Warped Passages - Lisa Randall [110]
However, if you did know your paint’s chemical composition, the rules of physics would allow you to deduce some of those properties. You don’t need this information when you’re painting (using the effective theory) but you would find it useful if you were making paint (deriving the effective theory’s parameters from a more fundamental theory).
Similarly, if you don’t know the short-distance (high-energy) theory, you won’t be able to derive measurable quantities. However, when you do know the short-distance details, quantum field theory tells you precisely how to relate the different effective theories that apply to different energies. It lets you derive the quantities of one effective theory, such as masses or interaction strength, from the quantities of another.
The method for calculating how quantities depend on energy or distance, which was first developed by Kenneth Wilson in 1974, has a fancy name: the renormalization group. Along with symmetries, two of the most powerful tools in physics are the effective theory concept and the renormalization group, both of which involve physical processes with very different lengths or energy scales. The word “group” is a mathematical term that stuck, although its mathematical origin is largely irrelevant.
Renormalization is not such a bad word, though. It refers to the fact that at each distance scale of interest, you pause to get your bearings. You determine which particles and which interactions are relevant at the particular energies that interest you at the moment. You then apply a new normalization—that is, a new calibration—for any parameters in the theory.
The renormalization group uses ideas that are similar to those set out in Chapter 2, where we discussed the feasibility of interpreting a higher-dimensional theory in lower-dimensional language and treated a two-dimensional theory that had a small rolled-up dimension as if it were only one-dimensional. When we curled up dimensions, we ignored all the details of what happened inside the extra dimensions and assumed that everything could be described in lower-dimensional terms. Our new “normalization” was the four-dimensional description that could be used when focusing on large distances.
We can use a very similar procedure to derive a theory that applies to long distances from any theory appropriate to short distances: decide the minimum length you care about, and “wash out” the physics relevant to shorter scales. One way of doing this is to find the average value of those quantities whose details would make a difference only at the shorter distances you have chosen to ignore. If you had a grid filled with grayscale dots, you would literally average the shade density of the smaller dots to find the shade for bigger dots that would reproduce their effect. Your eyes do this automatically when you view something with fuzzy resolution.
If you can see things only with a given level of precision, you don’t need to know what happens on smaller scales to make useful calculations that relate measurable quantities. Your most efficient course often involves choosing the “pixel size” in your theory to agree with your level of precision. That way, you can neglect heavy particles that you’ll never produce and short-distance interactions that will never occur. Instead, you can focus your calculations on particles and interactions that are relevant at the energy you can achieve.
However, if you do know the more precise theory that applies at smaller distances, you can use that information to calculate quantities in the effective theory that interests you—that is, the effective theory with lower resolution. Just as