Warped Passages - Lisa Randall [95]
I now want to consider a different kind of symmetry, known as an internal symmetry. Whereas spatial symmetries tell us that physics treats all directions and all positions as the same, internal symmetries tell us that physical laws act the same way on distinct, but effectively indistinguishable, objects. In other words, internal symmetry transformations exchange or mix distinct things around in a way that can’t be noticed. In fact, I have already given an example of an internal symmetry—the interchangeability of the candles on a menorah. The internal symmetry says that two candles are equivalent. It is a statement about the candles, not about space.
A traditional menorah, however, has both spatial and internal symmetries. While different candles are equivalent, which means that there is an internal symmetry, a menorah also looks the same if it is rotated 180 degrees about the central candle, which means that it has spatial symmetry as well. But an internal symmetry can exist even when there is no symmetry of space. For example, you can interchange identical green tiles in a mosaic even when the leaf they combine to portray has an irregular shape.
Another example of an internal symmetry is the interchangeability of two identical red marbles. If you hold one such marble in each hand, it wouldn’t matter which was which. Even if you’d labeled them “1” and “2,” you would never know whether I had somehow managed to interchange the two marbles. Notice that the example of the marbles is not tied to any spatial arrangement in the way that the examples of the menorah and the mosaic were; internal symmetries concern the objects themselves and not their locations in space.
Particle physics deals with somewhat abstract internal symmetries that relate different types of particle. These symmetries treat particles and the fields that create them as interchangeable. Just as two identical marbles behave in exactly the same way when you roll them or bang them against a wall, two particle types that have the same charges and mass obey identical physical laws. The symmetry that describes this is called flavor symmetry.
In Chapter 7 we saw that flavors are the three distinct particle types that have identical charges, one in each of the three generations. For example, electrons and muons are two flavors of charged leptons, which means that they have identical charges. Had we lived in a world in which the electron and the muon also had identical masses, the two would have been completely interchangeable. There would then have been a flavor symmetry, according to which the electron and muon would behave identically in the presence of any other particles or forces.
In our world the muon is heavier than the electron, so the flavor symmetry is not exact. But the difference in masses can be insignificant for some physical predictions, so flavor symmetries between light particles with identical charges, such as the muon and electron, are nonetheless often useful for calculations. Sometimes exploiting even slightly imperfect symmetries helps us to compute sufficiently accurate results. For example, the mass difference between particles is often so small (relative to energy or a large mass) that it doesn’t make a measurable difference to predictions.
But the most important type of symmetry for us at this point is the symmetry that is relevant to the theory of forces, which is exact. This symmetry is also an internal symmetry among particles, but it’s slightly more abstract than the flavor symmetry we just discussed. This particular type of internal symmetry is more analogous to the following example. As you might recall from high school physics, theater, or art class, three spotlights—generally one red, one green, and one blue—can shine together to produce white light. If we were to interchange the positions