me, serves as a prototype theory for the strong force.

The forces of nature III - The strong force (Part I)

2010-01-18T01:52:00.002-08:00

If one descends to smaller and smaller scales one always finds that larger things are built up from smaller things. When one looks to a human, she is made from cells. Each cell in turn is built from molecules, small and large.

Each of the molecules, in turn, is made from atoms. These atoms are rather small, like 0,0000000001 m each. There is one thing special about atoms which has not been encountered with molecules and cells: There is only a finite number of different ones of them observed in nature, while there appears to be an infinite number of different molecules and cells. In fact, atoms can be organized into a scheme (ok, this also applies to molecules and to some extent to cells also), the so-called periodic system of atoms. There are roughly hundred of them to be found in nature, and we managed to make a number artificially of them more over the years. Each of the atoms differs by its chemical properties.

So, it seems that atoms can be built, much like molecules. However, it is found that there are some atoms which behave in every respect essentially identical when it comes to chemistry, but they have a different mass. Both facts (and a number of others) suggest that atoms themselves are built from other things.

Indeed, it is found that atoms are made from two parts: Electrons and nuclei. The electrons orbit the nuclei, which is about 100000-times smaller than the atom (the electrons are even smaller as discussed previously). The are different electromagnetically charge with respect to each other, and there is always exactly one nuclei, but so many electrons that the total electric charge is zero.

It turns out that the charge is responsible for the chemistry, so the charge of the nuclei characterizes the atom. The mass of the atom is made essentially by the nuclei, which is about 2000 times heavier than the electrons. So different mass nuclei provide the same chemistry. Why?

Well, it turns out that the nuclei are composed from different objects themselves, the nucleons. That is the reason why new ones can be made and there are chemical identical ones with different mass. They nucleons come in two types, the neutrons and the protons. The latter carry the charge, making the atom chemical active, while the neutrons are chemically essentially inactive. On the other hand both have essentially the same mass. So chemically different atoms differ by their number of protons, but chemically identical atoms having different mass differ by the number of neutrons.

It is found that the nucleons have about the same size as the nuclei, so they are fairly densely packed inside the nuclei (in a typical atom there are a few dozen nucleons). What is keeping them together? It cannot be gravity alone, as it is too weak. If gravity alone should provide this, the nuclei would be much, much larger. It cannot be electromagnetism, as the neutron has no charge. So it must be something different. Indeed it is a new force, the so-called strong (or, since it was discovered in the context of the nuclei, nuclear) force. This force is binding the nucleons together to form the nuclei, and thus shapes the very word we live in as much as electromagnetism does.

The force between the nucleons is created by the exchange of mesons. These particles are usually not observable in nature as they decay too fast by the weak interactions to be discussed later. They can be observed in cosmic rays. The most important meson is the pion, having about an eighth of the mass of the nucleon. So, in contrast to the photon, it is massive. It also can carry charge, there is a positive one, a negative one, and a neutral one. There are also other mesons, the kaon, the rho, and the omega, playing a role in the nuclear force. In fact, as it was started to investigate this, more and more of these mesons have been found. Also, it was found that the nucleons are not the only of their kind. There are other, quite similar objects, like the delta or the cascade particles. Those nucleon-like particles have been termed the baryons, in distinction to the mesons. Both together are called hadrons, to distinguish them from the leptons. These mesons and baryons can again be put into a kind of periodic table, and we can produce new ones of them.

As the experience with atoms already told, this indicates that the baryons and mesons are themselves composites from other particles. Indeed, they are built up from quarks. Mesons consists of two, baryons of three quarks. If there are objects which are constructed from four or five or more quarks is not really known. If so, they are rathe short-lived and decay into mesons and baryons. During the recent years, conflicting experimental results made this a hot debate, and the judge is still out. These objects would be called tetraquarks (four quarks) or pentaquarks (five quarks).

In any case, there has to be a force holding the quarks together inside mesons and baryons. It turns out that this is again the strong force, but in another disguise.

