Coordinate Systems

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

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

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

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

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

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

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

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

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

The Higgs effect

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.