In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as $(M, \omega)$ where $\omega$ is a sympathetic form on $M$ corresponding to the Poisson bracket. Given a Hamiltonian $H$, its flow is the trajectory of our particle with initial conditions. Now, in local coordinates, we can write
$$
 \vec{z} = P\_t \vec{z}(0) \quad (1)
$$
where $P\_t$ is the flow evaluated at $\vec{z}(0)$, which is the propagator. Note that there is a differential equation that (1) also satisfies. In particular, just differentiate both side by $t$, then we get an ODE which we can solve the flow $P\_t$ if we only know the Hamiltonian vector field. Now, in quantum mechanics, I learned from class that we obtain an similar equation that looks like
$$
 | \psi \rangle = U\_t | \psi\_0 \rangle \quad (2)
$$
where $U\_t$ is unitary and satisfies $U\_{t+s} = U\_tU\_s$ for all $s,t$ defined. And (2) also satisfies a differential equation similar to that of (1). So my understanding is that every thing we did in my quantum class is just a local theory. My intuition tells me that globally, one which to have some configuration space $N$ that is in some sense "diffeomorphic" to the projective space of $l^2(\mathbb{N})$ locally. And the manifold we want to work with is $M:=TN$ whatever $TN$ means in this case. Then our states are just point on $M$, with Hamiltonian defined over $M$ to $\mathbb{R}$. And $U\_t$ would be a Hamiltonian flow. \


However, anything that goes infinite dimensional can go crazy. So I am looking for a global picture of quantum mechanics. In particular, a definition for 


1. What does smoothness mean (or even differentability) for $f: l^2(\mathbb{N}) \rightarrow \mathbb{R}$ or $l^2(\mathbb{N})$.
2. definition of locally $l^2(\mathbb{N})$ manifold
3. explaination/theorem on the global picture of quantum mechanics.
4. Or perhaps none of that makes any sense whatsoever, then what should I be looking into?


Thanks tonnes!



Infinite dimensional manifold
Have a look at section 48 of:
Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997,
[(pdf)](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf).
There is a mistake there that I did not yet correct online. 


