Warped Passages - Lisa Randall [96]
We will now see that this symmetry is closely analogous to the symmetries associated with forces, because in both cases you are not able to observe everything. The light setup exhibits symmetry only because we are not allowed to look at everything, only the combined light. If you could see the lights themselves, you would know they had been interchanged. As mentioned earlier, this close analogy between colors and forces is the reason for the terms “color” and “quantum chromodynamics” (QCD) in the description of the strong force.
In 1927, the physicists Fritz London and Hermann Weyl demonstrated that the simplest quantum field theory description of forces involves internal symmetries similar to that in the spotlight example above. The connection between forces and symmetry is subtle, so you won’t usually read about it outside textbooks. Because you don’t really need to understand this connection to follow the discussion of the issues about masses—including the Higgs mechanism and the hierarchy problem of the next few chapters—you can skip ahead to the next chapter at this point if you want. But if you’re interested in the role of internal symmetry in the theory of forces and the Higgs mechanism, read on.
Symmetry and Forces
Electromagnetism, the weak force, and the strong force all involve internal symmetries. (Gravity is related to symmetries of space and time, and must therefore be considered separately.) Without internal symmetries, the quantum field theory of forces would be an intractable mess. To understand these symmetries, we need first to consider the gauge boson polarizations.
You might be familiar with the notion of polarization of light; for example, polarizing sunglasses reduce glare by letting through only light that is vertically polarized and eliminating the horizontally polarized light. In this case, polarizations are the independent directions in which the electromagnetic waves associated with light can oscillate.
Quantum mechanics associates a wave with every photon. Each individual photon has different possible polarizations as well, but not all imaginable polarizations are allowed. It turns out that when a photon travels in any particular direction, the wave can oscillate only in directions that are perpendicular to its direction of motion. This wave acts like water waves on the ocean, which also oscillate perpendicularly. That is why you see a buoy or a boat bob up and down as a water wave passes by.
The wave associated with a photon can oscillate in any direction perpendicular to its direction of motion (see Figure 57). Really, there is an infinite number of such directions: imagine a circle perpendicular to the line of motion, and you can see that the wave is able to oscillate in any radial direction (from the center to the outside of the circle), and there are an infinite number of such directions.
But in the physical description of these oscillations, we need only two independent perpendicular oscillations to account for them all. In physics terminology they are called transverse polarizations. It is as if you had labeled a circle with x and y axes. No matter what line you draw from the center of the circle, it will always intersect the circle at a particular position—a particular pair of x and y values—and can therefore be uniquely specified by only two coordinates. Similarly (without going into the details of how this works), although there are an infinite number of directions that are perpendicular to the direction in which a wave travels, all of those directions can be obtained from combinations of polarized light in any two perpendicular directions.
Figure 57. A transverse wave oscillates perpendicularly to the direction of motion (in this case up and