Warped Passages - Lisa Randall [97]
The important thing is that, in principle, there could have been a third polarization direction, one that oscillates along the direction in which the wave travels (had it existed, it would have been called the longitudinal polarization).15 That is how sound waves travel, for example. But no such polarization of the photon exists. Only two of the three conceivable independent polarization directions exist in nature. A photon never oscillates along its direction of motion or in the time direction: it oscillates only along the directions perpendicular to its motion.
Even if we didn’t already know from independent theoretical considerations that the longitudinal polarization was spurious, quantum field theory would have told us not to include it. If a physicist were to make calculations using a theory of forces that mistakenly included all three polarization directions, the theory’s predictions of their properties wouldn’t make sense. For example, she would predict ridiculously high gauge boson interactions rates. In fact, she would predict gauge bosons that interacted more often than always—that is, more than 100% of the time. Any theory that makes such nonsensical predictions is clearly wrong, and both nature and quantum field theory make it clear that this nonperpendicular polarization does not exist.
Unfortunately, the simplest theory of forces that physicists could formulate includes this spurious polarization direction. That is not so surprising because a theory that would work for any photon can’t possibly contain information about one particular photon traveling in one particular direction. And without such information, special relativity would not distinguish any direction. In a theory that preserves the symmetries of special relativity (including rotational symmetry), you would need three directions—not two—to describe all the directions in which a photon could oscillate; in such a description, the photon could oscillate in any direction of space.
But we know that isn’t true. For any particular photon, its direction of motion is singled out and oscillation in that direction is forbidden. But you wouldn’t want to have to make a different theory for each and every photon, all with their own directions of travel. You would want a theory that works no matter which way a photon is travelling. Although you could try to make a theory that didn’t include the spurious polarization direction at all, it is far simpler and cleaner to respect rotational symmetry and eliminate the bad polarization in some other way. Physicists, aiming for simplicity, have recognized that quantum field theory works best when they include the spurious longitudinal polarization in their theory but add an extra ingredient to filter out the good, physically relevant predictions from the bad.
This is where internal symmetries enter the picture. The role of internal symmetries in the theory of forces is to eliminate the contradictions that the unwanted polarization would create without making us forfeit the symmetries of special relativity. Internal symmetries are the simplest way to filter out the polarization along the direction of travel that independent theoretical considerations and experimental observations tell us does not exist. They classify polarizations into good and bad categories, those that are consistent with the symmetries and those that are not. The way it works is a bit too technical to explain here, but I can give you the general idea by using an analogy.
Suppose you have a shirt-making machine that can make left and right sleeves in two sizes, short and long, but for some reason the inventor of the machine neglected to include a control to ensure that the left and right sleeves are the same size. Half the time you will make useful shirts—ones with two long sleeves or two short sleeves—but half the time you will make useless, unbalanced shirts with one short and one long sleeve. Unfortunately for you, this is the only shirt-making machine you have.
You have a choice of throwing your