Kkangliu said:
We all know that Einstein's theory apply to the big universe, and quantum mechanism apply to the nanoscopic scale, and both theory work perfectly in it's realm, but how could two theory contradict each other?
By "contradict each other", usually what is meant is that gravity has not so far been made into a quantum theory. All the other forces of nature, the strong force, the weak force, and the electromagnetic force admit a so-called "quantum field theory" (QCD, electroweak theory, and QED respectively). This means that we have been successful in turning the field equations of these forces into a quantum field (the details of which are far too broad to go over here).
We have not been able to do so with gravity. The difficulty mainly comes from the fact that gravity is a feature of the underlying space-time itself (although there are field-theoretic theories of gravity, but this is beyond my knowledge area). The procedure to quantize the gravitational field yields non-renormalizable integrals (integrals which diverge, and which we can't make converge with a finite number of adjustment parameters). Moreover, even the phase space of general relativity is very difficult to pin-point in the Hamiltonian formulation (which leads to canonical quantization as we are familiar with). The phase-space of three-metrics and their associated "super-momenta" has far too many degeneracies (since a large number of different three-metrics will describe the same three-geometry, all related to each other by coordinate transformations). And it is difficult to find the correct phase space (saying "the phase space is the space of 3-geometries" is well and good, but to mathematically formulate such a statement is highly non-trivial).
But, since these forces act on such different realms, it's not so easy to experimentally see where the contradictions lie. In the quantum regime, the force of gravity on Earth, for example, is so much weaker than all the other forces, that we can't experimentally see where the breakdown happens. To all energy scales that we have been able to test in the laboratory, the standard model of quantum mechanics has performed beautifully. In all astrophysical applications that we can observe, general relativity has performed beautifully (ignore for the moment possible problems with dark matter). We have not been able to experimentally observe a contradiction in either of these theories, but because the math doesn't work out for a quantum version of gravity, we expect a problem somewhere down the line. We could see such examples of contradiction, perhaps, near the event horizons of black holes, and such, but it's not like we have a black hole in some laboratory somewhere that we can study...
