JohnLuck said:
Actually why is the Planck length thought to be the smallest meaningful measure of length? I would also love to know this.
Basically it's just an order-of-magnitude estimate. If you have no idea how to calculate the volume of a ball, you can guess that it's the product of any three lengths that characterize a ball, because this gives you a result with the right unit. If you e.g. guess that it's the circumference to the third power, then you're wrong by a factor of factor of 6π
2, but when you do this sort of thing, you're not likely to be wrong by a factor of 10
100.
So once you have decided that the concept of length will be problematic at some scale, you can estimate that scale by combining relevant constants into a product that has the right unit. The relevant constants here are G,c and ##\hbar##. I will try to explain why.
To understand a concept like length, we need to find a theory of physics that defines it as a mathematical term, and then we need to do experiments that show that the theory's predictions are accurate. To understand "length" is to understand the mathematics of the theory that makes the best predictions about results of experiments, and to understand what sort of thing is considered a "length measurement" for the purposes of testing the accuracy of those predictions. The best theory we have about the properties of space and time is general relativity. It describes a spacetime in which there's an invariant speed, c. This is why c is relevant. GR is also a theory of gravity. This is why G is relevant.
GR describes a relationship between the properties of spacetime and the properties of matter. Unfortunately, it describes matter classically, and we know that there are situations where a classical description of matter fails miserably. Because of the relationship between matter and spacetime, we should expect GR to also fail miserably at describing the properties of space and time in those situations. Since GR is the best theory of space and time that we have, this means that
all the mathematical definitions we have of "length" are part of theories that are useless in those situtations. And this means that none of those definitions can provide us with any sort of
understanding of the real world in those situations. The concept of "length" as we know it (i.e. as it's defined by our best theories) has become irrelevant and useless.
It's possible that a better theory would include a definition of something we might want to call "length" in those situations, but that definition might be very different from the definition of "length" in GR. So maybe in a hundred years, people will be talking about lengths at sub-planck scales, but to them the word will mean something different from what it means to us.
In
some of the situations where a classical description of matter fails (specifically those situations where gravity can be neglected), we can use a quantum theory of matter instead. This is why ##\hbar## is relevant.
