The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [18]
The fly isn’t convinced. “I’m with you when you talk about moving a centimeter, or half a centimeter, or even a quarter of a centimeter,” the fly responds. “But when you speak of locations that differ by a ten-thousandth or a hundred-thousandth of a centimeter, or even less, you’ve lost me. To an egghead, those might be different locations, but it flies in the face of experience to say that here and a billionth of a centimeter to the left of here are really different. I can’t sense such a tiny change in location and so I don’t count them as different places. Same goes for speed. I can tell the difference between going a kilometer per hour and going at half that rate. But the difference between .25 kilometers per hour and .249999999 kilometers per hour? Please. Only a wise fly would claim to be able to tell the difference. Fact is, none of us can. So as far as I’m concerned, those are the same speeds. There’s far less variety available than you’re describing.”
The fly has raised an important point. In principle, he can occupy an infinite variety of positions and attain an infinite variety of speeds. But in any practical sense, there is a limit to how fine the differences in location and speed can be before they go completely unnoticed. This is true even if the fly employs the best of equipment. There is always a limit on how small an increment in position or speed can be and yet still register. And regardless of how fine those minimal increments are, if they’re not zero, they radically reduce the range of possible experience.
For instance, if the smallest increments that can be detected are a hundredth of a centimeter, then each centimeter offers not an infinite number of detectably different locations, but only a hundred. Each cubic centimeter would thus provide 1003 = 1,000,000 different locations, and your average bedroom would offer about 100 trillion. Whether the fly would find this array of options sufficiently impressive to keep away from your ear is difficult to say. The conclusion, though, is that anything but measurements with perfect resolution reduces the number of possibilities from infinite to finite.
You might counter that the inability to distinguish between tiny spatial separations or differences in speed reflects nothing more than a technological limitation. With progress, the precision of equipment always improves, so the number of discernibly distinct positions and speeds available to a well-funded fly will also always increase. Here I must invoke some basic quantum theory. According to quantum mechanics, there’s a precise sense in which there is a fundamental limit on how accurate particular measurements can be, and this limit can’t ever be surpassed, regardless of technological progress—ever. The limit arises from a central feature of quantum mechanics, the uncertainty principle.
The uncertainty principle establishes that regardless of what equipment you use or what techniques you employ, if you increase the resolution of your measurement of one property, there is an unavoidable cost: you necessarily reduce how accurately you can measure a complementary property. As a prime example, the uncertainty principle shows that the more accurately you measure an object’s position, the less accurately you can measure its speed, and vice versa.
From the perspective of classical physics, the physics that informs much of our intuition about how the world works, this limitation is completely foreign. But as a rough analogy, think about photographing that impish fly. If your shutter speed is high,