I was just thinking about how can Hawking radiation escape a black hole. And then I came up with this idea:
In quantum mechanics a wave function is a superposition of states.
So the wave function of real black hole will be a superposition of different geometries with a peak for the classical Schwarzschild geometry.
Then maybe(?) we can estimate the wave function of a black hole of mass M like this:
$$|BH(M)\rangle = \sum_{m=0}^{\infty} c_m |SW(m)\rangle$$
Where BH is the black hole state which is a superposition of different geometries, here SW is a Schwarzield geometry of mass m. Just to simplify things we only consider these states and not more complicated geometries. $c_m$ are some complex amplitudes which have a maxima around $M=m$ e.g. $c_m \approx e^{-i(m-M)^2}$
So when we take a measurement of the Hawking radiation, this will make the BH state collapse into one which peaks at a lower mass, since these are the only states at which the radiation could have left it.
In other words, the reason why radiation can escape a black hole is some of the states of a black hole superposition have geometry with a smaller event horizon. And in those histories (in the sum over histories) the radiation was never in a state where it crossed over an event horizon in the first place.
The idea is that as more radiation is detected from the BH, the BH wave function collapses such that $c_m$ peaks at lower masses until it is entirely gone.
Obviously this is just a rough argument, so I'd be interested if you know if this has been formulated properly with better mathematics? I think the main issue to formulate this properly is that we don't know which sum of geometries is correct and consistent.
