* The catch is that they have to be quasiprobabilities, which means they can be negative. There can be patches of the phase space, IE particular combinations of position and momentum, to which you assign a negative quasiprobability. The fact that you are limited by Heisenberg uncertainty means that you never actually predict anything you observe to occur with a negative probability.
Despite this the quasiprobabilities (or quasiprobability densities for continuous systems) are treated mathematically just like probabilities (probability densities).
I think that this alternative mathematical form for the same physics reveals that the “ughhh what?” that surrounds measurements in quantum mechanics is always there, but it takes on a different mathematical flavour in different formulations. If you are using Hilbert space you worry about the Born rule. If you are in phase space you are probably trying to work out what a negative quasiprobability means. Its not just ignorance anymore, something very odd is going on. The fact that these are (I think) the same problem seen from different perspectives is potentially useful in seeing the way to the solution.
If you really hate the L2 norm of the Born Rule it is actually possible to do quantum physics entirely with probabilities* in phase space (https://en.wikipedia.org/wiki/Phase-space_formulation).
* The catch is that they have to be quasiprobabilities, which means they can be negative. There can be patches of the phase space, IE particular combinations of position and momentum, to which you assign a negative quasiprobability. The fact that you are limited by Heisenberg uncertainty means that you never actually predict anything you observe to occur with a negative probability.
Despite this the quasiprobabilities (or quasiprobability densities for continuous systems) are treated mathematically just like probabilities (probability densities).
I think that this alternative mathematical form for the same physics reveals that the “ughhh what?” that surrounds measurements in quantum mechanics is always there, but it takes on a different mathematical flavour in different formulations. If you are using Hilbert space you worry about the Born rule. If you are in phase space you are probably trying to work out what a negative quasiprobability means. Its not just ignorance anymore, something very odd is going on. The fact that these are (I think) the same problem seen from different perspectives is potentially useful in seeing the way to the solution.