Weekly update (7/26/2026)

Not much progress this week, as I spent it helping my son move for his PhD.

Subtle issues with quantum observable. I am tackling one of probably the hardest problems: figuring out exactly what physical assumptions are equivalent to assuming quantum observables are self-adjoint. The issue is that the math is a mess. Position X is a self-adjoint operator if we consider wave-function on the whole R, but not we we restrict the space on a finite interval, like [0,1]. However, if we use periodic boundary conditions, then it is self-adjoint. You would like to say that symmetrized polynomials of position and momentum are observable. But, in general, they are not self-adjoint. For example, X^2 P + P X^2 is not self-adjoint. It is symmetric. If you use the Schwartz space, however, X, P and all the symmetrized monomials are self-adjoint. But then it is no longer true that self-adjoint operators are generators of one-parameter families. There is SO much intuition we get from the physics does not map to the math as it is currently formulated. This led me to the next insight.

Interpretations and reconstructions of quantum mechanics are not currently possible. An interpretation of quantum mechanics should tell you what ALL objects in the theory correspond to physically. A reconstruction of quantum mechanics should reconstruct ALL the mathematical objects from physical principle. That is, you can’t claim you have a complete interpretation/reconstruction if you can’t tell me what the topology means, what the sigma-algebra on top of which you define probability means, etc… You can’t say “I know what the probability means, but I do not know what the objects on which I define probability are.” I mean, you can, since basically everybody does that… But the worst thing is: the current mathematical framework is built on objects that cannot be given such tight correspondence. So, until those details are fixed, there is no real quantum interpretation or reconstruction possible.

So, no concrete final products this week… but the thinking progressed.