Trying to solve quantum gravity is frustrating. We have made tremendous progress in quantum theory, but it seems that every time we find a new quantum technique, there's a reason it doesn't quite work with gravity. Take, for example, the case of quantum fluctuations and renormalization.
There are many ways to calculate quantum interactions, but one of them involves quantum fluctuations. Suppose you wanted to determine the odds of an electron at point A being later observed at point B. It's possible that the electron could just travel from A to B. But because of quantum uncertainty, it's also possible that a virtual electron-positron pair appears, interacts with your electron, and shifts the odds. Less likely, but not impossible, you might have two virtual pairs, or three, or fifteen. If you want to calculate the odds of your electron going from A to B, you have to calculate the odds of all possible paths and interactions. You may have seen these represented as Feynman diagrams.
This works great for individual particles, though the calculations can be tedious. But when you use this approach to quantize things such as an electromagnetic field, you start to see problems. The possibilities grow and grow, and when you try to add everything up, you get infinity rather than a finite probability. Mathematically, your sum diverges. This was a big problem with quantum field theory until we figured out you could use a trick known as renormalization. Basically, the total sum isn't what's important. What matters is how that sum differs from the background. Thanks to renormalization you can cancel out the (infinite) background to get an accurate finite result.
So why not do the same thing for a gravitational field? Well, it turns out that quantum renormalization only works for Euclidean space. In general relativity, the mass-energy of a system warps space and time. So all those quantum fluctuations curve spacetime, and curved spacetime induces even more virtual particles, which warp space even more... oh no! It all breaks down, and we can't quantize gravitational fields the way we quantize the other fundamental forces.
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