Today started out with so much progress. I started early, made ample progress with programming and business contacts, bided on freelance projects, and was anxious to start a day of mathematics work. But when the time to do maths rolled around, I procrastinated, and procrastinated, and procrastinated. (Actually, for the past 3-4 days I've not done much maths work, for the same reason.)
I finally started doing some reading about 2 hours ago. It was vague, unfocused, distracted reading. More about states on a QMS, representations, KMS states, and quite a few things I don't understand (list at the end.) But I must keep good habits, so I read anyways, and am now writing something barely coherent about it.
Here's what I plan for this week:
1) KMS states, the structure of their sets, symmetries on QMS. 3-4 more hours and 1 more blog post.
2) Some algebra: group rings, group algebras, algebras over a field, et cetera. Just to get my definitions, examples, and intuition straight. 3-4 hours. 1 blog post should be enough, I think.
3) QMS from 1-dim Q lattices and the Bost-Connes system. However long it takes!
What I don't understand today:
Connes has a proposition (3.8) about $KMS_\beta$ states...they're extremal iff the GNS representation is factorial. I have an idea of what a GNS rep is, I have no idea what it means for one to be factorial
Same prob mentions that the set of $KMS_\beta$ states is a convex compact Choquet simplex. I've never seen "Choquet Simplex" before. I'll have to do some googling when I'm more awake.
"phase transition" is mentioned often. While I have an idea what this means for a classical thermodynamics system (changing states in matter) I have no idea what it means for a C* dynamical system. I did manage to find some stuff in R. Haag's Local Quantum Physics (around page 213) that I'll have to look at tomorrow.
I can barely keep my eyes open. Goodnight!