Some measurements confirm this statement?
Or this is a theoretical conclusion?
I'm not sure, i'd guess its just a theoretical conclusion based on math.
Why, what is the logical deduction?
I too would like to know how it is we know quantum effects become important in this regime. Other than knowing we need QG before we get down to a singularity, what in particular makes us think the planck scale is selected by nature as the transition scale?
It is a dimensional analysis argument.
Quantum mechanics gives us a time or length scale for an object of particular energy or momentum.
General relativity gives us a length scale given a mass (or energy).
Therefore we can look at where these length scales are comparible. The Planck units are units defined using G, hbar, c, etc.
Wheeler and others believe that the structure of space-time becomes a "quantum foam" on the quantum scale. The now-current Wiki article seems reasonable sane on the topic and has some references.
There is some discussion of the "quantum foam" idea in MTW's "Gravitation" as well. I would say at this point that this is not a theorem so much as a speculation, one that's significant enough to have a reasonable amount written about it, however.
This is based on the finding that quantum general relativity, although a good effective quantum theory at low energies http://arxiv.org/abs/gr-qc/9512024, is not perturbatively renormalizable at high energies (ie. short distances).
It has not been ruled out that quantum general relativity is non-perturbatively renormalizable, in which case it would not break down at the Planck scale. http://arxiv.org/abs/0709.3851
Theoretical. I don't know if there are any situations where they both make clear predictions that contradict each other, but see here for a discussion of one of the main problems in figuring out how to reconcile them, having to do with the fact that the uncertainty principle would seem to allow for huge uncertainty in energy at sufficiently small scales, but in GR big energies cause significant curvature of spacetime, and my understanding is that physicists only know how to make predictions in quantum field theory if they have a specific known background spacetime. I guess another more general conflict is that quantum field theories treat the other set of forces using a common set of rules, but if you try to apply these rules to gravity you get infinities which can't be "renormalized" as in the case of the other forces.
"quantum foam" means a difficult topology, but whats the problem. GR can describe any kind of topology. If Wheeler states that spacetime is a "quantum foam", this means that he uses GR at that scale.
No. You are assuming that there is still a smooth manifold in your statements there, as we need that for GR.
So "quantum foam" doesn't just mean a difficult topology.
Imagine for example a fractal landscape ( http://en.wikipedia.org/wiki/Fractal_landscape ). On very long scales it looks like a smooth flat 2-d surface. But as you zoom in, it appears to have tecture, and you can zoom in infinitely far and it will just be more and more roughness. You can't "zoom in" far enough that it appears smooth. It is fundementally rough, and "emergently" smooth.
This is very far from just a topology issue. The mathematical language of GR, differential geometry, assumes a manifold. Wheeler's point is that quantum mechanics hints that we may have to use a more base concept -- some kind of "pre-geometry" -- to even approach quantum gravity correctly.
For example, some attempts to quantize GR directly have found a fractal spacetime that only in length scales much greater than planck lengths does a smooth spacetime with 4 dimensions emerge ... on smaller length scales the fractal dimension appears closer to 2. So at least in some approaches, Wheeler's intuition is playing out.
But aren't these are approaches in which GR does not break down at the Planck scale?
There are two separate questions here: (1) Why do we believe that GR and QM are inconsistent? (2) Why do we expect that inconsistency to manifest itself at the Planck scale?
The simple answer to #1 is that nobody has succeeded in producing a theory of quantum gravity. We're not sure why they've failed. Possibly we need to give up some cherished feature of one of the theories, like background independence (cherished by relativists) or unitarity (cherished by quantum theorists).
The answer to #2 is that the Planck scale is the only scale you can construct out of the appropriate physical constants. This is not an absolutely secure argument. For example, if large extra dimensions http://en.wikipedia.org/wiki/Large_extra_dimensions exist, then we could see quantum gravity at the LHC.
The quantum foam can be smooth, only the scale is Planckian. I dont think that quantum foam is a real fractal, because particle scales (due two particle creation and annihilation) define the scale. Or not?
I think that quantum foam can be a fractal just in the case
when something is existed below quarks, what we don't know yet!
I like Ben Crowell's answer #1...His answer # 2 is appropriate for classical relativistic theory.
I also like Justin's observation
Quantum foam in fact is usually interpreted to mean the cessation of space and time as we know it...just as smooth waves of water dissolve into a spray..called spindrift at high winds velocities....Below Planck scale you can't even speak about "topology" because space and time dissolve into wild uncontrolled undulations....a type of quantum "ambiguity"
Two places where relativity and quantum mechanics each break down to infinities are the big bang and black holes...neither works at those type singularities and we have no theory so far that does.
This answer seems incomplete insofar as it doesn't have anything specifically to do with gravity. After all, before physicists had figured out how to quantize classical electrodynamics, no one expected that the characteristic scale of quantum electrodynamics should be the Planck scale, and of course it isn't! I found an interesting paper here which gives a series of basic arguments for the Planck scale should be the scale of quantum gravity, summarized on p. 3:
For example, argument (4) involves the idea that to probe what's going on in smaller and smaller volumes of space, you need probes of higher and higher frequency and therefore higher energy, and if the volume were as small as the Planck scale the energy density would be so high as to form a Planck-scale black hole.
I think that the main problem about Planck scale and quantum gravity is not the topology but the probability in the quantum theory.
The constants that Planck scale quantities are derived from are the speed of light, Planck's constant, and Newton's gravitational constant (which also appears in the GR field equations). The presence of Newton's constant is what makes these quantities refer to gravity and why no one should expect quantum electrodynamics to have any relation to the Planck scale (since Newton's constant doesn't show up anywhere in classical electrodynamics).
Still seems like an overly handwavey argument, the fact that you can construct length, distance and energy densities from some fundamental constants which include the gravitational constant doesn't give any clear reason why this should be the characteristic scale of quantum gravity, the arguments in the paper I linked to are more physical. For a different electrodynamics analogy, would any pre-QED physicists have argued that quantum electrodynamics effects don't become significant until we reach the Planck charge?
You can also get a set of dimensions by combining e, c and G. I believe these were first formulated by Irish Physicist George Johnson Stoney. The dimensions are different than Planck units - one of the reasons I have always considered Planck units as more or less metaphysical numerology - it all started as a dimensional analysis without physics and it still has no physics -
For a another take
Chronos posted this on another thread http://arxiv.org/abs/gr-qc/0601097
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