Since there are starting to be a number of people following the project and collaborating in various ways, it may be a good idea to start having a brief weekly update on what has happened in various fronts.

Automated list of open problems from the book. In a previous post I was musing over the idea of using the book as the master repository of conjectures and open problems. The more I thought about it, the more it made sense, so this week I started futzing around on a prototype, which is somewhat live here. I have a prototype LaTex syntax for an open problems environment, which LaTex expands but can also be recognized through a regular expression. I created a repo for tools in general, and a python script which is essentially a wrapper over a the regular expression and outputs JSON. Then a GitHub action runs after every push and updates a JSON file in another repo. This repo will essentially hold any other autogenerated files as well. Then, the page works like all the other pages in our website: gets the JSON and formats it. Missing features are links from the web page to the latest draft of the book, and maybe (in the future) a way to search through the open problem. But I’ll start practicing what I have first, to see how it feels to use it.

It reminded how much I used to love software engineering: take a big problem, chunk it up into independent pieces… Nail one after the other. Nice.

New video. Released a new video this week about the importance of statistical mechanics in physics. Decent turnout. Still trying to understand what to do next.

Reverse Physics for Quantum Mechanics. Tobias and I worked a bit on the general issues. One of the problems I had for some time is disentangling the big hairball that are the postulates of quantum mechanics. How many independent assumptions are there when defining quantum states? It seems we solved that, or at least we made good progress. The realization is that space of pure states/space of ensembles/Born rule are 3 increasing structures were the next implies the following but not the other way around. This means that I have a tentative structure for the chapter, and I can finally start working on it. At least, I think so.

Characterizing squares of distance functions. One of the problem I have in ensemble spaces is that the mixing entropy, that generalizes the Jensen-Shannon Divergence (JSD) for both classical and quantum mechanics, is not a distance. The JSD is the square of a distance function in both classical and quantum mechanics, but proving requires the actual functional form… which makes it impossible for me to understand what are the fundamental properties that generalize. I need a way to characterize squares of distances in the same way that the triangle inequality characterizes distances. While talking to Matt Insall, a mathematician I routinely speak to, he reminded me that Norman Wildberger has done a lot of work in geometry avoiding irrational numbers. If you do not know Norman’s work, he has a YouTube channel where he shares his ideas. Norman suggested that the quadrea, which is defined on top of the quadrance (the square of a distance), cannot be negative. This may give me an inequality on square distances! So, I’ll have to check whether the quadrea defined on top of the JSD is non-negative and understand what properties of the JSD guarantee that. May be messy… but it might just work!

There is probably other things. Maybe I should start taking notes throughout the week… Have a good week-end everybody!