Is There a Rigorous Proof of a Quantum of Space in Quantum Mechanics and GR?

[lokofer]

Is there a "rigorous" proof..

-That a "Quantum" of space (a minimum lenght, area or volume for any 4-dimensional Manifold) exist applying Quantum mechanics and GR?..both together or in the "Semi-classical" limit?...

- this "Planck lenght" should be obtained directly from math and never imposed by a "necessity" because if not theory would be wrong..this is why i don't believe much in "String Theory" you'll never be able to "see" (detect, prove) that there are 9+1 space-time dimensions or that "ojbects" called strings exist.

 

[TRodrigues]

Apparently, yes. It has to do with Heisenberg uncertainty principle. I reproduced it here during another discussion; actually, it's the derivation for the quantization of time there, but from there you can conclude that a signal (which travels at c, always) cannot travel less than a Planck length, because it would need to do it in a fractionary amount of Planck times, which cannot happen.

 

[Haelfix]

Note that the clock calculation is somewhat naive. It takes no account of how your *length* or *time* variables might change with GR and instead treats them as perfectly quantum parameters.

Lets just say the calculation is a ballpark guess at the regime where things start to break down, and where nasty GR like objects (like black holes) will tear apart usual notions of what things are like..

 

[lokofer]

-I don't agree much with the ¿hypothesis? \Delta E < mc^{2} since mass can be 0, the rest i can understand..

- Then is "Space-time " is Quantizied...¿what's the problem with "Path Integrals" and GR ?..since you could apply "Regge Calculus" (Numerical methods) and solve Quantum Gravity.. for my the biggest "obstacle" in Quantum Physics was that somehow space was "continouos" so the momentum could be oo but now that Space and time are quantizied then all problems should disappear.