One of the best approaches to quantum gravity is by means of the Euclidean path integral formalism.1 It considers all possible geometries to give a wavefunction of the universe; a sort of quantum superposition of spacetimes (whatever that means). The mathematical trick is to work in imaginary time to make (some but not all) spacetimes become static compact geometries (otherwise calculations blow up), as this imaginary time behaves like another space coordinate.
Up to now, the best we can do is to use a semi-classical treatment, where we only consider geometries that contribute most to the wavefunctions; those who are solutions to the classical Einstein equations. Then, we can compute, for example, the probablity that an initial spacetime tunnels to some final spacetime. We only need an Euclidean solution joining the two, called instanton. We can use this to compute what is the probability that a bubble appears out of vacuum!
I use instantons and tunneling between geometries to tackle the information loss paradox: the question of whether we can recover the information of anything that fell into a black hole (e.g how to reconstruct my favorite videogame that disappeared behind the black hole horizon) or it is simply lost (which would be in conflict with our knowledge of quantum field theory). For more details see this review of my collaborators.
- For a pedagogical introduction to the topic and the relation with the black hole temperature, check Hartmann’s lecture notes on the web. ^
