Bashing Our Heads Against the Planck Length

If we take Planck’s constant, h, Newton’s constant, G, and the speed of light, c, we can form a constant which has as its unit the unit of length. This is the Planck length: sqrt(Gh/c^3) or approximately 4 times 10^(-33) cm (I’ve not used hbar here for some silly reason.) It is often argued that the Planck length is the natural length at which an as yet undiscovered theory of quantum gravity will take over.
There is a nice argument where the Planck length emerges naturally from considering gravitational collapse. Consider a system of energy E. If this energy confined to a ball of radius c^4R/G<E , then the system will eventually collapse to a black hole (this is called the Hoop conjecture.) On the other hand, if the system has energy E, then it cannot be localized more than it’s Compton wavelength R=hc/E. What then is the minimum radius achievable? Well it’s just the Planck length!
So the Planck length arises naturally when we ask what is the minimal size object we can make which doesn’t collapse into a black hole and which obeys the uncertainty principle. But does this mean that the Planck length is the smallest length we can measure? I mean, just because the Planck length follows from the above argument doesn’t imply that we cannot make measurements which localize a particle to a distance less than the Planck length. However, a recent Physical Review Letter (vol 93, p.21101, 2004), “Minimum Length from Quantum Mechanics and Classical General Relativity” by Xavier Calmet, Michael Graesser, Stephen D.H. Hsu attacks exactly this issue (for the arXiv version click here.) And what do the authors discover? They discover that if they try to use an interferometer, or simple time of flight measurements to determine locality, they get the answer that the minimal distance measureable is the Planck length! So there really is a sense in which distance shorter than the Planck length has no meaning.