Warped Passages - Lisa Randall [155]
In fact, not only string theory, but all fields of physics are easier to understand when interactions are weak. That’s because if the weak interaction is only a small perturbation, or alteration, to a solvable theory—usually a theory with no interactions—then you can use a technique called perturbation theory. Perturbation theory lets you creep up on the answer to a question in the weakly interacting theory by starting from the theory with no interactions and calculating small improvements in incremental stages. Perturbation theory is a systematic procedure that tells you how to refine a calculation in successive steps until you reach any desired level of precision (or until you get tired, whichever comes first).
Using perturbation theory to approximate a quantity in an unsolvable theory might be compared to mixing paint to approximate a desired color. Suppose you’re striving for a subtle blue with hints of green that resembles the Mediterranean at its most beautiful. You might start with blue, and then mix in smaller and smaller amounts of green, alternating at times with a bit more blue, until you’ve achieved (almost) the precise color you’re after. Perturbing your paint mixture in this fashion is a way of proceeding in stages to obtain as close an approximation as you want to the desired color. Similarly, perturbation theory is a method for closely approximating the correct answer to whatever problem you’re studying by making incremental progress, starting from a problem you already know how to solve.
Trying to find the answer to a problem about a theory with strong coupling, on the other hand, is more like trying to reproduce a Jackson Pollack painting by randomly pouring paint. Each time you poured some paint, the picture would change completely. Your painting would be no closer to the desired goal after twelve iterations than it would be after eight. In fact, each time you poured the paint you would as likely as not cover up much of your previous attempt, changing the picture so much that you would essentially be starting afresh each time.
Perturbation theory is similarly useless when a solvable theory is perturbed by a strong interaction. As with futile attempts to reproduce a modern splattered masterpiece, systematic attempts to approximate a quantity of interest in a strongly interacting theory will not succeed. Perturbation theory is useful and calculations are under control only when interactions are weak.
Sometimes, in certain exceptional situations, even when perturbation theory is useless, you can still understand the qualitative features of a strongly interacting theory. For example, the physical description of your system might resemble the weakly interacting theory in gross outline, even though the details are likely to be rather different. More often, however, it is impossible to say anything at all about a theory with strong interactions. Even the qualitative features of a strongly interacting system are often completely different from those of a superficially similar, weakly interacting system.
So, there are two things you might expect for strongly interacting ten-dimensional string theory. You might believe that no one can solve it and you can’t say anything about it at all, or you might expect strongly interacting ten-dimensional string theory to look, at least in gross outline, like the weakly coupled string theory. Paradoxically, in some cases neither of these options turns out to be correct.