The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [62]
To get a feel for these breakthroughs, imagine that Ralph is planning to play the next two rounds of the weekly worldwide lottery, and he’s proudly worked out the odds of winning. He tells Alice that since he has a 1 in a billion chance each week, if he plays both rounds his chance of winning is 2 in a billion, .000000002. Alice smirks. “Well, that’s close, Ralph.” “Really, wise guy. What do you mean close?” “Well,” she says, “you’ve overestimated. Should you win the first round, playing a second time won’t increase your chances of winning; you would already have done so. If you win twice, we’ll have more money, sure, but since you’re working out the odds of winning at all, winning the second lottery after the first just doesn’t matter. So, to get the precise answer you’d need to subtract the odds of winning both rounds—1 in a billion times 1 in a billion, or .000000000000000001. That yields a final tally of .000000001999999999. Questions, Ralph?”
Minus the smugness, Alice’s method is an example of what physicists call a perturbative approach. In doing a calculation, it’s often easiest to make a first pass that incorporates only the most obvious contributions—that’s Ralph’s starting point—and then make a second pass that includes finer details, modifying or “perturbing” the first-pass answer, as in Alice’s contribution. The approach easily generalizes. If Ralph were planning to play the next ten weekly lotteries, the first-pass approach suggests that his chance of winning is about 10 in a billion, .00000001. But, as in the previous example, this approximation fails to account correctly for multiple wins. When Alice takes over, her second pass would properly account for instances in which Ralph wins twice—say, on the first and second lotteries, or the first and third, or the second and fourth. These corrections, as Alice pointed out above, are proportional to 1 in a billion times 1 in a billion. But there’s also an even tinier chance that Ralph wins three times; Alice’s third pass takes that, too, into account, producing modifications proportional to 1 in a billion multiplied by itself three times, .000000000000000000000000001. The fourth pass does the same for the even tinier chance of winning four rounds, and so on. Each new contribution is far smaller than the previous, so at some point Alice deems the answer sufficiently accurate and calls it a day.
Calculations in physics, and in many other branches of science too, often proceed in an analogous fashion. If you are interested in how likely it is that two particles heading in opposite directions around the Large Hadron Collider will bang into each other, the first pass imagines they hit once and ricochet (where “hit” doesn’t mean they directly touch, but rather that a single force-carrying “bullet,” such as a photon, flies from one and is absorbed by the other). The second pass takes into account the chance that the particles hit each other twice (two photons are fired between them); the third pass modifies the previous two by accounting for the chance of the particles hitting each other three times; and so on (Figure 5.1). As with the lottery, this perturbative approach works well if the chance of an ever-greater number of particle interactions—like the chance of an ever-greater number of lottery wins—drops precipitously.
For the lottery, the drop-off is determined by each successive win coming with a factor of 1 in a billion; in the physics example, it’s determined by each successive hit coming with a numerical factor, called a coupling constant, whose value captures the likelihood that one particle will fire a force-carrying bullet and that the second particle will receive it. For particles such as electrons, governed by the electromagnetic force, experimental measurements have determined that the coupling constant, associated with photon bullets, is about .0073.2 For neutrinos, governed by the weak nuclear force, the coupling constant