Rotations provide another familiar example of a symmetry. Imagine a laboratory as a windowless spaceship in free-fall, isolated from electric and magnetic forces, in which an experimentalist has some apparatus to determine a certain law of physics. Suppose that the experimentalist makes a measurement, the spaceship is rotated, and the same measurement is made again. The results of the two measurements are always found to be the same: When writing the laws of physics, there is no need to specify the orientation of the laboratory. Much less familiar, and more far reaching, it is not necessary to specify the speed of the laboratory—the laws of physics do not change if the experiment is done, for example, on an airplane. This last symmetry principle was the crucial one that led Einstein to develop special relativity. All of the astonishing results of special relativity—such as the equivalence of mass and energy and the inability to travel faster than light—follow from the requirement that physics be symmetrical under the operations discussed.

How are these symmetries of space and time—which are called space-time symmetries—relevant to particles and their interactions? First, there is a direct consequence for the very nature of elementary particles themselves. Rotational symmetry leads to elementary particles' possessing a new attribute, called spin. For example, electrons come in two varieties: left-handed and right-handed. The difference can be pictured in terms of the view of a football's spin as seen by the quarterback who threw the pass. The football spins clockwise if thrown by a right-hander and counterclockwise if thrown by a left-hander. Photons also come in the same two varieties of spin, but other particles have three spin orientations, and still others are spinless.

The symmetries of space and time also constrain the rules by which all particles interact. One should not forget why it is so important to understand these interactions. The explanation of every physical process, from the growth of plants to space shuttle lift-off, has its fundamental origin in these interactions. The properties of materials, from concrete to quicksand, ultimately depend on the properties of elementary particles.

Consider a collision between two electrons, shown in Figure 3.1. The two electrons labeled 1 and 2 approach each other, collide, and then leave as the two electrons labeled 3 and 4. Quantum mechanics says that there is no unique outcome: Sometimes the electrons are deflected by large angles, sometimes by small angles. In this quantum world the laws of physics can be phrased in terms of probabilities. If the laws determine the probabilities for all possible outcomes, then the description is complete. The problem is that the probabilities depend on the speeds, directions, and spins of each of the four electrons for which there are an infinite number of possible values. The importance of symmetry can now be appreciated: Probabilities depend on the speeds, directions,