The Planck length is an interesting one - it's the length you get if you combine three other constants: the speed of light, the gravitational constant and the Planck constant.
There's nothing intrinsically significant about it, but there are clues that it's a really important scale for trying to tie together various areas of physics - particularly, as you've noted, quantum mechanics and general relativity.
There is something called the Compton wavelength. This is the wavelength λ
at which a photon has the same energy as a particle of mass m
= ℏ / mc
where ℏ is the reduced Planck constant and c
is the speed of light. The Compton wavelength has a physical significance in that it's the shortest wavelength photon that can be used to measure the position of a particle accurately: a short wavelength must be used to create an accurate measurement, but as you use shorter wavelength photons their energy increases... and once this energy exceeds the energy of the particle you're trying to measure, you run the risk of creating another identical particle when making your measurement - and ruining it! When you get to this minimum wavelength, you're squarely in quantum mechanical territory.
Now, to a different area of physics.
There's a concept called the Schwarzschild radius. If a mass m
is compressed to the Schwarzschild radius r
, its gravitational escape velocity reaches the speed of light (for a non-rotating black hole this the event horizon). The radius is approximately*:r
is the gravitational constant. Once you reach this scale, you're very much in the realm of general relativity.
OK. So what?
Both the Compton wavelength and the Schwarzschild radius are measures of length. At what point does λ
? If we set:
ℏ / mc
and solve for m
And if we put this mass into either the Compton wavelength equation or the Schwarzschild radius equation we get a length:l
^3) = 1.6e-35 metres
The Planck length! So when we get down to this scale, it would seem that both quantum mechanics and general relativity are having important effects. So yes, any theory (string theory, quantum gravity, etc.) working on this scale has to incorporate quantum mechanics and GR neatly without any nasty infinities or division by zero. Interesting, huh? * it's approximate because it ignores a factor of two: when doing this type of calculation we're just interested in orders of magnitude, and so it's very common to just ignore numerical factors.