The forces of nature II - Electromagnetism

2009-12-21T09:32:00.000-08:00

Electromagnetism is another force that everyone of us experiences in their daily lives. Of course, everything that is electric is based on electromagnetism. But it is not only that. When we knock on the desk, the force we feel is actually also electromagnetism. The force which keeps a crystal together, or the desk, or make water not spontaneously evaporate: All this and much more is electromagnetism. In fact, even light is electromagnetism. Our eye registers electromagnetic radiation and turns it with the brain into visual information.

As the name already suggests - Electro-magnetism - electromagnetism is the unification of the electric force, which make electric things work, and magnetic force, which is the force making magnets act like they do.

Indeed, both forces are not fundamentally different. If one takes special relativity, the theory of movements close to the speed of light, into account, they turn out to be just two sides of the same coin.

If looked at from the quantum perspective, this is even more manifest: Both forces are mediated by the same particle, the photon. The photon itself is not feeling the force itself it mediates. This makes electromagnetism fundamentally different from the weak and strong forces, which will be described next.

In turn, almost all matter feels electromagnetism: All quarks carry an electric charge. A charge, like the mass in case of gravity, is the fundamental docking port for the exchange particles. Also electrons, and their heavier cousins the muon and tau, carry electric charge. On the other hand, the neutrinos and the ever-searched-for Higgs particle are both neutral, and do not interact with the photon (there is a subtlety in this statement which will be discussed much later).

An amazing difference between mass and electromagnetic charge is that it is quantized. There appears to exist an elementary charge of which the charged particles carry a certain number. For historical reason, this is counted in thirds, instead of integer. So, the up quark carry 2/3 of an electric charge, while the electron carries -1, and so on. This is different in two respects from mass, the charge of gravity.

One is that it is quantized. Though we understand how to describe this quantization, the standard model of particle physics actually gives no explanation for it. In particular, the fact that electrons and up quarks have electric charges in a ratio of whole numbers is not understood, though it is required for the standard model in its current form to work. There are many unified theories which try to explain it, but none of them has produced yet observable effects which are different from the standard model and could be accessed in experiments.

The other property is that there is a positive and a negative charge. Mass comes only in positive portions. Electric charge comes in two varieties, + and -, which can compensate each other. As a consequence, the net charge of an array of objects can be zero, which is not possible for gravity. Such an exact zero is only possible because of the quantization. In general, almost all objects have almost zero charge. As a consequence, electromagnetism is screened at long distances: Net zero charge objects can only pull at each other, if for some reasons their internal charge array is such that positive and negative charges are not evenly distributed. Then another such object can pull, say, at the top side and push at the bottom. But this is a weak effect compared to the electromagnetism of charges objects. Only close by this can be important, and yields many of the everyday experiences with electromagnetism. And that is pretty good: As stated before, electromagnetism is very much stronger than gravity, and if it would not be screened, everything would pull or push at each other with enormous strength, creating an immediate collapse.

The only electromagnetic objects not affected by this screening are photons. Therefore, they can travel freely. The most common experience with this is light, but also the cosmic microwave background or radio is based on this. If photons would also be charged, this would not be possible.

Hence, electromagnetism introduces two new concepts: Quantization of charge and different signs of the charge, yielding the possibility of screening. Both will become very important for the strong and weak forces as well.

The quantum theory of electromagnetism, in contrast to the case of gravity, is very well understood. It is called quantum electrodynamics (QED), and it is one of the best and most successful theories today. As such, it is also part of the standard model of elementary particles, and there it is called the electromagnetic sector.

Unfortunately, it is not as good as it could be. If you increase the speed, and thus energy, of the particles, QED starts to get into trouble eventually. It appears that at very high energies the theory collapses, and electromagnetic interactions ceases altogether. However, this is likely stabilized when QED is embedded into the standard model.

Still, QED is more than only a successful theory. In fact, the structure of the theory, a so-called quantum gauge theory, is prototypical, and more complex versions of it are the theories describing the strong and weak forces, and also many attempts for quantum gravity. To our knowledge, it is the most important type of theories. What such a theory is precisely will be described after the introduction of the other two forces of nature: The weak and the strong one.

