Yes. An example. General relativity says there are black holes that are points. Quantum mechanics says there is no point, but everything has a size, though very small. General relativity is deterministic. Quantum mechanics is probabilistic. General relativity can predict the future. Quantum mechanics can not, as there is some randomness. They contradict each other.
Relativity and quantum mechanics (QM) are both brilliant. They give excellent results that match what we see in the real world. They’re also rather simple and elegant maths. There’s a thing called ‘Occam’s Razor’ which says that the simplest idea is usually the right one.
The thing is, though, that relativity assumes a uniform, smooth universe, whereas QM assumes it’s all granulated and made of separate ‘bits’. It’s like relativity sees a big sand dune but QM sees the individual grains of sand. When you think about a situation where both the large, ‘dune-like’ structure and the small ‘grainy’ structure are both equally important, relativity and QM have a fight and can’t agree what’s happening. To solve this problem, we have to go against Occam’s Razor and make our ideas not so simple any more. That’s where the problem is.
Actually, a lot of quantum mechanics IS compatible with relativity. At CERN they send particles around so fast that you have to use relativity to understand what’s going on.
The problem is trying to get gravity to work. We really have no idea how to do this, and since we can’t really test gravity on quantum scales very easily (CERN might see stuff, but it’s unlikely), it’s difficult to know where to begin.
Scott’s answer below is an excellent description of our best idea of the ‘why’ though 🙂
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