Warped Passages - Lisa Randall [94]
We all are familiar with symmetries not only in science and sacred symbols, but in secular art as well. Symmetry can be found in most paintings, sculpture, architecture, music, dance, and poetry. Islamic art is perhaps the most spectacular in this respect, with its intricate and extensive use of symmetry in architecture and ornamental art, to which anyone who has seen the Taj Mahal can attest. Not only does the building look the same from any side, but when viewed from the edge of the long pool in front, it is perfectly reflected in the calm surface of the water. Even the trees have been planted to preserve the monument’s symmetries. When I was there, I noticed a guide who was pointing out some symmetry points, so I asked him to show me the others. I ended up viewing the building from funny angles and scrambling up rubble on the edge of the site in order to see all the symmetries the monument presents.
In colloquial usage people often equate symmetry with beauty, and certainly some of the fascination with symmetry arises from the regularity and neatness that it guarantees. Symmetries also help us learn, since repitition, either in time or in space, can create indelible images in our mind. The brain’s programmed response to symmetry and its sheer aesthetic appeal explain in large part why we surround ourselves with it.
But symmetries don’t occur only in art and architecture, but also in nature, without any human intervention. For this reason you often encounter symmetries in physics. The goal of physics is to relate distinct quantities to one another so that we can make predictions based on observations. Symmetry is a natural player in this context. When a physical system has symmetry, you can describe the system on the basis of fewer observations than if the system had no symmetry. For example, if there are two objects with identical properties, I would know the physical laws that govern the behavior of one of the objects if I’ve already measured the behavior of the other. Because the two objects are equivalent, I know that they must behave the same way.
In physics, the existence of a symmetry transformation in a system means that there is some definite procedure for rearranging the system that leaves all its measurable physical properties unchanged.* For example, if a system possesses rotational and translational symmetries, two well-known examples of symmetries of space, physical laws apply the same way in all directions and in all places. Rotational and translational symmetries tell us, for example, that it doesn’t matter which way you are facing or where you are standing when you swing a baseball bat at a aball: provided you apply the same force, the baseball will behave in exactly the same way. Any experiment should yield the same result if you rotate your setup or if you repeat your measurement in a different room or in a different place altogether.
It is difficult to overstate the importance of symmetry in physical laws. Many physical theories, such as Maxwell’s laws of electrodynamics and Einstein’s theory of relativity, are deeply rooted in symmetry. And by exploiting various symmetries we can usually simplify the task of using theories to make physical predictions. For example, predictions of the orbital motion of the planets, the gravitational field of the universe (which is more or less rotationally symmetric), the behavior of particles in electromagnetic fields, and many other physical quantities are mathematically simpler once we take symmetry into account.
Symmetries in the physical world are not always completely obvious. But even when they are not readily apparent or when they are merely theoretical tools, symmetries usually greatly simplify the formulation of physical laws. The quantum theory of forces, which will soon be our focus, is no exception.
Internal Symmetries
Physicists generally classify symmetries into different categories. You are probably most familiar with symmetries of space—the symmetry transformations that move or rotate things in the