The forces of nature I - Gravity

2009-11-13T03:48:00.000-08:00

After illustrating last time how a force can be created by the exchange of particle, it is about time to make a list of which forces there are in nature, and which of them are included in the standard model.

Actually, there is only one force in nature which we currently know and which is not included in the standard model of particle physics. This is gravity. That is the force which pulls one inevitably to the ground, as long as one is not actively working against it. And the one which makes it so hard to get up in the morning. Or so.

It is actually not only the ground, and thus the earth, that is pulling at you, but actually also the earth is pulled by you. However, since the earth is much heavier than you are, it is rather ignorant of your presence. However, it cannot ignore the pull of the moon, to which it reacts with the tides. Nor can it ignore the sun, and this makes earth orbiting around it. On a larger scale, the solar system feels the pull of the milky way, making the solar system orbiting the center of it. And our galaxy the center of the local cluster of galaxies.

In fact, any object which has mass pulls any other object towards it, which has also mass. Actually, this is not entirely correct: Mass is not necessary, it suffices if there is energy in the game. This will lead a bit too far astray now, as it is necessary to delve into the theory of relativity for why this is the case, and I will leave this to later.

However, the generic concept that some objects act a force on each other because they both have a certain property is far more general. It is the simplest example of a charge. Gravity is simple in that everything pulls everything else to itself. In other cases, which will be encountered next time, this is not always the case: Some charges pushes away other charges.

So, why is gravity not included in the standard model (yet)? The simple answer is that we do not yet know how to really do it. There are quite a number of ideas, going by the fancy names of string theory, quantum loop gravity, and many others. However, none of these ideas could have been yet made so precise that it would actually explain how gravity quantitatively fits into the standard model.

The major problem encountered is that it is very hard to make gravity a quantum theory. That has rather technical reasons, and there are some hot leads how we can possibly circumvent this in the future. But not yet. The basic problem is essentially that we do not yet know how to cope with a pileup of gravitons, the (hypothetical) particles carrying the gravitational force, which inevitable always occurs in a quantum theory. That is actually an involved technical problem. For that reason gravity is not yet part of the standard model of particle physics, but instead described by a classical theory, general relativity.

The question is whether this matters when we want to talk about particle physics. The fortunate answer is that it does not, in most cases. The reason is that gravity is a very weak forces. Compared to those described by the standard model, it is about 10000000000000000000000000000000000000 times weaker than the weakest other force of the standard model. Therefore, only if there is a large charge - thus mass or energy - gravity becomes important. That happens only at energy scales which are more than 10000000000000 times larger than accessible in any experiment so far. In nature, it only occurs very close to a black hole or very, very early in the history of the universe. So, for most purposes, and in particular the ones of this blog, gravity can be neglected.

However, there are a number of open questions related to our limited understanding of gravity which have to do with large scales rather than particles: E.g., why is the universe expanding today? Also these questions will not be discussed for the moment in this blog.

Particles

2009-11-03T07:22:00.000-08:00

In the previous post particles appeared which are said to be exchanged between other particles. These particles are also called 'force' particles, in contrast to those objects exchanging them, the matter fields.

To the matter fields belong the leptons, the neutrinos and the quarks.

The force particles are the photons, the gluons, and the W and Z bosons.

The Higgs takes a role in between. On the one hand it can exchange force particles, on the other it is itself exchanged.

But how can one imagine the 'exchange' of a particle?

It is a little bit like when two boats pass by each other on a quiet sea. If they move, the generate waves which travel from one to the other, and are very much felt by each other. Anyone having traveled in a boat can confirm that it can get quite rocky if another fast boat comes close by.

The situation in particle physics is somewhat similar. The boats are the particles. The water is essentially a medium made up of force particles. If a particle now crosses this medium, it generates disturbances in it, which can travel and can be felt by other boats.

