A state $\phi$ satisfies the KMS condition at inverse temperature $\beta$ if for all $a,b \in \mathcal{A}$ we can find a function $F_{a,b} (z)$ with the following properties:

- Its holomorphic on the strip $\{z \in \mathbb{C} : 0 < \Im(z) < \beta \}$
- continuous on boundary and bounded
- $F_{a,b}(t) = \phi(a \sigma_t(b))$ and $F_{a,b}(t+i\beta) = \phi(\sigma_t(b) a)$

for all $t \in \mathbb{R}$

So what's the structure and intuition behind $\xi_{\beta}$? How should I think of it? If I'm thinking just in terms of a physical thermodynamic system, what do the extremal equilibrium states represent? My guess is states of matter, or pure phase states (considering that a change in a state of matter is a "phase transition"). So does that mean there are "impure phase states", id est, states in a mixture of phases (so that $\Sigma_{\beta}$ is non-trivial). Back to the QSM: If we're working in a system with only one phase transition (I'm not sure what a phase transition means in this context) does that mean that there are only two extremal $KMS_{\beta}$ states?

Apart from the equilibrium states, we're also interested in symmetries of the QSM system. Rather frustratingly, I cannot talk much about the physics of QSM symmetries (but I resolve to be able to soon! as I do with statistical thermodynamics!), so I must stick to the mathematics. We're interested in both automorphisms (symmetries of the whole system) and endomorphisms (symmetries of only a part of the system....I think.) In either case, they must be compatible [commute] with our time evolution, id est, if g is our auto/endo-morphism, then $g \sigma_t = \sigma_t g$ for all t. for automorphism g, we say $g \in Aut(\mathcal{A}, \sigma_t)$.

A subgroup $G \subset Aut(\mathcal{A}, \sigma_t)$ we call a group of symmetries by automorphisms. It acts on KMS states via pullback, id est, $g^{*}(\phi) (a) = \phi ( g (a) ) $. Connes included a lemma stating that inner automorphisms (from a unitary state $u \in \mathcal{A}$, inner automorphisms acts by $a \mapsto u a u^{-1}$) act trivially on KMS states. Actually, the lemma stated something else, but that was in the proof. The proof essentially uses an analytic continuation of the map $t \mapsto \sigma_t(a)$ to the whole complex plane (I don't know how that's possible) and uses the fact that KMS states have the property that $\phi(b \sigma_{i \beta} (a)) = \phi (ab)$. It then follows that $\phi(u a u^{-1}) = \phi(a)$, and the action is trivial.

An endomorphism $\rho$ also has its pullback, $\rho^{*} (\phi) (a) = \frac{\phi(\rho(a))}{\phi(e)}$, assuming that $\phi(e) \neq 1$.

$e=\phi(1)$ (assuming $\mathcal{A}$ has unity, of course) and its not too hard to see that e is an idempotent ($e^2 = e$, because $e^2 = \phi(1)^2 = \phi (1^2) = e$). Inner endomorphisms also act trivially, by a similar proof.

Things I don't know this post:

Choquet Simplex

factorial GNS representation

primary KMS state.

The analytic continuation used in the proof of the triviality of the action of a group of inner automorphism.

Connes also talks a bit about multiplier algebras and essential ideas - I may look into those tomorrow, but as best I can tell I won't be needing those ideas any time soon. Marcolli doesn't mention them at all, for instance.

Connes also mentions that we can pushforward KSM states: I largely understood what I read there [or I think I did], but I choose not to write about it, largely because of time, and largely because I don't yet see how it relates.

In your project, are you pretty much using research papers or is there any books which you can read?

ReplyDeleteThe thing I find sometimes is that even when some result should be "clear" from the definition, it is hard to get a good understanding of the meaning of the result without a good intuition for what is going on. I guess it must help that this things relate to statistical thermodynamics/mechanics. It might even be worth reading some course notes in the subject to help in the intuition front.

I'm reading two main monographs [plus associated papers]. Connes & Marcolli's Noncommutative Geometry, Quantum Fields, and Motives. And Marcolli's Lectures on Arithmetic Noncommutative Geometry.

ReplyDeleteThere's not really any textbooks for the main subject. But I have been able to chase down textbooks and monographs for tangents. For instance, R. Haag Local Quantum PHysics and Bratteli & Robinson Operator Algebras and Quantum Stastical Mechanics contains a lot of the QSM stuff, especially on those KMS states. That said both of those books are pretty far above my physics background and reference a lot of statistical thermo that I've never seen. I think it would be worth going through some course notes on a lower level course, but I just don't have the time.