Infinite dimensional manifold

Question

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!

Answer 1

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). There is a mistake there that I did not yet correct online.

Answer 2

As far as the QM goes, I don't know enough to really say, but books like Quantum Mechanics for Mathematicians would give a mathematical overview of the large pieces of QM. However, infinite dimensional manifolds are out there in the literature. A quick googling brought up what looks like a good place to is arXiv:math/9202206. This paper is appears to be a 20 year old survey of infinite dimensional manifolds. For something a tad more recent and superficial you can try the wiki article on Banach manifolds which does not have many references but it will provide you with the keywords to do more googling.

EDIT: As mentioned by Kofi, what you need to extend the definition of a smooth manifold to an infinite dimensional setting is a notion of a derivative in an infinite dimensional setting. There is one that goes by the name of ``Frechet derivative.'' Using this derivative you just start mimicking finite dimensional spaces. For QM you will probably only ever need the Hilbert manifold case (locally diffeomorphic to a Hilbert space) and not the more general Banach or Frechet spaces but the idea for those is that as long as a notion of derivative is available you can define a smooth structure.