Why are then the force particles are called particles? The waves are indeed very much different from the boats. The reason is that the medium is very much different from water. If the waves in the medium are strong enough, they actually become very narrow, and look very much like a boat (a particle) themselves. Therefore, at strong waves, or if much energy has been invested in creating a wave, the wave looks like another boat (another particle). In fact, some of these can then exchange waves themselves, the force particles become matter particles in their own right.

So there is an interesting duality between the force carriers and those affected by the force, similar to the Higgs particle itself.

As a consequence, it has become common to talk even of the medium as an ensemble of particles, though this is not entirely right: If the waves are shallow, there is no structure which could be recognized as a particle. It is exactly this domain, which is least understood. The reason is that the medium is also in another respect different from water. If the waves are shallow, they affect the remaining medium much stronger than do water waves. In fact, only on the contrary, if the waves are strong, they more or less ignore the remaining medium, but only then.

To understand how this medium behaves is one of the central questions in my own research, but also one of the great unsolved questions in the standard model during its more that thirty year old history.

The standard model

2009-10-14T08:39:00.001-07:00

Let me introduce the players in the standard model. These are the so-called elementary particles. These elementary particles are the smallest objects we known in nature. And small means small: We are sure they are smaller than 0.0000000000000000000001 meters. That is really tiny. However, it would not be the first time that when we look closer they are actually consisting out of something even smaller. But let me for the moment assume that this is not so.

So, what are the players then. Well, they can be divided in a number of groups.

First, there are things we call leptons. An example for a lepton is the electron. Electrons are the things that make up electrical current, so we have to deal with them every day. There are two heavier copies of the electron: The muon and the tau, about 400 and 3600 times heavier than the electron. If they would be stable, we could use them also for electrical energy, but they are not: They decay in some other elementary particles after fractions of a second.

There is a second group of leptons, the so-called neutrinos. These are the lightest particles which have a mass. We are not yet sure what their mass exactly is, just that they are at least 500000-times less heavy than an electron. Again, there are three of them, one for the electron (called electron neutrino), one for the muon (muon neutrino) and one for the tau (tau neutrino).

The second group of particles are called quarks. Quarks make up composites of quarks, known as hadrons. The most prominent hadrons are the proton, the nuclei of a hydrogen atom, and the neutron. The latter two are composites of the up and down quarks. Besides these two, which have both approximately ten times the mass of an electron, there are four more. The strange quark, about 200 times heavier than an electron, the charm quark (neat names), 3000 times as heavy as an electron, the bottom quark (9000 times), and the big guy, the top quark (350000 (!) times). Again, the heavier quarks are not stable, they decay.

Then there is light. Yes, ordinary light (and X-rays, and infrared light, and so on) is made up out of particles, the photons. They are exchanged between anything that has an electric charge, e.g., a quark and an electron.

But they are not the only thing, which can connect particles. There are gluons, which connect quarks. Neither photons, nor gluons have a mass. And gluons are a bit strange, but this will be discussed much more in detail latter. Sufficies to say, we do not see gluons as we do see photons.

Then there are the W and Z, both of approximately half the mass of a top quark. They are important for radioactive decays, and they are also a bit strange. They connect both quarks and leptons. Also, because they are so heavy, they are not very stable.

Finally, there is an elusive guy, called the Higgs. We did not yet find it - perhaps we will at the next big particle physics experiment, the Large Hadron Collider LHC at the European Center of Particle and Nuclear Physics CERN. But it is important, because it seems to be connected with all the masses which have been floating around.

Introduction

2009-10-07T09:25:00.000-07:00

So, this is the blog in which I will discuss my research. I will try to be as general as possible, though at times it might get a bit tricky.

The general scope of my work is the standard model of particle physics - that is our current idea of how the smallest objects, the elementary particles, work. Very nice general introductions to this topic can be found at the large particle physics laboratories in Europe, at CERN or, in German, at DESY. Here I will only discuss what is of direct relevance to my own work.

An additional companion to this blog is my twitter account, on which I push some insights, some news, or some general remarks on my research, and on what is going on in the world of particle physics from my perspective.

Enjoy!