The limits of the standard model II

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

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)

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

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

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

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

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

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

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

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

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.