New video. Released a new, more complete, video on the classical limit. It’s also a way to participate in Curt Jaimungal's #CORE1 initiative, which I think it's a brilliant idea!
New brief. Wrote a new brief that tries to formalize the argument that I had already showed in a previous video. This is an attempt to “dress up” Reverse Physics, and make it look more like the type of things one does in mathematical logic and Reverse Mathematics. It will probably evolve, but it’s a decent first try.
More work on Reverse Quantum. Related to the above, I have started to work more on the Reverse Physics of Quantum Mechanics. I finally have a list of conditions with their likely logical relationships… now I have to “untangle the web” into a sequence that works for the linear exposition of the written word. It’s going to be a pain!
I am finally starting to understand more of infinite dimensional function spaces! It’s funny how you can hold things in your head, but not see the connections… Something clicked on Friday. While I was hoping that the entropy would give us not just a distance function, but it would, in a way, generate the topology of the space, decide what ensemble converges to what ensembles, I have now realized that that can’t be… In fact, I was able to create a sequence of ensembles for which the entropy converges, but the observables do not converge in a meaningful way. The same problem applies in Hilbert spaces… So, I have now another reason as to why Hilbert spaces don’t work, and I think the way one gets the right topology is how one does it in analysis: via the dual space. This is consistent with the idea that topology is experimental verifiability: all measurable statistical quantities (i.e. affine linear functional of ensembles) define the topology.
As other odds and ends, we still have to decide what to do for live-streams. Hopefully we’ll make a decision tomorrow. Thanks for the attention!
Some notes on 002-SuperpositionAndMultipleMixtures.pdf
This is a lovely observation. The way I read it, the brief is really identifying a single geometric fault line between classical and quantum theory: classical mixed states live in a simplex, so their decomposition into pure states is unique; quantum mixed states do not, and that non-uniqueness is the mixed-state shadow of superposition.
That framing is helpful because it makes superposition feel less like a mysterious operation on state vectors and more like one face of the convex geometry of quantum states. The qubit example is the perfect intuition pump: the same maximally mixed state can be decomposed in the z-basis or the x-basis, and the formalism treats both preparation stories as leading to the same density operator.
I would be interested in seeing this connected explicitly to the algebraic picture: states as positive normalized functionals on a C*-algebra, pure states as extreme points, and mixed states as convex combinations. In that language, the contrast with classical probability spaces becomes especially sharp: classical state spaces are simplex-like, quantum state spaces are not.
So perhaps the slogan is: superposition is what non-simplicial state-space geometry looks like when viewed from the pure-state side.