Happy Easter! Quick update for this week.
More imprecise probability. We met with Gert de Cooman who spent a few decade on the foundations of probability and has also, in the last few years, started to look at probability in quantum theory. He has a series of nice results in characterizing probability spaces from the space of utility functions. Effectively, the “safe bets” determine the probability space, and the symmetry of the safe bets are enough to identify the space. It is likely that these can be expressed as symmetry of the entropy in the context of ensemble spaces, in which case we should be able to homogenize the two perspectives.
A couple of results on ensemble space topology. I have integrated a couple of contibutions by Junde Sung, a recently graduated student at UMich. The first is, as conjectured, when two ensembles are fixed, the mixture function depends only on the probability and it is an homeomorphism (i.e. topology is the same). If the probability and one ensemble is fixed, instead, we do not have a homeomorphism because the resulting map is not an open map. However, it is still an open question whether that is an embedding. It is probably what we want to show to solve a number of problem, so any help from people that understand topology in the context of convex/vector spaces would be appreciated!
Born rule equivalence to von Neumann entropy. I have created a draft for a new brief that shows assuming the Born rule in quantum mechanics is equivalent to assuming the von Neumann entropy.
Standards for Reverse Physics. The work above is an attempt to do Reverse Physics in the most formal way possible, so that we start having more clear standards for the approach. Quantum mechanics forced this because a lot more difficult to disentangle the physical assumptions implicit in the mathematical structure. If you are interested in Reverse Physics, feedback is appreciated.