If you look in the mirror, you notice that your left side is very similar to your right side. It is symmetric from left to right. We call this a mirror symmetry. Nature has developed your body pattern over hundreds of millions of years. Body patterns with mirror symmetry have certain evolutionary advantages and so we find this symmetry throughout the animal kingdom.
However, this symmetry is not perfect. Your heart is slightly to the left side of your body, breaking the symmetry. There was no particular advantage of having your heart directly in the center of your body. At some evolutionary point, it ended up on one side or the other (the left side) and it stuck. There was no advantage to changing it.
The mirror symmetry we find in many animal bodies is sometimes advantageous, but it is not compulsory. Although nature has made use of this mirror symmetry to a certain extent it has the freedom to construct body patterns which are not perfectly symmetric. Our heart is one example.
There are, however, symmetries of nature that are exact and never violated. These symmetries are fundamental and are deeply and closely connected with physical laws. These symmetries have two properties: 1) the equations that govern the motion of objects are symmetric and 2) the vacuum is symmetric. Let me explain.
The equations that govern the motion of everyday objects are called Newton's laws. These laws do not distinguish between left and right. Imagine taking an enormous mirror and holding it up to the entire universe. What you would see is the mirror image of our universe. The equations that govern the motion of objects in this new mirror image universe would be exactly the same as in our own. Newton's laws would not change whatsoever. So, even though individuals are not mirror symmetric, Newton's laws are mirror symmetric.
If we were given two sets of equations, the first set were the equations of motion for our own universe, the second set were for the mirror image universe, we would not be able to distinguish between them. We would not be able to determine which ones were which.
The vacuum is also mirror symmetric. What I mean by the vacuum is complete and total emptiness of any matter or energy of any kind. We come close to this in outer space. However, to truly be a vacuum, we must get far away from any planets or stars or cosmic dust of any kind. If we go to the most remote part of the universe, as far away from any matter as possible, we will come pretty close to a vacuum.
Now, if we hold the giant mirror up to the vacuum, the image of the vacuum would look exactly the same. If we were shown two vacuums, one was our own vacuum and the other was the mirror image vacuum, we would not be able to distinguish between them and we would not be able to determine which one was our own and which one was the mirror image.
Another important, truly fundamental symmetry is 'translation symmetry'. A translation is a change in position or location. For example, if you move your furniture from one home to another, you translate it. You do not change the furniture itself, you simply move it from one place to another. When something is translated, it is moved to a different location without changing anything else about it.
Again, Newton's Laws do not change under translation. Newton's Laws are the same in East Lansing as they are in Paris. They are the same on the moon. They are the same on Alpha Centauri (our nearest star neighbor). The Laws that govern motion do not depend on position. They are symmetric under translation.
The vacuum is also translation symmetric. This means that the vacuum is the same here as it is in a distant part of the universe. We can not distinguish the vacuum at different locations.
As a result of this symmetry, linear momentum is conserved. If you ever had a physics class, you learned that when no external forces act, the total momentum is conserved. Such a conservation law exists when, and only when, there is a fundamental translation symmetry.
Another symmetry is rotation symmetry. If we rotate, (begin facing one direction and then rotate our bodies so that we are facing a different direction), we do not change the laws of nature. They are symmetric under rotation. The vacuum is also rotationally symmetric. Rotation symmetry is another fundamental symmetry of nature.
Symmetries have a special mathematical structure which has been studied extensively by both physicists and mathematicians. Whenever we see this mathematical structure, we realize that we are dealing with a symmetry. The translation and rotational symmetries are intuitive examples of this mathematical structure. We can imagine what it means to 'translate' and 'rotate' an experiment and our experience tells us that the laws of nature should not change. However, there are other symmetries which we cannot imagine so easily. Nevertheless, we know they are symmetries, because they share the same mathematical structure.
One of these symmetries is related to electricity and magnetism. For hundreds of years, electricity and magnetism were thought to be completely different and unrelated phenomena. When a static electric charge was built up on something it could shock a person. However, magnets were never known to shock people. Notwithstanding these differences, a symmetry was discovered that related them. We now call it the electromagnetic symmetry and it is a fundamental symmetry. The laws of nature are absolutely the same under this symmetry and so is the vacuum. A law deeply connected with this symmetry is conservation of electric charge. This law tells us that the total amount of electric charge never changes; it is conserved.
