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THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 12 - Beyond Strings: In Search of M-Theory
An Approximation Method
The limitations of the methods physicists have been using to analyze string theory are bound up with something called perturbation theory. Perturbation theory is an elaborate name for making an approximation to try to give a rough answer to a question, and then systematically improving this approximation by paying closer attention to fine details initially ignored. It plays an important part in many areas of scientific research, has been an essential element in understanding string theory, and, as we now illustrate, is also something we encounter frequently in our day-to-day lives.
Imagine that one day your car is acting up, so you go see a mechanic to have it checked out. After giving your car a once-over, he gives you the bad news. The car needs a new engine block, for which parts and labor typically run in the $900 range. This is a ballpark approximation that you expect to be refined as the finer details of the work required become apparent. A few days later, having had time to run additional tests on the car, the mechanic gives you a more precise estimate, $950. He explains that you also need a new regulator, which with parts and labor costs about $50. Finally, when you go to pick up the car, he has added together all of the detailed contributions and presents you with a bill of $987.93. This, he explains, includes the $950 for the engine block and regulator, an additional $27 covering a fan belt, $10 for a battery cable, and $.93 for an insulated bolt. The initial approximate figure of $900 has been refined by including more and more details. In physics terms, these details are referred to as perturbations to the initial estimate.
When perturbation theory is properly and effectively applied, the initial estimate will be reasonably close to the final answer; when incorporated, the fine details ignored in the initial estimate make small differences in the final result. But sometimes when you go to pay a final bill it is shockingly different from the initial estimate. Although you might use other, more emotive terms, technically this is called a failure of perturbation theory. This means that the initial approximation was not a good guide to the final answer because the "refinements," rather than causing relatively small deviations, resulted in large changes to the ballpark estimate.
As indicated briefly in earlier chapters, our discussion of string theory to this point has relied on a perturbative approach somewhat analogous to that used by the mechanic. The "incomplete understanding" of string theory that we have referred to from time to time has its roots, in one way or another, in this approximation method. Let's build up to an understanding of this important remark by discussing perturbation theory in a context that is less abstract than string theory but closer to its string theory application than the example of the mechanic.
Imagine that one day your car is acting up, so you go see a mechanic to have it checked out. After giving your car a once-over, he gives you the bad news. The car needs a new engine block, for which parts and labor typically run in the $900 range. This is a ballpark approximation that you expect to be refined as the finer details of the work required become apparent. A few days later, having had time to run additional tests on the car, the mechanic gives you a more precise estimate, $950. He explains that you also need a new regulator, which with parts and labor costs about $50. Finally, when you go to pick up the car, he has added together all of the detailed contributions and presents you with a bill of $987.93. This, he explains, includes the $950 for the engine block and regulator, an additional $27 covering a fan belt, $10 for a battery cable, and $.93 for an insulated bolt. The initial approximate figure of $900 has been refined by including more and more details. In physics terms, these details are referred to as perturbations to the initial estimate.
When perturbation theory is properly and effectively applied, the initial estimate will be reasonably close to the final answer; when incorporated, the fine details ignored in the initial estimate make small differences in the final result. But sometimes when you go to pay a final bill it is shockingly different from the initial estimate. Although you might use other, more emotive terms, technically this is called a failure of perturbation theory. This means that the initial approximation was not a good guide to the final answer because the "refinements," rather than causing relatively small deviations, resulted in large changes to the ballpark estimate.
As indicated briefly in earlier chapters, our discussion of string theory to this point has relied on a perturbative approach somewhat analogous to that used by the mechanic. The "incomplete understanding" of string theory that we have referred to from time to time has its roots, in one way or another, in this approximation method. Let's build up to an understanding of this important remark by discussing perturbation theory in a context that is less abstract than string theory but closer to its string theory application than the example of the mechanic.
Table of Contents
.......The Elegant Universe
.......The Elegant Universe