The noncommutative torus $A_\theta$ is a $C^*$-algebra corresponding to an irrational foliation of the torus $\mathbb{T}^2$ by lines of slope $\theta \notin \mathbb{Q}$.

As far as I am reading it is generated by two bounded linear operators $U,V$ acting on $L^2(\mathbb{T})$. $$ U(f)(z) = z f(z) \hspace{0.5in} V(f)(z) = f(e^{-2\pi i \theta}z) $$

This is cool... but **what quantum mechanical system does this correspond to?**

In particular,

- can we find a hamiltonian $\mathcal{H}$ corresponding to this system?
- Or is this a quantization of a different Poisson structure than the free particle?

In order to address this question, I tried writing the two formulas in bra-ket notation:

$$ \langle f |U | z \rangle = z \langle f| z\rangle \hspace{0.5in} \langle f |V| z \rangle = \langle f| e^{-2\pi i \theta} z\rangle $$

Here $\langle f | \in L^2 (\mathbb{T})$ and $|z\rangle \in \mathbb{T}$. In a sense it didn't matter which function $f$ we picked we can just write:

$$ U | z \rangle = z | z\rangle \hspace{0.5in} V| z \rangle = | e^{-2\pi i \theta} z\rangle $$

These look like the position and momentum operators and we get *an* representation of the Heisenberg algebra, since $UV = e^{2\pi i \theta} VU$.

Basically... $C^\ast$ algebras supposed to be the "**same**" as quantum mechanical systems. So what is the quantum mechanical system corresponding to the non-commutative torus $A_\theta$ ?

The article says define Poisson bracket on the phase space $C^\infty(\mathbb{T}^2)$ by $$\{ f,g \} = \theta \left( \frac{\partial f}{\partial x_1} \frac{\partial g}{\partial x_2} - \frac{\partial f}{\partial x_2} \frac{\partial g}{\partial x_1} \right)$$

Then build this *deformed* multiplication $\ast_\hbar$ such that $$f \ast_\hbar g - g \ast_\hbar f = i\hbar\{ f,g\} + O(\hbar^2) $$

Then quantizing we get $[ x_1, x_2] = i\theta\hbar$, which looks like $\theta$ could be absorbed into $\hbar$ ? This might be semiclassical analysis or something.

I don't understand this:

When $n=2$ the skew-symmetric matrix $\theta$ is just determined by a real number, again denoted $\theta$, and $A_\theta$ will be isomorphic to the

crossed-product $C^\ast$-algebra for the action of $\mathbb{Z}$ on the circle $\mathbb{T}$ coming from rotation by angle $2\pi \theta$. For this reason the algebras are often calledrotation algebrasorirrational rotation algebrasif $\theta \notin \mathbb{Q}$.

Notesby Marc Reiffel explain the deformation of the Poisson structure... $\endgroup$