Another example is quantum chromo-dynamic symmetry. Quantum chromo-dynamics is responsible for the force that holds protons and neutrons together in the nucleus of an atom. It is an extremely strong force, much stronger than electromagnetic force or gravitational force. There is a fundamental symmetry of quantum chromo-dynamics that has the same mathematical structure, even though we cannot imagine a quantum chromo-dynamical symmetry transformation. As a result, quantum 'color' is conserved.
In the recent past a new, very interesting kind of symmetry has emerged. Previously, we believed that a symmetry of the equations of motion would always be a symmetry of the vacuum and vice versa. However, we have discovered a new symmetry which is a symmetry of the equations of motion, but not of the vacuum. This symmetry is called electroweak symmetry and this bizarre arrangement turns out to be responsible for the mass of the electron among other things.
We call a symmetry that is a symmetry of the equations of motion but not of the vacuum a broken fundamental symmetry. We know that the electroweak symmetry is broken for the following reasons. If we try to make the electroweak symmetry a symmetry of both the equations of motion and the vacuum, then the electron is forbidden to have a mass. However, we know through countless experiments that the electron does has a mass. On the other hand, if we remove the symmetry altogether, so that neither the equations of motion nor the vacuum is symmetric, then relativistic quantum mechanics becomes sick. It no longer gives reasonable answers. For example, if we ask how often two particles collide, it tells us infinity. But, if we incorporate the electroweak symmetry as a symmetry of the equations of motion but not of the vacuum, then both of these problems are cured. So, we realize that the electroweak symmetry must be a broken fundamental symmetry.
To understand how the symmetry can be in the equations of motion, but not in the vacuum, imagine being on the top of a mountain peak surrounded by a circular valley. On the other side of the valley, the mountains rise up all the way to the clouds and out of view. This mountain peak and valley happen to be extremely symmetric. If you look North, it looks exactly the same as if you look West, or South, or East or any other direction. This symmetry is analogous to the electroweak symmetry.
On the other hand, if you hiked down to the valley bottom, you would not be able to see the symmetry. Suppose you hike down to the East side of the mountain peak. Now, if you look North or South, you see a flat path that winds around the mountain. But, if you look West, you see a an incline up the side of the mountain. If you look East, you see the steep cliffs of the higher mountains. The symmetry is hidden to you because of your location.
The equations of motion are like the hiker on the top of the mountain. They see the symmetry. But, the vacuum is like the hiker at the bottom of the valley. The vacuum does not see the symmetry. In fact, the mountain and valley are like the potential energy in modern theories. The higher you are up on the mountain, the greater your potential energy. The lower you are, the lower your potential energy. The lowest potential energy is obtained at the very bottom of the valley. The vacuum is always the state of lowest energy. Once the vacuum is down in the valley, the symmetry is broken.
We realize now that the electroweak symmetry is a broken fundamental symmetry, but we do not yet know what is responsible for that breaking. That is, we do not know what is responsible for creating that mountain peak in the middle of the valley. We have a number of candidate theories that are being studied intensively by theorists. In some, the mountain peak potential is created by one particle, called the Higgs particle. Sometimes, this particle comes by itself, other times it comes with a host of new particles that help stabilize this moutain peak potential. There is often one new particle for every particle we have already discovered. For example, the partner of the electron is called the selectron (the leading 's' differentiates it from the electron).
Another possibility is that the mountain peak potential is the result of the combination of two particles, usually called technifermions. In this theory, the technifermions are attracted to each other so strongly, that they bind into composite particles which then create the mountain peak potential. This theory has the advantage that it is related to the way protons and neutrons get their masses.
There are several others, but most of them are related to these two in some way or another.
Right now, a super collider is being built on the border of Switzerland and France, the Large Hadron Collider or LHC for short. When completed, it will accelerate protons to speeds very close to the speed of light. It will then smash them into each other in enormous collisions. These collisions will create new particles that we have never before seen. Because these new particles will only live for fractions of fractions of seconds, this machine also has a host of detectors built around the collision point, attempting to soak up every last detail of the collision. Each theory for electroweak symmetry breaking will finally be put to the test and we will answer the question of what breaks the electroweak symmetry. The next decade should be very exciting indeed.