The Planck length
Thanks to all those people who commented on the significance of the Planck length. Putting it all together, one plausible argument for the significance of the Planck length seems to be something like this.
First, suppose we have a particle of mass whose wavefunction is localized within a length , in each co-ordinate. For each co-ordinate the standard deviation in position satisfies:
The uncertainty principle tells us that the uncertainty in each momentum co-ordinate satisfies:
It follows that:
where now is the average of the square of the total momentum, i.e., not a single co-ordinate. Using the usual formula connecting energy and momentum for a free particle in special relativity, we obtain . If we assume that and can be identified then we obtain:
In short, strong localization in position forces a large momentum, which creates a large energy density.
It’s plausible that if we make the energy density large enough, we’ll create a black hole. Equating to for some notional black hole mass M and seeing at what radius that is, we get the Planck length, up to a small constant. I haven’t put the details in because this stage of this argument is even more hokey than the rest: energy density depends on what frame you’re in. However, it is at least plausible.
If you take the “no hair” theorem of general relativity seriously, then you’d believe that such a black hole should have no internal structure. But the detailed wavefunction would seem to be such an internal structure, and so we have a problem, which seems likely to signify a breakdown in one or both of general relativity or quantum mechanics. How that breakdown would manifest itself, I don’t know.
There’s lots wrong with this argument: the sign ambiguity for E, the use of the free particle formula for energy, and so on. No doubt many of these difficulties disappear in some more sophisticated approaches to quantum gravity; I’ve just been using undergrad quantum and GR.
Nonetheless, this argument does seem suggestive that particle having wavefunctions with structure on the Planck scale would, indeed, be very interesting objects, and that it is at least somewhat likely that general relativity, quantum mechanics, or both, would break down at that level.