Weekly update (5/10/2026)

Livestream on reverse quantum. We had a livestream on the progress on Reverse Physics for Quantum Mechanics. I hope we’ll keep having one every week, so if anyone wants to discuss particular topics or answer specific questions, let me know! Next week, the plan is to talk about the symplectic group, how I think it should have a simple characterization in terms of entropy and the work that a student has been doing to see whether it works.

Progress on function spaces for quantum mechanics. We have made some progress on questions that Tobias Thrien, David Carfi’ and I have been circling around. First, I was finally able to show that the sup semi-norms that generate the Schwartz topology are equivalent to similar families of semi-norms using Lp. So, for the purpose of topology and convergence, you are really requiring the continuity of the expectations of all the polynomials of position and momentum, and how you construct the semi-norms is not important. Second, I was able to find a counter-example that shows that the moment problems is not solvable in Schwartz space: you can have two distinct Schwartz functions that agree on all expectations of all polynomials of position and momentum. I’ll write a brief next week about it. So, really, the basic requirement for a well-behaved and meaningful state space is that all observables under considerations are defined over the whole space (i.e. the expectation exists for each state) and they are continuous (i.e. small state changes correspond to small expectation changes). It’s all becoming clear and simple.

Lots of briefs. Added a few briefs this week. One shows how convergence in Hilbert space may mean that states and expectations converge differently: state converges to the ground state but the expectation of N converges to the first level! This is because N, like any unbounded operator, is not continuous over its own domain. Another shows that linearity in quantum mechanics is a mathematical convenience, not a physical requirements: we can have non-linear representation of quantum states. Last one shows that the Schroedinger equation is equivalent to conservation of entropy. Writing these things is sooooo… much nicer than writing papers. Just the bare minimum to show the result. No fluff, just stuff. Yes, it’s a reference…😊

Github action problem. The problem is that the github action that I setup rebuilds every brief every time. It’d be nice if it only rebuilt what changed… Anyone wants to help me with that?

Spectrum of a quantity for ensemble spaces. I think I am starting to understand how to use AI. I am trying to clean up the open problems for ensemble spaces as much as possible, so that I am giving at least a well-posed question to mathematicians. This week, I cleaned up the problem of defining supports for statistical quantities, so that one recovers the structures of classical and quantum mechanics (roughly, the corresponding of the quantum logic lattice… which is not a logic, but just the lattice of supports). What happened was that, while I was cleaning up, I defined…

Componential and facial subsets. A set of ensembles is componential if it contains all the components of its ensembles and it is a face if it is both convex and componential. It turns out that if a set if both convex and componential then it is also affine (i.e. contain all affine combinations, not just the convex ones). The AI tools allowed me to write all these definitions much more quickly, together with their closure operations, and some nice properties. For example, the componential closure of a convex set is convex and therefore a face, but the convex closure of a componential set is not convex… unless you are in a simplex! So we have other equivalent conditions for classical ensemble spaces. This is still all preliminary, so it is not in the main part of the ensemble space chapter yet, but the point is that the AI allowed me to get this going in five hours instead of two weeks, so I went ahead and did it.

Discussion on substack. Last but not least, a substack users engaged with me this week, and posted progress on one of our open problems! So, maybe this idea of doing open research on substack is not totally crazy… 🤣🤣🤣

Well, maybe I forgot something as usual… but plenty of things for this week!